Cooperative control theory and integrated flight and propulsion control

Appendix

Copies of Publications

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 15, NO. 6, November-December 1992

Analysis of Airframe and Engine Control Interactions

and Integrated Flight/Propulsion Control

John D. Schierman" and David K. Schmidtt

Arizona State University, Tempe, Arizona 85287

A framework is pn_ted for tbe analysis of dyunl¢ cross-coupling between airframe and engine control systems. This approsels is developed for assessing tbe significance of airframe/engine interactions wttb regard to system stability, period, tnd critical frequency ranges wbere interactions are ¢speciaily problematic. The stability robustness against mkframe/engine internetJons are of parUcuinr interest, and s robustness snaiysis approach is devdoped and presented. Tbe difference betweeu systems exhibiting two*directJomd vs oue-diroc- tionai coupling is also d_. Two control configurations of • vehicle previously considered in several integrated flJgbt/prop_ ¢mttrol studies are then evaluated using the technique, end it is shown that the baseline configuration reflects little siguiflcant airfrmme/eugine interactions. Consequently, cisundeai deeentral- bed airframe and engine comroJ laws appear to he quite adequate. However, tnaiysis of the other system configuration shows significant performance degradation in the engine loop because of airframe/engine cou- pling.

Introduction Potential Sources of Airframe/Engine Interactions The airframe/engine interactions highlighted in this section DVANCED concepts for highly maneuverable fighter aircraft and those capable of short takeoff and vertical are elaborated on in Refs. 1-9. Consider for discussion pur- landing utilize the propulsion system for augmenting the lift poses the vehicle system in Fig. 1. Thrust reversing nozzles may and maneuvering capabilities of the vehicle. The integrated be considered for improving forward speed control of the flight and propulsion control (IFPC) problem addressed aircraft. Vectoring of the engine's aft nozzle may be used to herein and elsewhere _-s focuses on the interactions between augment attitude control power, and ventral nozzle thrust may airframe and engine systems, especially in control law synthe- augment aerodynamic lift. Left and right ejectors, drawing sis and analysis of such configurations.

primary thrust from the engine's mixed flow (core and bypass The main purpose of this paper is not to discuss any par- flow) and secondary thrust from intakes over the top of the ticular IFPC control law synthesis procedure but f'wst to fuselage may also augment lift and enhance pitch and roll present an analysis framework that will expose how the inter- control power. The lift and attitude responses of the airframe actions manifest themselves and second to determine if cross- will be influenced by thrust disturbances in these sources, and coupling dynamics between the airframe and engine are of effects of the ejector's secondary flow may significantly influ- sufficient "magnitude" to significantly affect stability and/or ence the airframe aerodynamics.

performance of the feedback systems. The analysis technique On the other hand, commands in thrust reversing, thrust also addresses the issue of the system's robustness against vectoring, and ventral and ejector thrust may cause pressure uncertainties in these interactions. Airframe/engine interac- disturbances in the augmentor or mixing plane. If the nozzle is tions are often a significant source of uncertainty in the sys- operating in an unchoked condition, these pressure distur- tem's dynamics.

bances may propagate through the fan bypass duct and cause Another objective of the paper is to use the analysis ap- engine transients such as a reduction in fan surge margin.

proach to evaluate airframe/engine czoss-coupling on a vehicle Reaction control system jets, used for airframe attitude con- that has been the subject of several studies in IFPC. The anal- trol, as well as upper wing surface blowing, used for lift aug- ysis reveals that critical cross-coupling is not present for this men•at•on, usually draw bleed air from the engine's compres- vehicle, as modeled, for the operating condition and control sor. Thus, core flow dynamics can also influence the lift and configuration evaluated. As a result, the classical control laws attitude responses of the airframe. Increased RCS thrust will considered in this example would appear to deliver adequate cause reduced core pressure due to compressor bleed flow stability robustness and performance. A second control con- demand, creating engine flow disturbances. Also, fright dy- figuration is then considered, and the analysis shows increased namic pressure, angle of attack, sideslip angle, and inlet flow cross-coupling due to an added reaction control system (RCS) distortions can influence the effectiveness of the RCS control causing a significant degradation in engine loop performance.

jets and cause reduced fan surge margin.

Roll

\ \ / /

Presented as Paper 90-1918 at the AIAA/SAE/ASME/ASEE 26th Joint Propulsion Conference, Orlando, FL, Jub" 16-18, 1990; received Oct. 3, 1991; revision received Feb. 28, 1992; accepted for publication March 6, 1992. Copyright © 1992 by John D. Schierman and David K. Schmidt. Published by the American lnstirule of Aeronautics and Astronautics, Inc., with permission.

*Research Asso_ate and Doctoral Candidate, Aerospace Research Center, College of Engineering and Applied Sciences. Student Mem- ber AIAA.

Tlmlm" VeN_ Wi_ Thnm Veoom_ tProfessor of Engineering and Center Director, Aerospace Re- Nm2k A Thrust RcverDng search Center, College of Engineering and Applied Sciences. Associate Fellow AIA.A.

Fig. ! Typical vehicle configuration.

PA_ _ NOT FILMED

SCHIERMAN AND SCHMIDT: AIRFRAME AND ENGINE INTERAC'TION$ 1389 f _ l_m_ Tlna RInqrmL l_mm ver.ml I " _ Aaat_m_a tA l:lt_m, rlvdu low, it is implied that the analysis is being performed for a specific operating condition and a specific engine control - I un -.,, c._ I- Ics 7_w ,,_ I =, B. 4 I _ W_*| Sw.rir_lUo_ml mode.

if it can be assumed here that any gain scheduling leads to slowly time-varying gains, then the particular feedback system being considered can be treated as (approximately) time invari- ant. In this case, the system nonlinearities reside primarily outside the feedback loop, and the purpose of feedback is to force approximately linear behavior between ._ and .to. The analysis framework that follows focuses only on the feedback portion of the system. However, this does not imply that the prefilters, gain scheduling, limit logic, etc., outside the feed- back loop are not important to the system design, but that stability and performance of the feedback loops are funda- Fig. 2 Example Interactions between airframe and engine subsys- mental to a successful design. Furthermore, since the feedback reins.

control loops for the airframe and engine are, under current practice, developed by different organizations, it could be ar- Flight Envek_c gued that interactions in these loops would constitute the most Iofo_tuon difficult design challenge.

Now, more specifically, consider the aircraft dynamics iso-

I J

lated from the engine dynamics, with input/output character- istics defined in terms of a matrix of transfer functions G$ (s), __ (Nmli_.ar Funcuo_) where Gain I Scheduling yA(s) =G#(s)uA(s) (I) Likewise, let the isolated engine's input/output characteristics

ry,..1 ,, P";I r,.1

be defined in terms of a matrix of transfer functions G_(s),

F

where y_(s) = G_(s)uE(s) (2) Fig. 3 Full nonlinear airframe/engine system represeulntion.

Consider that each of these systems will be acted on by feedback control compensation matrices KA (s) for the aircraft flight control system, and KE(s) for the engine control system.

It is important to note here that, for the type of vehicle The associated engine feedback system is shown in Fig. 4 [note being considered, the propulsion system not only affects the again that KE(s) and G_(s) are, in general, matrices].

(slower responding) transitional velocity of the vehicle but also The closed-loop quasilinear responses of this system are may be both a lift and moment "actuator," affecting the given by vehicle's (faster responding) attitude dynamics. All of these interactions just described between the airframe and engine are YE(S) = [I 4- G_($)K£-($)] -IG_- ($)KE(S)yEc($) (3) shown in Fig. 2.

and the closed-loop characteristic polynomial is Analysis Framework The technique to be presented is a quasilinear approach for _d(S) = q_(s) det [1 + G_(s)K_(s)] (4) assessing airframe and engine interactions) ° This procedure seeks to provide a better understanding of the effects of these where the roots of ¢bo_(S) are an aggregate of the poles of G_.(s) interactions. It is recognized that many of the interactions and g_(s).

discussed previously involve nonlinear phenomena, and de- tailed nonlinear simulations wig ultimately be required. How- ever, the justification for the quasilinear analysis and the treat- ment of engine limits is specifically noted herein.

Consider the airframe/engine nonlinear system similar to that discussed in Refs. 11-13 and shown in Fig. 3. Y.4c is the vector of commands to the flight control system, and y_ is the . vector of commands to the engine control system, uA is the Fig. 4 Block diagram of tbe isolated engine feedback loop.

vector of aircraft control inputs (flap deflection 6F, thrust vec- tor nozzle deflection brv, etc.), and us is the vector of engine control inputs (fuel flow rate wr, nozzle area AT, etc.). Fi- nally, YA is the vector of aircraft responses (angle of attack a, pitch rate q, etc.), and YE is the vector of engine responses (turbine temperature 7"4, fan speed N2, etc.).

Implicit in the feedback portion of this system is that the matrix G(s), the quasilinear input/output mapping of the ve- hicular system, is a member of a set of such mappings, g(s), and strongly depends on the particular flight and engine oper- ating condition. In fact, each such operating point manifests a particular quasilinear system model and control architecture, which define the matrices G(s) and K(s). Furthermore, these mappings may reflect a particular control mode, such as "rid- yr=d_s) ing an engine limit." In such a case, the controlled responses YE(S) depend on the operating limit. In the discussion to fol- Fig. $ Block dligrtm of Ihe coupled airframe/engine wslem.

1390 SCHIF.RMAN AND SCHMIDT: AIRFRAME AND ENGINE INTERACTIONS But since the airframe/engine system dynamics are in fact "_ Deviaeion dee coupled, their input/output characteristics are more accurately represented as c_ gEKE _ _• O,m

'""'l = r °""' °'"'1[""'1 =

yE(s)J LG_ (s) G_(s) J Lut(s)J " "LU_lsIJ C5) where, again, GA(s), Ge(s), O,tE(s), and O_(s) are, in gen- eral, matrices. Note also that GA(s) and Ge(s) may differ Margins _k_ _ from the decoupled subsystem models G,_($) and G_:(s) by some amounts &A (s) and Ae(S), respectively, due to the eross- coupling actually present between the airframe and engine systems. That is,

,.',4

GA(s) = G,_(s) + AA(S) Frequency (6) O_(S) = _2(S) + A_(S) FJI. ? Example nonJnleractJng (solid line) and interacting (dasbed line) systems' engine loop truLsfers.

Further, G,cE(s) and G_ (s) represent any input coupling that leads to the open-loop engine control inputs influencing air- frame responses or the open-loop ah'frame control inputs in- Conversely, from Eq. (8), note that the disturbance interac- fluencing the engine responses, respectively. Now, if both tion matrix DA(s) is independent of GAE(s). Hence, it may be G, cE(S) and G_ (s) are "large," the system is said to exhibit "large" if G_ (s) is "large," even though GAE(S) is "'small.'" two-directional coupling. If only one is "large," the coupling That is, the disturbance interaction matrix can be "large" even between the subsystems is primarily one-directionaL if only one-directional coupling is present.

The actual coupled system, under the influence of the air- Finally, both the additive and disturbance interaction ma- frame and engine control feedback compensation KA (s) and trices depend explicitly on the airframe control laws KA(s). If KE(s), is then shown in Fig. 5. In this figure the lower portion the airframe loops are not closed [KA (s) = 0], EA is) reduces to of the block diagram is the original engine loop, but it is no &_(s) and DA (s) reduces to zero. Consequently, the phenom- longer isolated from the airframe as in Fig. 4.

enon of interest here is fundamentally one involving feedback.

Figure 5 reveals how, for example, the coupling dynamics To reveal the import of the additive and disturbance interac- GAt(s) and G_ (s) and the airframe dynamics GA (s), aug- tion matrices, note that the quasillnear responses of the engine mented with the airframe compensator KA(S), interact with system in Fig. 6 are the engine loops. (Note that a dual exists for the effects of the coupling and augmented engine dynamics on the airframe

y..ts) = [i + (ol + E,,)KE] - '((:;Z- + E,, )K,y_ is)

loops.) Through block diagram manJpnlation, the system in Fig. 5 may be represented as in Fig. 6, where

+ [I +iC; +,EA)X_]-'D,,y,,,(s) (9)

EA(s) = % - C_ [I +X,,,,(C_ +/'-,,)]-'X,,C;_ (7) Comparison of the decoupled engine system's input/output relationship of Eq. (3) with the truly coupled system's input/ D,,(s) = (:;_ [I + X,,,(C_ + _,,)]-'X,, (8) output relationship of Eq. (9) reveals that the additive interac- tion matrix EA (s) can affect both stability and performance of (Note that functional dependence on s is not indicated in some the engine feedback system. However, the disturbance interac- of these terms to simplify notation.) Because of the manner in tion matrix DA (s) does not affect stability of the quasilinear which these terms affect the engine loop, E,,(s) will be re- system, since (as shown later) the characteristic polynomial of ferred to as the additive interaction matrix, and DA(s) will the closed-loop coupled system is independent of this matrix.

be referred to as the disturbance interaction matrix. Clearly, if Clearly, however, DA (s) has an impact on the engine control _A(S), At(S), Gm_(S), and GF_4(s) are not really zero, the system performance. Commands into the fright control system engine loop is not actually that shown in Fig. 2 but rather that YAc(s) disturb the engine responses through DA (s) and appear shown in Fig. 6 as output disturbances to the engine control loops. Thus, if The critical expressions of Eqs. (7) and (8) reveal several key DA (s) is large, the closed-loop engine performance will suffer.

facts. First, Eq. (7) shows that the additive interaction matrix Quite significant is the fact that E_(s) can affect the inter- EA (S) depends on the weighted matrix product of the input acting system's closed-loop stability. The closed-loop charac- coupling transfer matrices Gt._ (s) and GAE(S), the airframe teristic polynomial for the coupled system is dynamics G_ (s) + AA(s), and the change in the engine transfer function matrix due to coupling _E(s). E,4 (s) will therefore be Od(S)=r_ol(s)del[l + [G_(s) + EA(s)]K_(s)] (10) "small" (for example, small maximum singular value) if At is "small" and if either GAE(S) or Gr.4(s) is "small." Thus, if Here the roots of _o_(s) are an aggregate of the poles of only one-directional coupling is present, the additive interac- Kf(s) and the poles of the system with only the airframe tion matrix will tend to be "small."

loops closed with KA(S), or the values of s for which dell/ +GAis)Kais)]=O. These facts are derived in Appendix A.

Now it can be shown from Nyquist stability theory _4 that the closed-loop system in Fig. 6 is assured to remain stable if the y._s) --_ feedback loop is stable for EA(S)=0, and if urns)

(,l)

for all frequencies _>0. It can further be shown that this is _(s) assured if Fig. 6 Block dht|ram of tbe engine feedbnck loop internctJngwith

+ for an

the airframesubsystem.

(12) SCHIERMAN AND SCHMIDT: AIRFR.AME AND ENGINE INTERACTIONS 1391 BE.A kA dA(S) = (16) ! + kA (Z,_ + _,,I ) Equation (15) shows clearly that e,4 (s) is a strong function of the frequency-dependent (weighted) product of Z_(s) and &Ae(s). Hence, if either £Ae($) or f_(s) (or both) are small and 5e(s) is small at critical frequencies, then ca(s) will tend to be small at those frequencies.

The characteristic equations in Eq. (14) also show that if eA(s) is large, then gain and phase margins present in the decoupled engine loop transfer [ke(s)8_(s)] may be eroded 10-_ )0e lot in the coupled engine loop traasfer, as depicted in Fig. 7.

f-requcacy in Rad/Scc However, from Eq. (12), stability of the coupled system is assured if Fig. g Optn-loopnormalizedtransferfunction uugniludu.

leA(jo_)ke(j_)l<ll +e2(j_)ke(Jo_)l for all ,o>0 (17) ,to[ Stability garlimS which is the scalar form of Eq. (12).

Note that the focus of this analysis has been the effect of ..... ...... ........................ " to +a_%)i ...; airframe dynamkz on theengine loop.A dualanalysis reveals how theinteractiom affect the airframeattitude loops. That is, the dualof Eq. (9)gives the airframeresponses forthe interacting system as i yA(s) = [I+(G; + E_)K,,]-'(O_ +E_)X,,y_,(s) + [I + (G_ + Ee)KA] -'Dey_(s) (18) IOn 1oo r-mwm_ i_ I_dtS_ where the interaction matrices Ez(s) and D_(s), given below, are the duals of EA(s) and DA(s): Plot of Eq. (In, the scalarform of Eq. (12).

Fill. 9 Ee(S) ffi AA - GA_[I + Ke(GZ + Ae)]-' KeGF.,_ 119)

if De(s) = (2o)

The airframe loops are assured to remain stable in the presence

] <_.[i + -']

of interaction uncertainties as long as for all _o>0 (13) _[Ee(j,.,)KA(j_o)]<o_[l+G_(jw)K,q(jto)] for all _>0 where 8 and o denote the maximum and minimum singular

(21)

values of a matrix, respectively.

These key inequalities are measures of the overall system's stability robustness with respect to uncertainties in airframe/ engine interactions. In fact. the system's robustness can be to : : i i i iii_ : i _ i indicated by plotting both sides of Eq. (12) or (13). Itis evident that there will be loss of robustness at frequencies where EA (s) is "large" (i.e.. if its maximum singular value is large). At these critical frequencies, a stability robustness margin may be 2O defined as the distance between the left- and riOt-hand sides of Eq. (12) or (13). Since EA(s) is a strong function of the ........ : ....... .....................

-20 cross-coupling dynamics GAe(s) and GeA (s), small variations in elements of either GAE(S) or Ge_(s) at some critical fre- quency may reduce this margin to zero and thus lead to the failure of the aforementioned stability criteria.

-%

IOe IOn The significance of the preceding results may be seen more Pmqumc-y u. iUOfoec dearly by considering a single-input/single-output engine con- Fill. i0 F_m81_ performuee ..mdy_.

- trol system. Let the regulated engine response of interest be, for example, fan speed N2, and, for a fixed nozzle area, let the control input be the main burner fuel flow rate WF. In this case, the transfer function matrices G_(s), AA (S), Kz(s), and E,q(s), as well as DA(S), reduce to scalars, denoted by g_(s), 6A(s), kE(s), eA(s), and dA(S). Then Eq. (9) reduces to the q scalar relationship (g_+eA)kE ] 1+(g_+eA)

[ [

ye(s) = 1+ (e_ +----_A)kEJ y_ + dAY_

(14)

Also, if all system transfer functions are assumed to be scalars, Eqs. (7) and (8) reduce tO eA(S) = 6F. -- I +kA(g,_ +iSA) (15) 1392 SCHIERMAN AND SCHMIDT: AIRFRAME AND ENGINE INTERACTIONS ate the relative sizes of the input/output relationships of the which is the dual of Eq. (12). Also, "large" DE(s), for exam- airframe and engine, the system must be normalized by, for ple, will degrade the flying qualities of the flight control system example, estimates of the maximum values of the controls and due to disturbances arising from engine commands.

As a final note, this analysis does not necessarily require responses. The values used to normalize this plant are given in Table 1 and are taken from Ref. 9.

analytical models of the airframe and/or engine. Input/output mappings of the system could conceivably be experimentally Figure 8 reveals that the cross-coupling terms gAE(S) and obtained, and graphical data could be used exclusively to ob- g_(s) are both smaller than the diagonal elements in Eq. (22) tain plots of Eqs. (7), (8), and (12), for example. by approximately 40 dB for frequencies above I rad/s. (Re- call that the loop gain cross-over frequencies are around 3-5 rad/s.) Also, since there are no visible differences in the plots Two Case Studies of hA(s) and g_(s), and he(S) and g_(s), &A(S) and &f(S) are quite small. Hence, from Eqs. (7), (8), (19), and (20), eA(s), The techniques just presented will now be used in the anal- d,t(s), ef(s), and dE(s) should all be quite small, and it might ysis of an airframe/engine system that has been the subject be expected that airframe/engine interactions will be negligi-.

of several investigations of integrated flight and propulsion ble. However, the complete analysis requires knowledge of control) J.4... The baseline vehicle to be considered is repre- candidate control laws, since feedback compensation could sentative of a high-performance Short Takeoff and Landing increase critical cross-coupling.

(STOL) fighter aircraft equipped with a thrust-vectoring/ Shown in Fig. 9 are plots of both sides of the key inequality thrust-reversing nozzle. The operating point under consider- of the stability robustness analysis, Eq. (12) or (17). This figure ation is the approach-to-landing flight condition at an airspeed shows that ]eAkEJ for the baseline configuration is much of go -- 120 kt and flight-path angle _'e= - 3 deg. The quasilin- less than J I +g_k d throughout the frequency range shown.

ear vehicle system model is that given in Refs. 10, 15, and 16.

The stability margin, defined here as the minimum distance A second configuration will also be considered, which is iden- between the left- and right-hand sides of the inequality of tical to the baseline but with a high-pressure RCS added. Al- Eq. (12) or (17), occurs near 0.2 rad/s and is approximately though significant airframe/engine coupling may be expected, 40 dB for the baseline configuration. Therefore, the analysis the analysis will show that little critical interactions exist for indicates significant engine loop stability robustness against the baseline configuration, and only one-directional coupling uncertainties in airframe/engine interactions.

is present for the configuration that includes the RCS. Note Figure 10 presents the magnitude of the engine's fan speed also that, although the analysis herein involves only single-in- sensitivity function 11/[! + (hE +ea)kE]] along with the mag- put/single-output systems, the last section presented a multi- nitude of the engine loop disturbance interaction due to pilot variable methodology and thus is not restricted to scalar sys- input dA (jt_) [Eq. (8) or (16)] for the baseline configuration.

tems.

The spectrum of the engine response because of these distur- For both cases the airframe's dynamics are aerodynami- bances, or N2/6mck, is shown in Fig. 11, also labeled as the cally unstable. The airframe flight control design objective is baseline configuration. This response is, of course, the prod- to stabilize the airframe's dynamics and obtain classical pitch uct of the two terms plotted in Fig. 10. These plots reveal that rate and angle-of-attack responses from pilot pitch stick in- the fan speed loop will reject disturbances arising from pilot put/_. The objective of the engine control law is to regulate pitch inputs, since g_¢ (J"0 is small.

the fan speed. The control laws for both cases are given in In summary, the analysis of this airframe/engine system Appendix B.

description indicates that the additive and disturbance interac- Ca_! tion effects ea(s) and dA(s) are small [and although not shown, eF(s) and de(s) are small as well]. Hence, the coupling The open-loop system is described as in this vehicle will not significantly degrade the closed-loop performance of both the airframe and engine subsystems; the system is therefore robust against interaction uncertainties and

(22)

N= L&_(s) gE(s) J

decentralized control laws appear quite adequate.

where, for example, - 14(s + 0.03 ±O.07j)(s + 0.6)(s + 1.4)(s + 3.6Xs + 7)(s + 90)

g,4 (s) =

(s + 0.064-0.2jXs + 1.4)(s - l.SXs + 2Xs + 3.6Xs + 7Xs + 90) 1.3(s + 0.06 4- 0.2j)(s - 1.5)(s + 2)(s + 16 4- 6j)(s + 37) (23)

gHs) =

(s + 0.06 4- 0.2j)(s + 1.4)(s - 1.5Xs + 2)(s + 3.6)(s + 7)(s + 90) Note the unstable mode at 1.5 rad/s. From Appendix B, the control law is /tt, where w! : fuel flow rate, and the pitch attitude control 6p.ch, .,to the feedback gains K,. and K;¢, and the pilot stick gain K6= are given in Appendix B. These control laws lead to gain cross- over frequencies in the engine and aircraft pitch loops of ap- I_1 lot) 1OI proximately 3 and 5 tad/s, respectively.

Fnaluency mRed/See Shown in Fig. 8 are the magnitudes of the input/output mappings in Eq. (22), as well as the mappings for the decou- Fi s. 12 Open-loop oormsfized transfer function magnitudes with pied airframe and engine g_(s) and g,f(s). To properly evalu- pitch RCS control included.

SCHIERMAN AND SCHMIDT: AIRFRAME AND ENGINE INTERACTIONS 1393

Table i I_dmle; of mujmum over the frequency range shown. Hence, strong one-direc- values of controls and responses tionai coupling is indicated.

The large increase in the magnitude of g_ (j¢) causes the qm_u = 0.06 rad/s magnitude of da (j_) to significantly increase [see Eq. (16)], as amax ffi 3 des shown in Fig. 10. This figure indicates thai the engine loop can Nz _ = 570 rpm no longer effectively reject fan speed disturbances arising from 6TV mu -- I 0 des pilot pitch stick inputs. In fact, Fig. 1 ! shows the significant wfmax = 5000 Ib/h increase in the magnitude of the fan speed response due to pilot pitch stick input over the baseline case.

Table 2 Addlelve Furthermore, the increase in magnitude of ge.4 (j_0) causes perturbations or gAE (j_) an increase in magnitude of eA (jw) over the baseline configu- ration as well, as indicated in Fig. 9. Hence, stability robust- Case 6i&AeJ, (rad/s)/Ob/h) ness against uncertainties in airframe/engine interactions is 1 !.6 reduced. Figure 9 shows that the stability margin is reduced 2 3.2 from the baseline configuration to approximately 20 dB. again 3 4.7 measured at 0.2 rad/s, it is worth noting that this critical 4 6.3 frequency is well removed from the cross-over frequencies of 5 6.7 the airframe and engine loops (3 and 5 tad/s). Note that in The airframe/engine system's closed-loop airframe transfer functions [see Eq. (18)] are -O.l(s +O.O6*O.2j)(s + 30) d(__) s + -4e-4(s + 24-O.6jXs +4)(s + 5)(s-76) (_mm) a(s) = (s +O.OS±O.2jXs + 2.8±2.8j) Tl(s) 6akt( ) (s +O.O5±O.2j)(s + 2.8±2.gj) T2(s) N_(s) q(s)= - O.05s(s +O.07Xs + 0.5) rad/s -4e-5(s+2Xs+3Xs+7±2jXs-21) I-'_l (s + O.O54-O.2j)(s + 2.8 ±2.gj) Tt(s)iTI6'ed(s) + (s +O.O5-,-O.2jXs + 2.84-2.gj) T2(s) Nk(s) where (s + 0.4)(s + 2 ± 4j )(s + 8Xs + 90)

(25)

T,(s) = Tz(s) = (s + 0.4Xs + 2 ±4jXs +8Xs +90) (s + 0.4Xs + 2 4-4jXs +8)(s +90) and where Tt(s) is unity to the accuracy displayed, indicating that engine modes are essentially unobservable in the airframe responses. The transfer functions between the airframe responses and commanded fan speed N_ are also quite small since the disturbance interaction effect dE(s) is small.

The closed-loop fan speed response [see Eq. (9) or (14)] for the airframe/engine system is N2(s) = (s +'_RS+i_"4_S+90) Tt(s) N2c(s)+ (s_O.'_'+-2-_j'_s+8--"_90) T2($) 5ai_(s) where (s + 0.05 * 0.2jXs + 2.8 • 2.8J) s(s + 0.4Xs + 3 _-2j)

T,(s) = T=(S) = (26)

(s + 0.05 ± 0.2j)(s + 2.8 4- 2.8j) (s + 0.05 4- 0.2jXs + 2.8 -,-2.8j) As with the airframe responses, Tt(s) is unity, indicating that this situation small increases in the magnitude of g_(jt0) may cause a substantial increase in the additive interaction term airframe modes are essentially unobservable in the engine re- sponse. The fan speed response from pilot pitch stick input is eA (rio), since this term is a strong function of the product of quite small since dA (s) is small. g,4E(j_0) and g__A(jto) [Eq. (15)]. Hence, small variations in gAE(j_0) may therefore cause significant degradation in stabil- Case2 ity robustness and/or performance. For these reasons, a sensi- tivity study will be performed on gAe'(jt_).

Now consider the same vehicle with similar control laws but For this vehicle and control system configuration, the pitch with pitch attitude control power enhanced by a combination trim occurs at a small thrust-vectoring angle brv. Thus, engine . of thrust vectoring and pitch RCS jets. RCS jets, which draw bleed flow from the engine's compressor, will directly in- fluence the quality of airflow through the engine, thus increas- ing airframe/engine interactions. Models of the effects of bleed flow on the propulsion system were provided by the NASA Lewis Research Center. The control laws for this configuration are also detailed in Appendix B and are such zo _":_, .......... '"o) i..i..i.it that the airframe and engine control loops, cross-over frequen- cies, etc., are essentially the same as those for the baseline configuration.

The magnitudes of the elements of the plant transfer func- tion matrix [Eq. (221] are shown in Fig. 12. Again, the plant was normalized using the maximum values of control inputs -I tO) and responses given in Table 1, and the maximum value of the Fn_m_ i- lUd_S.c pitch RCS jet nozzle area A e was 1 in.=. When compared with Fig. 8, this figure shows that the addition of pitch RCS control !_1-13 IP_ of l_i. (17) for _.rtom maZsJ_Ies of _o_ir- increases the magnitude of gE.4 (fie) by approximately 50 dB frame Inter'actions, igAEU_)I.

1394 SCHIERMAN AND SCHMIDT: AIRFRAME AND ENGINE I_TERACTIONS exhibit few interactions. Classical decentralized control laws therefore appear quite suitable. However, the analysis revealed significant one-directional cross-coupling for a second control .............. , I .............. : ...... configuration with a reaction control system added. Inclu- sion of the RCS jets led to significant disturbances in the fan speed loop arising from pilot pitch inputs, and reduction in the stability robustness against variations in airframe/engine interactions was also recorded. The analysis accurately indi- cated the frequency at which instability would first occur i II • .4 _ , -0.2 0 0-2 due to these variations. Frequently, only engine-to-airframe -4 -2 0 2 4 interactions are thought to be of concern; however, this case clearly indicates strong alrframe-to-engine coupling. In some Fig. 14 Locus of the alrframe/englne system's dosed-loop poles as previous IFPC studies only engine-to-ah'frame interactions tbe magnitude of IAE(]_) is increased.

were thought to be of concern. Although this may have been a valid assumption for the vehicle configurations examined, - analysis methodologies should, in general, consider two-direc- thrust transients will not generate large pitching moments, and tional coupling.

this is the reason gAE(S) is small in this case. If the vehicle configuration was such that the trim thrust-vectoring angle Appendix A: Derivation of Eq. (10) were large, thus increasing the component of the thrust vector perpendicular to the airframe's longitudinal axis, engine thrust Let a state-space realization of the input/output mapping transients would create larger pitching moments. In such a for the fully coupled aircraft/engine system be defined as case, gA_(s) would be larger.

±A AA A,ce 1 x,4 Figure 13, like Fig. 9, shows the inequality of Eq. 07). This

[:]=[ , if. i

figure, however, displays leAkEJ for various values of the mag- Jc A_ AE] xe B_ BeJ[uL] nitude of gAE(j_). Here,

["]- olr .l

yE CEJ Lx_J fAD gAE(Jw) = (Jg_loou_ + 81g/zl) e,'_ (2"/) Table 2 lists the additive perturbations of the magnitude of and the mapping given as gAE(j_) corresponding to thedashedcurvesin Fig. 13.

Figure 13 shows that leAkal is much less than 11+gEkal

'"("1 : [ =

throughoutthefrequency range forthe nominal magnitude of

Ye'(s)J LG_(s) G.,-(s) J LuE(s)J L e( )J

gAE(J_), and stability of the system isnot in jeopardy. How-

(A2)

ever,thestability margin reduces tozero(le, ck_[ = [ I+g2k_[ ) at -0.2 rad/swhen themagnitudeof gAE(j"_) isincreased by withsystemcharacteristic polynomial only 6.'/ (rad/s)/(Ib/h) (case5).From Fig. 12, note that - AA_ gAz(j_), thusincreased, would become comparable inmagni- detrsZ-AA sI-A_] (A3) tude to the othertransfer functions in thesystem.

_s(S) = L - AF.,4

Figure 14 shows how the closed-loop eigenvalues of the system varyas themagnitudeof gAE(J_) isincreased. Higher Also let the state-space descriptions of the aircraft and engine frequency enginepoles arenot shown and do not varyto any compensation KA (s) and K_(s) be, respectivdy, great extent. However, this figure shows that a low-frequency Cohugoid mode) instability does indeed occur at a frequency x,_ = AkAXkA + B,AeA,, uA = C*,XkA of 0.2 rad/s. Further, this instability occurs precisely for the increase in magnitude of gA_(jco) corresponding to case 5 in ±xe= Atcxtt + B_eea_, ue = Ctrxt¢ (A4) Fig. 13. It is also significant that the critical frequency of instability (0.2 rad/s) is not near the engine or airframe loop where eAc(S)=YAc(s)--YA(S) and e_(s)=y&(s)-ye(s) are cross-over frequencies where phase margin is measured and theinputs to the an'craft and engine compensators. The char- that Eq. (17) correctly indicated that instability will first occur acteristic polynomials of thesecompensators are at this critical frequency due to variations in akframe/engine interactions.

¢_,_(s) = det(sl- A,_) Conclusions ¢_te(s) = det(sl - A ,a) (AS) Expressions were derived for additive and disturbance inter- action matrices that may be used to quantify the significance Sought now isthestate-space description of G_(s)+ EA (s), of airframe/engine interactions on either the engine control as presented inFig. 6.UsingEqs. (AI)and (A4), and referring loops or, for the dual analysis, the flight control loops. A to Fig.5,yields thedesh'ed result, or technique for determining the stability robustness of the sys- tem against uncertainties in these interactions wa¢ presented.

The size of the interaction matrices in critical frequency _ce = A_4 A e B_._C, A xE ranges, measured, for example, by their singular values, quan- tifies the effect of airframe/engine coupling on closed-loop Xta -- BIt_CA 0 A k_ Xt,_ stability and/or performance. The critical interaction matrices were shown to depend on the control compensation as well as the input/output characteristics of the airframe/engine sys- YAc tem. If the system exhibits two-directional coupling, stability as well as performance may be compromised. Systems with one-directional coupling may preserve adequate stability ro- bustness, although performance can be seriously affected.

This analysis was then applied to an alrframe/engine system y_--[O C_ O] x_ (A6) considered in previous integrated control studies, and two

I..X,__J

cases were presented. The baseline configuration was shown to SCHIERMAN AND SCHMIITT: AJRFRAME AND ENGINE IN'FI_LACTIONS 139._ Table BI Airframe eoalrol law galm Denoting this system as Gain 5_ch = 6-rv 6_ch = 6re -- 8A ¢ ±1 = Aix, + B_uE + B2YA¢ K/=, deg/deg - 2.9 - 4.6 Kte, deg/deg - 3.7 - 0.1 YE = Cix, (A7) Ktq, deg/rad/s - 56.5 - 2.3 Ki=. deg/Ib - 0.7 - 0.5 it can be shown I_ that the characteristic polynomial of this system [G_(s) + EA (S)] is The engine control input is w/= main burner fuel flow rate,

¢,(S) = det(sl - A l) = ¢s(s)ckt_ (s) det [I + GA (s)K,4 (s)]

Ib/h.

(AS)

The aircraft responses are a ffiangle of attack, deg, and q = pitch rate, rad/s.

Appending the state equation for the engine compensator The engine response is Nz = engine fan speed, tom.

KE(s) to the state equation for O_:(s)+ EA (s) gives the state- Two cases are presented with different control architectures space description of the open-loop system of Fig. 6 { or [G2(s) for pitch attitude control, defined as 6_. For the first case, + EA(s)]KE(S)I as pitch is controlled only by thrust vectoring, thus, 6_ = 5re.

For the second case, pitch is controlled by a "blend" of both thrust vectoring and pitch RCS jet nozzle area, def'med as

[:j: r-,1 +.,..., ,x..., L-...,r: It]""

6_=6rv-SA¢.

The airframe's short period mode is unstable, and the con- trol objective is to stabilize the short period mode and obtain o,[:j <.,, a desired modal frequency near 4 rad/s and a damping ratio of 0.7. This is achieved by feeding back angle of attack and pitch The characteristic polynomial of this system is rate to pitch control. The other airframe control objective is to increase the flight-path time constant (usually denoted as [sI-AI - Bt Ctt] I/Tr_) to approximately 0.5 rad/s. This is achieved by feed- 02(s) = det L 0 = Ol(s)0tz(s) (AI0) ing back angle of attack to the flaps. Finally, the pilot stick sl - A t,J force gain is adjusted to give an approximate Bode gain on Closing the (engine) loop in Fig. 6, the state-space equation for q(s)/tf,_(s) of 0.03 (rad/s)/lb. In summary, the airframe the entire closed-loop system is then control laws are _flwa = - Kfaot [.]={ .,, JC -- BazCt At_ J kXt_

_1)

y_=[C, 0] [ ::,] (All) The values of the gains for both pitch control case are given in Table BI. Note that increased control power in using RCS and the characteristic polynomial for this closed-loop system is jets led to the reduced feedback gains.

Finally, to regulate fan speed, fan speed is fed back through proportional plus integral compensation, with gains of -6 . rsl-A, -B,c,,] _d($) = aetL B,rc, sl-AteJ 0b/h)/rpm and - 3 [0b/h)sl/rpm, respectively.

The effects on system stabifity of the low gain flap loop are = ¢bt,(s)C)t,(s) det [I + (G_ + EA)KE] (AI2) minimal. Therefore, the two-by-two system shown in Eq. (22) is obtained by first dosing the flap loop and then combining or the two aircraft attitude responses (a and q) to form one blended aircraft response.

¢kd(S) = Os(S)_tA (S) det[I + GA (s)KA(S)]

(AI3)

X 0rE(s) det [I + (G_* + E,q)KE]

Acknowledgments This work was sponsored by the NASA Lewis Research De f'ming Center under Grant NAG3-998. Peter Ouzts and Sanjay Garg have served as technical program managers.

Ool(s) = _bs(s_t_ (s) det [I + GA (s)KA (s)] 4,rE(s) (A 14) gives References Od(S) = _,_(S) det [I + (GZ + EA)KI] (AI 5) ISmith. K., and S_ewgtt, C., "A Survey of Control Law Options for Integrated Flight/Propulsion Control for Fighter STOL Approach," which is the result presented as Eq. (10). Note that det[l Proceedings of the AIAA Guidance and Control Conference (Seattle, + GA (s)KA (s)] is a rational function with denominator equal WA), AIAA, New York, 1984 (AIAA Paper 84-1900).

to _bs(S)cktA(s). Thus, the roots of O,_(s) are the roots of 'Shaw, P., Blumberg, K., Joshi, D., Anex. R., Vincent, J., and Skim, C., "Development and Evaluation of an Integrated Flight and _t_(s), which are the poles of KE(s), and the values of s for which det [I + GA (s)KA (s)] equals zero. Propulsion Control System," Proceedings of the AIAA Joint Pro- puhion Coqftrem_ (Monterey, CA), AIAA, New York, 1985 (AIAA Paper 85-1423).

3Smith, g. L., "Design Methods for Integrated Control Systems," Appendix B: Case Study Control Laws Aao Propuhion Lab., Air Force Wright Aeronautical Laboratories.

The following defines the controls and measured responses AFWAL-TR46-2103, Dayton, OH, Dec. 1986.

for the case study vehicular system used in the analysis.

_Rock, S.M., Enmmi-Naeini, A., and Anex, R. P., "Propulsion The aircraft control inputs are: _-v = nozzle thrust-vector- Control Specifications in Integrated Fright/Propulsion Control Sys- tems," Procmdin_ of the AIAA IA SME/SAE /A SEE 24th Joint Pro- ing angle, deg; A¢ =pitch RCS jet nozzle area, in.Z; and 5n_ pubion Co_tferenoe (Boston, MA). AIAA, Washington, DC, 1988 = trailing-edge/leading-edge flap deflection angle, in.2 1396 _HIEIU_AN AND SCHM[DT: AIRFRAME AND ENGINE INTERACTIONS (AIAA Paper 88-3236).

tIPeczkowski, J., and Stopher, S., "Nonlinear Multivariable Syn- SShaw, P. D., Rock, S. M., and F'tsk, W. S., "Design Methods thesis with Transfer Functions," Proceedings of the 1980 Jams Auto- for Integrated Control Systems." Aero Propulsion Lab., Air Force matic Control CoherencY. Vol. 1. Pt. WAg-D, 1980.

Wright Aeronautical Laboratories, AFWAL-TR-88-2061, Daylon, 12Sain, M., and Peczkowski. J., "Nonlinear Multivariable Design OH, June 1988.

by Total Synthesis." Dept. of Electrical Engineering, Univ. of Notre 6Gar$, S., Mattern, D. L., and Bulhu'd, R. E., "Integrated Flight/ Dame, Control System Technical Rept. 36, Notre Dame. IN, March Propulsion Control System Design Based on a Centralized Approach, 1985.

Proceedings of the AIAA Guidance, Navi&ation, and Control Con- t3Schmidt, D., and SchJerman, J., "A Framework for the Analysis ferenee (Boston, MA), AIAA, Washington, DC, 1989 (ALAA Paper of Airframe/Engine Interactions and Integrated Flight/Propulsion 89-3520).

Control," Proceedings of the American Control Coqference (Boston.

7Berry, D., and Schweikhard, W., "Potential Benefits of Propul- MA), June 1991, pp. 761-766.

sion and Flight Control Integration for Supersonic Cruise Vehicles," 14Doyle. J., and Stein. G.. "Multivariable Feedback Design: Con- based on SAE Paper 740478, 1974.

cepts for a Classi_odern Synthesis." IEEE Transactions on Auto- STape, R., Hartill, W., et al., "Vectoring Exhaust Systems for matic Control, Vol. AC-26. No. 1. 1981. pp. 4-16.

STOL Tactical Aircraft," Journal of Engineerinffor Power, Transi- IsSchmidt, D., and Schiermaa, J., "Extended Implicit Model Fol- tions of the American Society of Mechanical En&ineer$, July 1983.

lowing as Applied to Integrated Flight and Propulsion Control."

CGarlh S., Mnttern, D., Bright, M., and Ouzts, P., "H-Inf'mity Proceedings of the AIAA Guidance, Navigation, and Control Confer- Based Integrated Flight/Propulsion Control Design for a STOVL Air- ence (Portland, OR), AIAA, Washington, DC, 1990 (AIAA Paper craftin Transition Flight," NASA TM 103198, Aug. 1990.

9O-3444).

'°Schmidt, D., Schierman, J., and Garg, S., "Analysb of Air- 16Schierman, J., and Sdunidt, D., "Robust Control Synthesis for frame/Engine Interactions--An Integrated Control Pergxx_ve," Integrated Flight and Propulsion Control," IEEE Conference on De- Proceedings of the AIAA/ASME/SAE/ASEE 26th Joint Propulsion cision and Control, Honolulu, HI, Dec. 1990.

Conference (Orlando, FL), AIAA, Washington, DC, 1991 (AIAA I?Kwakernnak, H., and Sivan, R., Linear Optimal Control Sys- Paper 90-1918).

tems, Wiley-intcrscience, New York, 1972.

Recommended Reading from the AIAA Education Series

Orbital Mechanics

V. A. Chobotov The only text specifically structured for the teach- ing of astrodynamics, this book selves the needs of senior-level undergraduate and graduate students as well as the practicing engineer.

The book reviews the fundamentals of kinematics, Kepler's and Newton's laws; addresses the applied, or engineering, aspects of orbital mechanics; reviews the solution of Kepler's equation along with orbital maneuvers; discusses relative motion in orbit and the various perturbative effects, including the mathematical foundations; examines orbital systems of satellites and "frozen orbits'; presents the basic concepts of interplanetary trajectories; and, finally, summarizes the current hazards associated with space debris.

1991, 375 PO, illus, Hardcover * ISBN 1-56347-007-1 AIAA Members $47.95 • Nonmembers $61.95 * Order #: 07-1 (830) your order today; can1-8001fAI2-AIAA elNAA, Ames'into In_ilme a4Aemnauli_ and ,_tmmulim Sales Th'-CA ,Jwadents.. 8.2S%: DC. 6% For shq_mO and i,,andlmO adld $4.75 tor1.4books icat Publicatgx_s Cu_omerService,9 Jay Gould Ct.,P.O. Boa753.Wai0od, MD 20604 lot mu lotIvDhw _anObes}. O_en une_$5000 must bepre_ P'lease altow 4 _ets b' Phone 3011645-5643, Oe_. 415. FAX 301/843-0159 daMry. Prims m sui01ecl Ioc_moe wm_o_ notre l_tums m_t beaccepti_ w_,m150ayx ANALYSIS OF AIRFRAME/ENGINE INTERACTIONS - AN INTEGRATED CONTROL PERSPECTIVE David IC Schrrddt ° and John D. Schicrman" Department of Mechanical and Aerospace Engr.

Arizona State University Tempe, AZ 85287-6106 Sanjay Garg* Lewis Research Center Group Sverdrup Technology, Inc.

Cleveland, OH large flow disturbances around the airframe is another example of airframe/engine interactions. Flight data was used to estimate that a double engine unstart, experienced by an XB-70 during a Techniques for the analysis of the dynamic interactions turn at Mach 3, would have produced a 2.5g acceleration and 30 between airframe/engine dynamical systems are presented.

deg/sec roll rate if the pilot had not taken corrective action. Data Critical coupling terrns are developed that determine the from a YF-12 airplane showed a yaw acceleration due to engine significance of these interactions with regard to the closed loop unstart was approximately 88% of the acceleration produced by stability and performance of the feedback systems. A conceptual maximum rudder deflection. The engine's bypass doors ('BPD) model is first used to indicate the potential sources of the were also seen to be as effective as the aileron and rudder coupling, how the coupling manifests itself, and how the controls in producing rolling and yawing accelerations. The magnitudes of these critical coupling terms are used to quantify rolling and yawing acceleration derivatives with respect to the effects of the airframe/engine interactions. A case study is bypass door opening (measured as % of maximum opening) of also presented involving an unstable airframe with thrust the YF-12 at Mach 3 are 0.35 deg/sec2/(percent max BPD) and vectoring for attitude control. It is shown for this system with 0.11 deg/sec2/(percent max BPD), respectively. These classical, decentralized control laws that there is little derivatives with respect to aileron and rudder deflections airframe/engine interaction, and the stability and performance (measured as % of maximum deflection) are 0.295 with these control laws is not e.ffected. Implications of parmr_ter uncertainty in the coupling dynamics is also discussed, and deg/sec2/(percent max 8aileron) and 0.073 deg/sec2/(percent max effects of these parameter variations are also demonstrated to be 8rudder), respectively.

small for this vehicle configuration.

In the design of the control systems for such aircraft, as well as for the propulsion system, one must properly account for lnlroduclion the dynamic coupling between the airframe and the engine, [2,3]. The purpose of this paper is not to discuss the design of a In the design of highly maneuverable fighter aircraft, or particular control system, but toindicate how the cross coupling for those capable of short take off and vertical landing (STOVL), dynamics between the airframe and engine can effect the the propulsion system is frequently being considered for airframe/engine system stability and performance, and to present augmenting the lift and the maneuvering capabilities of the methods for determining their significance from the perspective vehicle. Some designs include thrust vectoring to affect the of control system design. This problem is similar to that attitude of the airframe and thrust reversing to quickly change discussed in [4], for example. However, this cited reference flight velocity. Reaction control jets, drawing high pressure air does not fully explore the problem of two-directional coupling from compressor bleed, can be used for attitude control of the between the airframe and engine systems. In this paper, the aircraft. Compressor bleed can also be used for upper wing more general problem involving two-directional coupling is surface blowing to effect the boundary layer, thus the specifically m_ated. In the next section, the systems theory to be characteristics of the lifting surface. Changes within the engine used will be developed and presented, followed by a that either effect the thrust or compressor pressure will therefore demonstration of the theory and further discussion on the effects effect the airframe dynamics. Variable inlet geometry, used m of the coupling. Finally, a classical decentralized control law, effect the airflow through the engine can also effect the drag, developed for a vehicle that has been the subject of several pitch and yaw characteristics of the airframe.

studies on integrated flight/propulsion con_'ol, will be evaluated, On the other hand, the attitude dynamics of the airframe and it will be shown that for this vehicle system, critical cross can effect the airflow at the inlet to the engine, thus effecting the coupling is not present.

quality of airflow the fan and compressor receive. Deflecting the thrust-vectoring nozzle angle or thrust reverser port can effect System Analysis Preliminaries the back pressure on the turbine, especially if the nozzle is not choked. This change in back pressure propagates through the Let the aircraft perturbation dynamics defined in the engine, and constitutes an unwanted disturbance.

neighborhood of the relevant flight condition be described in Reference [1] presents particular examples of terms of a matrix of transfer functions GA(S), where, airframe/engine interactions from actual airczaft. This reference records that an F-104 airplane executing a rolling maneuver at

y^(s) =G^(s)u^(s) (1)

Mach 1.87 experienced sideslip, which precipitated an engine surge. This sudden reduction in airflow caused the inlet shock to move forward, which then caused a diverging yawing with y^(s) the vector of aircraft responses ( angle of attack, a, motion. The phenomenon of engine unstart, in which the pitch rate, q, etc.), and UA(S) the vector of aircraft control inputs, normal shock at the throat "pops" out of the inlet and causes (flap deflection, _F, thrust vector nozzle defle/:tion, 5rv, etc.)

" Professor, Assoc.F.cllow,AIAA Likewise. let the engine dynamics defined in the neighborhood *" Doctoral Candidate, Student Member, AIAA of the relevant operating condition be described in terms of a Aerospace Technologist. Senior Member, AIAA matrix of transfer functions GE(s), where,

ZAA f'Aeee, No. qO-ICllg)

/ RESe 7 O A'I ThE

Again, d(s) represents any disturbances acting on the system, yE(s) = GE(s)uE(s) (2) such as atmospheric turbulence. This closed loop system is governed by the following input-output relationship, [5], with yit(s) the vector of engine responses( turbine temperature, T 4, fan speed, N2,etc.), and uit(s) the vector of engine control inputs, (fuel flow rate, wF, nozzle area, AT, etc.)

y^(s)l= [I+ G(s)K(s']'IG(s)K(s f Y_:))] + [l+ G(s,K(s)]'ld(s)

yit(s) J Each of these subsystems will be acted upon by feedback (5) systems with control compensation matrix KA(s), for the flight control system, and KE(s), for the engine For a tracking and regulation feedback system, the control system, as shown below, for example, where y_ it the closed loop performance is defined in terms of how well the vector of desired or commanded responses. system's responses follow the commanded inputs and, at the same time, reject unwanted disturbances acting on the system.

d(s) Thus, from the above equation, the perfomumce objective of the control design is to make the matrix [I + G(s)K(s)]'tG(s)K(s) approximate the identity matrix in a certain frequency range, and make [I + G(s)K(s)] -t - 0 over the frequency range where dis) has significant power. Finally, the characteristic polynomial, A(s), for this closed loop system is, [5], - Block Diagram of the Engine Feedback Loop (6) A(s)= eel(S) det[l + G(s)K(s)] Here dis) represents any outside disturbances acting on the where the open loop system, G(s)K(s), has the characteristic system.

polynomial 0ol(S), which has roots equal to the poles of G(s) More generally, however, the aircraft/engine system and K(s).

input/output dynamics are The above expressions represent a very general case.

Typically, theapproachused in the control design issimpler, in that control cross-feeds may be absent. (i.e. KAE(S) and Kitx(s)

r _ru...(s:n

= . = [G(s)J

= 0). This implies the compensator matrix, K(s), is block diagonal. This situation may be represented as shown in the following block diagram, and will be the configuration considered in the remainder of this paper. The case with cross- where GA*(s) and GE'(s) are different from GA(s) and Git(s) feeds, although more complex algebraically, may be addressed above by the amounts AA(S) and &it(s), respectively, due to in a manner similar to that presented here.

dynamic cross coupling between the engine and airframe subsystems. That is, GA* = G^ + AA Git* = Git + Ait (3a) Further, GAE(s) and GL_(s) represent input coupling also due to ainerarne#engine dynamic interactions. Specific examples of these coupling effects will follow. Note that GA*(S), Git*(s), GAIt(s) and GEA(S) all have the same characteristic polynomial, denoted as ¢o1"(s).

In the most general case, the control compensation matrix may have the form.

r KA(S)

K(s) = LKEA(s ) Kit(s) J (4)

Etg_ - Block Diagram of the Airtn'aft/Enginc Loop With where the off-diagonal terms, KAE(S) and KEA(S), represent Diagonal K(s) control cross-fccds between the airframe and engine Each of the terms arising from the effects of the airframe/engine subsystems. The entire system is then representable in the coupling are apparent in the above figure.

following block diagram.

Note that if the compensation KA(s) and Kit(s) are synthesized assuming that the system is decoupled" the engine loop would be as shown in Fig. 1. For this system, the responses would be given by yit(s) = [I + G£KE]'IGEKE y_(s) + [I + GitKr_-td(s) (7) which of course differ from those given by Eqn. (5) if coupling is present. Also, the closed loop characteristic polynomial for the system in Fig. 1 is -Block Diagram of theAirframe/Engine Feedback System

(8)

A(s)= 0ol'(s) det[l + G#S)KE(S)] where theroots of eel'(s) arethe polesof GE(S) and KE(S).

Generally, the roots of this polynomial would notbe a subset of those for ¢o3(s)of Eqn. (6).

yE(s) = [1 -,-(GE+EA)KE_I(GE+EA)KE yE_S) Figure 4 showshow the coupling dynamics, GAE(S) and + [I + (GE+EA)K_'1(d^(s)+d(s)) (II) GEA(S), and the airframe dynamics, G^*(s), augmented with the airframe compensator, KL(S), can all be grouped together to and the closed loop characteristic polynomial for this system is form a u'ansfer function that will be denoted as EL(S).

A(s) = Oo1"(s) det[l + (GE + E^)KE] (12) / EA(s) Here the roots of ¢_ol'(s) are the poles of KE(s) and the poles of the system augmented with KA(S), or the values of s for which det[I + KLGL'] = 0. Eqn. (12) and this result are derived in Appendix A. Note here that E^(s) effects the characteristic polynomial, but dL(s) does not.

Eqn. (11) reveals how EL(s) can degrade engine conu'ol system performance. Also note how commands into the flight control system, YAe(s), are transmitted to the engine responses through d^(s). Eqn. (11) also shows that this term enters into the engine responses the same way as any other disturbances, d(s). Thus, the commanded inputs into the aircraft (from the pilot) act as additional disturbances to the engine..

y_s) Perhaps more significant, Eqn. (12) shows that the system's closed-loop characteristic polynomial is affected by E^(s), thus EL(S) can clearly effect the stal_lity. It can be shown from Nyquist stability theory, [5,6], that the closed loop system in Fig. 5 is assured to remain stable if the loop is stable for E^(s)=O, and if - Block Diagram of Aircraft/Engine Loop With the Airframe's Influences on the Engine Loop Grouped as EL(S) det[I + ((3 E + ¢.E^)KF.] _ 0, [0<£<11 (13) for all frequency. It can further be shown that Eqn. (13) is In other words, since AL(S), AE(S), GLE(S), and GEA(s) are not assured if really zero, the engine loop is not that shown in Fig. 1, but rather that shown in the following figure.

Omax (ELK E) < Omm(I+GEK E) (14) _IdA(s) +- d(s) for all frequency, where o denotes the singular value of a matrix. Thus, it is evident from this inequality that there will be loss of stability robustness for "large" EL(S ) , (i.e., if its maximum singular value is large.)

Consider now a single-input/single-output engine control system. For example, the engine response of interest may be fan speed, N2, and, for a fixed nozzle area, the input to control the fan speed may be the main burner fuel flow rate, w F. In this case, the transfer function matrices GE(s), KE(S), and EL(S ), as - Block Diagram of the Engine Loop for the Coupled well as d^(s), reduce to scalars, denoted by gE(s), kE(S) and Ah'craft/Engine System e^(s), etc. Then Eqn. (11) reduces to the scalar relationship Here the effects of the actual coupling present are grouped into the terms EL(s) and d^(s). These expressions, given below, are (gE+eA)kE YEt(S) + . I . (dL(s) + d(s)) yE(S) = 1 + (gE+ eQkE 1 + tgE+ eA)XE obtained by block diagram manipulation of Fig. 4.

(]5) EL(S) = AE - GF.A[I + KA(GL + AA)]'IK^GAE (9) If all transfer functions are assumed for the moment to be scalars, Eqns. (9) and (10) reduce to, d^(s) = GEA[I + K^(G L + AL)]-IKAYAe(S) (10) e^(s) = _E- gEAgAEkL The critical closed-loop coupling mau'ix E^(s) depends I + k^(g^+SA) (16) most importantly on the product of the input coupling transfer functions GEL(S)GLE(S), as well as on the airframe control law, dA(S) = gEAkA YLc(S) KL(S), the airframe dynamics, GL(S) + IlL(S), and the change in 1 + kL(gA+_A) (17) the engine transfer function, fiE(s). Therefore, ffA E is "small', and ff either G,,a_(s) or GEA(S) or both are "small," then EL(S) is "small." Also note that if the airframe subsystem includes no Note again that if 6E is small, and ff either gEL(S) or gLE(S) O1" both are small, then eL(S) is small.

feedback (KL(s):=O), E^(s) simply equals fiE(s), and dL(s)=0.

Eqn. (15) shows that if e^(s) is large, then gain and Note further that the disturbance d^(s) is independent of GA_(s).

phase margins present in the kE(s)gE(s) loop transfer may be Hence this disturbance may be significant even though GLE(S) is eroded, as depicted in Fig. 6. But from Eqn. (14), this will not small.

occur if It can also be shown that the input/output characteristics leLkgl << II + gEkEI (Ig) of the system in Fig. 5, including the coupling effects, is for all frequencies.

Example Frequency Response for the Loop dA(S) Transfer Function, k_(s)_(s)

- -. Ma_tude

_-.. "-.../ofk_s)ge(s)

d(s) s) -EngineLoop withCoupling and Model Uncertainty Here any additional model uncertainty is represented by the block U(s). The basic effect of this uncertainty is to merely i''-" " " _ . Phase of [ increase the "effective" additive dynamics from Ex(s) to E^(s)+U(s). But the problem of obtaining a model, or even a bound on U(s) is difficult, and currently is the subject of research.

Sources of Cross Couvlln_ Consider fast a vehicle to be controlled with thrust vectoring, as shown in Fig. 8. The pitch attitude, 0, is to be m e^(s)k_s) - _ ...... _ ] con.rolled by the nozzle deflection angle 8rv.

Frequency Example Loop Transfer Frequency Response or Open Loop Bode Plot In alltheabove discussion, thefocushas been on the effect of the airframe on theengineloop.Of course, a dual situation is present in that the engine also affects the airframe loop. In designing K^(s), the flight control designer must obtain airframe responses to pilot inputs that meet the flying quality specifications. These specifications require a pure aircraft-like modal response, and certain frequencies and dampings for these modes. Consider the dual of Eqn. (15), that is, the equation for the aircraft response, T yA(S) = (g^+ CE)k^ + 1 l+(g^+ eE)k^ y,_(s) l'#,h cry)k^ (dE(s) + d(s)) - Vehicle Pitch Attitude andNozzleThrust Vectoring where the coupling term eE(s) models the effect of the engine on

Angle

the airframe attitude loop. If eE(S) is small, the aircraft response transfer functions will exhibit almost perfect pole-zero Assuming some aerodynamicpitch damping, Co', is present, cancellations of the engine modes. Thus, only airframe modes and ignoring the plunge degree of f_.edom, the attitude equation will be dominant, as desired. This cannot be assured if eE(s) is is large. Furthermore, if the disturbance from the engine, dE(s), is significant, it will degrade the flying qualities.

The final topic is that of model uncertainty, or x + c'06: -Tl,sin(s ,) (]9)

uncertainty in all the system model transfer functions. Returning the focus to the engine loop, uncertainty can be modeled as where I... is the mass moment of inertia of the aircraft about its additive dynamics just like E^(s). Hence, uncertainty just adds center o_(gravity.

directly to EA(S) and then:fore has the same effect on the 1ooo.

With modeling uncertainty, the engine loop in Fig. 5 may be Now consider the following block diagram for considered changed to that shown below. describing the possible interactions between the airframe and engine.

Ct = constant (assume.d) Airframe Piu:h N = small perturbation engine fan speed

r

Note that the term C E reflects the engine's influence on the airframe's dynamics.

manded __ Rate, I(3 Under the assumption that no interactions between the airframe's attitude dynamics and the engine dynamics exist, or C A, CE and CO areallzero, theinput coupling dynamics, gAE(S) and gEA(S), are both zero, and theairframe transfer function Vectoring "_" - " reduces to

Speed, N

- Block Diagram of theEngine Influenced By the gA(s) s(s+co) (24) Airfran_ Likewise, the engine transfer function reduces to The thrust vectoring isused to control theair:raft's pitching motion. However, if the nozzle is not choked and the augmentor pressurechanges due to the re-directed engine gE(s) = (s-vtE) (25) exhaust, this may change the back pressure on the turbine, whichwill affect thefanspeed. Thesedynamicsare represented Now take the following for the airframe pitch compensation: in theabovefigure by G2(s). The airframe pitching motion will effect the airflow at the inlet, which willeffect the flow conditions at the compressor face. This effect is represenwxl by k^(s) = (mco/K_ _ (26) Gi(s). In this example, it is considered that the nozzle area is fixed so thattheenginefan speed would be controlled by the This leads to an augmented airframe transfer function that is a fuel flow rate, wf.

Let the fan speed dynamics be modeled here as a first first order lag, with a pole at -race. Finally, let the engine compensator, kE(S), be simply a gain kE.

order lag with a time constant of l/z_, and let the fan speed With the model and these conu'ol laws, equation of motion be -m_CE(`t F._K_)( s+Co_C^K _i+C_s (s+C-_)] lq = -'q_N + `t_wr + CO_v + C^O

(20)

eA(S) = *:1(S)*Ot(S_ 1 _c°ce(1/K6)(s+Ce_ K_(s+'tit)+CEc')] where the parameter C A reflects interactions from the pitching

L ¢o;(S) J

dynamics and the parameter Cs reflects the effect of _rv on the

+ `t_ECA

fan speed. Although for this model these coupling terms are

e:lfs)(s+_e)

considered constants, these effects may actually turn out to have (27) ....

dynamics.

Considering the outputs of interest to be pitch angle and Clearly, if CE is small, or if g_,(s) is small, then e^(s) is small.

engine speed, the linearized model leads to the following Note further that as the airframe crossover frequency race is increased, e^(s) is increased. Hence for all other things equal, a tighter airframe control loop can have a dilaterious effect on the

[+.,11+lr l

engine loop.

N(s)J-Lg (s) g s) JL wf j (21)

Using the fonowing numericaJ values, the system's open-loop transfer function magnitudes are shown in Fig. 10.

where, Model parameters: K_t= -0.08, Co = 1.1, '%, = 0.78, 'tit = 1.4 • .. K_s(s+`tE) + CEC#s

gAtS) -- . g_(S) =

Design paran_ters: ooce = 6, k E = 9

Col(S) ¢_,(s)

Coupling parameters: CE = 1.e-05, CA= 10, C_ = 0 CAK_ + C_ s(s+C-. 0) . 't_ s(s+C¢) g_(s) = g_(s) --- These parameter values were chosen by approximately matching

¢ol(S) ¢_1(s)

(22)

the frequency responses of the more complete system model to be discussed in the case study of the next section. Note that and the open loop characteristic polynomial is g.E(S) is very small, as are A^(s) and AE(s).

The size of the product of the input coupling transfer

¢_l(s) = s(s+co)(s+_it) - CACE

(23) functions, and the coupling transfer function e^(s) are shown in Fig. 11. Clearly both are small as expected, and hence, the where: coupling effects will not be significant. The size of kEe^(s), compared to the loop transfer, or open loop Bode, for the engine loop is shown in Fig. 12. The fact that the loop transfer is much

Co =

larger at all frequencies, along with Eqns. (15) and (18), assures that the coupling effects are truly small. The gain and K s = -(To It cos(Bw.))/I_ phase margins of the engine loop are unaffected, as are the closed loop transfer functions, as shown below. For the

CE = -(G 1_ sin(_.))/l>v

completely decoupled system, the closed-loop transfer functions To = _ thrust _. = u-ira thrust vectoring nozzle angle t = C_ N = small perturbation thrust rio 0(s...__.! = 6{s+l.l) 6p(s) (s+l.l)(s+6) ° iii!:ZiilZiii; ..........

N(s)= 7.O2

.S0 ........ :....._..._..: ¢ .: _*c_)la "...

No(s) (s+8.42) while with the parameter values given above, the closed loop "_ i i : ! !iii ":-.:...__/'_Is)_¢_) : : : :: : - : . ,_ ,: transfer functions for the coupled system are • _oo........ i(:,Hii: ...... ! ..!. :-. i.:.: :........ h-.,, .....

0(s) 6(s+l.l)(s+8.42) i : ::!i:: : - '- : _(s) (s+l. IXs+6)(s+8.42) ""_'0-_ lOe I0_ ltn N(s) 7.02(s+ I. I)(s+6) No(s) (s+l.l)(s+6)(s+8.42) 0 Clearly, for the parameter values selected forthis system, the -20 coupling is not significant.

s0 _ _ i i i _ii!!i i i iiii

0_f:_ i i............ iiiiii .............. LL_iil

.......

g -50

_ iiiiiii_ i ........ i ........ !iiii

• . : -: :::: : : :': , . : i " _, -loo 10a I0_ 10a r-n:qeeacy ta Rad/S_ -150 EigtIlr,._ - OIx:n Loop BodcPlot For the Engine Loop -2OO 10-1 10o 10t 10: : : : ::::.: : : : ::::;: : : ......

• _ _" : i _i_ _ ::! ::iii:;i 0 .... _-_-_..-, ...... _----:-.-:...:. .,_ ......... :...--_--._..h,) 3_J, - Open Loop Tmns'fer Function Magnitudes _r_s)_s) :_ _" i i i i:

2o ........ _ _._ ..... i_:t-Lii ........ __

.so

• i _iiiii i i!iiiiii i i ii!i!i

& .4o........ :,.... _...-_..-." ._-i.i..':_ ................

:"i i i!::Ti ii Ti::::T_i ........ i!iT_iT

:: ! i!ii::::! i i i ::::i::!i i i i i::i_ : : : 'r :::: : : : ';: . : : : • "1_00_ 10_ 10_ 102

.=_. ........ i-_-!i!i ! T;!_, ............. _

-3 _ 10o I0_ 10_

-_o :-:-; ........ i.....i..$..i.i.i.i.li, !.....!...::..i.i.i.i

V_Ir,..U -Magni-mdcs of CA(S) and theProduct gl.-.A(s)gAI-(S) Now consider increasing the coupling parameters to - _j ...... ,.._.:.:.._: ........ :.. ..:.i.i,.; ........ !.....L.;..:.i.j: C E = 0.025, CA = 12, CS = -3.5 The closed loop wansfcr functions become 10o i0_ r-n_,_cy i_ R_t,S_ l](s) 6(s+l.l)(s+9.514) - Open Loop Bode Plot For the Engine Loop _v(S ) = (s+1.093)(s+7.214+-2.267j) N(s) 7.02(s+ I. !)(s+6) No(s) = (s+1.093)(s+7.214"I'9.267j) Now, there isno longeraccurate cancellations of theengine modes inthe airframe response, and airframe modes in the engine response. However,in this case the stability robusmess isstill not greatly cffccted. As shown in Fig.13, CAkE(S ) is still quite small nearthe engine loop cross-over frequency region (= 7 rad/scc,) so gain and phase margins of this loopwould not bc eroded.

to stabilize the short period mode, and achieve a modal frequency of 4 tad/see and a damping ratio of 0.707. This may be achieved by feeding back angle of attack and pitch rate with Using the techniques just presented, anention will be feedback gains of 3.934 (deg)/(deg) and 57 (deg)/(rad/sec), directed to the analysis of an airframe/engine system that has respectively, using the thrust vectoring control. Also, the flight been the subject of several studies of integrated flight and ca .#he path time constant, lets, must be increased to approximately control, [e.g. 7]. The vehicle to be cgnsideted is representau.ve 0.52 rad/sec. This may be achieved by feeding back angle of of a high performance fighter aircraft with 2-D thrust vectonng attack, with a feedback gain of -2.897 (deg)/deg), to the flap.

and thrust reversing. The vehicle dynamics arc linearized about the Short Take Off and Landing (STOL) approach-to-landing For the aircraft decoupled from the engine, these feedback gains reference condition at an airspeed of Vo = 120 Knots and flight give the following closed-loop aircraft transfer functions for path angle To = -3 °. The system states arc pilot input, 8p(s), to pitch rate and plunge acceleration, = [u, w, q, 0, N 2, N2..s, P6, T41B] T q(s) =. -0.05369s(s+0.07352)(s+0.5231) (_sec)lbs tip(S) (s+0"03065-+0" 1703j)(s+2"885Y'2"893j) when:, the "aircraft" states are: u = body axis forward velocity (ft/sec) N,(s) 0.01687s(s+0.003983)(s+ 10.46) (g's) 8p(s) = (s+0.03065_+0.1703j)(s+2.gg5:k2.g93j) 1-_ w ffi body axis plunge velocity (ft/sec) q = pitch ram (rad/s_) The objective of the engine control design is to regulate 0 = pitch angle (radians) the fan speed. Proportional plus integral compensation will be used, with gains of -6 flb/hr)/rpm and -3 (0b/hr)sec)/rpm, respectively. With these control laws, the loop gain crossover and the "engine" states are: frequencies for the thrust vectoring and the engine loops are both N 2 = engine fan spe_ (rpm's) between 1 and 10 rad/sec, and gain and phase margins of the N2..s = engine compressor speed (rpm's) fuel flow rate loop appear adequate. The open-loop Bode plots P6 = engine mixing plane pressure (psia) for these loops are shown in Figs. 14 and 15 below.

T4ZB = high pressure turbine temperature (°R) The control inputs used in this study are: o i ......

iiiii iiiiiiiiii

10.= 104 100 lot Ifla where, 8n,p, = trailing edge and leading edge flap deflection (deg) _v = nozzle thrust vectoring angle (deg) wf = main burner fuel flow rate (#/hr) ...... !i! ...... ....

10-2 IO-z lOO lOi 102 The state space description for the system is given in Appendix B. The aircraft's attitude dynamics are to be controlled by.thrust r-nq,mey ia radaee vectoring. For this model the nozzle throat area ts not -Fieure 14 - Thrust Vectoring Loop Transfer Frequency Response considered as an input, therefore, only the fuel flow rate is used to control the engine fan speed.

Classical feedback control laws were synthesized for the airframe and engine by considering each subsystem separately,

o .... ii!; i :,:--:.if.:.:.; ......i...i.:._.i-;:i

and treating them as non-interacting. The open loop thrust ) vectoring angle to pitch rate and airframe plunge acceleration transfer functions are _ .so 10.2 }0._ I0o 10: 102 q(s) -0.0797s(s+0.1894:L'0.101j) ( dct_g ) &,(s) = (s+0.05681:L'0.2154j)(s- 1.065)(s+1.472) • 50 ......... : . [ -too" ..... ".-i--- ..... i- ....... --.i--i.i.!,!_!- ......... i._._.ii.:_: ...... -.: :'.i.ii_.- Nz(s) 0.02127s(s+O.O2456)(s+6.984) (g's) 8iv(s) = (s+0.05681_+0.2154j)(s-l.065)(s+l.472) d-_

.,,o !:i i:il i

10.= 10-z fOe 101 102 The airframe dynamics are aerodynamically unstable. The flight r-mlo_-y in md/t_ control design objective is to obtain classical longitudinal aircraft responses given by, -Fucl Flow Rate Loop Transfer F-axquency Response With this dcoouplcd design, the actual coupled aircraft/engine system will now be evaluated in terms of the q(s) Kq s(s + 1/'tel) (s + l/xoa) significance of the subsystem interactions. First, the magnitudes of the four transfer functions, analogous to those in Eqn. (21), 8u(s) (s 2 + 2_phgOph S + _ph)(S 2 + 2_,V(OS p s + (O_sp) are shown in Fig. 16. Note, as with the simpler model discussed previously, the pitch-rate-to-fuel-flow transfer function, gAE(S), Nz(s) = KNz(S + l/'tN I) (s + I/ZN_)(s + l/'tN)) is small, along with AA(s) and AE(s). Shown in Fig. 17 is the 8st(S) (S 2 + 2_phfJ0ph S + _2h)(S2 + 2_spC_p S + 0_sp) critical cross-coupling wansfcr function CA(S), and its small magnitude is apparent. Finally, the size of *^(S)kE(S) is (Phugoid Mode) (Short Period Mode) (28) compared to the engine loop transfer in Fig. 18. Clearly, the gain and phase margins in this loop will not be degraded due to this small term. Furthermore, the effects of cross coupling on ,he closed-loop performance is likewise not expected to be The closed loop transfer functions for the awframc significant for this system.

responses, for the fully coupled system and the decentralized control laws above are q(s) -0.05369s(s+O.07279)(s+0.5232) T(s) 8p(S) (s+0.03029"_. 1704j)(s+2.gg5:l:2.893j) Nz(s) = 0.01687s(s+0.003275)(s+ 10.46) T(s) _p(S) (s+0.03029"I'0.1704j )( s+2.885.'f.2.g93j) where, (29) :E 150 (s+0.4198)(s+ 1.99"X.3.535j)(s+8.014)(s+89.67) T(s) =

.,00 ' !!ill !i!!'

(s+0.4198)(s+1.99+_.3.535j)(s+8.014)(s+89.67) IO.l IOo lOJ IO2 r-_uen_, m l_VSea: (Note that these transfer functions are 9th order. The additional pole is due to the integral control of engine fan speed.) The - Open Loop Transfer Function Magnitudes transfer function T(s) is essentially unity. The transfer function for the engine fan speed response to a commanded engine fan speed for the decoupled system is -20 : ::: : . : :::: : : : N2(s)_ = 0.1469s(s+ 16.4Jl:f.5.89j)(s+36.93) Na(s) (s+0.4195)(s+l.99+_3.535j)(s+8.014)(s+89.67)

........ :-:i,;-! i!! ........ {i...... {41

The companion transfer function for the same control laws on qoo .......... .....: ...... Z..):::;:.=........:: ......i.i.Z.:2 ........ i.....:...i..i.i.i...

| the coupled system is

]

N2(s) = 0.1469s(s+ 16.43:f.5.g9j)fs+36.94) T(s) N2_(s) (s+0.4198)(s+ 1.99"_.535j)(s+8.014)(s+89.67) "_'°I........+., .-_-.:..!,._,.i.i_i ........i ...-:--_..i._._._..+ ...... i,...,:....:...i.i._.i.

-160 ................ i"" where, (30) 10o 101 Frtqutaq, ia Xad/S_ T(s) = (s+0.0303:k0.1704j)(s+2.g85:f.2.893j) (s+0.030291-0.1704j)(s+2.g85.'L.2.893j) _ - Magnitude of the e^(s) Transfer Function Clearly for this system, no significant coupling between the _ i i :: ::::iiii ! i ;iiii;i i i i ili!; engine and the airframe attitude dynamics is present, and the control systems suffer little performance degradation.

If model uncertainty is considered in the coup.ling dynamics, large effects of course are possible. This will be evaluated briefly below. Let the coefficient on the mixing plane pressure in the pitch-rate state ex]uadon of motion be varied from -50 ....... 7.-.T-;,-"---'!'r-? ........ v-"T'7"T?'!-?';7 ........ 7'"7 ........ 7"7!'

._ -0.03 to 0.01 (radJsec2)/(psia). Also let the coefficients on the plunge-velocity in all the engine stale equations be varied :1:20 -loo .......... r {--i--:i ................. :_,:-:i ............. :............... times their nominal values. Finally, let the coefficient on the thrust vectoring angle input to the mixing plane pressure . : - - { { i : ". : a :i equation of motion be varied from -25 to 25 (psia/sec)/(deg).

The ranges of these parameter variations are only first "l-fo-2 IOO I0_ IOa order approximations, based on studies of other coupled aircraft/engine models as well as simple engineering -7O considerations [8,9]. For example, the parameter C_: in the -s0 conceptual model of the last section is analogous to the coefficient on the mixing plane pressure in the pitch rate state .90 equation. Since CE is a function the trim thrust vectoring nozzle .100 angle, it can therefore change sign depending on the trim value.

These variations form a "three-dimensional parameter space" in = -II0 which all the parameters arc varied at the same time. Root loci -120 of the full 8'th order closed loop system due to all these parameter variations shows that this range of variations will not -130 cause instability, and the variation in the magnitude of the -140 coupling transfer function is still quite small (though not shown).

-150 IO-1 tO0 lOl IO: The effect on the closed-loop system transfer functions r--r_q_ney in will now be assessed. For example, selecting the following set of parameter variations,

- Open Loop Bode Plot for the Engine Loop

-0.03 (rad/sec2)/(psia), -20 times, and -25 (psia/sec)/(deg) (31) the closed loop transfer functions become L q(s____) = -0.05369s(s+O.0656)(s+0.5346) T(s) _p(s) (s+0.02gl l:k0.1646j)(s+2.523+7.6gj) 0 '''" " ' -_" _":'_"'.'"",""_"-';'--'_- ........... ":'--'i-:-:'_: ........ - ........ - ..... . " "_ m (s+0.4317)(s+2.009-_3.485j)(s+8.055)(s+g0.17) "o .E T(s) = (s+0.4122)(s+2.124+_3.S85j)(s+7.990)(s+90.16) • _0 iiiiiii! _ i-' i i!i" i!iiiii!i!ii: ........ _....... ' "'" _""" "" '_ "C':" ""!"" "_"*" '" "' i_i__----..: ...... "* -(sO ................:",":!' i! ........ .'"":'":"'v_.

}

N,(s) -0.09057s(s+0.001526) T(s) -gO : • .ii'i ....... i..._..a..:.i.:.:.= ...... _.-..._: ........ :..

_,(s) (s+0.0281 hk0.1646j)(s+2.523:k2.68j)

....... : :t

i....... i.: .......... i.... i...... ...... i

(s+0.4175)(s+2.011_.3.558j)(s+7.2661"0.35 lj)(s-24.05) -12_ I_ 10_ IC_ T(s) = (s+0.4122)(s+2.124:k3.585j)(s+7.999)(s+90.16) -70 .

N2(s) 0.1469s(s+ 15.87_6.145j)(s+38.64) T(s) N_(s) (s+0.4122)(s+2.124:_.3.S85j)(s+7.999)(s+90.16) (s+0.02777!'0.1649j)(s+2.598:_.2.719j) T(s)= (s+0.02811i-0.1646j)(s+2.523:_.2.68j) which differ from the transfer functionsof Eqns. (29) and (30), and T(s) is now no longer unity. However, the effect of these

"° I ........ i-!i/,: ........ i-X ...... /_

parameter variations on the flying qualifies has been evaluated, ,,o_ :: .......... _:, and they are minimal.

For the parameter variations selected as in Eqn. 01), the I00 I0] open-loop airframe/engine u'ansfer functions are shown in Fig.

19, and the magnitude of the cross-coupling, t_ransfer function is shown in Fig. 20. Finally, cAkE(s) is again compared to the

- OpenLoopBodePlot for the Engine Loop

engine loop transfer in Fig. 21. These plotsshow that although the coupling has increased from the nominal system, as presented in Figs. 16 through IB, itis still quite small The performance, as measured by the closed-loop wansfcr funcnons Conclusions issomewhat effected, but the stability mbusmess is not,forthis case.

Two coupling transfer function man'ices were derived that quantify, in a meaningful way, the significance of _o ........... : . . :; .

: : : : : :::: ...... airframe/engine interactions on _ engine controlloop. (These matrices each have dualsforquantifyingthe effects on the flight control loop.) The size of these matrices, measured, for

.... -i-iii!i ........ i_i-iii_

o example, by their singular values, quantifythe effect of coupling on closed-loop performance and stability robustness. These cross coupling terms we_ shown to depend on the control

_ _:-ii!-_

compensation Iransfcr functions and the n'ansfer functions for the airframe/engine system.. In particular, they are functions of the off-diagonal transfer functions in the system's transfer -I00 function matrix. When the critical coupling terms are small compared to the magnitude of the loop transfer function (matrix), crosscouplingeffects are minimal. A conceptualmodel "l"10-1 10O I0] |[_ was offered to demonstram the method. A case study of an F-iz_ucazy in Rad/r_ airframe/engine system used in earlierstudies of integrated conn-ol techniqueswas thenpresented.This study revealed that - Open Loop Transfer Function Magnitudes thisparticular vehicle, as modeled, exhibited very little critical interactions. A classical decentralized control system synthesized .20, assuming the airframe and engine subsystems are to_ally non-

i i:i, i i;i:ill

interacting was quite suitablein this case. Other vehicle _0 ......... =:i_iiiii ........ ! u!_,_,=,., i i ....... _ configurations, and/or morn accurate models of the cross- i i i i !iii i ! .,jl :: coupling effects may reveal much more significant airbamc/engine imeractions.These interactions, however, may be evaluated with the analytical framework presentedher_n.

.so .............. i ...... _ Ap.endix A- Derivation of Eouation (121 _ : i _:ii i _ : i!! " I Let the state space description of the fully coupled aircraft/engine system Wesented be defined as .......... iiiiii ........ ii:ii 'ii.......

"_

10o 10] 10_ r-mqumcy i_ Rad/S_ [XA 1 [ AA A_I[xA] - [ BA BAE] [u^] xz =LA_A AEJtXEJ_'[BEA B__ JLu_J F_JgtllY,._ - Magnitude of the cA(s) Transfer Function o lr,A, [ 0 CEJLXEJ (AI) leadingto . 0 +I,,:l>,,,.

I.OJ

LyE(s)J LG_(s)G__(s) luE(s)J LuE(s)J (A2)

>,,,:I c, o1[:']

(A111 with characteristic polynomial, and the characteristic polynomial for this closed loop system is (A3) A(s) = det [ sI-A1 -B iChl (GE+E,OKE] B_C1 sl-Ah ] = 01(s_ka(s) det [I + Also let the state space descriptions of the aircraft and engine (A12) compensators, K^(s) and KE(s), be, respectively, or.

A(s) = OA -- CkAXk.

*o,(S)_(s) det [I + G;(s)K^(s)] iI_t._(s) dei [I + (GE+E_OKF.]

_ = A_x_ + B_y_ UE = C_x_ (A4) (A13) l:_.fming where y^c'(S) = y^c(s) - y^(s) and yEc'(s) = y_(s) - yE(S) are the inputs to the aircraft and engine compensators. The (l);i(s) -- 4);i(s)41,(s)det [I + G,_(s)K^(s)] ii_(s) (AI4) characteristic polynomials of these compensau_'s are gives 0k.(s) = det (sI - Ak.)

_(s) = det (sI - Aka) (AS') A(s) = _'ol(S) det [I + (GE+EA)KE] (A15) Sought now is the state space description of which is the result presented as Eqn. (121. Note that GE(S)+EA(S), as presented in Fig. 5. Using Eqns. (A1) and (A4), and referring to Fig. 3, yields det [i + G_(s)KR(s)] is • rational function with denominator equal to 0ol(s)_(s). Thus, the roots of _ol"(s) art the roots of 'll r A,,

_E/--/ ._ ._ B_C.,,,//_E/+ BE uE+ o y_

¢_(s), which are the poles of KE(S), and the values of s for

o A_, .IL_ I. 0 J LB_J

_k.,] L-B_C^ which det [I + G_(s)K^(s)] equals zero.

YE=[ 0

cE o _

Annendix B- State Snace Model for the Case Study

Xxd

Using Eqn. (A1), the airframe/engine system is modeled in the following form Denoting this system as ifAx+Bu xl -- Alxl + BlUE + B2YAc YE = Clxl (A7)

[ BA B_I

where, Affi[/_E_ ] and B=LBE_ BEJ it can be shown, [10], that the characteristic polynomial of this system, (GE(s)+E^(s)), is with states, x ffi [u (ft/sl_), w(f[/scc), q(rad/sec), O(radians), N2(rpm's), Nz.s(rpm's), P6(psia), T,_m(°R)] "r 01(s) = det (sI - AI) = 0"ol(s)q_(s) det [I + G_(s)KA(s)] (AS) and inputs, u = [Snaw(deg), 5Tv(deg), w_(#/hr)] "r For the vehicle in question, the model is given below, [7].

Appending the state equation for the engine compensator, KE(s), to the state equation for GE(S)+E^(s) gives the state space description of the open loop system of Fig. 5, -2.6590e-01 -2.6650e-01 1.9480e+02 -4.5990e+00 (GE(s)+EA(s))KE(S), -5.8930e-02 1.0670e-01-3.8600e+01-3.1840e+01" A^= -1.5410e-0037.8060e-O3-1.949Oe-Oll.oOO0e+O0 -4.8180e-040 .,, x_ 0 ,_ jtx_j -1.5780e-05-2.106(O-061.8260e-04-2.9570e-06 AAE= 9.4600e-07 3.7440e-07 3.6680e-05 2.6760e-06 3.1,140e-042.5990e-043.8190e-022.2500e-03 0 0 0 0 YE=[ C1 O] [xX_] (A9) r7.7820e-01 1.5420e-01 0 __ It can further be shown that the characteristic polynomial of this / 1.5180e--01 3.0080¢-02 0 AE._ =/7.93X,0c-01 1.5720c-01 0 system is L -1.0050c-01 -1.9920c-02 0 0ol(S) = det [ sl-Ai -BIC_l= F-4.1910c+00 6.0220c.,,.00-3.4340c+02 1.1600c+011 0 sI-Az, j t (A10) | 4.2630¢-01-5.7070e+00 2.7160e+01 1.0400e+Ol/ AE=/ 2.2950e-01 1.1550¢-01-9.0240e+01 8.4760e-01/ 3.7400e-02-1.0360e-01-7.9540e+00-1.0680e+OOJ Closing the (engine) loop in Fig. 5, the state space equation for the closed loop system is -4.1830¢-04 -8.4280e-02" 1.2380e-08 -5.4520e-01 -2.1475e-01 -7.9700e-02 8.8132e-03 B_ = 5.5070e-08 I 3.4360e-05 "

B^= I

0 0 5.3600e4Y_ BEA= 0 BE = 1.8130¢-02 [ 1.,_i90e-01 "

I ° ii

1.6430e-01.

Acknowledgements This work was sponsoredby the NASA Lewis Research Center under Grant # NAG3-998. Mr. Peter Ouzts is the technical programmanager.

References [I] Berry,D., Schweikhard,W., "Potential Benefits of Propulsion andFlight Control Integration forSupersonic Cruise Vehicles," basedon SAE paper 740478, 1974.

[2] Shaw, P.D., Rock, S.M., and Fisk,W.S., "Design Methods for Integrated Control Systems," AFWAL -TR- 88-2061, Aem Propulsion Laboratory, Ah" Force Wright Aeronautical Laboratories, Dayton, Ohio, June, 1988.

[3] Smith, K.L., "Design Methods for Integrated Control Systems," AFWAL-TR-86-2103, Aero Propulsion Laboratory, Air Force Wright Aeronautical Laboratories, Dayton, Ohio, December, 1986.

[4] Rock, S.M., Emami-Naeini, A., Anex, R.P., "Propulsion Control Specifications in Integrated Flight/Propulsion Control Systems," AIAA Paper No.

88-3236, AIAA/ASMF_,/SAE/ASEE 24thJoLt Propulsion Conference, Boston, Mass.,1988.

[5] Doyle,J.,Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Transactions on AutomaticControls, Vol.AC-26, No. I, pp. 4-16, Feb.,1981.

[6] Rosenbrock, H., "The Stability of Multivariable Systems," IE.EETransactions on AutomaticControls, Vol.AC-17, pp.105-107, Feb.,1972.

[7] Garg, S., Mattern, D.L.,and Bullard, R.E.,"Integrated Flight/Propulsion Control System Design Based on a Centralized Approach," AIAA PaperNo. 89-3520, AIAA Guidance, Navigauon and Control Conference, Boston, Mass.,1989.

[8] Lancaster, E., .let Propulsion Engines, Princeton University Press, Princeton. N.J., 1959.

[9] Tape,R.,Harfill, W., et aI., "Vectoring ExhaustSystems forSTOL Tactical Aircraft," Journal of Engineering for Power, Transactionsof the American Society of Mechanical Engineering, July, 1983.

[10] Kwakernaak, H., Sivan, R., Linear Optimal Control Systems, Wiley-lnterscience, New York,1972.

II Extended Implicit Model Following As Applied To Integrated Flight and Propulsion Control" David K. Schmidt t and John D. Schierman tt Department of Mechanical and Aerospace Engineering Arizona StateUniversity Tempe, AZ 85287-6106 propulsion system was considered to bca component of these generalizedactuators, and its controlsystem was later designed, Abstract off lineor independent of the airframe, to mcct theserequired bandwidths. However, this approach can only directly account An extended model following control synthesis methodology, for one-directional dynamic interactions between the airframe including loop transfer recovery, is presented and applied to and engine. It cannot directlytake into account how the synthesize control laws for integrated flight and propulsion airframe's dynamics influence those of the engine. The control (IFPC). The vehicle considered is representative of an allowable unmodclcd or ignored interactionsthesedesigns can unstable modem fighter aircraft; with a 2D thrust-vectoring and tolerate was the subject, for example, of Ref.3. Finally, thrust-reversing nozzle. The linearized design model includes although the resulting control laws were evaluated in a manned both airframe and engine dynamics. The fact that it is necessary simulation, an analytical validation of the flyingqualities was not to regulate some responses as well as dynamically shape others pexfonne_ so compliance with themilitary specification was not is discussed, thus leading to a hybrid-control-problem considered.

formulation. A previously developed model-following In Rcf. 4 a centralizedapproach was exercised that formulation of the LQR problem is extended to handle this directly applied Linear Quadratic Gaussian/Loop Transfer hybrid problem. Compensators are then obtained to realize an Recovery (LQG/LTR) methodology, using a linear fully output-feedback control law, by using a loop-transfer-recovery integrated airfranw./propulsion dynamical model. This approach procedure. The airframe and engine responses are decoupled, can accoum for two- directional dynamic coupling. The r_sulting and porfect airframe response following is obtained. The loop control laws wcrc not evaluated analytically in terms of the transfers also reveal good stability robustness and reasonable resultingflyingqualities, duc in part to the complexity of the loop cross-over frequencies that would not lead to excessive actuation requirements. The approach also yields compensators closed-loop systems obtained via thismethod. Italso tends to of dynamic order lower than the plant, thus easing their result in high order compensators that may be difficult to implementation. When compared to the results for a classically implement. Also, in both these studies, simulations revealed designed control law, the performance of the muhivariable high actuationrequirements,indicative of high loop-crossover frequencies.

design was superior to that of the classical, while the loop shapes were quite similar. The issue of simpler feedback compensation was the subicct of Ref.5, in which LOG/LTR was again applied to Introduction synthesize full-order compensation.These compensators wcrc then partitioned and simplifiedvia order reduction. Except for Enhancement of maneuvering capabilities of high the resultingloop shapes, these control laws have not been performance aircraft by propulsion systems capable of delivering furtherevaluated.

forces and moments to the flight control process is considered a In thispaper a new synthesisapproach isoffered, and viable engineering approach. For aircraft such as those capable explored via a case study. The design objectives will bc of short take-off and vertical landing (STOVL), significant presented at the outset, thejustification isgiven forconsidering dynamic interactions between the airframe and the engine arc this synthesis approach in light of these design goals, the present, and some configurations may lead to interactions in synthesis methodology is presented, and the case study is critical frequency ranges. Recently, Schmidt and Schiermant addressed. A pseudo-classical design isalso developed for the discussed the difference between the more common one- purposes of comparison. The resultsof this study will be directional coupling between airfran'_ and engine,and the critical discussed visa vis the aforementioned design goals, and two-dimensional variety. A measure of the critical interaction conclusions presented.

was developed that was expressed in terms of the sizeof an Itwillbe shown that the two controllaws so developed interaction matrix compared to the magnitude of the loop both satisfy, thegoals stated, and in fact lead tosimilar results for transfer. For aircraft/enginesystems which do not have the vehicularsystem considered. This is considered a positive significant dynamical interactions, separate designsof the flight rcsuh since one goal of developing the new technique was to and propulsion control systems have bccn quite adequate.

obtain somewhat classical-like control laws. The fact that the However, ifthis coupling is large and not taken into account rcsuhs for both controllaws arc similarisalso duc to the fact when designing the control laws, then these dynamical that, as shown in Rcf. I,this particular vehiclemodel possesses interactions willlead tolossof system performance and stability little of the critical two-directional coupling. This model was robusmess, or in scvcrc cases toinstabilities.

selected hcrc in spiteof thisfactbecause itwas used in Rcfs. 2 This problem isreferredto here,and elsewhere, as the and 5, and further comparison of rcsuhs istherefore possible.

Integrated Flight and Propulsion Control (IFPC) problem.

During the past several years, design integrationmethods 2-5 Design Goals and Melhodology Motivation have been proposed that were intendedto synthesize integrated control laws, while in a variety of ways dealing with the The goals or design objectives forcontrol laws that are potential dynamic interactions.

aimed at addressing the IFPC problem involve system In Ref. 2 a decentralized off-line approach was performance,robusmcss, and implementation issues.

considered, by which the flightcontrol laws for the airframe, Performance - Foremost among the performance issues plus the required generalized actuation bandwidths were is the fact that the control systems must deliver excellent obtained via Linear Quadratic Regulator (LQR) theory. The handling qualities, in spite of the potential airframe/engine dynamic coupling. The handling qualities criteria arc quantified in terms of specified time constants, damping ratios and " As Presented at the 1990 AIAA GN&C Conference. Portland, OR frequencies for the airframe modes, as well as closed-loop l:_ofe,ssor of Engineering; Associate Fellow. AIAA.

frequency from pilotinput. Control laws thatproduce closed- 1"1. Doctoral Candidate;Student Member. AIAA.

Copyright (_)1990 by David K. Schmidt Published by American Inst.of Aeronautics and Astronautics by permission

AZAA /#0. qo-

loop airframe responses that reflect classical airframe dynamics as depicted in Fig. 1. Since such LTR procedures recover the are desirable. In fact, how well the resulting airframe responses state-feedback loop shapes at the input to the plant, the robustness properties of the state-feedback control law are approximate certain frequency responses of a conventional recovered there. Further, since the state-feedback control law is aircraft with the desired modal characteristics is one step in meeting the military specifications t0. One implication of this obtained via an LQR formulation of the model-following problem, compensators with the robustness propenies of the design goal is that the control system should decouple the LQR solution result.

airframe and engine responses. If the engine's dynamics are observable in the aircraft responses, then classical airframe Case Study Vehicular System dynamical properties are not obtained. Note that these design goals are not those of a regulator.

The vehicle to be considered in this investigation is the Engine control, on the other hand, requires regulation of same as in Refs. 2 and 5. It is representative of a high responses about an operating point, with gain scheduling and performance fighter aircraft with the capabilities of 2-D thrust transition control from one point to the next within the operating vectoring and thrust reversing. The vehicle dynamics are envelope. For example, in order to maintain stable combustion, linearized about the Short Take Off and Landing (STOL) it is important that the fan and compressor do not exceed their approach-to-landing reference condition at an airspeed of Vo = surge limits. For structural considerations, the main burner and 120 Knots and flight path angle _ = -3". The states, controls the high pressure turbine should not exceed specified pressure and responses are listed below. This model, with the same and temperature limits. Therefore, stable, robust regulation of control and measurement vectors is used for both the classical responses such as fan and compressor speeds, temperatures, design and the EIMF/LTR design presented in the nest sections.

and pressures, is a primary goal in the control design of the The state vector is en_ne.

Finally, these performance objectives must be met with = [u, w, q, 0, N 2, N2..s, Pt, T41B] T minimum actuation requirements, such that rate and deflection where, the aircraft states are limits are avoided. Not only are high actuation requirements taxing on the hardware, rate and deflection limiting degrade both u = body axis forward velocity (ft/sec) performance and stability by introducing unmodeled non-linear w = body axis plunge velocity (ft/sec) effects into the loops. Therefore, control bandwidths or q = pitch rate (rad/sec) crossover frequencies must be as low as possible.

0 = pitch angle (radians) Robustness - The system must possess adequate stability margins so that it is robust against unmodeled or inaccurately modeled dynamics. Usually, this requires minimum gain and and the engine states arc phase margins in all loops, although singular-value-based 6 N 2 = engine fan speed (rpm's) robusmess analysis is currently popular. Also, the loop transfers NL5 = engine compressor speed (rpm's) must roll off sufficiently to handle high-frequency unmodeled dynamics or non-linearities. P6 = engine mixing plane pressure (psia) Implementation - The compensation should be easily T4t B = high pressure turbine temperature (*R) implementable. This implies that it should be of low dynamic order, and preferably should be similar to classical control laws.

The control inputs to be considered are If so, the results can yield additional insight with regard to the control system's interactions with the overall airframe/engine

= [Avs, _rv, 8n,v,, wf]

system. Furthermore, the existing techniques for control law validation and verification, as well as the necessary gain where, the aircraft controls are scheduling may still be utilized.

A78 = thrust reverser port area (in 2) The synthesis approach to be presented will be referred to as the Extended Implicit Model Following/Loop Transfer iSTV = nozzle thrust vectoring angle (deg) Recovery (EIMF/LTR) technlquet, v. Model following is an 5n_ = trailing edge flap deflection angle minus leading integral part of the formulation so that the closed-loop airframe edge flap deflection angle (deg) - see Reference [3] responses may be shaped to take on the desired dynamics. This method does not yield a regulator, and may not necessarily give and the single engine control to be considered here is loop transfers with classical (k/s) loop shapes. However, the design goals were not those for a regulator, and classical wf = main burner fuel flow rate (#/hr) stability augmentors (e.g., pitch dampers) do not yield regulator loop shapes either. Implicit model following rather than explicit (Note that the main nozzle throat area control used in Ref. 2 is model following is utilized to eliminate the dynamic prcfiher that not used in this study.) The aircraft's forward velocity is to be is a integral part of the latter control structure. This leads to essentially regulated with the thrust reverser, while the attitude closed-loop airframe responses of lower dynamic order that are dynamics are controlled by thrust vectoring. The flaps are direct simpler and easier to evaluate in terms of handling-qualities lift devices which are used to control the flight-path-to-attitude assessments, and simpler to implement. Also, perfect model- response, and the fuel-flow rate is used to control the engine fan following concepts 6 arc exploited to minimize loop gains and speed. The measurements used for feedback arc crossover frequencies.

The implicit-model-following formulation of Refs. 6 and 7 are herein extended to address the hybrid problem of model = [ u, w, q, N2]T following for some responses and regulation of others. As noted earlier, engine responses, as well as aircraft velocity in some The vehicle model, partitioned in the following manner, cases, must be regulated. Consequently, for an integrated is given in Appendix A, synthesis approach to the IFPC problem, regulation as well as model following must be admitted in the formulation.

Loop-transfer recovery is employed to synthesize the , ]CxAI [BA. B, }[oA 1

XE _ At xE Bp.A BE UE compensators, utilizing the state-feedback gains obtained from the solution to the EIMF problem. This may be accomplished by where the subscript A denotes aircraft subsystem and controls, exploiting the asymptotic properties of the Kalman filter, as in and the subscript E denotes engine subsystem and controls.

the standard LQG/LQR approach 8, or by using a direct recovery Results from a modal analysis are shown in Table 1. This table technique as presented in Ref. 9. Either technique yields the presents the open loop poles and the responses dominated by compensators necessary to realize an output feedback structure, these modes.

that it should not be increased above this value due to excessive Table 1 - Modal Analysis of the Open Loop System flap deflections.

The requirements on the phugoid mode will be met by Open Loop Poles Mode Shapes achieving some modest damping for this mode, and by ......p..huEoid mode (u) rendering this mode essentially unobservable in the attitude .-o-o_.5_ *-_.o_.2!5..9_ .........

response. The desired attitude response may be defined in terms short period mode (w,q,O) -1.472 of the following dynamic model.

+1.065 qm(S) _ Ms(S + I/'tO_) highly coupled engine modes 8p(S) S2 + 2_spfDspg + (b_ip involving all the engine states .-6.9_. ................................................................................... _s___! = 7_ ._9+2R mosth' associated with Pfi 8p(S) S 2 + 2_sp_spS + O_sp (2) The open-loop thrust-vectoring-angle-to-pitch-rate, airframe or, in state space form: plunge-acceleration (at the center of rotation), as well as the fuel flow-to-fan-speed transfer functions are o , -CO]sp -2¢,pC0,pJ t x2 q(s)= -0.0797s(s+0.1897---'-'-_. 1013)) T(s) (I_1 &,,(s) (s+0.05709.20.2153j)(s- 1.065)(s+ 1.472) ct(s) -0.1542(s+0.04249_0.1957)Xs+28.65). T(s)pegl

01[:: ]

(3)

_= s 1 065 s+i 47i" _degl &,(s) (s+0.05709"20.2153j)( - • X • ) T(s) = (s+ 1.401)(s+3.569)(s+6.958)(s+89-28) Here, 8p is the input from the pilot (e.g., stick deflection). The (s+ 1.401)(s+3.569)(s+6.958)(s+89.28) remaining terms to be selected are N2(s) - 0.1469(s+16.43-+5.89))(s+36-9'1) T(s) (RPM I Z.a = -4.42 deg/(slug-ft/sec) ws(s) (s+l.401)(s+3.569)(s+6.958Xs+89.28) _ #/hr I M b -- -0.0797/lbs T(s) = (s+0"051M6-+0"2155))(s'l'065)(s+l'472) These terms are obtained from the short-period approximation (s+O.05709"L,0.2153j)(s- 1.065)(s+ 1.472) for the study vehicle. With this approximation, the model in From the above transfer functions and Table 1 it can be Appendix A yields seen that the short period mode is unstable. Note that the poles of T(s) in the airframe transfer functions are predominantly those c_(s) -0.1542(s+28.67) -4.42 (for s-O) for the engine modes, and the engine dynamics are essentially Sty(s) " (s-l,OO3)(s+l.464) " (s- 1.003)(s+1.464) unobservable in these airframe responses. The converse is true in the engine transfer function.

and q(s._....._) = -0.0797(s+0.3199) Performance Objectives Sty(s) (s- 1.003)(s+I.464) The flight control synthesis objective is to obtain The objective of the engine control design taken here is classical longitudinal aircraft responses to pilot stick input, given to regulate the fan speed. However, quantitative specifications by, on disturbance responses of the fan speed, such as maximum overshoot allowed or desired settling time, have not been q(s) = Kq s(s + 1/_%) (s + 1/'te_) formulated at this time. The response characteristics will be _p(s) (s2 + 2_ph0,_ph s + (02ph)(S 2 + 2_s-pC..Orp S + O_sp) selected to yield engine-loop crossover frequencies close to those in the attitude loop, thereby maximizing the potential for dynamic interactions, the basic issue in this research c_(s) = Ka(s + l/'t m) (s + l/_az) The following block diagram presents the closed loop 8p(S) (S 2 + 2_ph0)ph S + f_ph)(S 2 + 2_sp(Osp S + _p) system and shows the measurement and control vectors.

(Phugoid Mode) (Short Period Mode) (I) Responses of Interest The short-period mode must be stabilized, achieving a specified Controls, u _ [q' l_. _ etc.]

frequency and damping ratio. Also, a desirable value forthe real

-

flight-path time constant,l/'te2, (not present in the open-loop transfer function)should be obtained.Table 2 lists the desired _ -'_ - _---1 _ [Measurements'Y values selectedfor these parameters in this analysis, and are / believedtobe consistent with the military specification I0.

- Block Diagram of the Feedback Control Structure Table 2 - Desired Attitude Modal Parameters where, (Osp 2 Rad/Sec _,p 0,707 St,, [= k2t k_ k23 k24 u " k3t k32 k33 k34 q -

fkll kl k3k 1[ t

l/q:o2 0.52 Rad/Sec u =-K(s) y -K_Sp The value for the flight path time constant is driven by handling (4) requirements, but is also consistent with Ref. 5, which states Note that the structure of the compensator, K(s), will be the a(s)= -4.066(,*0.1067--#0. l ? 14)) T(,) (7 } same for both the classical and EIMF/LTR designs presented in 6_ds) (s÷O.08664±0.0943$jXs+ 1.408± 1.409j) the next sections. Also note, K6p will be a 4xl vector of T(s) = (s+0.4117Xs¢ i .985*--3.53ii)($÷8.015 )(s+89.65) (s+0.4104)(,+ 1.985±3.53 lj)(s+8.016)(s+89.67) (5) constant gains (for both designs) on the pilot stick input, 6p.

Finally, because the plunge velocity, w, is a state used in the where T(s), which reflects theeffects of theengine dynamics, is vehicular model, it is used, instead of angle of attack, a, in the approximately unity for each response. Therefore, the engine measurement vector. The response of interest, a, may be response is decoupled from thatof the airframe's attitude and obtained simply by the relationship, ct = w/V o. flight-path response. The closed loop I/'[e2 achieved is about 0.63 (I/s), and the short-period damping and frequency Classical Control La, Synthesis achieved are 0.7075 and 1.99 rad/sec, respectively. Thus these design goals all appear to be adequately met.

First, the desired i/'t0: can be obtained via augmenting Figs. 3 and 4 present the closed-loop frequency the lift effectiveness of the airframe, or by increasing Za. This responses for angle of attack and pitch rate from pilot stick may be achieved by feeding back angle of attack to the flaps, input. Also plotted in the dashed lines are the responses of the with a gain corresponding to k32/Vo in Eq. 4. The necessary Za desired dynamics presented earlier.

is obtained with a feedback gain of 2.9 (deg/deg). Next, to stabilize the attitude response, angle-of-attack (or w/V o) will be 20, - ;-.-. -_- .,.- ..... : : : .

fed back to the thrust-vectoring nozzle, with gain k22/Vo in Eq.

4. A root locus of this transfer function (with the flap loop closed) would reveal that such a loop closure would yield the desired short-period frequency with a gain of 1.32 (deg)/(deg).

Then, to augment the damping of the resulting short-period mode, pitch rate will also be fed back to the thrust-vectoring t_J 10o 10t 102 nozzle, with gain k23. Again, a root locus for this loop closure would reveal that the required gain is 24.7 (deg)/(rad/sec).

Finally, feeding back forward speed with a small gain to the thrust-reverser port area can be used to help regulate forward speed, which will help damp the phugoid mode, force a front- side response, and eliminate a non-minimum phase flight path.

At the slow flight velocity, the vehicle's trim condition is "on the back side of the power curve," as shown in Fig. 2.

.... ..i..i.iii.i i ..... i..i iiii.!ii, i i.i !!i iiiii

lOq I0o lOl 102 F'_lUeacy i. Rad/S_ (Dem_ Model - --) ,."Back side" - Closed Loop Frequency Response of Angle of Attack-to-Pilot Stick Input (l_g/lbs) Power _,,,_ of the -30 .........................

-40 ":........ _•..: .......... :.. ,.

/_ Power Curve

.so__: i :: i:: iii

Velocity 10q !_ 101 10; - Example Power-vs-VelocityCurve This leads to nonminimum phase behavior associated with a right-halfplane transmission zero in the attitude transfer function, or a right-half plane I/z0], (seeEq. (I)). Itturnsout

-i- iii-i!ii ...... !-i :-i!-iii

that the airframe transfer function matrix considered later in the multivariablc case also has a transmission zero at the same location.This featurelimits robusmcss recovery in the LTR procedure s.tl. Regulation of forward velocity eliminates this problem. A speed-loop gain on the thrust reversing loop, or kit 10o 101 I r_ of 0.5 (in2)/(ft/sec) is selected here.

Finally, the engine response must be regulated to reject - Closed Loop Frequency Response of PitchRate-to- disturbances. A simple proportional-plus-integral loop is used, PilotStickInput ((Rad/Sec)/Ibs) with gains of -6 (lb/hr)/rpm and -3 ((lb/hr)sec)/rpm, respectively, to close the loop on engine speed to fuel flow. So These responses show good agreement, especially inthe critical k,_(s) is 6(s+.5)/s. Designs with additional engine loop closures frequency range between 0.5 and I0 rad.scc.

are currently under investigation.

Fig. 5 shows the disturbance rejectionperformance, in The closed-loop airframe response transfer functions terms of the closed-loop sensitivity function relating engine using this c.onmal law are speed to a speed disturbance,or mag[I/(l+kg)] at the engine speed output.

q(s) _ -0.0795s(s+0.2048)(s+0.6266) Engine speed disturbances will be rejected below about 4

us) s÷0

tad/see.Fig. 6 shows the response of the fan speed to a one RPM step disturbance. This plot shows good regulation T(s) = (s+0"4052)(s+l'985-+3"532))(s+8'016)(s+89"68) performance with a settling time to 2% of the finalvalue of (s+0.4104)(,+ 1.985±3.53 ljXs+8.0 i 6)(s+89.67) approximately 5 seconds.

IO

-5

i -I0 _ii ___ _ o ...... i:i.......

.20 10-1 10o 101 ll_ .30 10.1 I0o I0_ 102 F_xlUmC ) tnRad/Sec -Ioo ............... ;,_- -.-"_' ..

-12o ........ .... -.-,'J ......

- Sensitivity Function of Measured Fan Speed-to-Fan -.o I

Speed Disturbance (RPM/RPM )

.,,o..... I

| -180 || " i I 10-_ IOo I0] I f_ IItl .... i....... ?....... i ...... i....... i............ i .... i .....

F-mquency mR_/Sec - Thrust Vectoring Angle Loop Transfer - With All Other Loops Closed

. .i ...... i...... i...... i....... :: ! ! i i

a._........... ._ ...... i ...... ; ...... : ..... 2 ...... i ............ 2 ......

.-40 - : i : -60 ....... i _ 10-1 10o 101 10l 1.5 0 O.J _$ • 4J 200 .

- Measured Fan Speed Unit Step Disturbance Response The open-loop Bode plots for these control laws are shown in Figs. 7 through 10, where each loop transfer shown reflects the fact that all other loops are closed.

10-t 10o I0_* 102 F.n_t,mc_ in _ - Flap Angle Loop Transfer - With All Other Loops _0 ' _.ii i _ i ¸ i !!

Closed 40, ::::; " : i i!iiil i i i iii:i I ._o____._o . • ! ..... i i _ • !i ....... !-.

10.1 10o 101 11_ 10-1 I0O 101 lOa |00 : :: ,.]

,_ 5o ..... ........... i..... i : i o -.-: ...... ------ii- ..... ..-i- -i : :

i i :ii:ii ::...i..i.!-i.::!.::.. i./.;..i..i. ::iii

-.1oo"5° .. .i ._:.i.ii .... _ .. i .... i ii__ ..

10-1 100 10: IC_ 10-z 10o 10_ 10z - Thrust Reverser Port Area Loop Transfer - With All F-mqucncy inRsd/Sec Other Loovs Closed - Fuel Flow Rate Loop Transfer - With All Other The thrust-reverser loop has a gain cross-over frequency Loops Closed of 0.2 rad/sec, a phase margin of 1100, and an infinite gain rateloop, for example, that design had a 15 dB gain margin and margin. The thrust-vectoring loop has a gain cross-over 50 ° phase margin. The cross-over frequencies of the thrust- frequency of 2.2 tad/see, a phase margin of 45 °, and a low-gain r_verser, thrust-vectoring and fuel-flow-rate loops from the margin of approximately -6 dB. The flap loop has a magnitude same study were 1.7,6.2 and 3.2rad/sec, respectively.

less than one for all frequency, and a gain margin of approximately 6 dB. Finally, the fuel-flow-rate loop has a EIMF/LTR Control Synthesis Methodology6, 7 cross-over frequency of 3 rad/sec, a phase margin of 64 °, and infinite gain margin.

Consider the control of the linear time-invariant These results can be compared with those for the aircraf_engine dynamic system modeled as LQG/LTR control design recorded in Ref. 5. For the fuel-flow- x=Ax+Bu frequencies down. If perfect model following is not achievable y = Cx (6) the performance is achieved via arbitrarily high gains.

The synthesis approach just described must now be extended to allow regulation of some of the system's responses.

The model of the desired dynamics to be followed is represented Regulation is incorporated into the model following synthesis by as simply defining the desired model to be followed by the regulated responses as the constant zero. For example, if _'m ----" AmXrn + Bm_p responses Yl and Y2 are to follow a desired model with Ym = Cmxm (7) responses Ym, while responses Y3 and Y4 are to be regulated, then the error vector becomes simply _Sp = -100. _p where _Spis the stick input from the pilot.

= Y2 - Y2,, The error vector to be chosen is I Yt - Y_, e=y-ym

(8) y4

(13) and the error dynamics to be selected in the synthesis are Otherwise, the formulation and solution to the LQ problem proceeds as above.

= -Gee (9) Once the EIMF state-feedback gains are found from this procedure, compensators may then be synthesized using the Defining the quadratic loss function to be: loop-transfer-recovery procedures of Ref. 5, 9 or 11. The approach of Ref. 9 yields a closed-form solution and exact recovery, while the more familiar approach of Ref. 5 or 11 yields asymptotic recovery. Proceeding as in Ref. 9, a singular J =I" { (e+Gee)TQ(e+Gee) + uTRu}dt value decomposition of the control input matrix for the plant, B, (10) is used to formulate a reduced order observer, described as: the solution of this linear quadratic problem is the state-feedback control law (14) u =-Kmx - Kffxm- KruPp (11) from which the LTR compensator matrix is obtained as shown below.

Implicit model following results when the gains on the model states are zero. This can be assured if Cm is chosen to be K(s) = Kro(C(sI - _)-1_ + _)

square and invertible, and the error dynamics are chosen to be (15)

The algorithm to obtain this compensator is presented in

G, = -CmAmO,_ (12)

Appendix A. Note that via standard LQG/LTR, the compensator is of the same order as the plant and order reduction may be Perfect model following results when the error vector is exactly considered. In this LTR procedure, a reduced order observer is zero for all time, and is achievable when CB is full rank. If obtained directly. However, it does not guarantee any high- perfect model following is achievable and the system has no frequency roll off, so this would be added, if necessary, as the non-minimum phase transmission zeros, the above LQ final step in the synthesis.

formulation will asymptotically approach the perfect model With the compensation K(s) so obtained, and the pilot- following result as R --o 0. Fig. 11 presents the closed-loop input gains taken from Eq. 11, the augmented system becomes system implied by Eq. (11), for implicit model following.

that shown in Fig. 1.

EIMF Control Law Synthesis _ (st-A)" 1 B The desired dynamic model to be followed by the

1'

aircraft's attitude response is, consistent with Eq. 2, qm(S) = M_(S + 1/%_) Fim]_ 11 - Model Following State-Feedback Control Block Diagram _p(S) S2 + 2_,pO_S + ¢_sp Although the matrices Q and R in the above loss function

o_(s) = z_

can be used to adjust the gains, it must be emphasized that the _p(S) S2 + 2_spC.OspS+ tO_sp choice of desired dynamics to be followed and the error vector to (16) be minimized is the most critical pan of the synthesis. The transmission zeros of the system are determined by the choice of or inputs and followed responses, thus, by the choice of the error vector. As Reference [6] states, for a square system, some of the closed-loop poles approach the finite open loop transmission zeros, and, under the conditions of perfect model following, the X2 "(_Sp "2_,p(1):p rest of the closed loop poles approach the poles of the error dynamics, Ge. Further, for implicit model following Eq. 12

° ]Ix'?

qm Mg'to= M_, x2 reveals how the error dynamics are directly related to the desired (17) model dynamics. The choice of desired dynamics and error vector can also greatly influence the shapes of the loop transfers, with Finally, formulating the problem such that perfect model- following is achievable keeps the loop gains and crossover o_, v = 2 rad/sec Q = lxlOS(diagl0.4,1,100,0.1]) and _sp = 0.707 R = I x 10 "a (diag[ 1,0.2,0.2,1 e-031 ) llxe2 = 0.52 7-,0 = -4.42 These values were chosen primarily on the basis of the resulting M_ = -0.0797 Bode loop shapes, with special attention to stability marDns and loop cross-over frequencies. The resulting EIMF control gains, With regulation of forward speed and engine fan speed also K_ and Kfb, are listed in Appendix C.

desired, the error vector is: Figs. 12 through 15 show the individual loop transfers.

U W - W m _= q - qm

N2+f N2

I0-I I0o lOI I0_ (18) Note that integral of fan speed is added in the above. Addition of this term is associated with the fact that integral action on N2 is desired. Again, note that plunge velocity is used, where w=a/V o. With this error vector, the finite transmission zeros of the open-loop system are shown in Table 3.

.io0'° i!:: i i i i

Table 3 - Finite Transmission Zeros of the O cn Loop System IO-t 10o 103 I fin Transmission Zeros -68.612 - Thrust Reverser Port Area Loop Transfer - With All Other Loops Closed -13.0491 + 5.5632j -1.0 0.0 The transmission zero at -1 is due to the inclusion of the integral of engine fan speed in the error vector, as explained in Appendix "_IJ, lOo 10_ 10a C. The transmission zero at the origin is due to the fact that pitch-rate is used in the error vector. If pitch rate plus integral of pitch rate, or e, were used, this zero would move into the left 3O0 half of the complex plane.

The error dynamics are now selected to be G, = 0 -CmAmC._ 0 g_ 0 0 0 0 (19) 'l_z I0o lO] I0_ r-nsqumey in RadtS_ This choice of error dynamics reflects the desire to decouple the - Thrust VectoringAngle Loop Transfer- With All attitude dynamics from the engine speed and forward speed, as Other Loops Closed well as implicitly model follow the desired short period model.

Finally, the forward-speed and en[_ine-speed responses will The thrust-reverser loop has a cross-over frequency of 0.15 include a mode with time constants g_ and g_,_, respectively.

rad/sec, a phase margin of 90 °, and an infinite gain margin. The Values for these time constants were chosen to be 0.1 and 1 thrust-vectoring loop has a cross-over frequency of 2.1 rad/sec, rad/sec, respectively.

a phase margin of 55 °, and a gain margin of -I0 dB. The flap loop has a magnitude lessthan one for all frequency, and a gain EIMF Results margin of approximately I0 riB. The fuel-flow-rate loop has a Before synthesizing the dynamic compensation via the cross-over frequency of 10.2 rad/sec, a phase margin of 70 °, LTR procedure, the frequency responses of the loop transfers, and a gain margin of 12 dB. These resultsshow thatthe EIMF using state feedback gains obtained from the EIMF control laws, design gives loop shapes, loop cross-over frequencies and Krb(sl-A)-]B, are investigated. This is done, for example, to stability margins thatare very similar to those of the classical check the performance, controller bandwidths, and stability design presented earlier.

robusmess. Since loop transfer recovery will be used later, the bandwidths and robustness of the state-feedback control laws will be recovered, by definition. Also, for the control laws implemented as in Fig. 1, it can be shown z2 that the responses to pilot input are unchanged due to the inclusion of estimation in the manner described herein.

For the results presented below, the values of Q and R in the loss function of Eq. (10) are s(s+O, l)(s+ l)3(s+ 13.05+_5.563j) t T(s) = ((s+0.000151 l_/ (s+ i 3.05__.5.563j)(s+68.62): I

)

is+ 13.__ i )_-'_+68.62) J _r Near-perfectmodel following isevidentin theseresponses.

lOq I0o 101 I02 Figs. 16 and 17 present the closed-loop frequency responses for pitchrate and angle of attack to pilotstickinput.

200 Also plotted arc the desiredfrequency responses.Since they are essentially the same, and considering the closed-loop transfer functions given above, one must conclude that the desired handlingqu.'flifics would bc achieved.

-I00 ]0q 10e 10z 102 Fmq,imcy m m_.+ - Flap Angle Loop Transfer - With All Other Loops Clos_

i: iiii iiilii

._o[ i i i iiiiii i _i i!iii!i i _ii!i_i

10+I 10e 10* IOZ ==

.__2o " : :.ii.i.i.i ...... i...i..i._.i.ii.l.i ...... _i...i..i..i.i.:_::i

.is0. : ...... i i :i::ili : :::1

.2oo..... ;...+..+.:->_,.> ..... .,,q,i+.i.ii.l.i::i ...... ........i...

10-1 too 101 10g -50 ..................

I0-I I0o 101 102

: :: ..... >_iiiii! i i ii!i!!

FnxlWnCy m Rld/Sec {Iksm:d Model = --) qoo " : i_...... i. ,.i.,!..i+i.Lii - Closed Loop Pitch Rate-to*pilot Stick Input II Fratucncy Response _. -150 50 . . .

-200 : : : :::::: : : ......... : • • 10q I00 10i 102 Frequency in Rad/Sec - Fuel Flow Rate Loop Transfer- With AllOther Loops Closed .i ............. ii .................... ii! .......... i .....

EIMF/LTR Results lOq 10o 101 102 With the state-feedback gains now available, the compensation issynthesized as outlinedin Appendix D. The responses taken forfeedback are u,w, q and N 2, identical tothe 200 .........................

classical case. Again, thisleads to a 4x4 compensator matrix,

iiiiiiii ..... i .+.I ii!iii i

K(s), as in Fig. l,which describesthe closed-loop system.

Recallingthatthedesiredpitch rateand angle-of-attack- to*pilot inputtransfer functionsare qm(S) -0.0797(s+0.52) 10.t ]0o 101 10_ Frequency m Rad/Se_ _ Mede.l = --) _p(s) = (s+ 1.414+I.414j) (#I_) - Closed Loop Frequency Response ofAngle of ecru(s) _ -4.422 {deg] Attack from Pilot StickInput (Deg/Ibs) _p(s) - (s+l.414+IAl4j) t # / Shown in Fig. Ig isthe performance of the control law inrejecting fan speed disr, n'banccs, again expressed in terms of the closed-loop transfer functions obtainedusing this control law the magnitude of the sensitivity function for fan speed ale, mag[I/(l+gk)]. It is noted that speed disturbances will bc q(s) -0.0797(s+0.52) rejected below about 15 tad/see. This performance is better than that shown for the classical control law.

8p(S) = (s+l.414+1.414j)T(S)(#1_) Fig. 19 shows the response of the fan speed to a one RPM step fan speed disturbance.

Very accurate pole-zero cancellations in the closed-loop transfer 8p(s) (s+l.414+_lA14j) function leads to the following transfer function for this disturbance response.

N2(s) s(s+l.251)(s+3.518)(s+6.805)(s+97.64) RPM d(s) " (s+l)(s+l)(s+13.05:LS.563j)(s+68.61)(RPM) IG Table 4 - ELMF/LTR Compensation Mamx Numerators for the -10 Individual Compensator Transfer Units of -20 Functions Bode Gain Compensator

i

-30 NK11 = -122(0) (0.6X!)[0.94,9.9l -0.6 in2/(fVsec) ..4O 10-t 1_ 10, I_ NKI 2 = 37(0) (-0.6)(1)[0.94,101 .0.2 tn2/ffVscc) F"rr.@z:ncy m R,xtJScc NKI3 = -89(I XI.5)(14)(29X32) +121 in2/i.r,d/.'_ec) NK21 = -0.3(0) (1)[0.92,12](-33) +0.1 dcg/(ft/sec) - Sensitivity Function of Measured Fan Speed-to-Fan Speed Disturbance (RPM/RPM) NK22 = -0.2(0) (I)[0.92,14](133) .0.3 dcg/(ft/sec) NK23 = -2](-0.1)(1)[0.92,14}(67) +1.9 dcg/(rnd/sec) NK31 = 2.1(0) (1)10.89,17](23) +0.4 dcg/fft/sec) NK32 = -0.7(0) (I)10.91,16]{29) -0.4 deg/{ft/scc) 0.8 ........................ . ............ . ............ :........

NK33 = 59(0.3)(1)[0.92,141(68) +16.0 deg/(rad/scc) NK41 = -4.9e5(I)[.0.42,1.5110.95,4.11 -1284 #Rulfft/sec) 0.6 ....................................................

e_ NK42 = 1.5c5[0.07,0.78l(I)(3.3)(3.8) +82.5 #/hrltft/scc) ,¢ ! NK43 = -!.03c6(0.6)(1)(1.7)(4.6)(47) -16.488 #/hr/_r'ad/sec) 0.4 ............................................................. NK44 = -85(1X-3.7)(4XIOX-92) -85.1 #/hr/(RPM) z Characteristic Polynomial of Compensator : 0.2 .......... i..................................................

A(s) = (0) (1)[0.92,14.2](68.6) Note: (a) = (s+a), and [a,b] = complex mode with damping 1 2 3 4 ratio = a, and frequency = b Tune inSecmds F,.LgtlIg..._- Measured Fan Speed Unit Step Disturbance the new technique were somewhat superior to those for the Response classical design, the individual loop transfers of the two control laws were quite similar. Both of these are considered to be positive attributes of the new procedure offered. The airframe Clearly the steady state value of N2(t) to a step disturbance goes responses with the new control law were exactly those desired, to zero. One pole at -1 is the error pole, ge_2, as discussed in thus demonstrating the performance achievable, subject to Ref.6. The remaining poles are transmission zeros arising due actuation bandwidth, with this approach. Finally, engine control to the loop-transfer recovery. Note that all these parameters were laws were simultaneously synthesized, along with those for the chosen directly or indirectly in the synthesis, and therefore may airframe, and would appear to deliver good disturbance-rejection be adjusted as desired.

performance. This was also accomplished with reasonable Comparisons of singular value plots of the loop transfers crossover frequencies. The simplicity of the classically designed using the state feedback control law, Kfb(SI-A)dB, and the LTR compensators was superior to the new controller, the latter being compensation, K(s)C(sI-A)-tB, as well as the loop transfers of fourth-order while the former consisted primarily of constants.

each loop, with all other loops closed, revealed complete If different vehicle configurations ultimately exhibit more bi- robustness recovery, as promised by this exact recovery directional coupling than that considered here, a classical control method, Ref. 9. Consequently, the loop transfers for the loops synthesis may, however, cncoumer considerably more difficulty broken at the control input are identical to those for the EIMF than that demonstrated here. Whether the difficulty involved state-feedback control law. Specifically, the individual loop with the newer approach is significantly increased as well is an transfers are as shown in the prc_ous section.

open question.

The state-space realization for the compensators is given in Appendix E. The compensator transfer-function matrix, or Appendix A. Linear Model for the Case-Study K(s) in Fig. 1, is given in Table 4. Note that Kl4(s), K24(s ), Vehicle and K34(S ) are essentially zero, so they are not listed. These transfer functions are all fifth order, with poles at the The states are defined as transmission zeros of the plant.

x = [u (ft/sec), w(ft/sec), q(rad/sec), 0(radians), Conclusions N2(rpm's), N2.5(rpm's), Pt(psia), T41e(*R)] T A control law synthesis technique was presented that with inputs, was developed to achieve excellent handling qualities, u = [ATs(in2), 6n,_(deg), _>rv(deg), wf(#/hr)] T decoupling the engine and airframe dynamics, with modest control bandwidths or crossover frequencies. The robustness For the vehicle in question, the model is properties of the LQR solution were exploited by formulating the implicit model following problem in the LQ framework, and utilizing a novel loop transfer recovery procedure to obtain the A A = feedback compensation. The methodology was applied to the -5.8930c-02 1.0670e-01 -3.8600c+01 -3.1840e+01 integrated flight and propulsion control problem in the form of a -2.6590e-01 -2.6650c-01 1.9480e+02 -4.5990e+00 case study, utilizing the linear model of an unstable fighter - 1.5410e-03 7.8060e-03 - 1.9490c-01 -4.8180e-04 aircraft, with engine dynamics and a 2D thrust-vectoring and 0 0 1.0000c+00 0 thrust-reversing nozzle. A classically designed control law was developed for comparison.

The results revealed that both control laws would appear to deliver adequate performance, as defined herein, with modest gain crossover frequencies, thus keeping actuation requirements to a minimum. Although the airframe responses obtained using

AAE = Appendix C. Gains From EIMF Synthesis 3. ! 440e-04 2.5990e4)4 3.8190C-02 2.2500e-03 - 1.5780¢-05 -2.1060e-06 1.8260e-04 -2.9570¢-06 Kib = 9.4600e-07 3.7440e-07 3.6680e-05 2.6760e-06 (Columns 1 through 5 ) 0 0 0 0 -5.8088e-01 -3.6615c-01 1.6462c+02 1.4829e+02 -4.0719e-03 1.2147e-01 -3.1857e-01 -2.0023e+01 1.9819e+00 -1.4838c-05 9306(O-01 -3.7123¢-01 5.6713e+01 1.6195e+01 1.1593c-04 AEA = 5.2975e+00 1.0497e+00-3.2672e-05-3.4348c-05-IA915e+01 7.7820e-01 1.5420e-01 0 0 1.5180e-01 3.0080e-02 0 0 7.9340e-01 1.5720e-01 0 0 (Columns 6 through 9) K6p = - 1.0050e-01 - 1.9920e-02 0 0 5.6152e-03-5.7903e-01 2.2713e-03 1.1439c-03 -8.1261e.01 2.4741e-05-2.2436e-03 2.0137e-05 4.7736c-06 -7.8155e-01 -5.8529c-05 5.5120¢-03-3.6180e-05-I.3315e-05 1.9853e*00 A E = 4.0994e+01-2.3376c+03 7.8965e+01 6.8074e+00 2.2454e.08 -4.1910¢+00 6.0220c+00-3.4340¢+02 1.160(O+01 4.2630e-01 -5.7070e+00 2.7160c+01 1.0400e+01 Note that the gains in the 9th column are the gains on the integral 2.2950¢-01 1.1550c-01 -9.0240e+01 8.4760e-01 of fan speed.

3.7400e-02-1.0360c-01 -7.9540e+00 -1.0680e+00 Appendix D. Algorithm for Obtaining the EIMF/LTR [B^ BAE] = Compensator of Fig. 1 -2.0550c-01-4.1830c-04-8.4280c-02 3.4360e-05 -2.9360e-04 -5.4520e-01 -2.1475e-011.2380c-08 Under the assumption that CB is of full rank, obtain the 1.0680e-04-7.9700c-028.8132e-03 5.5070c-08 singular value decomposition of B, 0 0 0 0 ZB ,'1" [BE,, Bd = B=[Ut U2][ 0 ]Vt 0 0 1.4690e-01 0 0 0 5.3600e-02 Defining, -4.3020e+01 0 0 1.8130e-02 0 0 1.6430e-01 Appendix B. Transmission Zeros Given an output to a linear system as: the state space matrices for the LTR compensator of Eqns. (14) and (15) are, y = clx 1 + c2x 2 with, /.

= UIA L1 B = U_A L2 = KroLt D = Kn,L2 x 2 = J Xldt K(s) = Kro(C(sI - _)-lfi + D) then one of the finite transmission zeros of the system is: Appendix E. EIMF/LTR Compensator State Space Z : -C2/C 1 Realization For the following system 5.7226e+04 2.0882e+04-4.2212_44M 6,4(X)2e,d_ 9.4118e+00 0 -2.292_+04 -8.3294e.,O3 1.6822e-d_ -2.5505e+04 -1.3850e+01 o 2.3632e+01 8.6205e.+00 -1.1212e+01 2.6424¢+01 5.5470e-03 0

,,lr,.1.,.r -4.3788e+04 -1.5978e.*O4 3.27.86e+04 -4.8980e+04 -3.9126e-aDO 0

x2J-i 1 0 JLx2J 1. 0 o o 0 o -l.OOOOe,_o o 0 0 0 0 0 0

--tc, c l[x;]

g_- it can be shown that the transmission zero, z, solves the -4.1565¢+04 1.2836e-tO4 -8.7611e+04 -9.4118e*00 1.6689e-*-1)4 -5.1168e+03 3.4906e+04 1.3850¢+01 following generalized eigenvalue/eigenvector problem, -1.6144¢+01 4.9111e-*00 -1.4672e+01 -5.5470e-03 (Reference [13]): 3.2034¢+04 .9.g216e+03 6.6991e*04 3.9126e.d)0 0 0 0 1.0000e+00 o 0 o 1.0000e+00 z m2 = m2

[lOol 0° llml][ a2 b liml0 0 0 0 0 clc2 0

(Columns 1 through 5) v v 2.0736e+02 7.5653e+01 1.1425e+01 2.318ge+02 4.3601e-02 1.1963e.tO0 4.3648e-01 1.3142e+00 1.3379e+00 1.5458e-0a from which it can be seen that: 2.9408e+00 1.0731e+00 1.5779e+01 3.2895©+00 ..4.7652e-04 6.6995e+05 2.44.46e+05..4.9409e+05 7.4930e+05 9.1941©+01 ml =zm2 Clml + c2m2 = 0 (Column 6) 5 = -4.2457e-02 -1.2198e+02 3.6856e+01 -8.9368e-_1 -4.2457e-02 -1.4980e-04 -3.4894e-01 -1.7434e-01 -2.1007e+01 -i.4980e-04 which implies that z = -c2/cl. Note that this proof can be 4.6320e-04 2.0863e.d)0 -7.2555e-01 5.9131e-_01 4.6320e-04 extended to a general nth order system.

-8.5134e+01 -4.9011e+05 1.5027e._05 -1.0254e+06 -8.5134e+01 l0 Acknowledgements This work was sponsored by NASA Lewis Research Center under Grant No. NAG3-998. Mr. Peter Ouzts is the technical monitor.

References Schmidt, D.K., Schierman, J.D., "Analysis of [1] Airframe/Engine Interactions - An Integrated Control Perspective," AIAA Paper No. 90-1918, presented at the 26th AIAA Joint Propulsion Conference, Orlando, FI, July, 1990.

Shaw, P.D., Rock, S.M., and Fisk, W.S., "Design [2] Methods for Integrated Control Systems," AFWAL -TR- 88-2061, Aero Propulsion Laboratory, Air Force Wright Aeronautical Laboratories, Dayton Ohio, June, 1988.

Rock, S.M., Emami-Naeini, A., Anex, R.P., [3] "Propulsion Control Specifications in Integrated Flight/Propulsion Control Systems," AIAA Paper No.

88-3236, AIAAIASMEISAEIASEE 24th Joint Propulsion Conference, Boston, Mass., 1988.

[4] Smith, K.L., "Design Methods for Integrated Control Systems," AFWAL-TR-86-2103, Aero Propulsion Laboratory, Air Force Wright Aeronautical Laboratories, Dayton Ohio,December, 1986.

[5] Garg, S., Mattem, D.L., and Bullard, R.E., "Integrated Flight/Propulsion Control System Design Based on a Centralized Avvroach," AIAA Paper No. 89-3520, AIAA Guidance, Navigation and Control Conference, Boston, Massachusetts, 1989.

[6] Anderson, M.R., Scbmidt, D.K., "The Significance of Error Dynamics in Model-Following For Flight Control Design," AIAA Paper No. 87-2311, AIAA Guidance, Navigation and Control Conference, Monterey, California, 1987.

[7] Schmidt, D. K., "Flight Control Design Research to Meet Handling Qualifies Requirements, Phase II," final report for the McDonnell Aircraft Co., performed at Arizona State Univ., Dept. of Mech and Aero Engr., Tempe, AZ, July, 1989.

[8] Doyle, J., Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Transactions on Automatic Controls, Vol. AC-26, No. 1, pp. 4-16, Feb., 1981.

[9) Bacon, B.J., "Closed-Form Solution for Loop Transfer Recovery Via Reduced-Order Observers," AIAA Paper No. 89-3455, AIAA Guidance, Navigation and Control Conference, Boston, Massachusetts, 1989.

[IO] Anon., MIL-8785C, Flying Qualities for Piloted Airplanes, USAF, Flight Dynamics Laboratory, WPAb-B, Dayton, Ohio.

[11] Ridgely, D.B., Banda, S.S., "Introduction to Robust Multivariable Control," AFWAL-TR-85-3102, Flight Dynamics Laboratory, Air Force Wright Aeronautical Laboratories, Dayton Ohio, February, 1986.

[12] Schmidt, D.K., and Foxgrover, J.A., "Muhivariable Flight Comrol Synthesis Approaches to Meet Handling Qualifies Objectives," A1AA paper no. 84-1831, GN&C Conf., Seattle, August, 1984.

[13] Sinha, P.K., Multivariable Control, Marcel Dekker, Inc., New York, 1984.

II Robust Control Synthesis for Integrated Flight and Propulsion Control John D. Schierman and David K. Schmidt Aerospace Research Center College of Engineering and Applied Sciences Arizona State University Tempe, AZ 85287-8006 Abstract Two control synthesis methodologies are presented and applied to synthesize control laws for integrated flight and propulsion control (IFPC). The vehicle considered is representative of an unstable modern fighter aircraft equipped with a 2D thrust-vectoring and thrust-reversing nozzle. The lineadzed model of this vehicle includes both airframe and engine dynamics. It is necessary to regulate some responses and dynamically shape others, thus leading to a hybrid control problem formulation. A linear quadratic (LQ) model following formulation is the first approach to this hybrid problem.

Compensators are then obtained to realize an output-feedback control law, by using standard loop-transfer-recovery procedures.

An H"* formulation is also presented. For the LQ formulation, near- perfect airframe response following can be obtained while good stability robustness and reasonable loop cross-over frequencies are found in the individual loop transfers. The trade-off between model following performance and multivariable stability robustness, as measured by singular value tests, is specifically addressed. Results obtained via the H** control formulation are shown to be similar to those from the LQ formulation.

As Presented at the December 1990 IEEE Conference on Decision and Control Honolulu, Hawaii.

Robust Control Synthesis for Integrated Flight and Propulsion Control*

John D. Schiermant and David K. Schmidt i'* Aerospace Research Center College of Engineering and Applied Sciences Arizona State University Tempe, AZ 85287-6106 1. Introduction Conventional aircraft typically do not experience significant dynamical interactions between the airframe and propulsion subsystems. Separate control designs of these subsystems are quite adequate. However, new aircraft configurations are under development in which the propulsion systems are capable of delivering forces and moments to the flight control process to enhance the maneuvering capabilities. For such aircraft, significant dynamic interactions between the airframe and the engine can occur and some configurations may experience interactions in critical frequency ranges. If this coupling is large and not taken into account when designing the control laws, then these dynamical interactions can lead to loss of system performance and stability robustness, or to instabilities, as discussed in Ref. 1.

This problem is referred to here, and elsewhere, as the Integrated Flight and Propulsion Control (IFPC) problem. During the past several years, design integration methods 2-5 have been proposed that were intended to synthesize integrated control laws, while in a variety of ways dealing with the potential dynamic interactions.

In this paper a design approach different from those in Refs. 2-5 is offered, and explored via a case study. This new approach will be referred to as Extended Implicit Model Following (EIMF). Two design methodologies will be presented which implement this new approach.

First, EIMF control laws will be synthesized by linear quadratic (LQ) with Loop Transfer Recovery, (LTR) techniques, designated as the EIMF/LTR design 6. Then, a unique H** formulation will be developed and used to synthesize a second set of control laws, referred to as the EIMF/H** design.

The design objectives will be presented at the outset, the justification is given for considering this design approach in light of these goals. Then the synthesis methodologies are presented, and the case study is addressed. The results will then be discussed vis a vis the aforementioned design goals, and conclusions presented.

2. Design Goals and Methodology Motivation The design objectives for the IFPC problem involve system performance, stability robustness, and implementation issues.

Performance - Foremost among the performance issues is the fact that the control systems must deliver excellent handling qualities, in spite of the potential airframe/engine dynamic coupling. The handling qualities criteria are quantified in terms of specified time constants, damping ratios and frequencies for the airframe modes, as well as closed-loop frequency responses from pilot input. Control laws that produce closed-loop airframe responses that reflect classical As Presented at the December 1990 IEEE Conference on Decision and Control, Honolulu, Hawaii.

"_Doctoral Candidate and Research Associate.

1"i Professor of Engineering and Acting Director; Member, IEEE.

Copyright © 1990 by David K. Schmidt airframedynamicsare desirable.In fact, how well the resulting airframeresponses approximate certainfrequencyresponses of a conventional aircraft with thedesiredmodalcharacteristics is one stepin meetingthe military specifications 7.Oneimplication of this designgoal is that thecontrol

system should decouple the airframe and engine responses. If the engine's dynamics are

observable in theaircraft responses to pilot inputs,thenclassicalairframedynamicalpropertiesare not obtained.Notethatthesedesigngoalsarenot thoseof a regulator.

Engine control, on the other hand,requiresregulation of responses about an operating

point, with gain schedulingandtransitioncontrol from one point to thenext within the operating

envelope. For example,in orderto maintainstablecombustion,it is important that the fan and

compressor do not exceed their surgelimits. For structuralconsiderations, the main burnerandthe high pressureturbine should not exceedspecifiedpressureand temperaturelimits. Therefore,

stable, robust regulation of responsessuch as fan and compressorspeeds,temperatures,and

pressures, is a primary goalin thecontroldesignof theengine.

Finally, theseperformance objectivesmust be metwith minimum actuationrequirements

suchthatrate anddeflectionlimits areavoided.Not only arehigh actuation requirements taxingon

the hardware,but rate and deflection limiting also degradeboth performanceand stability by

introducing unmodelednon-linear effects into the loops. Therefore, control bandwidths or

crossoverfrequencies mustbeaslow aspossible.

Robustness - The system must possess adequate stability margins so that it is robust against unmodeled or inaccurately modeled dynamics. Usually, this requires minimum gain and phase margins in all loops, although singular value basedS, 9 robustness analysis can be performed as well. The results in this paper include both single-loop and multivariable robustness margins.

Also, the loop transfers must roll off sufficiently to handle high-frequency unmodeled dynamics or non-linearities.

Implementation - The compensation should be easily implementable. This implies that it should be of low dynamic order, and preferably should be similar to classical control laws. If so, the results can yield additional insight with regard to the control system's interactions with the overall airframe/engine system. Furthermore, the existing techniques for control law validation and verification, as well as the necessary gain scheduling may still be utilized.

Motivation - Model following is an integral part of the formulation considered here so that the closed-loop airframe responses may be shaped to take on desired dynamics. Model following design goals are not those for a regulator and this method may not necessarily yield loop transfers with classical (k/s) loop shapes, just as classical stability augmentors (e.g., pitch dampers) do not yield regulator loop shapes. Implicit rather than explicit model following is utilized to eliminate the dynamic pre-filter that is present in the latter control structure. This leads to closed-loop airframe responses of lower dynamic order that are easier to evaluate in terms of handling-qualities assessments, and simpler to implement. Also, perfect model-following concepts 10 are exploited to minimize loop gains and crossover frequencies.

For an integrated synthesis approach to the IFPC problem, regulation as well as model following must be admitted in the formulation. Typically, engine responses, and perhaps aircraft velocity must be regulated. The implicit-model-following formulation of Refs. 10 and 11 are herein extended to address the hybrid problem of model following for some responses and regulation of others.

The standard LTR procedureS, 9 is employed to synthesize compensators necessary to realize an output feedback structure, utilizing the state-feedback gains obtained from the LQ solution to the EIMF problem. This LTR procedure recovers the state-feedback loop shapes, and hence robustness, at the input to the plant.

Compensators are also directly synthesized by a new H** formulation to realize an EIMF/H** control law design. It will be shown that the control laws developed by the EIMF/LTR and EIMF/H** methods have similar characteristics. It will be shown that with both synthesis techniques there is an explicit trade-off between model-following performance and stability robustness. This interesting result is one of the more significant theoretical aspects of this research and is currently under further consideration.

3. Case Study Vehicular System

The vehicle to be considered in this investigation is the same as in Ref. 6. It is representative of a high performance fighter aircraft with the capabilities of 2-D thrust vectoring and thrust reversing. The vehicle dynamics are linearized about the Short Take Off and Landing (STOL) approach-to-landing reference condition at an airspeed Vo = 120 Knots and flight path angle 3,o = -3 °. The states, controls and responses are listed below. This model, with the same control and measurement vectors is used for both the EIMF/LTR and EIMF/H** designs.

The state vector of the model, and the control inputs to be considered are, respectively, _' = [u, o., q, 0, N 2, N2. 5, P6, T41B] T and _ = [ATs, _v, 8naps, wf]T These variables are defined in the following table.

Table 3.1 - States and Controls of the Case Study Vehicular System The aircraft states are: The enfine states are: u = body axis forward velocity (ft/sec) N 2 = engine fan speed (rpm's) a = angle of attack (deg) N2. 5 = engine compressor speed (rpm's) q = pitch rate (rad/sec) P6 = engine mixing plane pressure (psia) 0 = pitch angle (radians) T41B = high pressure turbine temperature (°R) The aircraft controls are: The singl¢ _ngine control is: A78 = thrust reverser port area (in 2) wf = main burner fuel flow rate (#/hr) 5rv = nozzle thrust vectoring angle (deg) 8trap s = trailing edge flap deflection angle minus leading edge flap deflection angle (deg) - see Ref. [5] The aircraft's forward velocity is to be regulated essentially with the thrust reverser, while the attitude dynamics are controlled by thrust vectoring. The flaps are direct lift devices which are used to control the flight-path-to-attitude response, and the fuel-flow rate is used to control the engine fan speed. The measurements used for feedback are = [ u, ct, q, N2] T The vehicle model, partitioned in the following manner, is given in Appendix A, XE AEA AE BEA BE tiE (3.1) where the subscript A denotes aircraft subsystem and controls, and the subscript E denotes engine subsystem and controls. Results from a modal analysis are shown in Table 3.2. This table presents the open loop poles and the responses dominated by these modes. Note that the short period mode is unstable.

Tab]c3_ - Modal Analysis of the Open Loop System Mode Shapes Open Loop Poles -0.0571 _+0.2154.j_ phugoid mode (u) ,.,..n.....,..,.°,...°....,.,°.....,o_. °._...N,....,°°.,o.,°°, ,,,.,,_,°,Q.,._o,.°,.,,.,..,.*..°°°,O_*.*°*.,N*..°,_**_,.°**.*.°...**°.

-1.472 short period mode (w,q,0) + 1.065 -1.401 highly coupled engine modes -3.569 involving all the engine states .:..6.:.9...5...8. .................................................................................................................................

-89.28 mostly associated with P6 4. Performance Objectives for the Case Study The flight control synthesis objective is to obtain classical fourth-order longitudinal aircraft responses to pilot stick input, given by, Kq s(s + 1/'to,) (s + l/xe,)

q(s) _

_p(S) (S 2 + 2_phCOphS + (.02ph)(S 2 + 2_sp(.OspS + COs2p)

ct(s) Ka(s + 1/'ta,) (s + 1/'ta_)

Sp(s) (S 2 + 2_phOOph S + 032h)(S 2 + 2_spOOsp S + C02sp)

(Phugoid Mode) (Short Period Mode) (4.1) This implies the engine modes should not contribute to these responses. The short-period mode must be stabilized, achieving a specified frequency and damping ratio. Also, a desirable value for the real flight-path time constant, 1/7:o2, should be obtained. Table 4.1 lists the desired values selected for these parameters in this analysis, and are believed to be consistent with the military specification 7.

Table 4.1 - Desired Attitude Modal Parameters t0sr, 2 Rad/Sec _sr, 0.707 1/7:02 0.52 Rad/Sec The value for the flight path time constant is driven by handling requirements, but is also consistent with Ref. 5, which states that it should not be increased above this value due to excessive flap deflections.

The requirements on the phugoid mode will be met by achieving some modest damping for this mode, and by rendering this mode essentially unobservable in the attitude response. Therefore, the desired attitude response may be defined in terms of the following dynamic model in state space form:

-2 .pO .pjtX2

[q ml--[Mh, lI:;l

(4.2) which yields the following transfer functions: am(S) = _p(S) s2 + 2_sp_spS + ¢..02sp qm(S) _ Ms(s + 1/X_) _p(S) S 2 + 2_sp_OspS + C02p (4.3) Here, 8p is the input from the pilot (e.g., stick deflection). The remaining terms to be selected are Z a and M_. These values are obtained from the short-period approximation for the study vehicle.

This approximation yields o_(s) -0.1526(s+28.67) -4.376 (for s--0) (s- 1.003)(s+ 1.464) -- (s- 1.003)(s+1.464) q(s) -0.0797(s+0.3199) gtv(s) (s-l.OO3)(s+l.464) from which, 7__ = -4.376 deg/(slug-ft/sec) M_ = -0.0797/lbs Therefore, Eqn. (4.3) becomes: or.re(s) _ -4.37_ {deg / 8p(s) (s+l.414+l.414j) _lbs!

qm(S)_-0.0797(s+0.52) {_cl/sec] _p(S) (s+l.414+l.414j)_ lbs t (4.4) The objective of the engine control design here is to simply regulate the fan speed.

Quantitative specifications on the disturbance response of the fan speed, such as maximum overshoot allowed or desired settling time, have not been formulated at this time. So, the response characteristics will be selected to yield engine-loop crossover frequencies close to those in the attitude loop, thereby maximizing the potential for dynamic interactions, the basic issue in this research.

5. Control Law Structure for the Case Study The following block diagram represents the closed-loop system, Responses o f Interest etc.]

¢ Controls, u r---"a [q, l',k.

_ Measurements, y Figure 5.1 - Block Diagram of the Feedback Control Structure where, the control law is u = -K(s) y -K6p_ p or, (5.1) Ksp_p

rAT1 111 k12 k13 14 II 1.1 - .32 -,33 .24

Note that the structure of the compensator, K(s), will be the same for both the EIMF/LTR and EIMF/H"* designs. Also note, Ksp will be a 4xl vector of constant gains (for both designs) on the pilot stick input, 8p.

6. EIMF/LTR Control Synthesis Methodologyl0,11 Model Following - Consider the control of the aircraft/engine modeled as linear time- invariant dynamical system, or = Ax + Bu y= Cx (6.1) The model of the desired dynamics to be followed is represented as Xm = Amxm + Bm_p Ym = Cmxm 8p = Ap_ (6.2) where 8p represents the (unknown) stick input from the pilot. Since 8p is not known a priori, it is modeled as low-pass white noise.

The model following error vector is the difference between the vehicle's responses and the responses of the desired model, e = y - Ym (6.3)

with errordynamics

e = -Gee (6.4)

The error dynamicsmatrix, Ge,is selectedby the designer. The quadratic loss function to be

minimizedis

J = { (6+Gee) T Q (%+Gee) + uT R u }dt (6.5) where the weighting matrices on the error dynamics and control inputs, Q and R, are also to be selected.

The solution of this LQ problem is the constant-gain control law, u = -Kn,x - Kffxm - K%_ (6.6) Perfect model following results when the error vector is zero for all time, and is guaranteed achievable when CB is square and of full rank. The perfect model following control law can be obtained by algebraically solving for u which yields (_ + Gee = 0. However, this control law will resuh in closed loop pole-zero cancellations of any right half plane transmission zeros. The above LQ control law will asymptotically approach the perfect model following control law as R approaches zero, if perfect model following is achievable and the system has no right half plane transmission zeros 12. If right half plane transmission zeros are present, the LQ formulation will give closed loop poles located at the stable mirror images of the right half plane transmission zeros.

Implicit model following results when the gains on the model states, Kff, are zero. This can be assured if C m is chosen to be square and invertible, and the error dynamics are chosen to be Ge = -CmAmCl (6.7) The matrices Q and R in the above loss function can be used to adjust the control law design, but the choice of desired dynamics to be followed and the error dynamics are the most critical part of the synthesis.

The synthesis approach just described must now be extended to allow regulation of some of the system's responses. Regulation is incorporated into the model following synthesis by defining the desired responses to be "followed" by the regulated responses as the constant zero.

For example, if responses Yl and Y2 are to follow a desired model with responses Ylm and Y2m, while responses Y3 and )'4 are to be regulated, then the error vector becomes: "l Yl " Yl. / Y2- Y2,, Y3 y4 (6.8) Otherwise, the formulation and solution to the LQ problem proceeds as above.

Robustness - In Appendix B, one form of the LQ guaranteed singular-value-robustness margin is presented. Unfortunately, the model following linear quadratic design does not deliver such a guarantee. The solution of the state-feedback gain matrix, Krb, of Eqn. (6.6) is, Kfb = R't[(CB)TQCI + BTp1] (6.9) where, A R = R + (CB)TQCB, CI = CA + GeC and P1 is the solution to the following matrix Riccati equation, 0 = P,AI + ATpt " PIBR"BTp: + CTQ,CI (6.10) with, AI = A- BR':(CB)TQCI and QI = Q- QCBR':(CB)TQ Now, just as Kalman's Inequality, Eqn. (B.6), can be derived from the associated LQR Riccati Eqn. (B.5), the following inequality can be derived from the above Riccati equation: [I + R:t2Kfb_(I+Z)BR'I/2]T [I + R:tZKI-o¢(I+Z)BR':/2] > I (6.11) where, (6.12) Z = P'I:cTQC = Pi1(CA + G,C)TQC Note, ¢ = (sI-A) "1 is the resolvent matrix of the system of Eqn. (6.1) evaluated at s=jo (co = frequency), and ¢ is its complex conjugate. The following guarantee results from this inequality: _[P, 112(I + (Krb0(I + Z)B)'I)P, "1/2] > 1/2 (-6 dB) for all co (6.13) where, cr = minimum singular value.

A, lthough the guaranteed stability robustness of this system is less than that for LQ regulators, when R = roI, where r o = scalar, and _(Z) << 1, Eqn. (6.13) will approach the LQR robustness guarantee of Eqn. 03.7).

The above reveals the trade-off between multivariable robustness and model following performance. Model following performance may be improved by either increasing Q or decreasing R. Increasing Q will directly increase Z. Decreasing R will decrease PI, increasing P1-1, thus also increasing Z. As Z gets larger the guarantee offered by Eqn. (6.13) moves further away from the LQR guarantees. Recall that if R is set to zero, and if there are no right half plane transmission zeros, then perfect model following results. In this case, Q1 in the above Riccati equation becomes zero. Hence, P: = 0, Z becomes infinite, and no guarantees can be given by Eqn. (6.13).

If R is chosen to be roI, and Q is decreased, then Eqn. (6.13) will approach the LQR robustness guarantees. However, reducing Q degrades the model following performance.

Scoling Effects - Since the loss function J, of Eqn. (6.5) is a scalar, the minimization of J must be formulated so that it will appropriately minimize the model following errors and control efforts according to their relative sizes of units. This may be achieved by normalizing or scaling the control inputs and system responses by dividing each by their maximum value, and choosing Q = %I, and R = roI. For example, nominal values of fan speed are of the order of 10,000 RPM's, and nominal values of angle of attack are of the order of less than 10 °. Therefore, a unity change in fan speed is insignificant, whereas a unity change in angle of attack can be a large perturbation. By scaling, a unity change in fan speed will be equivalent in size to a unity change in angle of attack.

It can be shown that the following choice of weighting matrices is equivalent to scaling the control inputs and system responses.

Q = qoQ', qo = scalar, Q'= S_ R = roR', ro=scalar, R'= S_ (6.14) where, S_ = diag{ 1/(el )re.x} Su = diag[ 1/[ui )mix} (6.15) (el)m, x is the maximum allowable magnitude of the i'th model following error, and (Ui)m,. x is the maximum control effort available from the i'th control. This choice of Q and R is effectively Bryson's rule 13 for choosing weights in the LQR quadratic loss function. It may be a difficult task to choose the matrices S e and S u from a trial and error approach. These values should be chosen in an intelligent manner from an understanding of the system.

Once Q' and R' are fixed, the only design "dial" left is the ratio qo/r o. It has been found that only the ratio, not the individual values of qo and r o, determines the robustness/performance trade-off in the design. If this ratio is decreased, the guarantee given by Eqn (6.13) approaches the LQR robustness guarantees, but model following performance degrades.

LTR - Assuming that weighting matrices Q and R can be found that give a satisfactory trade-off between performance and robustness using the EIMF state-feedback gains, compensators may then be synthesized using the standard LTR procedureS, 9 to obtain output feedback control laws. With the compensation K(s) so obtained, and the pilot-input gains, K s, taken from Eqn.

(6.6), the augmented system becomes that shown in Fig. 5.1.

7. EIMF/LTR Control Law Synthesis for the Case Study With the desired attitude model to follow, presented previously, and the desire to regulate forward speed and engine fan speed, the error vector is: D IX-tim q " qm U

N2+f N2"

(7.1) Note that inte_al of fan speed is added in the above. Addition of this term is associated with the fact that integral action on N2 is desired. With this error vector, the finite transmission zeros of the open-loop system are shown in Table 7.1.

Table 7.1 - Finite Transmission Zeros of the O _en Loop System Transmission Zeros -68.612 - 13.0491 + 5.5632j -1.0 0.0 The transmissionzeroat -1 is due to the inclusion of the integral of engine fan speed in the error vector, as explained in Ref. 6. The transmission zero at the origin is due to the fact that pitch-rate is used in the error vector. If pitch rate plus integral of pitch rate, or 0, were used, this zero would move into the left half of the complex plane.

The error dynamics are now selected to be GC = 0 g_ 0

0 0

0 0 g_,a (7.2) This choice of error dynamics reflects the desire to decouple the attitude dynamics from the engine speed and forward speed, as well as implicitly follow the desired short period model, (A m and C m are given by Eqn. (4.2).) Finally, the forward-speed and engine-speed responses will include a mode with time constants g_, and g_r_., respectively. Values for these time constants were chosen to be 0.1 and 1 rad/sec, respectively.

Some design results are given for two different values of the ratio qJr o. Note that for the vehicular case study, the scaling matrices Sy and S u, used in the weighting matrices Q and R, have been chosen, from Ref. 5, to be Sy = diag[0.05, 0.3, 17.189, 1.7446x10 -3] (7.3) S u = diag[0.02, 0.1, 0.1, 2.0x10 "4] 8. EIMF/LTR Design Results The first results presented are for the ratio qo/ro = lxl04. Using the EIMF state feedback gains, Kfb, the compensator is obtained from the standard LTR procedure 9. Comparisons of singular value plots of the loop transfers using the state feedback control law, or Krb(SI-A)-IB, and the LTR compensation, or K(s)C(sI-A)-1B, revealed complete robustness recovery. The compensator transfer-function matrix, or K(s) in Fig. 5.1, is given in Table 8.1 for this control law after some straight forward order reduction. The transfer functions presented in this table are all fifth order, with poles at the finite transmission zeros of the plant, plus one additional pole at the origin due to integral control on fan speed. Note that many of these compensators can be simplified further.

For qo/r o = lxl04, near-perfect model following performance is achieved. The closed-loop transfer functions obtained using the above feedback compensation are, a(s) _ -4.389 Tl(s) {deg/ 8p(s) (s+l.414+l.414j) _lbsJ q(s) -0.0797(s+0.521) T2(s)( lbi_s ) 6p(s) (s+l.414+l.nlnj) (8.1) where the poles Tl(s) and T2(s) are the poles of the phugoid mode and all engine modes. Both Tl(s) and T2(s) are very close to unity due to accurate pole-zero cancellations. Comparing these results to the desired responses given by Eqn. (4.4), near-perfect model following is evident.

Figs. 8.1 and 8.2 present the closed-loop frequency responses for angle of attack and pitch rate from pilot stick input. Also plotted are the desired frequency responses, which are not visible, since they are essentially the same as the closed-loop responses.

Thefan speed disturbance rejectionperformanceis indicatedby the magnitude of the engine

loop's sensitivity function, shown in Figure 8.3. It can be seen that disturbances with frequency content below 20 rad/sec will be rejected.

Table 8.2 summarizes the cross-over frequencies, phase margins and gain margins for all four individual loops, with each loop broken at the input to the plant, and all others closed. Note that the magnitude of the flap loop is less than one throughout the frequency range.

Table 8.1 - EIMF/LTR Compensation Matrix Con- Numerators for the Individual Bode Measure- Units of merits trois Compensator Transfer Functions Gain Compensator u sq-in/(ft/sec) -123.3(0)(0)(0.90)[0.93,9.5l -0.7 A78 142.3(0)(- 1.8e-04)(-0.61) [0.93,10.1 ] -0.6 sq-in/deg sq-in/(rad/sec) _- 110.3(0)(1.5)(13.2)(-25.2)(31.1) 120.0 q N2 -0.04 [-0.2,0.41[-0.06,8.5](I 3.4) -3.8e-4 sq-in/(RPM) U -0.4(0)(0)[0.9,11.8](-29.5) 0.I deg/(ft/sec) -84.8(0)(0)[0.92,13.6] -I.I deg/deg -2I. I(0)(-0. I)[0.92,14.0] (67.2) 2.I deg/(rad/sec) q N2 1.2e-05(-0.5)[0.7,1.0](I 7.2)(26. I)(35.6) -6.5e-6 deg/(RPM) u 1.6(0) (0) [0.9,13.9] (50.4) 1.1 deg/(ft/sec) a 6flap - 1.9(0)(- 1.2e-05) [0.9,13.9] (51.3) - 1.4 deg/deg 58.8(0)(0.3) [0.9,14.2](68.3) 17.9 dee,/(rad/sec) q deg/(RPM) -3.4e-05(0.4)[0.01,2.3][0.9,13.8](63.4) 5.7e-5 U (#/hr)/(fL/sec) -4.6e+05(0)(2.9e-05)(-0.06)(2.2)(4.3) -20.0 Wf Gt 5.4e+05(0)(--4.0e-05)(0.7)(2.2)(4.3) 250.0 (#/hr)/deg q (#/hr)/(rad/sec) - 1.0e+06(0)(0.3)(2.2)(4.3)(46.9) - 1. le+4 (#]hr)/(RPM) 8269(0.005)[1.0,1.11(10.4) 7.1 Characteristic Polynomial of Compensator : A(s) = (0"_(0_r0.9,14.2](68.6) Note: (a) = (s+a), and [a,b] = complex mode with damping ratio = a, and frequency = b

.,00 i i i i iiiii ! i iii!iii ......i'i"

10-1 ,00 ,01 ,02 lOq 10o 101 102 Frequency m R,d/Sec (Deaired Model = --) Figure 8.1 - Closed Loop Frequency Response of Angle of Attack from Pilot Stick Input (De-Jibs)

70i ......... i!!iii! ......... i!! ii ..... i iili

10-1 I0O lOt 102 ._oo..._..._.-._ __..i.._..i:.i_ii...... _ .....i....

-250 ................................

-300 lO-I I0O I0_ 102 Frequency in Rad/Sec (Desired Model = ---) Figure 8.2. - Closed Loop Frequency Response of Pitch Rate from Pilot Stick Input ((Rad/Sec)/lbs) -5 ............. f. ..........

= -lo...... :---!--i-i.i.!-ii! ...... :---i-: !-:-i.i-ii ...... !---i-.i.i-i.?i _ ii___ ii!!i!)i! ! i -15...... i...i., i..!. i.i.i.i! ..... i-.. :..-i.-i..i.i-i. i i...... i...i.-!..i-i.i, i iii If# 101 102 Frequency in Rad/Sec Figure 8.3 - Sensitivity Function of Measured Fan Speed-to-Fan Speed Disturbance (RPM/RPM) Table 8.2 - Individual Loop Characteristics Cross-Over Phase Gain Margin Frequency Margin Loop (rad/sec) (degrees) (dB) Thrust 0.18 90 0o Reversing Thrust 2.0 50 -10 Vectoring -10 Flaps Fuel Flow 10.2 75 12 The cross-over frequencies and stability margins in all the loops are quite good. The thrust vectoring and fuel flow loop transfers are presented below, for example.

*'0 Thms_ Vectoring _ Transfer - With All Other Lo_.s Closed : _ : : :-:.:': : : : : ::::: ." : "': :::::

= :01 ...... iiii ')i ...... iiiiiii_ ....i

10-: I00 101 102 ...... ', ..... ',.ii !i',', 10.1 100 101 102 Frequency in RKI/Sec 40 Main Burner Fuel Flow Rate Loop Transfer - With All Other Loops Closed '_ i i i ! i!iii i i i i :i:ii i i : :i i:: I |O-I I(}O lOl 102

",_0 ...... ::_ i ! ii i'ii_i ...... ii i'iii ¸i_i ii ii i_

10-1 I{3O 101 102 Frequency inRad/Scc Figure 8.4- IndividualLoop Transfers-Loops Broken One ata Time, With All Other Loops Closed The following figure presents o(I + (KG)I), scaled at each frequency to obtain the least conservative results, as discussed in Appendix B. Again, since full robustness recovery was obtained, this plot is the same whether implemented with full state-feedback or LTR compensation.

so.....i..i.i.i.iii.il ....i..i.i.l.!iii.i ....!..i.i.!,!.!ii.! ....i..i.i.!.i_

= ,5.....i-i-!i-!i::i-i ..... i--i-i.!-iiiii ....i-.!i-i-i::::i-i ....i.-_,_-!ii-

> ..... :' i" :.':: L:': ..... :' _ : :':f::': .... :." '. ' i ":':" :.'. :.... _"_': ::_:':

i iiii!i iiiiiiii i!iiii

I _:.::::iiii:: ...._::i::iiii:: ....i":!::i::i:::: ° ..........

10-1 100 10_ 10_ 10_ Frequency in R_I/Sec Figure 8.5 - Scaled Multi.variable Singular Value Robustness Test qo/r o = 10,000 This plot showsthat this system has "LQ-like" muldvariable robustness for frequencies above = 0.3 rad/sec. For piloted aircraft, loss of robustness in the low-frequency range, or the phugoid mode is not as critical as loss of robusmess at higher frequencies.

It is noted that the results (not shown) for the unscaled singular value test are quite poor.

The original units led to widely separated singular values of the loop transfer, and previous work 14 has shown that these multivariable robustness tests work best when plant and loop transfer singular values are closely spaced. Recall that scaling the controls by the matrix Su gives approximate equivalence in the sizes of the units on the controls. This produces singular values of the scaled loop transfer that are much closer together, and the singular value robustness test, which plots 2(I + (SuKGSj)d), shows much improved results compared to the unscaled singular value test.

Decreasing the qo/ro ratio from lxl04 to 2/3 leads to improved low-frequency robustness, with singular values greater than the LQ guarantee. However, the high-frequency robustness decades. The results using frequency dependent scaling are shown below.

6 _ : _'.::T:'. _ : : :::::: : : : :::::: '. : : ::':_ 4 "'-"."?_ii!! .... !";?':'?;.".": .... ?'::'!_::!?"" : : ::z:: ,_ i iii!iii! ! iii.ii!ii i iiii!iii ! _'" ._ 2 .... .-.-:-:.::-:-:.... :.-:-:-:-:.-.--.... :.-:.:.:--:::: .... :-.-.-..:: : :::::::: : :::::::: i _'._:: !!!.ii aid °'_ .......................

• ii!i!ii i i l!i i iii!i

-4 " - °:- " -2--;- : , ,'.?; .... C- " '. ": "2-:::'7. .... 2 ' ":" "2"C -;'2-::: .... 2- - -:" -;- _: : ,2 -6 t04 10o 101 102 103 Frequency in Rad/See Figure 8.6 - Scaled Multivariable Singular Value Robustness Test qdro = 2/3 The model following performance also degrades• The corresponding closed loop responses are, cx(s) -0.1376(s+32.27) So(s) (s+1.352+_l.323j) (Note: 0.1376x32.27 -- 4.44) q(s) -0.07897s(s+0.2738_+0.1479j) /rad/sec I 8p(s) = (s+0.1241+0.1477j)(s+l.352+l.323j) T2(s) t lbs t (8.2) and Tl(s) and T2(s) are only approximately unity. Comparing these results with the desired responses (Eqn. (4.4)) and the responses of the previous case (Eqn. (8.1)) it can be seen that the short period and phugoid modes are no longer decoupled, there is no longer a real 1/'t02 zero, and the desired short period mode's frequency and damping are not achieved. Note, however, the engine's disturbance rejection performance, as measured by the fan speed sensitivity function (not shown), remains approximately the same as that shown in Fig. 8.3.

The individual thrust-vectoring and flap loop transfers also remain approximately the same.

The cross-over frequency of the fuel flow loop, on the other hand, decreased to 0.5 rad/sec.

In summary, the first control law gives near perfect model following performance at the expense of low frequency multivariable robustness, and larger cross-over frequency in the fuel flow loop. The second case led to improved low-frequency multivariable robustness at the expense of both model following performance and high frequency multivariable robustness.

9. H** Theory 15 An EIMF control synthesis technique can also be formulated using an H**-norm nainimization framework. The following figure displays the general H** control block diagram structure.

Exogeneous Outputs of Inputs, w Interest, z Ira..._ v Control Measurements, y Inputs, u K (s) Figure 9.1 - General Block Diagram for the H** Control Problem Here, the plant P(s) represents the plant dynamics, plus any frequency dependent weighting functions. The exogenous inputs, w, represent external inputs to the system, which may include commanded inputs, low frequency disturbances, and high frequency measurement noises. The outputs of interest, z, are those variables to be controlled, which may include plant responses as well as control inputs, u.

The design objective is to find a compensator, K**(s) that stabilizes the closed loop system and minimizes the H*O-norm of the transfer function matrix from the outputs of interest, z, to the exogenous inputs, w. The H**-norm of a matrix T(jo_) is defined as I"11..= sup{_(T(jo3))} (9.1) Typically, the H** control methodology is used as a multivariable control approach to meet classical control design objectives, namely, loop shaping. Weightings Wl(s) and W2(s), in the figure below, are chosen, for example, to shape the singular values of the sensitivity and complementary sensitivity matrices.

Plant. P(s) Figure 9.2 - Example H** Control Block Diagram Once all weighting functions are defined, the "plant," P(s) is defined and the H** compensator can be obtained from Ref. 15. This involves iteratively solving two Riccati equations.

10. EIMF/H** Control Law Methodology and Synthesis In Section 6 the EhMF control law design methodology utilized LQR theory to minimize the quadratic loss function involving model-following errors, regulation errors, and control inputs.

The EIMF design objectives can also exploit H** theory to minimize the H**-noma of a transfer function, again involving model-following errors, regulation errors, and control inputs.

Here, an approximate equivalence will be developed between the EIMF/LTR and the EIMF/H** procedures so that comparisons can be made. The following block diagram presents just one EIMF/H _ formulation.

e=_ -Z_

Iw l

Figure 10.1 - EIMFB--I** Control Design Block Dia_am G,,(s) represents the airframe/engine system, Eqn. (6.1), and Gin(s) represents the desired model to follow, Eqn. (6.2). A vector of fictitious measurement noise inputs, w n, must be included in the vector of exogenous inputs, w. This noise is weighted by some small number, rio.

The matrices Ip and Ipl are used so that the only exogenous input into both the desired model and the vehicle model is the pilot stick input, 8p. For the case study, since the pilot stick input is a scalar, and there are four measurements, (y = [u, 0t, q, N2]T ) and four controls, (u = [A78, St, ,, 8flaps, Wf] T) then, 10000 Ip= 10000 1 0000 Ipl=[10000] 1 0 0 0 0 (10.1) The model following error is formed by subtracting the responses of the vehicle and desired model,

e

Thus, the intermediate output vector is the model following errors and the control inputs, or, Z' = [Z I' Z2'] T = [ e u ]T (10.3) Note from the block diagram that implicit model following is a result. This formulation will not, however, produce stick gains, such as K_ in Eqn. (6.6). Stick gains could possibly be incorporated into the matrices Ip and Ipl above. To date, these have been simply chosen to be unity "gains."

The intermediate output vector, z', is then weighted as shown in Fig. 10.1 to form the final output vector, z. Note that qo and r o are scalars, and Wl(s) and W2(s) are matrices which may contain frequency dependent weighting functions. Some parallel can be drawn between the weighting scheme used in the EIMF/LTR design method (see Eqns. (6.14) and (6.15)) here, by choosing qoWl(s ) = qoSy2, roW2(s ) = roSu 2 (10.4) However, for the results presented in the next section, the control inputs and responses are scaled according to: Ynew = Syy, Une w = Suu (10.5) where Sy and S o are given in Eqn. (7.3). Then the following weightings are used in conjunction with this scaled system: qoWj(s) = %I, roW2(s ) = roI, 1"1o= lxl0 "8 (10.6) This implementation is closely related to that of Eqn. (10.4), and, for numerical reasons, gives improved results.

From the block diagram, the state space description for this H °° model following formulation is,

Ezl]:i cv Cm]I::l I01u

o o i (lO.7)

Y=[ Cv

0 ]Xm

Frequency-dependent weighting functions can also be augmented to this realization, as desired, and the H *° compensator can then be obtained from Ref. 15.

11. EIMF/H** Results Some results are presented below for two different values of qo/ro. For the first case, qdro = lxl0 6, and the compensator transfer-function matrix for this case is given in Table 11.1, after some straight-forward order reduction. The transfer functions presented in this table are seventh order, with some poles at the finite transmission zeros of the plant. Again, inclusion of integral control on fan speed leads to one additional pole at the origin.

Table l 1,1 - EIMFAI** Compensation Matrix Numerators for the Individual Bode Measure- Units of Corl- Gain ments Compensator Transfer Functions trois Compensator -0.3 u 0.01 (0)(0.08)(i .0)(- 1.4)(8.4) [0.9,14. I ](69.6) sq-in/(ft/sec) -0.2 ct 0.05(0)(0.2)( 1.0)(I .3)(-8.0)[0.9,15.7] (6I. I) sq-irgdeg A78 1.4(0)(2.9e-03)( 1.0)[0.2A .9] [0.9,14.3](68.8) 76.0 q sq-in/(rad/sec) -6.9e-3 N2 -3.0e-04(0.04)(0.9)(I)[1.0,3.0][0.9,13.7](82.0) sq-in/(RPM) -0.04 u 2.0e-O3(O)(8.0e-03)(l)(-l.8)(4.9)[0.9,14.21(68.6) deg/(ft/sec) 0.18 ot _lv 8.3e-03(0)(-0.02)(0.4 )( 1)[0.9,14. I] (28.3)(68.3) deg/deg 0.2(0)(9.3e-04)(I)[0.6,2.4][0.9,14.2](68.6) 2.70 q deg/(rad/sec) -4e-3 N2 -4.8e-05 (6.8e-03)( 1.0)( l)[-0.04,2. I][0.9,14.2] (68.9) deg/(RPM) 4.7e-5 u 4.6e-05(0)[0.8,0.2](I)[0.9,14.21(-14.5)(68.6) deg/(ft/sec) 0.02 a 0.02(0X-0.02)(0.5)(I)[0.9,14.2](68.2) deg/deg _flap 5.Ie-03(0)[0.5,0. I ] (I)(2.2) [0.9,14.2](68.6) 4.4e-4 q degl(rad/sec) 4.5e-7 N_ -I. Ic-06 [-0.2,0.3] (I .0)(I)(3.0) [0.9,14.2] (68. I) deg/(RPM) 0.6 u 39.9(0)[-0.5,0.3]( 1.0)(6.5)[0.5,15.31 (#/hr)/(ft/sec) 6.8 o_ 3.3e+04(0)[1.0,0.3](1.0)[0.9,4.4] wf (#/hr)/deg 41.9(0)(-0.02)(0.8) [0.8,1.9] [0.9,13.4](85.3) 276.0 q (#/hr)/(md/sec) 0.2 N2 -5.2(-0.02)(0.7)( 1 )( 1.0)(5.9) [0.7,8.03 (#/hr)/(RPM') Characterisuc Polynomial ol Compensator : A(s)= (0)(0)(0.5)(1)[0.9,14.2](68.6) Note: (a) = (s+a), and [a,b] = complex mode with damping ratio = a, and frequency = b Comparing this table of compensators with Table 8.1, the Bode gains of the compensators for the thrust reverser and thrust vectoring controls are quite similar for both designs. However, the Bode gains above are much smaller for the flap and fuel flow compensators. The poles at (0)[0.9199,14.19](68.61), the transmission zeros of the plant, are present for both designs.

However, they are approximately cancelled in all but the fuel flow compensators above, whereas they are only cancelled in the fan speed-to-thrust vectoring and flap compensators in the EIMF/LTR design. Also, the above design contains the additional poles at (0.5)(1).

These differences in compensation lead to differences in the individual loop transfers. For this design, the thrust vectoring loop transfer has large gain at high frequencies. However, the other three are all low-gain loops. The cross-over frequencies, and stability margins of the other three loops are summarized in the table below.

Table l 1.2 - Individual Loop Characteristics Cross-Over Phase Gain Frequency Margin Mar_n Loop (rad/sec) (degrees) (dB) Thrust 1.08 80 -6 Reversing Hap +11 -35/to=O.13rls Fuel Flow 0.2 4O + 10]_o=0.35r/s Further research involving other weighting schemes may help add roll-off to the thrust vectoring loop shape and increase the magnitudes of the other loops.

This value of qJr o gives near-perfect model following and the results match those of the EIMF/LTR design given by Eqn. (8.1) and Figs. 8.1 and 8.2.

Although the performance is excellent, the muhivariable robusmess is quite poor, as seen in the next figure.

......... -- ..... -__i_.' :--." _-: i i !_i'7""i i i ! !iiii .................

...' :_. ...... . ........ : : : :::::: : : : :::;

"- " '"/iiiiiiii....... i!iiiiil.... ii"iii'iiil.... iiiYi

/".. iiii i ;!iiiiiii iiiiiii! ii!i!

"J -10 .....................................

).

: : : :::::: : : : :::::: ................

OU -15 ........................................................................

: : : : ::[:: : : : :::::: : : : :::::: : : : :::: "[0_ l 10o iOl 102 I0O F_qaency in _ Figure 11.1 - Scaled Muhivariable Singular Value Robustness Test q Jr o = 1x 10 6 Similarities in the design results between the EIMF/LTR method and the EIMF/H** method have been found. Just as in the EIMF/LTR design method, once the weightings Wl(s ) and W2(s) are fixed, the ratio qo/ro determines the model following performance and multivariable robustness achieved. Decreasing this ratio will increase the multivariable robustness. If this ratio is made small enough, the robustness can be made as large as the LQR guaranteed margins, however, the model following performance degrades.

Reducing the qo/ro ratio to a value of 0.05 dramatically improves the multivariable robustness, as shown in the figure below.

Minimum Singular Value of 0+inv(KG)) m 40 .-7 > _ 20

i ii:i ¸

.e 10

i iiiiii_ !)/_ii!i i i iiiiii! i i i ill!

""i."".i'.i_i"'!'i':'i?:i'i .... :"i'i'!'i'i?!! .... i":':!':?i" -10 10-1 I0O 101 102 103 Frequency in Rad/Sec Figure 11.2 - Scaled Multivariable Singular Value Robustness Test qJr o = 0.05 it can be seen that, not only does the robustness satisfy the guarantees of LQ regulators, but the high frequency robustness (roll-off) is excellent. Thus, unlike the EIMF/LTR design (Fig. 8.6), decreasing the qdro ratio here seems to improve the multivariable robustness for all frequencies.

Also, the individual loop shapes all have low cross-over frequencies and good gain and phase margins.

Unfortunately, improvement in the robustness comes at the cost of the model following performance, as seen in the next two plots.

50 .......................

10-1 100 101 102 200 .... ---.D-'.,.-r ............ "-I e_ ..... :.,.."Z_.'_._.'....,. ,_.; _;:: .... : - . :.; :-;; _ 100 ................ i i i i iiiii" __• .......

0 ; i i '. : .......... _'.,- ...............

10-! 100 101 102 F_iueney in _d/See _ Model = --) Figure 11.3 - Closed Loop Frequency Response of Angle of Attack from Pilot Stick Input (Deg/lbs) ........ i .1 i ....................

-40 .-__--:''77 .

10-t I0O 10t 102 0 : : :: ::,.:: : : :: ::::: : : :::::: _B 8 -10o : .i..i.i.i_."_i ...... i...i..i.i.i.'iii ..... _...:..:.:.:.::!

-300 _-- 10.1 10o 10t 102 Frequency in R_I/Sec (Desired Model .... ) Figure 11.4 - Closed Loop Frequency Response of Pitch Rate from Pilot Stick Input ((Rad/Sec)/lbs) Conclusions Control law synthesis techniques were presented that were developed to achieve excellent handling qualities, decoupling the engine and airframe dynamics. However, a clear trade-off between performance and multivariable robustness has been recognized and discussed. The methodology was applied to an integrated flight and propulsion control case study.

The EIMF/LTR approach led to control laws that deliver excellent model following and regulation performance with modest gain crossover frequencies, thus keeping actuation requirements to a minimum. The airframe responses were exactly those desired, thus demonstrating the performance achieved. The engine control laws were simultaneously synthesized, along with those for the airframe, and would appear to deliver good disturbance- rejection performance. The results also indicate reasonable multivariable robustness, as defined herein. However, an increase in low frequency robustness comes at the cost of decreases in both model following performance and high frequency robustness.

The EIMF/H** approach led to control laws that also deliver excellent model following performance. However, the multivariable robustness was poor. As with the EIMF/LTR design, the multivariable robustness can be improved, yet this reduces the model following performance.

Other H _ formulations which may, for example, take advantage of loop shaping techniques, offer future areas of research.

Appendix A. Linear Model for the Case-Study Vehicle The states are defined as x = [u (ft/sec), ct(deg), q(rad/sec), 0(radians), N2(rpm's), N2.s(rpm's), Pr(psia), T4tB(°R)] T with inputs, u = [A78(in2),/Snaps(deg), 5rv(deg), wf(#/hr)] T For the vehicle in question, the model is AAE = A A = 3.1440e-04 2.5990e-04 3.8190e-02 2.2500e-03 -3.6523e-02 3.8161e-01 -3.8600e+01 -3.1840e+01 -2.2924e-05 -1.5892e-05 -2.1976e-03 - 1.3331 e-04 -8.7843,--02 -2.8897e-01 5.6739e+01 5.8886e-01 9.4600e-07 3.7440e-07 3.6680e-05 2.6760e-06 9.8260e-05 2.7918e-02-1.9490e-01 .-4.8180e-04 0 0 0 0 0 0 1.0000e+00 0 AEA = A E = 8.1058e-01 5.5150e-01 0 0 -4.1910e+00 6.0220e+00 -3.4340e+02 1.1600e.+O1 1.5812e-01 1.0758e-01 0 0 4.2630e-01 -5.7070e+00 2.7160e+01 1.0400e+Ol 8.2641e-01 5.6223e-01 0 0 2.2950e-01 1.1550e-01 -9.0240e+01 8.4760e-01 -I.0468e-01 -7.1244e-02 0 0 3.7400e-02 -l.0360e-OI -7.9540e+00 -I.0680e+00 [B A BA£] = [BEA BF..]= -2.0550e-01 -4.1830e-04 -8.4280e-02 3.4360e-05 0 0 0 1.4690e-01 1.2018e-02 -1.5241e-01 -5.5082e-02 -2.0197e-06 0 0 0 5.3600¢-02 1.0680e-04 -7.9700e-02 8.8132e-03 5.5070e-08 -.4.3020e+01 0 0 1.8130e-02 0 0 0 0 0 0 0 1.6430e-01 Appendix B. Multivariable Singular Value Robustness Several singular value tests are often used to measure the stability robustness of muhivariable systems 8.9. For example, the following test may be used to measure the robustness of the system to multiplicative uncertainty at the plant input. First, it is assumed that the nominal closed loop system is stable, and multiplicative perturbations, E, in the loop do not change the encirclement requirements of the critical point in the Nyquist plot. Under these assumptions, if _(E) < _(I + (KG) "l) for all frequency, 03 (B.1) then the closed loop system is guaranteed to be stable in the presence of E, at the input to the plant, where the true plant is G(I+E). Note that _ = maximum singular value, and _ = minimum singular value.

Linear quadratic regulators guarantee a minimum value for the right hand side of the above inequality 9. Given the following linear time-invariant system, = Ax + Bu y = Cx (B.2) minimization of the quadratic loss function, j = [yTQy + uTRu] dt (B.3) leadsto thefollowing state-feedback control law,

u = -Keox, Keo= R'IBTp (B.4)

where Kfb is the matrix of regulator state feedbackgains,andP is the solution to the algebraic Riccatiequation,

0 = ATp + PA - PBR-1BTp+ CTQC

(B.5)

Kalman'sInequality,

[I + Rtt2Kfb_BR-1/2]T [I + RXt2KroCBRlt2 ] _ I

(B.6)

is derived from this Riccati equation. Note, ¢ = (sI - A) "1 is the resolvent matrix of the plant, evaluated at s =jc0, and _ is its complex conjugate. Under the assumption that R is diagonal and that the inputs can be scaled such that R = pI, the guaranteed singular value robustness margin for LQ regulators can then be derived from the Kalman Inequ',dity, and is given as _(I + (KmCB) -I) > 1/2 (-6 dB) for all co (B.7) Thus, in the absence of a model for the uncertainty, E, it may be desirable to find control laws that make the right hand side of Eqn. (B.1) as large as possible, and LQ regulators guarantee the above minimum value.

Furthermore, singular values of a transfer function matrix are not independent of the units of that matrix. Therefore, the choice of units for the system will directly influence the results of the singular value test of Eqn. (B.1). The following block diagram shows the inclusion of a scaling matrix S u at the input to the plant, with input multiplicative uncertainty.

yc _ Uout I _n Figure B.1 - Addition of Control Input Scaling to the Loop If S u is diagonal, this is equivalent to defining a new set of units for the control inputs. Breaking the loop at the point shown in the figure, the following scaled robustness test may be derived. If, O(SuZ Su 1) < _I +(SuKGSu') 1) for all co

(B.8)

then the system is guaranteed to be stable under the same assumptions stated for Eqn. (B.1).

The conservatism of the robustness test of Eqn. (B. 1) can therefore be reduced by finding diagonal scaling matrices, equivalent to finding a new set of units for the control inputs, such that the left hand side of the above inequality is made smaller, and the right hand side is made larger.

The robustness test is made independent of the units of the system by finding a diagonal scaling matrix at each frequency that maximizes the distance between the left and right hand sides of the inequality. In the absence of any models of the uncertainty, it may be desirable to find scaling matrices at each frequency that just make the right hand side as large as possible. This technique is referred to as the "scaled multivariable singular value robustness test," shown in Figs. 8.5, 8.6, 11.1, and 11.2.

Acknowledgements

This work wassponsored by NASA Lewis Research Center under Grant No. NAG3-998.

Mr. Peter Ouzts is the technical monitor.

References [1] Schmidt, D., Schierman, J., "Analysis of Airframe/Engine Interactions - An Integrated Control Perspective," AIAA # 90-1918, 26th AIAA Joint Prop. Conf., Orlando, July, 1990.

Shaw, P., Rock, S., Fisk, W., "Design Methods for Integrated Control Systems," [2] AFWAL-TR-88-2061, Aero Propulsion Laboratory, Air Force Wright Aeronautical Labs, Dayton, June, 1988.

Rock, S., Emami-Naeini, A., Anex, R., "Propulsion Control Specifications in Integrated

[3]

Flight/Propulsion Control Systems," AIAA # 88-3236, AIAA 24th Joint Prop.Conf., Boston, 1988.

[4] Smith, K., "Design Methods for Integrated Control Systems," AFWAL-TR-86-2103, Aero Propulsion Labs, Air Force Wright Aeronautical Labs, Dayton, December, 1986.

[5] Garg, S., Mattern, D., Bullard, R., "Integrated Flight/Propulsion Control System Design Based on a Centralized Approach," AIAA # 89-3520, AIAA GN&C Conf., Boston, 1989.

[61 Schmidt, D., Schierman, J., "Extended Implicit Model Following As Applied To Integrated Flight and Propulsion Control," AIAA # 90-3444, GN&C Conf., Portland, August, 1990.

[7] Anon., MIL-8785C, Flying Qualities for Piloted Airplanes, USAF, Flight Dynamics Lab, WPAFB, Dayton.

[8] Doyle, J., Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modem

Synthesis," IEEE Trans. on Automatic Controls, Vol. AC-26, No. 1, pp. 4-16, Feb., 1981.

[9] Ridgely, D., Banda, S., "Introduction to Robust Mulfivariable Control," AFWAL-TR-85- 3102, Flight Dynamics Lab, Air Force Wright Aeronautical Labs, Dayton, February, 1986.

[10] Anderson, M., Schmidt, D., "The Significance of Error Dynamics in Model-Following For Flight Control Design," AIAA # 87-2311, AIAA GN&C Conf., Monterey, 1987.

[11] Schmidt, D., "Flight Control Design Research to Meet Handling Qualities Requirements, Phase II," final report for the McDonnell Aircraft Co., performed at Arizona State Univ., Dept. of Mech and Aero Eng., Tempe, AZ, July, 1989.

[12] Kreindler, E., Rothschild, D., "Model-Following in Linear-Quadratic Optimization," AIAA Journal, pp 835-842, Vol. 14, No. 7, July, 1976.

[13] Bryson, A.E., Ho, Y., Applied Optimal Control, John Wiley & Sons, Inc., New York, 1975.

[14] Garg, S., "Turbofan Engine Control System Design Using the LQG/LTR Methodology," NASA CR-182303, June, 1989.

[15] Doyle, J., Glover, K., "State-Space Formulae for All Stabilizing Controllers That Satisfy an H..-Norm Bound and Relations to Risk Sensitivity," Systems & Control Letters 11, pp.

167-172, North-Holland, 1988.

A Framework for the Analysis of Airframe/Engine Interactions and Integrated Flight/Propulsion Controlt David K. Schrnidt ! and John D. SchienTuan 2 _/7//_ Aerospace Research Center College of Engineering and Applied Sciences _//'7 ,_"_TZ'_ '_ Arizona State University Tempe, AZ 85287-8006 Abstract delivered instantaneously by the engine, introducing time delay in the airframe's forward speed response. Thrust reversing may be used to Potential sources of airframe/engine interactions are explored improve the speed of response, but disturbances in engine thrust may for aircraft subject to the study of integrated flight/propulsion control then be more significant in the forward speed dynamics.

A quasi-linear framework for the analysis of these dynamical Thrust vectoring of the aft nozzle can produce moments to interactions between the airframe and engine systems is presented.

control the attitude of the airframe. Thrust from a ventral nozzle can This analysis can be used to quantify, in a meaningful way, the produce pitching moment as well as lift. The primary thrust for left magnitude of the interactions between the airframe and engine systems, determine if these interactions are significant to warrant and fight ejectors may come from the mixed flow (core and by-pass further consideration in the control law synthesis, and if so, what are flow) of the engine and is used to produce not only lift, but rolling moments as well. Effects from disturbances in the mixed flow that the critical frequency ranges where problems may occur due to these interactions. Justification for the use of this method, along with the produce the engine thrust will therefore be seen in the lift and attitude assumptions, conditions and restrictions that apply are discussed.

responses of the airframe. On the other hand, commands in thrust Sample results of this analysis are used to illustrate issues brought reversing, thrust vectoring, ventral and ejector thrust may cause forth in its development. Also, a comparison is made between pressure disturbances in the au=_-nentor or mixing plane. If the nozzle another framework for analysis in integrated flight and propulsion is operating in an unchoked condition, these pressure disturbances control, reported elsewhere, and the framework presented in this may propagate through the fan by-pass duct and cause a reduction in paper. fan surge margin (margin between normal operating fan pressure ratio and stall pressure ratio) or possibly a fan stall itself. This, in turn, effects thrust disturbances by disturbing engine flow. Therefore, 1. Introduction commands to control the airframe responses may influence the engine In the design of highly maneuverable fighter aircraft, such as dynamics.

those capable of short take off and vertical landing, the propulsion The secondary flow of the ejectors is produced when air is system is frequently being considered for augmenting the lift and the drawn through the ejector intakes by the primary flow from the maneuvering capabilities of the vehicle. Some designs include engine. Secondary flow effects may significantly influence the vectoring of the engine's aft nozzle to control the attitude of the airframe aerodynamics.

airframe. ! Thrust from a reaction control system (RCS) may also be The thrust from both RCS jets, used to control the pitch, roll used for attitude control of the aircraft. 2 The engine may be equipped and yaw of the aircraft, as well as upper wing surface blowing, used with a vena'al nozzle to enhance pitch control and augment lift.3 Left to augment lift, is usually bleed air from the engine's compressor.

and right ejectors, drawing primary thrust from the engine and Thus, the dynamics of the core flow can affect the lift and the attitude secondary thrust from intakes over the top of the fuselage can augment responses of the airframe. However, commands in RCS thrust will lift and enhance pitch and roll control. 2 Thrust reversing nozzles can cause reduced core pressure due to compressor bleed, effecting engine be used to improve forward speed control of the aircraft. 4 Upper flow disturbances. Also, airframe aerodynamic parameters such as wing surface blowing or blown flaps can be used to alter the boundary dynamic pressure, angle of attack and sideslip angle can influence the effectiveness of the RCS control jets, possibly calling for increased [ayer. thus the lifting characteristics of the wing. 5 control power, thus, increased compressor bleed flow.

In the design of the control systems for such aircraft and their Pressure disturbances at the inlet to the engine can alter the propulsion systems, the significance of the interactions between the drag characteristics of the airframe. Sudden reduction in airflow airframe and the engine must be assessed. This is a fundamental issue caused by fan or compressor surge can cause the inlet shock to move in the so-called Integrated Flight and Propulsion Control (IFPC) or pop out of the inlet which can produce rolling or yawing moments.

problem. 1 Variable inlet geometry used to control the position of the inlet shock The main objective of the paper is to present a quasi-linear can affect the drag and produce pitching and yawing moments. On the system analysis framework for assessing the significance of the cross- other hand, the attitude dynamics may significantly influence the coupling dynamics between the airframe and engine, to justify that this airflow at the inlet causing flow disturbances throughout the engine.

analysis produces meaningful results, and to state the conditions and The coupling between the airframe and engine may be viewed restrictions that apply to this methodology. The other objectives of the as in the Fig. 2.1. This figure indicates the engine can influence the paper are to contrast this approach to another in the literature, and to airframe, which, in turn, influences the engine.

describe potential sources of airframe/engine interactions.

The discussion on airframe/engine interactions is given next, 3. Justification for Quasi-Linear Analysis of Nonlinear in Section 2. In Section 3 the justification for why a quasi-linear Airframe/Engine Systems analysis is valid, given that the airframe/engine system dynamics are Airframe and engine systems are highly nonlinear. 10-13 In nonlinear, is presented. The quasi-linear analysis is described in Section 4. Sample results of this analysis axe then presented in light of this, the validity of quasi-linear analysis procedures, along Section 5. Section 6 is devoted to presenting a different analysis with the applicable conditions and restrictions for such procedures are explored in this section.

framework used in several studies3. 6 and how it is related to the Many points of operation for the airframe/engine system occur framework presented in Section 4.

at some steady state trim or equilibrium condition where accelerations 2. Potential Sources of Airframe/Engine Interactions are small or zero, and rates or velocities are constant.10,14 Large The purpose of this section is to detail dynamical interactions numbers of these reference or operating points can be defined throughout the flight envelopes of the airframe and engine. Usual between the airframe and propulsion systems. In particular, these practice involves feedback control design and stability and new designs used to improve the maneuvering abilities of the aircraft performance analysis at each operating point via quasi-linear or linear may impart significant coupling between the systems. Discussed is methodologies. Why linear methodology at certain operating points is both how engine dynamics can influence the airframe, and how a viable approach, and how nonlinearities are accounted for in airframe dynamics can influence the engine. Refs. 2, 3, 4, and 6 through 9 also elaborate on these interactions. wansitioning between operating points is discussed first. Then, quasi- linear methods are investigated for use at operating conditions and In conventional aircraft, changes in aft thrust cannot be during transitional phases of operation where linear assumptions are * To be presented at the AmericanControlConference, Boston, June, 1991. not strictly valid.

1Acung Director and Pmfes._r of A_ Engineenng. Given that feedback gains are synthesized by quasi-linear or 2Research Associat_ andDoctoral Candidate.

linear methods, they can be scheduled on parameters that define the Vdm_ _ _ TT_u. Rr,_nmL _ Veo_ the small perturbations remain within a certain domain of validity the stability of the linear system implies stability of the nonlinear plSCl,Roll,Ya. _ I . L_ Vemr_ _ 1 I I .. I c_ _ o,_ I _ a_ l_,,,a,_ III system. 15-17 At operating points where linear analysis is performed, I =';_q I I-- _,,_ s,,_=:.n_.,e I I II the system responses, Y, control inputs, U, and commanded inputs.

Yc, in Fig. 3.1 consist of the sum of the reference values and small perturbations. Sain, Peczkowski, and othersl2,13 give a similar , , E,/_a:"[a'h'e F''_f

-JI ,t - Ill

description for nonlinear engine systems.

Fig. 3.2 considers only the linear time-invariant small perturbation system model and control laws, G(s) and K(s), at a particular operating point. The assumptions implied here are that the feedback portion of the system behaves in a linear time-invariant fashion, and that the system responses, y, control inputs, u, and commanded inputs, Yc, are all small perturbation quantifies. Note that the objective of the feedback loop is to regulate the error signal, ¢, or to keep it small. Linear control synthesis and analysis is frequendy justifiable given that: (1) the error signal is kept small so that the small perturbation assumption is not violated, (2) the gain scheduling leads to slowly time-varying gains so that the system can be considered - Some Coupling Paths Between Airframe and Engine dme-invariant at each operating point, and (3) observing from the Systems figure, that nonlinearities of the system are outside the feedback loop, reference points. Some examples of flight steady state operation are thus cannot affect its stability.

constant speed-wings level-forward flight, climb-at-constam climb rate, steady-coordinated-banked turn, and approach-to-landing. Thus, Flight E_vel_e lnfon_auon feedback gains may be, in part, scheduled on pitch, roll and yaw rates, and their integrals, which define the atdtude of the airframe.

The gains may also be scheduled on parameters that determine the _l (NonLinear FuJnc_om) aerodynamic forces on the airframe, such as Much number, dynamic pressure, and aidmde, or ambient temperature and pressure.

Feedback gains on the engine may be scheduled using highly nonlinear tabulated data that take flight envelope information such as power lever angle, which defines the requested power level, and Mach number and ambient temperatures and pressures, which define inlet flow conditions. 14 However, the system must be able to transition from one operating point to the next in a smooth and stable fashion without great loss of performance. The design of the gain schedules during these transitions can be a difficult and time consuming process. One - Small Perturbation Linear Feedback System At One example may be to use linear or nonlinear on-line interpolation Particular Operating Point procedures. 12-14 The transition gain schedules may open some feedback loops An important use of linear control synthesis is that stability and close others depending on the control objectives at the reference robustness will be provided to the actual nonlinear system as control point in question. For example, during steady state operation, engine laws are designed to provide more robustness for the linear system control is one of regulating thrust for performance and fan speed to approximation.

keep the engine at the operating point. Engine switching logic, using Much experience exists using this approach in airframe control accel/deccel schedules, is used during transitions through power level synthesis and analysis.10 Linear airframe models are considered in operating points to regulate on limit variables, such as main burner design specifications given in Ref. 18. This document gives, for pressure or compressor turbine inlet temperature, to avoid engine example, natural frequencies, damping ratios and time constants of limits, at the expense of engine performance. 19 various modes that should be met at different phases of operation so Figure 3.1 shows the airframe/engine nonlinear system viewed that the airframe dynamics reflect good classical flying qualities.

in the manner just described. Here, G and Krepresent the family of Linear airframe control objectives typically require stabilizing or augmenting the stability of these various modes.

quasi-linear or linear airframe/engine models and control laws defined throughout _e flight envelope. This linear approach has also often been considered for control synthesis and analysis of the nonlinear engine system, as discussed Flil_ En_)= In/oramion: in, for example, Refs. 12-14. Sain, Peczkowski, and others 12,13 ^intaCt md _ Ramg offer a systematic control law synthesis procedure for the total Ai,,p=,_, V_m_ P'=s.m_.

o¢ M aCh No,. Pow_ Lm¢_ Ang]=.

nonlinear engine system by utilizing a linear control law synthesis Abler)| Teml:a_'atmv_. l:_._utc.a.e.w...

procedure at each operating point. Here, nonlinear plant and plant inverse models generate scheduled control inputs and response commands into thelinear feedback loop.

Often, however, the small perturbation assumptionmay Ix: too restrictive. For example, as statedin Refs. 3 and 19, the engine system is usually nonlinear during transient operation. Ref. 2 investigates a configuration involving RCS jets which lead to an GLin S_linlt absolute value nonlinearitydue to the fact that an increase in compressor bleed flow is required for both positive and negative pitch, roll and yaw moments. The dynamics that couple theairframe and engine, discussed in the last section, may also include nonlinearities. Quasi-linear analysis, using the describingfunction technique, may bc espcciaJly useful when nonlinearities inthe systom cannot bc ignored,yet are "small,"or can be isolated, such as in Fibre 3.1 - Airframe,_ngine Nonlinear System Viewed as a Family saturated actuators, or components with thresholds or hysteresis. IS of Operating Points In this case, theinpurJourput relationship of thenonlinearities are modeled as lineardescribing functions plus a remnant. Unlike Linear time-invariant analysis at particular operating points can linearmodels, which arc independent of the type of input to the accurately predict the stability of the equilibrium point of the nonlinear system,quasi-linear modcls of nonlinear systems may differ for each system given that transient motions from the steady state consist only input intothe system. Step, sinusoid, and statistical inputsarc often of small perturbations. Lyapunov stability theory states that as long as used in describing function analysis. Thus, sinusoidal input with YE(S) the vector of engine responses ( turbine temperature, 1"4, describing functions may accurately model the nonlinear input/output fan speed, N 2, etc.), and uE(s) the vector of engine control inputs, behavior of systems subjected to nearly sinusoidal periodic inputs, but are invalid for systems subjected to, for example, step inputs. What (fuel flow rate, w F. nozzle area, AT, etc.)

Each of these subsystems will be acted upon by feedback type of input is used in the analysis depends on [he important nonlinear features that need to be accurately modeled. The sinusoidal systems with control compensation matrix K^(s), for [he aircraft flight input describing function, used in limit cycle analysis, is equal to [he control system, and KE(S), for the engine complex ratio of the fundamental frequency component of the output control system, which is shown below, for example.

to [he input. The remnant models the effects of all higher harmonics.

d(I) Higher order quasi-linear approximations must be performed until the remnant is small enough to be considered negligible. First or second order quasi-linear approximations are usually acceptable due to [he attenuation characteristics of physical systems.

An important advantage of quasi-linear analysis is [he ability to obtain describing function models by experiment. If accurate math models of the dynamics of the system being analyzed are not Fi2ure 4.1 - Block Diagram of the Engine Feedback Loop available, describing function models of the system can be experimentally derived by measuring and tabulating the outputs of the Here YEc is [he vector of desired or commanded responses, and d(s) system for given inputs. For example, sinusoidal describing represents any exogenous disturbances acting on the system. If the functions of the system can be generated by varying the frequency of above system were linear, the responses would be given by the input sinusoid and measuring [he response of the system. The results may then be analyzed to obtain, for example, "transfer yE(s) = [I + GEKE]'tGEKE yE_(S) + [I + GEKFj'td(s) (4.3) function" like models or "Bode plots." It must be recognized, however, that, unlike linear systems, the resulting models obtained Note that often [he compensation K^(s) and KE(S) are here are dependent on [he amplitude of the input sinusoid.

As discussed in Refs. 15 and 20, equivalence can be drawn synthesized and implemented while essentially treating the subsystems as decoupled. Such control laws are defined here as decentralized between robustness analysis involving limit cycles in quasi-linear controllers.

approximations to nonlinear systems and stability robustness analysis More generally, however, the aircraft/engine system dynamics using linear tools based on Nyquist stability theory. That is, margins may be defined at a particular flight condition as shown in the to limit cycles for quasi-linear systems can be measured in [he same following matrix of sinusoidal input describing functions: way as gain and phase stability margins in, for example, Bode or Nyquist plots for linear systems.

Because of this, it is believed [hat the linear analysis to study the airframe/engine interactions presented in Ref. 21 can be direcdy

""'+'1 [ °''+' °"+'

extended to a quasi-linear analysis of nonlinear systems. That is, it is y_(s)J LG_(s) G_(s) LUE(S)J LuE(s)J (4.4) believed that the analys.is of Ref. 21 is not resmcted to those operating points wnere me airframe/engine system's dynamics axe where GA*(S) and GE'(s) are different from G^(s) and GE(S) above linear. The next section will present [he quasi-linear viewpoint of this analysis. Thus, from now on, the coupled airframe/engine system by [he amounts A^(s) and AE(S), respectively, due to dynamic cross- and control laws, G(s) and K(s), shown in the block diagram of Fig.

coupling between the engine and airframe subsystems. That is, 3.2, are considered to be quasi-linear systems.

In summary, implicit in this representation is that only one GA* = G^ + AA operating point is considered, and is not intended to embody the system's characteristics throughout the entire flight envelope. That is, GF.* = GE + AE (4.5) each operating point manifests a particular control architecture and system model. Note also that, although quasi-linear analysis is not Further, GAE(S) and GE^(s) represent input coupling between the restricted to the small perturbation assumpuons of linear analysis, for airframe and engine. This situation describes two-directional each class of inputs to be analyzed, a different quasi-linear coupling. That is, the airframe control inputs affect the engine representation of [he system must be obtained. For limit cycle responses, and, likewise, the engine control inputs affect the airframe analysis, sinusoidal input describing functions are used to define the responses.

quasi-linear system.

Note that [his representation of the fully coupled system may One final note is that the analysis to be presented is not not be strictly valid depending on the particular configuration under intended to replace the high order complex nonlinear integration study. It can be seen in Fig. 2.1 that the coupling, in general, is techniques involved in any final analysis and design iterations of the manifested due to airframe responses entering as inputs to the engine aix'ffame/engine control laws. These complex techniques must be used system, and engine responses entering as inputs to the airframe for certain flight phases where the nonlinearities are extremely large, system. The analysis should have analagous derivations for the such as encountered in violent combat maneuvering. However, linear different frameworks of [he coupled airframe/engine systems. This and quasi-linear control synthesis and analysis techniques are topic is discussed further in Section 6.

invaluable tools in obtaining control laws for a large portion of the A centralized synthesis decentralized implementation flight envelope, as well as in acquiring more physical understanding approach is defined here as one in which control laws are synthesized of the complex nonlinear system.

with some knowledge of the coupling that exists between the airframe and engine subsystems, yet contain independent conm_! compensation 4. The Quasi-Linear Analysis Framework for each subsystem. That is, this approach is defined as one in which The following analysis closely follows that presented in Ref.

K^(s) and KE(s), discussed previously, are designed with knowledge 21. The analysis is conceptually extended hem to include quasi-linear of the system given by Eqn. 4.4.

approximations to nonlinear systems. Let the quasi-linear aircraft Finally, control laws both designed with knowledge of model, defined at a particular flight condition be described in terms of airframe/engine interactions and implemented using cross-feedback the mau'ix of sinusoidal input describing functions, GA(S), where, paths between [he airframe and engine loops are defined here as centralized controllers. The following control law is one such y^(s) =G^(s)u^(s) (4.1) oentralized approach: with y^(s) the vector of aircraft responses ( angle of attack, or, pitch rate, q, etc.), and u^(s) the vector of aircraft control inputs, (flap uE(S)J L K_(s) K_(s) yECs)- yF.c(s) J (4.6) deflection, 5 F, thrust vector nozzle deflection, 51-v, etc.) Likewise, let the matrix of sinusoidal input describing functions defining [he engine The off-diagonal terms, K_(s) and KEA(S), represent control cross- dynamics be described as GE(s), where, feeds between the airframe and engine subsystems. It is argued in Ref. 3 that it may be desirable to implement [he airframe and engine yE(S) = GE(S)UE(S ) (4.2) control laws separately because a fully centralizedcontrol law Ref. 21 points out that, for a linear analysis, EA(S) can affect implementauon may be quite difficult ro perform. However, the the stability of the engine's closed loop system. This can be seen by question of the best approach to take in the IPFC problem is still under comparing the engine's nominal response, given by Eqn. (4.3), and debate.

the true system's response of Eqn. (4.7). It can be shown from For simplicity, the analysis will assume the control cross-feeds Nyquist stability theory 22.23 that, for a linear analysis, the closed loop are absent (i.e. KAE(S) = KEn(S) = 0.) This situation may b¢ system in Fig. 4.3 is assured to remain stable if the loop is stable for represented as shown in Fig. 4.2. For the linear analysis, the case EA(S)=O, and if with control cross-feeds, although more complex algebraically, may be addressed in a manner similar to that presented here, and it is det{I + ((3 E + eEA)KF_.] _= O. [0<_<I ] (4.8) believed that extensions to quasi-linear analysis may also be derived.

for all frequency, which is assured if o,_CEAKE)< O.,=(I+(3EKE) (4.9) for all frequency, where o denotes the singular value of a matrix.

Thus, it is evident from this inequality that there will be loss of stability robusmess for "large" EA(s), (i.e., if its maximum singular value is large.) As stated in the previous section, an equivalence can be drawn between limit cycle analysis for quasi-linear systems and stability analysis for linear systems. It is the contention here that as the _size" of the describing function E^(s) grows larger, the closed loop engine system will approach a limit cycle. Rigorous justification of this assertion is currently being addressed.

The utility this analysis is that the results can be used to determine if significant cross-coupling between the airframe and _ - Block Diagram of the Coupled Airframe/Engine System engine systems exists, atthe referencepoint under study,and if it needs to be addressed when synthesizing control laws. Another Each of the terms arising from the effects of the benefit from thisanalysis should be to determine the amount of airframe/engine coupling am apparent. This figure suggests that the coupling introduced into the system by the addition ofdevices,such coupling dynamics, GAE(S) and G_,(s), and the airframe dynamics, as RCS jets, thatuse thepropulsion system to enhance the airframe GA*(S), augmented with the airframe compensator, KA(S), can all be attitude control power. Also, more physical insight into the system's grouped together to form the describing function matrices En(s) and coupling dynamics may bc obtained by observing the critical the DA(s). In other words, since An(S), AE(S), GAE(S), and GEA(S) frequency ranges where En(s) grows "large."

arc not really zero, the engine loop is not that shown in Fig. 4.1 where Ref. 21 alsodiscusses how the effects of coupllng can degrade airframe/engine interactions ar_ ignored, but rather that shown in the the engine system's performance. Similarperformance analysis for Fig. 4.3. Here the effects of the actual coupling present arc grouped quasi-linear systems iscurrently under investigation.

into the terms EA(s) and DA(S)yAc. Ref. 21 gives, through block Note too, that the focus of this analysis has bccn the effect of diagram algebra, expressions for both EA(s) and Dn(s), and this the airframe dynamics on the engine loop. A dual analysis ispresent representation of the system is valid for linear systems. The validity inthatthe engine alsoaffects the airframe loop.

of this representation is still under investigation for analysis of nonlinear systems. However, at this point, it is assumed that 5. Sample Results of the Quasi-Linear Analysis Procedure describing functions, En(s) and DA(S), Can be found by some manner Using the techniques just presented, attention will be directed to the analysis of an airframe/engine system that has been the subject so that the input/output relationships of the systems shown in Figs.

4.2 and 4.3 are equivalent.

of several studies of integrated flight and engine control. 1.3.4 The vehicle considered is representative of a high performance fighter aircraft with 2-D thrust vectoring, thrust reversing and RCS jets at the Y*=(S)r==.__--_+ d(s) approach to landing flight condition. A more complete description of the vehicle and the control laws used can be found in Rcf. 21.

UF..(s) s Although obtained from a linear analysis, the results presented in this section will be considered quasi-linear input/output relationships to underscore the aspects of the quasi-linear analysis of the last section.

The _cngine plant is defined as the matrix of sinusoiciai input describing functions given by Eqn 4.4. The al.rframe response - Block Diagram of the Engine Loop for the Coupled is a linear combination of angle of attack and pitch rate, and the engine Airframe/Engine System response is fan speed. The control inputs arc thrust vectoring angle and fuel flow rate. The control law considered here is dccentraJized.

Fig. 4.2 shows that the critical closed-loop coupling matrix That is. no control cross-feeds are present and the airframe and engine En(s) depends most importantly on the input coupling sinusoidal control laws, kA(S) and kE(S) arc designed only with the knowledge of describing functions GEA(S) and GAE(S), as well as on the airframe gA(s) and gE(s). Note that lower case g is used to signify that these arc scalar describing functions.

control law, KA(S), the airframe dynamics, GA(s ) + AA(S ), and the Fig..5.1 shows the magnitudes of the four describing change in the engine sinusoidal describing function, AE(S).

functions of the plant. This figure shows that the cross-coupling Therefore, if A E is "small," and if GAE(S) and/or GEA(S) axe "small," dynamics are both smaller than the main diagonal describing functions then En(s) is "small." by approximately 40 dB for frequencies above one tad/see. Therefore, since cA(s) is a function of the cross-coupling dynamics, the size of For the linear analysis, the input/output characteristics of the system in Fig. 4.3, including the coupling effects, is cAkE will be quite small compared to the nominal engine loop describing function, gEkE, and airframe/engine interactions, as modeled here, will not instigate a limit cycle in the engine loop.

yE(s) = [I + (GE+En)KE]'I(GE+EA)KE y_(s) Fig. 5.2 compares the size ofenk E to the nominal engine loop + [I + (GE+EA)K_'I (DA y_+d(s)) (4.7) describing function, gEkE, when RCS jets arc added to the system to aid in pitch control. Although not shown, this produces an increased Note how commands into the flight control system, ync(S), are magnitude in the gEx describing function. In Section 2 it was transmitted to the engine responses through Dn(s), and this term discussed that RCS jets draw bleed flow from the engine's enters into the engine responses the same way as any other compressor, hence, control of the pitch attitude of the airframe direcdy disturbances, d(s). Thus, the commanded inputs into the aircraft influences the quality of the airflow through the engine. Although the (from the pilot) act as additional disturbances to the engine.

system would not experience a limit cycle due to the addition of pitch

RCS control, itcan beseen that the critical frequency in which a limit

for their analysis used varying magnitudes of aft and ventral thr_t cycle could first occur from additional changes in the system dynamics (engine responses) to effect pitching moments. Thus, a natm-al would be at approximately 0.2 rad/sec. Note also that the phase angle viewpoint for their model of how the airframe and engine interact is :_ of the u'ue system begins to differ from the nominal engine system in consider that the engine responses are control inputs to the airframe this region. system. That is, that the engine act as an attitude actuator to t2_e airframe, (as well, of course, as a forward speed actuator.)

2O Fig. 6.1 displays the airframe/engine system framework as viewed by Refs. 3 and 6. Here, R(s) represents generalized actuaw."s, that is, both airframe actuators and the engine system. Ks represe=:.s -20 the airframe actuators and engine compensation, or the "subsystem _ _ -40 control laws. u s, then, is the "subsystem" control inputs, and ume is the commanded inputs into the closed loop actuator/engiae _-_ subsystems. P(s) models the "mission level" airframe system, and the _ -80 "mission level" control laws are denoted as K.=. As defined in Section 4, y^ represents the airframe responses, and Y^e represents the -100 airframe commands to follow.

-120 - |4]0_ l I0o 101 10z Frequency in Rid/See ) - Open Loop Describing Function Magnitudes 5O - Airframe/Engine System Framework of Refs. 3 and 6 i _iiiiii ! i i liii ..... iiii Fig. 6.2 shows this framework with the engine system separated from airframe control inputs. Note that, for simplicity, the : ¢ : ....... . . . ....._... • ...

airframe actuator dynamics are modeled as the identity matrix. In .6 -50 comparing the system in Fig. 6.2 with the system of Fig. 4.2, it : : : :::::: : •¢-. :::':: : : : :::: follows that ! i iiiiiii ! i_i'te_-Lii i i i iii!

.lO0 i ! ! iiii!i .......... "'1 .... i!

-1.$0 ....... ;'":'"'"!'!!'?? ...... 7""7"!'"!'?1"77 ...... !"" !".:"4".'" !!"

l = -Kin (y^ - (6.2) tO0 10i 102 These equations can be used to draw the block diagram in Fig. 6.3, which shows more clearly the relationships between the two frameworks.

- Airframe/Engine System Framework With Engine System Explicitly Shown tO0 10t 102 Frequency in Rad/Sec - Engine Loop Describing Function With Pitch RCS Control Added These resuhs show that this system, as modeled, will not be significandy affected by airframe/engine interactions and decentralized conta'oI synthesis may be adequate. However, note that the question of performance degradation due to these interactions has not yet been addressed for quasi-linear systems. The linear analysis for this configuration showed that the disturbance rejection performance of the engine was seriously degraded due to the additional disturbances from aircraft commanded inputs through DA(S). Analogies to quasi-linear performance analysis are under study.

_ - System Framework of Refs. 3 and 6 as it Relates to the Framework of Section 4 6. A Related Analysis Framework This section relates the framework of the analysis developed The path from the engine responses to the aircraft responses by Rock, Emami-Naeini, Shaw and others in Refs. 3 and 6 with the framework for the quasi-linear analysis of Section 4. In Section 4, it through GAEGE -t is equivalent to the path from engine control inputs is mocleled that the airframe control inputs _fect the engine responses to airframe responses through GXE alone, as given in Fig. 4.2, for and the engine control inputs affect the airframe responses. This Gl_^ = 0. Notice that for this framework, the commands into the viewpoint seems natural if considering such interactions as RCS thrust closed loop engine system are no longer independent commands, commands (airframe control inputs for attitude control) drawing modeled in Section 4, but rather a function of the airframe responses engine compressor bleed air, thus affecting engine flow (engine and commanded inputs due to KmE. Also note that differences responses.)

between the nominal dynamics and the dynamics that include coupling However, in Refs. 3 and 6 the example vehicle under study effects of the airframe and engine, A^ and A_, are assumed zero here.

[2] Garg, S., Mattem, D., Bright, M., Ouzu, P.. "H-Infinity Based lntegrate_ as this issue was not addressed in Refs. 3 and 6. More significantly, Right/Propulsion Control Design for a STOVL Aircraft in Transiuon however, is that the input coupling dynamics from the airframe to the Flight," NASA TM 103198, NASA Lewis Research Center, Ohio. August.

engine, GEA, is assumed to be zero. Because of this, this framework 1990.

only considers one-directional coupling. Fig. 6.3 shows that the [3] Rock. S.M.. Emami-Naeini. A.. Anex. R.P.. "Propulsion Control airframe dynamics cannot affect the stability (if linear) or susceptibility Specifications in Integrated Right/Propulsion Control Systems." AIAA to limit cycles (if quasi-linear) of the engine loop. As discussed in Paper No. 88-3236. AIAA/ASMF_JSAE/ASEE 24th Joint Propulsion Ref. 3, two-directional coupling was not considered. From the Conference, Boston, Mass., 1988.

viewpoint of their framework, two-directional coupling would be modeled as engine responses-to-airframe inputs/airframe responses- [4] Smith, K.. Stewart. C., "A Survey of Conuol Law Options for Integrated to-engine inputs.

FlighCPropulsion Control for F_ghter STOL Approach." AIAA Paper No. 84- For quasi-linear analysis of nonlinear systems, it is important 1900CP, AIAA Guidance. Navigation and Control Conference, Seattle.

to realize that block diagram manipulation of systems may not keep the Washington. August. 1984.

input/output relationships of the actual system. Therefore, it is [5] Shaw, P.. et at., "Development and Evaluation of an Integrated Flight and imperative to model the coupled airframe/engine system properly when deriving critical coupling terms, such as E^(s). The Propulsion Control System." AIAA Paper No. 85-1423, AIAA Joint Propulsion Conference. Monterey, California. July. 1985.

frameworks presented in Section 4 and Refs. 3 and 6, are two possible models of how the engine and airframe couple. Which [6] Shaw. P.D., Rock, S.M., and Fisk. W.S., "Design Methods for Integrated framework should be used may depend on the configuration under Control Systems," AFWAL .TR-88-2061, Aero Propulsion Laboratory, Air study.

Force Wright Aeronautical Laboratories, Dayton, Ohio, June. 1988.

7. Conclusions [7] Smith. K.L.. "Design Methods for Integrated Control Systems," AF'WAL-TR- The linear analysis of Ref. 21 was conceptually expanded here 86-2103. Aero Propulsion Laboratory, Air Force Wright Aeronautical to embody quasi-linear approximations of nonlinear systems. A Laboratories, Dayton, Ohio, December, 1986.

sinusoidal input describing function matrix was derived that [8] Berry, D., Schweildaard, W., "Potential BenefiLs of Propulsion and Flight quantifies, in a meaningful way, the significance of airframe/engine Control Integration for Supersonic Cruise Vehicles." based on SAE paper interactions on the engine control loop. The size of this matrix 740478, 1974.

quantifies the effect of airframe/engine coupling on the susceptibility of the closed loop system to encounter a limit cycle. It was shown [9] Tape. R., Hartill. W., et at.. "Vectoring Exhaust Sygems for STOL Tactical that the off-diagonal describing functions in the system's describing Aircraft." Journal of Engineering for Power, Transacuons of the American function matrix play a significant role in determining any critical cross- Society of Mechanical Engineering, July, 1983.

coupling between the airframe and engine. When the critical coupling terms axe small compared to the magnitude of the nominal engine (10] Roskam, J., Airplane Flight Dynamics and AuJoraatic Flight Controls. Part system's describing function, for which cross-coupling is ignored, I1. Roskam Aviation and Engineering Corp., Ottawa. Kansas, 1979.

effects of airframe/engine interactions are minimal. A dual analysis [111 McRuer, D., Ashkenas, I., Graham, D., Aircraft Dynamics and Automatic exists for determining the coupling effects of the engine dynamics in Control. Princeton University Press. Princeton, New Jersey, 1973.

the nominal airframe loop.

Sample results of this analysis from a case study of an [12] Peczkowski, J., Stopher, S., "Nonlinear Muhivanahle Synthesis With airframe/engine system used in earlier studies of integrated control Transfer Functions," Proceedings of the 1980 Joint Automatic Control techniques was then presented. This study revealed that the vehicle, as Conference. Vol 1, PL WAg-D.

modeled at that particular operating point, exhibited very little critical interactions as fax as encountering limit cycles. A classical [13] Sain. M.. Peczkowski. J.. "Nonlinear Multivariable Design By Total decentralized control system synthesized assuming the airframe and Synthesis." Control System Technical Rcport #36. Department of Electrical engine subsystems are totally non-interacting was quite suitable in this Engineering, University of Notre Dame. Notre Dame, Indiana. March. 1985.

case. However, the analysis shows how the inclusion of pitch RCS [14] DeHoff, R., et. at, "FI00 Multivariable Control Synthesis Program."

control jets in the model does increase the amount of cross-coupling.

AFAPL-TR-77-35, Air Force Aero-Propulsion Lab. Wright Patterson Air Not examined, at this time, is the effect cross-coupling has on the Force Base, Dayton, Ohio, June, 1977.

closed loop performance for nonlinear systems. Previous studies involving linear analysis show that coupling can have a significant [15] Graham, D., McRuer, D., Analysis of Nonlinear Control Systems, John detrimental effect on the performance, and it is believed that this will Wiley & Sons, Inc., New York, 1961.

be the case with a quasi-linear analysis approach to study nonlinear system performance, if possible.

[16] Minorsky, N., Introduction to Nonlinear Mechanics, J. W. Edwards, Ann Comparison of the framework for the analysis presented in Arbor, Mich., 1947.

this paper with the framework developed in Refs. 3 and 6 showed that their framework does not consider two-directional coupling between [17] Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall Inc., New the airframe and engine. In their analysis, the airframe dynamics Jersey, 1978.

cannot affect the engine loop. This assumption may lead to erroneous [18] Anon., MIL-8785C, Flying Qualities for Piloted Ai_lanes, USAF, Flight conclusions if the system in question has significant two-directional Dynamics Laboratow, 'W'PAF'B, Dayton. Ohio.

coupling between the airframe and engine. From the discussion on potential sources of airframe/engine interactions it can be observed that [19] Concs_ndence with Mr. Duan¢ Mattem, Sverdrup Technology, Inc., LewLs two-directional coupling may be present in the configurations under Research C.¢nte, r Group, NASA Lewis Resea_h C.e..nl_', Cleveland Ohio.

study for the IFPC problem.

[20] Thaler. G., Pastel, M., Analysis and Design of Nonlinear Feedback Control Acknowledgements Systems, McGraw-Hill Inc.. New York, 1962.

This work was sponsored by the NASA Lewis Research Center under Grant # NAG3-998. Mr. Peter Ouzts is the technical program [21] Schmidt. D.. Schierman, J., Garg. S., "Analysis of Airframe/Engine Interactions - An Integrated Control Perspective." AIAA # 90-1918.

manager. Appreciation is expressed to Mr. Brett Newman for his material regarding nonlinear systems, and to Mr. Duane Mattem for presented at the 26th Joint Propulsion Conference. Orlando. Ft.. July, 1990.

his expertise in engine dynamics.

[22] Doyle, L, Stein, G.. "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis." IEEE Transactions on Automatic Controls.

References Vol. AC-26, No. !, pp. 4-16, Feb., 1981.

[1] Garg. S., Manem, D.L., and Bullard, R.E.. "Integrated Flight/Propulsion Control System Design Based on a Centralized Approach," AIAA Paper No.

[23] Rosenbrock. H., "The Stability of Multivariable Systems," IEEE 89-3520. AIAA Guidance, Navigation and Control Conference. Boslon, Ma., Transactions on Automatic Controls, Vol. AC-17, pp. 105-107, Feb..

1972.

Analysis of Airframe/Engine Interactions in Integrated Flight and Propulsion Controlf John D. Schierman I and David K. Schmidt 2 Aerospace Research Center College of Engineering and Applied Sciences Arizona StateUniversity Tempe, AZ 85287-8006 First, the basic analysis framework is reviewed in Abstract Section 2. Then, two case studies of a vehicle with different control configurations are presented in Sections 3 and 4. This An analysis framework for the assessment of dynamic airframe and engine was considered in several earlier studies of cross-coupling between airframe and engine systems from the the integrated airframe and engine control problem. 1.2,'1.5 In perspective of integrated flight/propulsion control is presented.

both control configurations the airframe's influence on the This analysis involves to determining the significance of the engine is shown to be significant, but it is also shown to interactions with respect to deterioration in stability robusmcss constitute coupling in only one direction. Then a sensitivity and performance, as well as critical frequency ranges where analysis of the system's stability and performance is performed problems may occur due to these interactions. The analysis on critical interaction effects identified by the analysis. Finally, illustrated here investigates both the airframe's effects on the Section 5 extends the analysis methodology to control laws with engine control loops and the engine's effects on the airframe cross-feeds between the airframe and engine systems.

control loops in two case studies. The second case study involves a muhi-input/multi-output analysis of the airframe.

Sensitivity studies are performed on critical interactions to 2. Review Of Analysis Framework examine the degradations in the system's stability robustness and performance. Magnitudes of the interactions required to A framework to analyze airframe/engine interactions was cause instabilities, as well as the frequencies at which the introduced in Ref. 8. Although the key features of the instabilities occur are recorded. Finally, the analysis framework framework are reviewed here, more emphasis is placed on some is expanded to include control laws which contain cross-feeds between the airframe and engine systems. aspects of the analysis that are pertinent to the case studies presented in the next sections. This analysis framework focuses on the feedback portion of the nonlinear airframe/engine system.

1. Introduction Each operating point of the system elicits a particular quasi-linear system model and control architecture, G(s) and K(s). Ref. 8 The Integrated Flight and Propulsion Control (IFPC) presented one viewpoint of how the airframe and engine systems problem addresses interactions between airframe and engine at one operating point interact. The treatment of nonlinear systems in control law synthesis and analysis for configurations effects, such as engine limits, is presented in Ref. 9. The that use the propulsion system to augment the lift and improve airframe/engine feedback system is considered as shown in Fig.

maneuvering capabilities of the vehicle. 1-7 These 2.1.

configurations may give rise to significant coupling between the systems. Formulation of methods for assessing the significance of interactions between the systems, from the perspective of control design is to be addressed.

Ref. 8 initially presented an analysis framework to assess if cross-coupling dynamics between the airframe and engine are of sufficient "magnitude" to cause significant loss in stability robustness and/or performance, and thus warrant y__._s) __(s_J=_ A(s) special consideration in the control law design.

The purpose of this paper is fourfold: ,_(s): I _ + (1) Present case studies that not only analyze the airframe's DA(s) effects on the engine, but also consider the dual analysis of the engine's effects on the airframe.

yE(s) (2) Perform a multivariable analysis of the airframe control loops.

(3) Investigate the system stability and performance Figure 2.1 - Block Diagram of the Coupled Airframe/Engine sensitivity to increases in critical coupling terms System identified by the analysis.

In this figure, YAc is the vector of desired or commanded (4) Expand the analysis framework to include control cross- airframe responses, perhaps from pilot inputs, and YEc is the feeds.

vector of commanded (or limited) engine responses, u^ is the vector of aircraft control inputs and u E the vector of operative engine control inputs. Finally, YA is the vector of aircraft tAs presentedat the 1991 AIAA GN & C Conference, New Ortmns responses and YE is the vector of engine responses comparable I Doctoral Candidate,Student Member,AIAA with Y_c.

2Acting Director, Prof. of Acro.Eng., Assoc. Fellow, A/AA Under theassumption that no coupling exists between Copyright © 1991 by John D. Schierman and David K. SchmidL the two systems, the airframeand engine input/output Published by American Institute of Aeronauticsand Astronautics, Inc. characteristics are defined intermsof the matrices GA(S ) and with permission.

responses, yxc(s), are transmitted to the engine responses GE(s), respectively. These will be referred to as the nominal through D^(s), and act as additional disturbances to the engine.

systems. The systems in which dynamic cross-coupling Thus, a key result of this analysis is that if D^(s) is large, the between the engine and airframe systems is considered differ closed loop performance will suffer.

from the decoupled nominal system models by the amounts Note that large EA(s) can degrade the performance as A^(s) and AE(s), respectively. The following notation will be well. However, since E^(s) is present in the returndifference used to relate the plant descriptions : matrix, italsoaffects the system'sclosed loop stability. Itcan be shown,10, II for example, that the closed loop system in Fig.

GA* = GA + A^ GE* = GE + AE (2. I) 2.2 is assured to remain stable if the loop isstablefor E^(s)=0, andif In the coupled system airframe responses are affected by engine control inputs either indirectly, or directly through G^E(s), and Omax(EAKE) < GmI,(I+GEK E) V co (2.5) engine responses arc affected by airframe control inputs either indirectly, or directly through GEA(s).

where w = frequency, and o denotes the singular value of a Finally, the system is acted upon by feedback control matrix. It is evident from this inequality that there will be loss of compensation matrices K^(s), for the aircraft flight control stability robustness for "large" EA(S).

system, and KE(s), for the engine control system.

Note that the focus of this analysis so far has been the Refs. 8 and 9 suggest that the coupling dynamics, effect of airframe dynamics on the engine loop. A dual is GAE(s) and GE^(S), and the airframe dynamics, G^*(s), present and the engine loops clearly also affect the airframe augmented with the airframe compensator, KA(S), be grouped loops, as shown in Fig. 2.3.

together to form the matrices E^(s) and the DA(S), which capture the effects of the actual coupling present on the engine loop.

This new representation of the coupled system is shown in Fig.

2.2. Note that in this figure d(s) represents any additional exogenous disturbances acting on the system.

u s yr=__....._ + T(s)--_ d(s) ) _ _Cs_ _ + v^fs) + d(s) Figure 2.3 - Block Diagram of the Airframe Feedback Loop for the Coupled Airframe/Engine System The dual of Eq. (2.4) gives the airframe responses as Figure 2.2 - Block Diagram of the Engine Feedback Loop y^(s) = [I+ (GA+EE)KA]'t(G^+EI-')K^ y._(s) For the Coupled Airframe/Engine System + [I+ (GA+EE)KA]'I(DE y_(s)+d(s)) (2.6) If coupling does not exist between the airframe and engine, EA(S) and DA(S) are zero. Thus, if E^(s) and DR(S) are where "small," as measured, for example, by singular values, this would indicate weak interactions between the airframe and (2.7) EE(s) = A^ - GAE[I + KE(G E + A_]']KEGEA engine systems. These expressions, given below, can be obtained by block diagram manipulation of Fig. 2.1, and are DE(S ) = GAE[I + KE(G e + AO]'tKE (2.8) principal to this analysis.

Large I_(s) and/or EE(s) can degrade the flying qualities of the EA(S) = AE - GF.A[I + KA(G^ + A^)]'tK^GAE (2.2) airframe control system. Further, the closed loop system in Fig. 2.3 is assured to remain stable if the loop is stable for DA(S ) = GEA[I + KA(GA + AA)]'tK^ (2.3) Es(s)=0, and if Eq. (2.2) shows that the "size" of the product Orm_(EEKA) < OmIn('I+G^K^) V ¢.o (2.9) GEA(S)GAE(S) is critical in determining the "size" of EA(S).

which is the dual of the key result of Eq. (2.5).

EA(s) will probably be "small" if AE is "small" and if either Two airframe/engine system configurations _ill now be GAE(S) and GEA(S) are "small." However, as illustrated in the considered in the next sections to assess the effects of cross- case studies in the next sections, when GEA(S) is "large," the coupling between the airframe and engine systems.

"size" of EA(s) becomes sensitive to small changes in the "size" of GAE(s). Note further that the "size" of DA(s) is independent of GAE(S), but may be significant ff GEA(s) is "large." Finally, 3. First Case Study - Scalar Airframe and Engine note that if loop closures on the airframe are not present Systems (K^(s)=0) then EA(s) = AE, and DA(s)=0.

The airframe/engine system used for this analysis has The input/output characteristics of the engine system been the subject of several studies of integrated flight and including coupling effects are propulsion control. 1.2.4,5 The vehicle to be considered is representative of a high performance Short Take Off and yE(S) = [I + (GE+EA)Kr_ "t(GE+EA)Kv y_(s) Landing (STOL) fighter aircraft equipped with a thrust + [I + (GE+EA)KF_'I(D^ y,_(s)+d(s)) (2.4) vectoring/thrust reversing nozzle and a reaction control system (RCS). The operating point under consideration is the approach- to-landing flight condition. At this operating point the airframe This reveals how airframe/engine interactions can affect the dynamics are aerodynamically unstable. The vehicle model was stability and performance of the system. Airframe commanded obtained from ReL I, and this particular system plant and control architecture was fgrst presented in Ref. 9. The following table defines the controls and measurements used for this configuration.

Table 3.1 - Controls and Measurements For The Case Study Vehicular System The aircraft controls are:

ii il .... iii ...... i...... iiiiiii

GTV = nozzle thrust vectoring angle (deg) Aq = pitch RCS control jet nozzle area (in2) 6n,p= = wailing edge - leading edge flap deflection angle (in 2) -100 ............

The en_ne control is: lift 100 10t w r= main burner fuel flow rate (#/hr) Frequency in Rad/Sec The aircraft me_Lsurements are: Figure 3.1 - Open Loop Normalized Input/Output Mappings a = angle of attack (deg) q = pitch rate (rat/see) This figure shows that since there are little visible differences in the plots of gL(S) and gL*(S), and gE(s) and The engine measurement is: gt:*(s), AA(S) and AE(S) are quite small. However, although gAE(S) is smaller than the diagonal terms in F_.q. 3.1 throughout N 2 = engine fan speed (rpm's) the frequency range plotted, gEA(S) is larger than both diagonal terms below 2 rad/sec. This is due to the RCS pitch attitude The vehicle's leading and trailing edge flaps are direct control. Recall that gEa(S) reflects how the engine responses are lift devices which are used to control the flight-path-to-attitude affected by airframe control inputs. The pitch RCS jets draw response. A combination of thrust vectoring and pitch RCS jet bleed air from the engine's compressor to enhance pitch attitude nozzle area is used to control the pitch attitude dynamics. This control power, gEA(S) will be even smaller than gLE(S) shown control "blend" is defined as ck. Only the fuel flow rate is used above when RCS jet control is not used. 9 to regulate engine fan speed.

Analysis of the airframe/engine interactions requires Classical feedback control laws were synthesized. The some knowledge of candidate control laws since the feedback flight control design objective is to stabilize the airframe compensation (KL(s) or KE(S)) appears explicitly in the dynamics and obtain classical pitch rate and angle-of-attack interaction matrices (for example, EL(S).) However, even responses from pilot stick input, G_, that meet flying qualities . . . 1." .

without knowledge of the control laws, investigation of the open requtrements. The objecuve of the engane control law is to hold loop plant can still reveal the nature of the airframe/engine the operating point by regulating the fan speed. The control interactions. Large gEA(S) in critical frequency ranges where design is detailed in Refs. 8, 9, 12 and 13.

cross-over is anticipated indicates the potential for significant With this decentralized design, attention will now be airframe/engine interactions. From Eq. (2.3), dA(s) may directed towards evaluating the coupled system. The effects on therefore be large. Fig. 3.2 presents the engine's fan speed system stability of the low gain flap loop are minimal.8 sensitivity function along with the magnitude of d^(s) for this Therefore, a two-by-two system can be obtained by closing the system, and dA(S) is indeed large due to large gEL(S). This flap loop and combining the two aircraft measurements to form figure shows that the fan speed loop will not effectively reject one blended aircraft pitch response. This open loop system is disturbances from pilot pitch stick inputs. Fig. 3.3 shows the significant fan speed disturbance due to pilot stick input. Thus, cross-feed compensation between the airframe and engine may be required to reduce this effect.

N2 g_(s) g_(s) wf (3.1) where Ktm and K_ are feedback gains on angle-of-attack and 80 : i i _ i!_i_ i i i !ii pitch rate, respectively.

In order to properly evaluate the relative sizes of the input/output relationships of the airframe and engine, the system must be normalized by, for example, estimates of the maximum i values of the (small perturbation) controls and responses, l The following estimates of these maximum values were used to 20 -- Disturbance to Engine Loop .... "---'.'--'--'.i.--.'.

normalize the plant. 13 e- From Pilot Input=dL ! i i _ ii Table 3.2 - Maximum Values of Controls and Responses -20 _ i ! i i"!"i'!'! ......... _'" _enssuvaty:" : : : : : : :: : Function qmax = 0.06 rad/sec 5t_max = 10 deg -40 ..... , . , ....

_,x = 3 deg Aq max = 1 in2 10-s 10o 10t N2max = 570 RPM's wf m,a = 5,000 lbs/hr Frequency in Rat/See Fig. 3.1 shows the magnitudes of the four normalized Fiture 3.2 - Fan Speed Sensitivity Function and Engine Loop input/output mappings in Eq. 3.1, as well as the nominal Disturbance From Airframe Commanded Responses airframe and engine models, gAG) and gE(s)- flower case letters indicate scalar transfer functions.)

200 ......

Short Period Mode Phugoid Mode

i °

........................... _" .....................

,-_ -200 o -4OO -600 0 1 2 3 4 5 Time in Seconds

% o 5

FiLmre 3.3 - Fan Speed Response From A One Pound Pilot Step Input (RPM's/lbs) 0.2 ......i............ i...... ..............

For this vehicle and control system configuration, the 0.1 trim point occurs at a small thrust vectoring angle, 8r,, thus ..... :............................. Phugoid ...............

engine thrust transients will not generate large pitching Mode moments, and this is the reason gAE(S) is small. If the mm thrust vectoring angle is larger, thus increasing the component of the thrust vector perpendicular to the airframe's longitudinal axis, engine thrust transients would create larger pitching -0.1 moments. In such a case, gAE(S) will be increased. The plant : Increasing gAE input/output mappings in Fig. 3.1 indicate that g,_,(s) is large -0.2 below 10 rad/scc. Thus, small increases in gAE(s) in this L frequency range can increase the size of gAE(s)gEA(s). From -0.2 -0.1 0 0.1 0.2 Eqs. (2.2) and (2.7), CA(S) and e_(s) may therefore be large, thereby degrading stability robustness and performance. For R L, Ur_ 3.4 - LOCUS of the Airframe/Engine System's Closed these reasons a sensitivity study will be performed on gAE(S).

Loop Poles As gAE(s) IsIncreased Figure 3.4 shows how the closed loop eigenvalues of the system vary as the magnitude of gAE(S) is increased. Higher frequency engine poles are not shown and do not vary to any 5O great extent. It can be seen, however, that the short period eigenvalues vary significantly. Although not shown, critical ,, In= asing g,= ;,:: zeros also vary as gAE(S) is increased. This reflects a degradation in the flight control system's closed loop performance. Fig. 3.4 also shows the locus of phugoid roots, "_"° 0 -_,,'_-.,,' ..... : ..-- .... ! . !!

from which it can be seen that increasing gAE(s) will cause a low

frequency instability. /

._ V; : ,.J.,, i:_ ; ; : ; ; _. ;_ : : : .*',2 • '.

Fig. 3.5 shows plots of both sides of the key inequality in the engine loop analysis, Eq. (2.5). This figure shows that -5o ........

leAkEI is indeed much less than II+gEkEl throughout the frequency range for the original value of gAE(S), and stability of I(eAkE){ For Originalg_ .: ..: ::: the system is not in jeopardy. A stability margin for this : : : : : : : :: : : : : : : : analysis is defined here as the minimum distance between leAkEi -100 ................

and ll+gEkEl. For the original value of gAE(S) this margin is IO-t I0O 101 approximately 20 dB, and the minimum distance occurs at 0.2 Frequency in RadgSec rad/sec, the frequency at which the phugoid mode goes unstable when gAE(S) is increased, (see Fig. 3.4.)

Fibre 3.5 - Plot of Eq. (2.5) Fig. 3.5 also shows le^kEI as the magnitude of gAE(S) is increased. First, the original value of g^E(s) was multiplied by magnitude of the loop transfer (gE-I-CA)kE approaches 0 dB as its 20 dB, and leAkEI and II+gEkEI touch at 0.2 tad/see causing the phase approaches -180" A similar result is indicated in Fig. 3.g stability margin to reduce to zero. At this point the stability test (the dual of Fig. 3.6) which shows the Bode plot for the of Eq. (2.5) can no longer guarantee the closed loop system is airframe loop. It is considered significant that the critical stable. Instability actually occurs when gAE(S) is increased by a frequency of instability (0.2 rad/sec) is not near the nominal loop factor of approximately 40 dB. From Fig. 3.1, note that gAE(S), cross-over frequency (3 tad/see) and that Eq. (2.5)correctly thUS increased, takes on a magnitude comparable to the other indicated that the minimum stability margin occurs at this transfer functions in the system. frequency.

Fig. 3.6 displays the Bode plots for both the nominal Unfortunately, however, this stability test was (i.e. decoupled) engine loop transfer, gEkE, and the engine loop conservative in thata stability margin of 20 dB was indicated, transfer for the coupled system, (gE+eA)kE. For the original whereas the actualmargin was approximately 40 dB. However, value of gAE(s) there is almost no difference in these plots. This shown in Fig.3.7 is the dual of thisstability test fortheairframe loop has an infinite gain margin and a 60" phase margin loop, namely Eq. (2.9). Note that the various plots of leEkAI occurring at a cross-over frequency of approximately 3 tad/see. correspond to the same values of gAE(S) as in Fig. 3.5. Again, As gAE(S) is increased, it can be seen that at 0.2 rad/sec the the minimum stability margin distanoe occurs at approximately 0 ...........

40 : : : :::::: i i i ii!

i ! i !!if!! _ i i iii:i i Inc_asinggAe i ! i i i!!i .c 20 "............. :":":'": ......... !.... ?"?'i"?'!'?i

i

: : ::::ii ! :

........ ! !

10-I lOO 101 lO-I I0O lOl 0 ...... : :" ; . ; : : :;: .... ......

5iiiiiii :. ::i !::ii:: 10-I I00 101 10-t _ 10o 10_ Frequencyin Rad/Sec Frequency in Rad/Scc Figure 3.6 - Engine Loop Transfer FrequencyResponses of Figure 3.8 - Airframe Loop Transfer Frequency Responses of Nominal(gEkE) and Coupled((gE+c^)kE) Systems Nominal (gxk^) and Coupled ((gA+cF.)k ^) Systems 0.2 rad/sec, and when IeEkA[ is increased by 40 dB it just touches ll+gAk^l. Thatis, thestability test for the_c loop i i i i!iii_ : : ::...

gives a more accurate indication of the stability margin of approximately 40 dB. Thus,thestability test must bcperformed 0............ ............

for boththe airframe and engine loops, and thesystem's actual stability ismore accurately predicted by thelarger ofthetwo stability margins asindicated by Eq. (2.5) or Eq. (2.9).

-10 ....

_ .,_.,.:.,.:__ _: L':i::iii i i ! _iii ._..- ". _.Z _ 6. .... : : ....

¢"" ..... ...... " ...... "_'":-*" ._'_ .... :2.'-'2"" • • : • _ -20_i.._..i..i-i._._. : i _ i ! ! i ' _ S " ; _ " I _ _ g _ _ ' _ " _ " _ - - _ - - _ -20 _i ::::i:: l(l+gAkA)[ i_i::i 10-1 10o I0_ .... - ............. _-.:i_-' _ -.40 ! : / : : : : 2: : . . ,\.J..

Frequency in Rad/Scc -60 Figure 3.9 - Closed Loop Pitch Rate Response From Pilot Stick

.........i..../i: :.._--i:--- !..:...i...i,i._-

Input -80 k.:,! .... .......

........ .

,_ i i i i!ii At this time, only the angle-of-attack and pitch rate -100 responses to pilot inputs have been evaluated with regards to lOx flight control performance. However, from F_z 1. (2.8), note that as gAr(S) increases, d__(s) will certainly increase, hence Frequency in Rad/Sec disturbance rejection performance in the airframe loops will also Figure 3.7 - Plot of Eq. (2.9) degrade.

In summary, the analysis revealed: Finally, closed loop "flight control" performance is 1) Disturbances to the engine loop from airframe evaluated in Fig. 3.9. This figure shows the magnitude of the commanded responses are large - due to large g_(s).

closed loop pitch rate frequency response from pilot stick input.

For the original value of gAs(S) the response of the coupled 2) Sensitivity to gA_(S) in terms of stability robusmess.

airframe/engine system closely resembles that of the nominal system which is considered to possess good flying qualities. As 3) The frequency at which instability occurs due to increased gA_(S) is increased, the response significantly deviates from the gA_(S) nominal near 0.2 rad/scc, as the damping in the phugoid mode approaches zero. Note that pcrforrnancc requirements, not 4) Closed loop airframe performancedegradation duc to closed loop stability, may be much more limiting.

increased gAE(S) 4. Second Case Study - Multivariable Airframe System The airframe/engine system considered in this case is the same as in the last section• However, RCS jets are no longer included and only thrust vectoring is used to control pitch 0 attitude. Flying qualifies requirements are better met by feeding m back forward speed, u (ft/sec), to thrust reverser port area, A-/g .o -20 (in2), as first discussed in Ref. 12. The control law design for "_ o this configuration is presented in Refs. 8, 9, 12 and 13. Again closing the flap loop and combining angle-of-attack and pitch "_^ -40 rate measurements to form one blended aircraft pitch response, gives the following three-by-three system: -60 U

:: i i iii g12 :!iii

-80 = g2t g22 (K_/K_) a+q g23 Sty

:: iiii::ii

gtl gx2 g13 I ATs N2 g31 g32 g33 wf (4.I) -100 104 10o 101 With the exception that pitch control no longer includes RCS Frequency in Rad/Sec jets, the following physical "equivalence" in notation can be drawn between this system and that of the last section.

Fieure 4.1 - Open Loop Normalized Input/Output Mappings [ g22(s)g23(s) 1¢_.)[ g_(s) gAE(S) ] Disturbance to Engine Loop g32(S) g33(S) gEA(S) gE(S) (4.2) Now, however, the airframe has two control inputs and two --"-__mm Pilot Input = d ^

o ......... i.... i .... i i ill i!

responses. Thus, the plant input/output descriptions are now expanded to

i i ii iiii i i!ili

GA(S)=I gllg2, g22g12 I GAE(S)=[ g131g23 -20 _ i i i i i i i! Sensitivity!'

GEA(S) =[ g31 g32 ] g_(s)=[g33] (4.3) ........ Function : Fig. 4.1 displays the magnitudes of the plant -40 .............

10-1 100 101 input/output mappings• Again, the control inputs and system responses are normalized to their maximum values. The system Frequency in Rad/Sec maximum perturbation forward speed and thrust reverser port area are taken as Figure 4•2 - Fan Speed Sensitivity Function and Engine Loop Disturbance From Airframe Commanded Responses umz x = 20 ft/sec A78m¢ x = 50 in 2 (4•4) both elements of GAE(S) at the same time. Physically, g_(s) is The other values were given in Table 3.2.

equivalent to gAE(S) (pitch response-to-fuel flow rate) of the Fig. 4.1 shows that since RCS jets are no longer used, previous case study, g13(s) models the effects of fuel flow rate g32(s) is quite small, as expected. Also, both gl3(s) and g23(s) on the forward speed. Consequently, this term is sensitive to are small, hence, GAE(S) is "small". However, g31(s) is quite the vehicle's thrust-to-weight rating.

Fig. 4.3 shows plots of both sides of the key inequality large and of the same order of magnitude as GA'(s) and gE'(s).

for the engine loop analysis, Eq. (2.5). This figure shows that Thus, GEA(S) is not "small" for this configuration either, g31(s) the stability margin with the original value of GAE(s) is is large due to the fact that changes in thrust reverser port area can influence the back pressure on the engine fan through the approximately 20 dB measured at 0.2 rad/sec.

by-pass duct• Thus, closing the loop on forward speed to thrust Fig. 4.3 also shows leAkEI for "larger" GAE(S).

reversing leads to large GEA(s) and perhaps significant Instability actually occurs for the increase in GAE(S) leading to airframe/engine interactions. the largest le^kEI shown in the figure. In this case, both g13(s) Again, from Eq. (2.3), dA(S) can be expected to be large and gz3(s) were increased by 46 riB. Although this gain margin since GEA(S) is "large." Fig. 4.2 presents the engine's fan speed may seem large, at the frequency in which the system goes sensitivity function along with the magnitude of dA(S) for this unstable, this is equal to an addidve (rather than multiplicative) case. This figure shows, however, that dA(S) is not as large as perturbation of only 3.6 (ft/sec)l(lbs/hr).

Fig. 4.4 displays the frequency responses of both the in the previous case due in part to different airframe feedback nominal and coupled system's engine loop transfers. As in the compensation, KA(s).

previous case study, this loop has infinite gain margin and 60" Attention is now directed towards a sensitivity analysis of phase margin occurring at a cross-over frequency of similar to that presented in the first case study. An investigation approximately 3 rad/sec. As the magnitude of GAE(S) is of Eq. (2.2) would show that g31(s) multiplies both gl3(s) and increased, the phase margin is reduced to zero and system g23(s), while investigation of Eq. (2.7) also indicates that g31(s) instability occurs. Note here that instability in this case occurs multiplies g13(s) in the (1,1) element of EE(s), and multiplies near 3 rad/sec.

g23(s) in the (2,1) element of EE(s). Hence, the system is potentially sensitive to deviations in g13(s) and/or g23(s). Thus, the following results present the sensitivity analysis increasing 50 Now consider the Bode plot in Fig. 4.6 showing the speed-to-thrust reverser loop with the pitch response-to-thrust !_r:==='_='---_In=_:cr_: 7' : : : : : ::: vectoring loop closed. Note in this loop, stability margins are decreasing as the "magnitude" of GAE(S) is increased, but in the e_ frequency range near 0.2 tad/see rather than near 3 rad/sec •_ 0 indicated in Fig. 4.4. This is not unusual for a multivariable ._= : ::_!i!: : _ ' :: :: ":"_:"_£-_ : : system, and underscores the need for singular value analysis, : : : : : . _ ': : ._. _"-:,%.

• : : .' .' _ . . _. ._.,..'_ along with consideration of individual loops. What Fig. 4.5 .2_ indicates is that the "smallest" EEK ^ for which stability is assured is of the order of 30 dB, and the instability for this a -50 worst-case matrix should occur at a frequency near 0.2 rad/sec.

Hence, although sensitivity to gt3(s) and g23(s) is indicated in KeAkE)JForOriginal_,c : : : : : : : Eqs. (2.5) and (2.9), the worst-case combination of changes in gt3(s) and g23(s) was probably not found in the above analysis.

i ! _ ii!_!i ! i!!!_!j

-100 I0-I 10o I0] Frequency inRad/Scc Figure 4.3-Plot of Eq.(2.5)

-so ' _ : _ i..!.i.i

C_

]oo : : :" :::_ _ ! i iiiii I

-100 10-t 10o 10t

i

5Oo !!i }.i..i..iil.ii ......... i ....

i

_5oi !!i!

; ...............

10-t 10o 101 -200 104 10o I0_ Frequency in Rad/Sce n. _2001 I Figure 4.6 - Airframe Forward Speed-To-Thrust Reverser Loop 104 10o pt 101 Transfer Frequency Responses of Nominal and Coupled Systems (With Pitch-To-Thrust Vectoring Loop Closed) Frequency in Rad/Sec Fizure 4.4 - Nominal and Coupled System's Engine Loop S. Analysis Framework With Control Cross-Feeds Transfer Frequency Responses The control laws K^(s) and KE(S) in the case studies just The dual stability test for the airframe loop is shown in presented are defined here as decentralized controllers in that Fig. 4.5. Note here that singular values are plotted since this is they involve no cross-feeds between airframe responses and a multivariable system analysis. The stability margin indicated engine control inputs, or between engine responses and airframe in this figure is approximately 30 dB measured at a frequency of 0.2 rad/sec. Thus, it can be seen that this test is less control inputs. The method of analysis presented in Ref. 8 considered only systems with decentralized control laws.

conservative in that a larger stability margin is guaranteed.

Centralized control laws may arise, for example, from application of multivariable synthesis approaches, and may well include control cross-feeds between the two systems. The 40 t' : : :::: i i i !!ii purpose of this last section is to extend the analysis framework to allow for these cross-feeds.

....... //i.?._. _ ._. i.. i..::..i.i, ii......... _O(I + GAKA) i.i Fig. 5.1 displays the system analogous to that in Fig.

20] z_/: "" i:i_:i54_ i/i ! i i iii "U 2.1, but with the control cross-feeds KAE(s) and K_A(s ) present.

_"" _creasingG_ :: --:. :-:. ::r..

i

-2o ..__: .{ ......... i--.:-_- _;¢--2..!: _-i

tZ -40 r_..: _ ._.!.. i _;.il. i. i:i ........ ._ [ o(F__K^) For Original G_ i ! i i i i i _60 _ -- : : : : ::: 104 100 10] Frequency in Rad/Scc Figure 4,_ - Plot of Eq. (2.9) Figure 5.1 - Airframe/Engine System With Control Cross-Feeds This system may also be represented as shown in Fig.

KEA =-(G_ "l GEA (5.9) 2.2 and 2.3. However, the complexity of the coupling expressionsincreasessignificantly, as shown below. Note that Hence, this cross-feed minimizes the disturbance from the the indication of functional dependence on s is not carried airframe to the engine loop. By duality arguments, DE(s)---O throughon theright hand sides ofsome of these expressions for when: simplicity of notation. It can be shown that, for thesystem in Fig. 5.1, the expressions forEA(s)and DA(S ) inFig 2.2 be.come KA£ = -(G,_) "t GAE (5. I0) EA(S)= AE + EAt(S) + EA2(S) + EA3(s) + EA4(S) (5.1) Note that this solution requires inversion of the airframe and engine plants, which is not advisable if right half plane DA(S) = DAI(S) + DA2(S) transmission zeros arc present. Also, the above solutions unfortunately do not lead to E^(s)=O and E£(s)=0.

EAt(S) = -GEA _bAKA GAI_ EA2(S) = GEA ¢PA KAE 6. Summary and Conclusions EA3(S) = -TA _bA GAE, EA4(s) = -T^ GA0^ KAE (5.2) Two case studies were presented in this paper that addressed the analysis of airframe/engine interactions. For both open loop airframe/engine configurations considered, the DAI(S) = GEA _bA KA, DAZ(S) = TA _'^ (5.3) airframe's influence on the engine loop was significant.

Commands to the flight control system resulted in significant where, undesirable fan speed disturbances. The engine's effect on the airframe loop, however, was "small" in both case studies, and _^(s)--(I + KA GA_ 1, 0A(S)=(I +G A KA_ 1 thus the interactions between the airframe and engine were one- directional Consequently, analysis revealed good stability robustness and closed loop flight control performance.

TA(S) = (G_ + EA1) Oh KEA KA (5.4) However, the analysis also indicated the system's . 1 potential sensitivity in engine-to-airframe interactions. The a,^(s) = {I +Kz, K, stability test used in the analysis of the airframe loop fEq. (2.9)) more accurately predicted the actual coupling "stability margin" for both cases considered. This underscores the need for AE(S)+EAI(S ) is identical tothe original E^(s) givenin analyzing both the airframe and engine systems to accurately Eq. (2.2). Thatis, EA(s)inEq. (5.1) reduces to this when the evaluate the significance of their interactions. For the second case study, which involved a multivariable airframe system, cross-feeds KAE(S)and KEA(S) arc zero.EA2(S ) arises from the sensitivity to engine-to-airframe coupling was also explored.

"KAE-GEA"pathintheblockdiagraminFig. 5.1.Thatis, EA(s) Again it was shown that instability could occur. However, a reducesto EA2(S) when KEA(S ) and GAE(S) are zero. EA3(S) more extensive sensitivity study is required with multivariable arises from the "GAE-KEA" path, or ER(s) reduces to EA3(S) systems, and worst-case combinations of plant variations is when KAE(S ) and GEA(S) are zero.Finally, EA4(S ) arises from sought.

the"KAE-KEA" path, or EA(s) reduces to EA4(S) when GEA(S ) Finally, extension of the analysis method to allow for and GAE(S) are zero. Dual results arise when considering the cross-feeds between the airframeand engine systems was effects on the airframe loop. In this case, theresults arethe presented.

same as those inEq. (5.1) - (5.4), butwith all subscripts (A and E) interchanged. Thus,thedual expressions are Acknowledgements EE(S) = A^ + EEl(S) + EFA(S) + E_(s) + EFA(S) (5.5) This work was sponsoredby the NASA Lewis Research DE(S) = DEI(S) + DE2(S) Center under Grant # NAG3-998. Dr. Sanjay Garg is the technical programrnanagcr.

EEl(s) = -GAE _bE KE GEA, EFA(S) = GAE 0E KEA References EE3(S) = -TE q)EGEA, EEa(S)= -TE GEOEKEA (5.6) [I] Garg, S., Mattern, D.L., and Bullard, R.E., "Integrated Flight/Propulsion Control System Design Based on a Centralized Approach," AIAA Paper No. 89-3520, AIAA DEI(S) = GAE _E KE, DF.7.(S) = TE 0'E (5.7) Guidance, Navigation and Control Conference, Boston, Ma., August, 1989.

where, Rock, S.M., Emami-Nacini, A., Ancx, R.P., [2] "Propulsion Control Specifications in Integrated G* K _-I 0E(S) =(I + KEG_)'I _(S) =(I + E El Hight/Propulsion Control Systems," AIAA Paper No.

88-3236, AIAA/ASME/SAE/ASEE 24th Joint Propulsion Conference, Boston, Mass., July, 1988.

TE(S) = (GA + EEl) OE KAE KE (5.8) [3] Smith, K., Stewart, C., "A Survey of Control Law

*E(s) = +K^E KE t

Options for Integrated Hight/Propulsion Control for Fighter STOL Approach," AIAA Paper No. 84-1900CP, AIAA Guidance, Navigation and Control Conference, Seattle, Washington, August, 1984.

Note that solving for the control cross-feed that will force DA(s)=O gives: [4] Shaw, P., ct al., "Development and Evaluation of an Integrated Flight and Propulsion ControlSystem," ALA.A Paper No. 85-1423, AIAA Joint Propulsion Conference, Monterey, CaJifom_ July,1985.

[5] Shaw, P.D., Rock, S.M., and Fisk, W.S., "Design Methods for Intcgrat.ed Control Systems," AFWAL -TR- 88-2061, Acro PropulsionLaboratory,Air Force Wright AeronauticalLaboratories, Dayton, Ohio, June, 1988.

[6] Smith, K.L., "Design Methods for Integrated Control Systems," AFWAL-TR-86-2103, Aero Propulsion Laboratory, Air Force Wright Aeronautical Laboratories, Dayton, Ohio, December, 1986.

[7] Berry, D., Schweikhard, W., "Potential Benefits of Propulsion and Fli_t ControlIntegration for Supersonic Cruise Vehicles,"based on SAE paper 740478, 1974.

[8] Schmidt, D., Schierman, J., Garg, S., "Analysis of Airframe/Engine Interactions - An Integrated Control Perspective," AIAA # 90-191g, presented at the 26th Joint Propulsion Co.'fference, Orlando, FI., July, 1990.

[9] Schmidt, D., Schierman, J., "A Framework for the AnalysisofAirframe/EngineInteractions and Integrated Flight/Propulsion Control," presented at theAmerican Control Conference, Boston, Mass., June, 1991.

[i0] Doyle, J.,Stein, G., "MultivariableFeedback Design: Concepts for a Classical/Modern Synthesis," IEEE Transactionson Automatic Controls,Vol. AC-26, No. I, pp. 4-16, Feb., 1981.

[11] Roscnbrock, H., "The Stability of Muhivariable Systems," IEEE Transactions on Automatic Controls, Vol. AC-17, pp. I05-I07, Feb., 1972.

[12] Schmidt, D., Schierman, J., "Extended Implicit Model Following As Applied To Integrated Flight and Propulsion Control." AIAA Paper No. 90-3444, AIAA Guidance, Navigation and Control Conference, Portland, Oregon, August, 1990.

[13] Schierman, J., Schmidt, D., "Robust Control Synthesis For Integrated Flight and Propulsion Control," IEEE Conference on Decision and Control, Honolulu, Hawaii, December, 1990.

F/A

Aircraft Analysis Of Airframe/Engine Interactions For A STOVL With Integrated Flight/Propulsion Control* John D. Schierman _t,T. Alan Lovell* and David K Schmidt** Aerospace Research Center College of Engineering and Applied Sciences Arizona State University Abstract robustness. The analysis also quantifies disturbances encountered in each loop due to the interactions between the This paper presents new results from a multivariable airframe and engine. Analyzing these interactions should analysis technique applied to an advanced STOVL help to further understand how they should be addressed in configuration with highly interactive airframe and the context of integrated control of the flight and propulsion subsystems and uncertainty in the interactions propulsion subsystems.

between the subsystems. This analysis method is used to The main focus in the IFPC problem is control assess the effects of the dynamic cross-coupling between the airframe and engine subsystems. The analysis synthesis and analysis of advanced concepts of highly maneuverable aircraft which utilize the propulsion framework addresses two-directional dynamic cross- subsystem for enhancing the lifting and maneuvering coupling, and also allows for cross-feeds between the capabilities of the airframe [1]-[8]. Fig. 1 illustrates some subsystem controllers. The issue of stability and of these new design concepts such as aft and ventral nozzle performance robustness is addressed, and the utility of vectoring, Reaction Control System (RCS) jets, and left singular value stability robustness criteria is presented.

and right ejectors. Vectoring of the engine's nozzles The configuration analyzed includes a thrust generates moments that enhance the attitude control of the vectoring/thrust reversing aft nozzle, powered lift through airframe. A ventral nozzle is located underneath the the use of a ventral nozzle and ejectors, and Reaction Control System jets. Investigation of the open-loop fuselage and redirects the engine's thrust for both pitch attitude control and lift augmentation. Thrust from RCS dynamics indicates that significant interactions between the jets is drawn from engine compressor bleed flow and is also airframe and engine are generated as a consequence of the used to enhance attitude control. Primary ejector flow is propulsive augmentation. A critical frequency range where due to the mixed flow of the engine (core and bypass flow) instability would first occur due to small variations in the and secondary flow is generated by ejector intake doors over coupling dynamics is also indicated by the analysis. A the top of the fuselage. If the ejectors act in unison, they stability sensitivity analysis reveals that the interactions provide propulsive lift at low speeds and hover. However, between the engine and the airframe's flight path response are critical with regard to stability and performance differential use of the left and right ejectors can enhance roll control of the aircraft.

robustness.

Pitch. Roll Introduction The main objective of this paper is to present new results of an analysis method that examines the effects of interactions between airframe and engine subsystems. This analysis technique was first introduced in [1], and further developed in [2] and [3]. The procedure is applied for analysis of a particular vehicle configuration that has been the subject of several studies involved in the Integrated Thtu_ Ventral With Thrust Vec_rmg Flight and Propulsion Control (IFPC) problem [4]-[6].

Nozzle & Tlan_t Rcvc_ing The central issues of the airframe/engine interaction - IFPC Vehicle Configuration analysis methodology presented herein are to reveal how the interactions between the airframe and engine are manifested, Traditional aircraft only utilize engine thrust to affect and to assess their significance. The "size" of the forward velocity, and there is little need to address dynamic interactions are quantified in a meaningful way to indicate interactions between the airframe and engine subsystems.

their effect on reductions in stability robustness, and Conversely, for these new aircraft design concepts, the degradations in closed loop performance. The analysis potential two-directional interactions between the airframe method presently developed has proven useful in identifying and engine subsystems are of major concern. Engine thrust critical frequencies where the system is lacking in stability will not only affect the forward velocity of the airframe, but will also influence the lift and attitude motion of the 1"Presented At The AIAA GN&C Conf., Hilton Head, 1992.

airframe as well. However, the inlet flow to the engine, _tDoctoral Candidate, Student Member, AIAA.

which affects the thrust produced by the engine is, in turn, * Graduate Fellow.

affected by the the dynamic motion of the airframe.

** Director, Prof. of Aero. Eng., Assoc. Fellow, AIAA. Although the airframe and engine subsystem dynamics are usually reasonably well modeled, the dynamic interactions Copyright © 1992 by D. Schmidt.

between these subsystems are frequently difficult to Published by American Institute of Aeronautics accurately predict and model early in the design cycle, and and Astronautics, Inc. with permission.

PAPEK ':l .-q623

presented in these references was to develop a centralized

are often a significant source ofuncertainty in themodel of

control law synthesis technique with a decentralized

thesystem's dynamics.Therefore, a key focusof the

implementation methodology. The centralized control

analysis presented in thispaper is system stabilityand

laws are obtained by various multivariable control law

performancerobustnessto uncertaintiesin the

synthesis methods. Then, decentralized control laws are airframe/engine interactions. The stability and performance developed that will "approximate" the centralized control in robustness of the system is most sensitive to certain some manner to yield approximately the same closed-loop critical interactions, and the analysis seeks to identify these perforn'lance, critical interactions.

The hierarchical decentralized control law architecture is defined here as System Description And Control Law Architecture The overall system's input-output characteristics are defined at one operating point by the matrix of transfer functions (3)

[uAl E KA °lrYAc YA 1

UE KEA KE L yEc yE One-directional control cross-feed is utilized in the = , or y(s)=G(s)u(s) (1)

[y'] E °"

hierarchical decentralized controller, brought about by the yE G_ G-E presence of K_ (s). The term "hierarchical" conveys that the airframe is viewed as the "higher level" subsystem, and where GA(s) represents the airframe dynamics, and G E (s) the "lower level" engine subsystem is a "thrust actuator" represents the engine dynamics. Two-directional dynamic generating forces and moments on the airframe. Fig. 2 interactions between the airframe and engine are modeled by displays the airframe/engine system framework viewed in the off-diagonal transfer function matrices, GAl.(S) and this manner. It can be seen that the airframe controller is G_A (s). GAE(S) will be referred to as the engine-to-airframe responsible for not only generating aerodynamic control coupling or interaction matrix, and GEA(S) will be referred surface inputs, UA(S), but also for generating engine thrust to as the airframe-to-engine coupling or interaction matrix.

commands, yT_(S), tO the engine subsystem. This invokes The responses of each subsystem are affected by the control the one-directional control cross-feed. However, the inputs of the other due to the presence of these interactions.

decentralized propulsion system controller is designed and YA(S) is the vector of airframe responses, and yE(s) is the built separately. YT(S) is the vector of engine thrust vector of engine responses. Likewise, UA(S) is the vector of responses, such as RCS, ejector, ventral, and aft thrusts.

airframe control inputs, and UE(S) the vector of operative These responses act as control inputs to the airframe. YE(S) engine control inputs.

is the vector of internal regulated engine responses, such as It is considered that the system is acted upon by either fan and compressor speeds, and pressures and temperatures centralized or decentralized controllers. Centralized at various stages of the engine. The objective of the controllers are synthesized to address the design objectives closed-loop propulsion system is to deliver the required of the overall system, and employ two-directional cross- thrust responses to the flight control loops for attitude and feeds between the interacting subsystems to aid in this lift augmentation.

effort. Decentralized controllers are designed, built and tested separately for each subsystem. Therefore, utilization of control cross-feeds is limited.

- • UA YA The centralized control law architecture is defined here

as C.ntrollerl

I I (Interactions)

iuA]i KA yA]

u_ K_ I_ [y_ yE (2) y _ Engine UE . _ ___ - Controller_--_E-4_ Engine YE or u(s) = K(s){yc(s) - y(s)}) KA(S) and KE(s) are the feedback control compensation - Hierarchical Decentralized Control Law matrices associated with the airframe and the engine control Architecture subsystems, respectively. Note the presence of the two- directional control cross-feeds indicated by KAE(S) and Finally, another class of decentralized controllers that Kr_(s ). yA¢(S) is the vector of desired or commanded employ no cross-feeds between the subsystems can also be addressed by the analysis technique. In this case, both airframe responses, perhaps from pilot inputs, and ym(s) is KAE(s ) and KEA(s ) are zero, and the matrix K(s) is block the vector of commanded (or limited) engine responses, diagonal.

from either pilot inputs or commands from an outer-loop In summary, the analysis methodology may address system.

Hierarchical decentralized control law architectures systems with two-directional dynamic interactions between were all proposed in [5]-[8]. The objective of the work the airframe and engine, and which employ either y^(s) = [Qv, q, 0, y, V v. V. V] T centralized or decentralized control laws.

(4) YE(S) = N2 Description Of The Vehicle Dynamics And Control Law In [5] it is noted that the blended responses V v and Qv are The vehicle configuration to be considered is representative of an E7-D delta wing supersonic aircraft, utilized by the controller to provide good handling qualities powered by a high bypass turbofan engine, with STOVL in transition flighL The plant and controller wansfer function matrices were capabilities. The linear dynamic model and control law normalized by estimates of the maximum allowable were provided by the NASA Lewis Research Center, and further details of the vehicle configuration are presented in perturbations of the responses and controls from their [5] and [6]. The control law to be investigated in this reference values. The maximum allowable perturbations in these responses and controls were provided by NASA analysis is documented in [5], which provides a detailed Lewis, and are also presented in Table 2. The units of all account of the design methodology and the system inputs and outputs are normalized so that the magnitudes of requirements. The focus of this study is on the the transfer functions could be meaningfully compared.

longitudinal dynamics of the vehicle. The reference point Unless otherwise stated, all results are presented in these about which the nonlinear model is linearized is the steady- normalized units.

state wings-level decelerating transition while approaching the hover landing flight phase. The forward flight speed is Table 2(a) - Airframe/Engine System Responses And 80 knots. At this slow speed the forces and moments Estimates Of Their Maximum Values controlling the aircraft are transitioning from those Estimate Of generated by the aerodynamic control surfaces to those System Responses Maximum Value generated by the propulsion system. Table 1 presents the open-loop eigenvalues of the engine dynamics and Qv = q + 0.30 6.3 deg/sec longitudinal airframe dynamics. Note that the airframe's q - pitch rate (deg/sec) 6.3 deg/sec short period mode is unstable for this configuration and flight condition.

0 - pitch attitude (deg) 21 deg _,- long. flight path Table 1 - Airframe/Engine Modes 4.0 deg anlsle(deg) Modes Eigenvahes /ra,:t/secl Vv =_/+0.1V 7.6 fffsec2 V - total ac_k:rafion -200 Pressure Mode 7.6 ft/sec 2 (fl/sec a) -38 Temperature Modes V - ta-ueairspeed (ft/sec) 76 ft/sec -29 -7.1 Rotor Speed Modes N2 - fan speed (rpm's) 120 rpm's -4.1 Table 203) - Airframe/Engine Control Inputs And 1.3 Unstable Short Period Estimates Of Their Maximum Values -2.1 Stable Short Period System Control Estimate Of Inputs Maximum Value -0.1+ 0.3i Phugoid Mode -0.1 - 0.3i _E - devon deflection 5.0 deg (deg) Aerodynamic pitch control is provided by collective Aq - pitch RCS area (m21 0.7 in 2 elevon deflection. Pitching moments are also provided by 4 - aft nozzle vectoring aft and ventral nozzle vectoring, and Reaction Control angle (deg) 10 deg System (RCS) jets. The vehicle is also equipped with left rI - ejector butterfly valve 8.0 deg and right ejectors which act in unison, and along with angle (degl ventral nozzle thrust, provide propulsive lift at low speeds -/79 - ventral nozzle and hover. An ejector butterfly valve angle controls the vectoring angle (deg) 10 deg amount of engine flow to the ejectors, thus the amount of ejector thrust. A7 s - ventral nozzle area 45 in 2 (in 2 ) The state space descriptions of the linear dynamic As - aft nozzle throat model and control law are given in Appendix A. The 20 in 2 responses and control inputs are defined in Table 2. The area _i n:_ first seven responses are airframe responses, while the fan w t - fuel flow rate 1000 lbm/hr speed, N z, is a critical engine response. Therefore, the (lbm/hr) airframe and engine response vectors are (see Eq. (1)) The engine's fan speed responses are shown in Fig. 3, and the magnitude of the airframe's pitch attitude, flight path angle, and forward velocity frequency responses are shown in Figs. 4 through 6. (The airframe responses Qv, rate may significantly affect the airframe's pitch attitude and flight path angle responses. In turn, due to the two- q, Vv, and V, are not shown but are directly related to the directional coupling, Fig. 3 shows that the engine's fan pitch attitude and forward velocity responses (Table 2(a)).)

speed may be significantly affected by the vehicle's It is clear that the airframe/engine system is quite utilization of the propulsion system to enhance attitude multivariable in nature in that each response is control and augment lift.

significantly influenced by several controls. With the response vector defined in Eq. (4), it is desirable to select o 0 control input vectors such that the plant transfer function matrix in Eq. (1) be approximately diagonally dominant.

ee However, due to the significant multivariable nature of the system, this could not be fully achieved. Fig. 3 shows that the engine's fan speed response from fuel flow rate is

approximately 2 dB larger in magnitude than its response i

from ventral nozzle area, A78, below 7 rad/sec. However, both w r and A78 may be considered primary fan speed -80 t \'_'-.

controls. Figs. 4 and 6 show that the airframe's pitch o o attitude and forward velocity responses from A78 are z79 _0 generally larger in magnitude than their responses from fuel flow rate. Therefore, for this initial study, the fuel flow rate will be considered the single engine control, and the -4o -Tf airframe control vector will be ordered as listed in Table 2(b). Hence, -60 -2o _,_ 0 UA(S) = [_Y_,Aq, Z s , r I, Z79, ATg. As] T -80 (5) 10-2 10-1 100 10l 10_ u E(s) = wf Frequency in Rad/Sec - Pitch Attitude Frequency Response Magnitudes t_ "20/0 lit / // // / _ N2w---f' ' ' ' '' ''_A-TsN2 A--_N2' ' A"-q' ' '"--"N2 --N2' ' ' ' '"" ' ' ' ' ' '" t_

il

-80 ....

10-2 10.1 100 101 102 Frequency in Rad/Sec t_ - Engine Fan Speed Frequency Response -20 Magnitudes -40 wf _7_tt With this selection, referring to Eq. (1), the airframe

§

As 7 f transfer matrix, G^(s), is 7x7, and the engine transfer -60 function, GE(s ), is a scalar. Thus, the engine-to-airframe coupling transfer matrix, G_(s), is 7xl, and the airframe- -80 10-2 10-1 I00 101 102 to-engine coupling transfer matrix, GE^(s), is lx7. (Note Prequency in Rad/Sec that the results of the analysis to follow are dependent on the selection of airframe and engine controls, and different - Flight Path Angle Frequency Response selections have not been fully explored for this vehicle.)

Magnitudes The partitioning of the control law matrix follows from the response vector (Eq. (4)) and the selection of airframe and Stability Robustness Analysis engine controls (Eq. (5)). K^(s) is therefore 7x7, and KE(s ) It is assumed here that the airframe and engine plants is a scalar. The control cross-feed matrices, K^E(s ) and are reasonably well modeled, and hence any uncertainties in K_ (s) are 7xl and lx7, respectively. G^(s) and GE(s) at the design point are negligible. Recall, however, that the dynamic interactions between the airframe With the choice of fuel flow rate as the engine control, and engine are difficult to accurately model, and may the engine-to-airframe and airframe-to-engine coupling contribute a considerable source of uncertainty in the model transfer functions in G_ (s) and GE^ (s) are comparatively of the system's dynamics. Because the plant uncertainty is large in magnitude. Figs. 4 and 5 show that the fuel flow structured in this manner, the structured singular value and the matrix P relates the off-diagonal uncertainty matrix, stability robustness criterion [9,10] may be utilized. A(s), to the block-diagonal uncertainty matrix, AD(S). That is, A(S) = AD(S) P. P =

i

m o (1o) -20 .__/8 2 yc(s) -40 -60 U' yB -80 _(s)

° I

-20 _ v iEigutg_7_ - System Description With Block-Diagonal -40 J ATB Uncertainty From [9] and [10], the system of Fig. 7 remains stable if and only if 10-2 lO-I 10o lOI 102 Frequency in Rad/Sec 1 IIADII. < IIQ22110 " (11) - Forward Velocity Frequency Response Magnitudes where, Additive uncertainty in the coupling dynamics is [[AD[[. ----sup [O'max(AD0(I)))] IIQ2211. = sup [_(Q22(j(o))] defined here as 11) (12) GAE = G'AE + AAE and GEA = G*EA + AEA (6) Here g(Qz2(j(o)) is the structured singular value of Q22(j(t)), where G'AE and G'EA represent nominal models of the and Omax(AD(j(t))) is the maximum singular value of interactions. Therefore, with these uncertainties, the "true" AD(j(O).

plant description is Fig. 8 presents the inverse of I.t(Q22(J(o) for the vehicle and control laws described in the previous section. This G(s) = G'(s) + A(s), where figure indicates that C/) 1 --19dB (13) G'(s)=[ G_GA G_]GE ' and A(s)=[ 0AE^ AA_]0 IIQ22111_ The feedback loop for the overall interacting system Hence, closed-loop stability is assured if and only if IIADU..

may now be represented as shown in Fig. 7, with the is less than -19 dB. Due to the structure of AD(j(o), this uncertainties in the coupling dynamics expressed in the also implies that stability is assured if both IIA^Ellooand following block-diagonal form: IIA_AII,,, are less than -19 dB [10]. But without a model or estimate of the uncertainty matrix, Ao(jm), Om,(Ao(jC0)) cannot be calculated, and hence particularly critical (8) frequency ranges cannot be identified more precisely.

Another stability robustness criterion was developed Q(s) in Fig. 7 represents the nominal closed-loop system, and presented in [1]-[3] and will also be utilized here. As and it can be shown that presented in [1]-[3], the airframe/engine system with the control law architecture of Eq. (2) may be described as shown in Fig. 9. Note that with reference to Eq. (2) and this figure,

[y]:[ol o2]Iyo 1

u' Q21 Q22 y' where, (9) KAE(s ) = crvo/(s)KE(s ), KF_.A(S ) = 9C_,A(S)KA(s ) (14) Qn = G'K(I+G'K) "1, Ql2 = (I+G'K) "1 Manipulating the block diagram of Fig. 9 into that shown Q21 = P (I+KG') qK, Q22 = -P ( I+KG')'tK in Fig. 10 gives rise to what [1]-[3] define as multiplicative Therefore a necessary condition for stability is that the and disturbance interaction matrices, MA(S) and DA(S).

det[I+GE(I+M^)KE] is nonzero for all frequency, to. This These interaction terms capture the effects of the airframe's is assured if the engine control law K E (s) stabilizes the influence on the engine's control loops.

(non-interacting) engine loop (in which case the delII+GEKE];_O ), and if [1],[12] • . -I Om, xOVIA(jto)) < OmmCI+(KE(jto)GE(Jto)) ) 2o __k__ for all to > 0 (17) Therefore, if Omax(MA(Jto)) is equal to or greater than -2£ Omin0+(KE(jto)GE(Jto))'I), the detII+GE(I+MA)K E] can no longer be assured to be nonzero. If this determinant is in

-!

fact zero, then the closed-loop system is unstable.

10-2 10.1 10o 101 102 For the airframe/engine system in question, the fuel Frequem'y in Rad/Sec flow rate is the single engine control, and the engine dynamic model, GE(S), is simply the fan speed-to-fuel flow - Structured Singular Value Stability Criterion rate transfer function, Nz(s)/wf(s). KE(S) is then, fuel flow rate-to-measured fan speed, or, wt(s)/N2(s). Also, the non- yA_S) interacting engine system (I+GEK E) is stable here. Fig. 11 shows the plot of the stability robustness criterion of Eq.

(17) for this system.

Although Fig. 11 shows that the criterion of Eq. (17) is satisfied for all frequencies, it can be seen that the y_s) magnitude of MA(Jto) is only approximately 2 dB below the magnitude of I+(KEGE) "1 in a critical frequency range between 0.4 and 1.0 rad/sec. Note further from Eq. (15) that MA(S) is a strong function of the coupling matrices Fi_Lglgg.__ - Fully Interacting Airframe/Engine System GAE(S) and GEA(S), which are considered uncertain here.

Hence, a significant amount of uncertainty arises in the multiplicative interaction matrix, and Fig. 11 indicates that frequencies between 0.4 and 1.0 rad/sec appear to be critical.

yA_(S) , , ,, ..... , ........ i |.4.(KL_G _). | ] _ ' .... ,,,, , , ,,,,, cfiacal / Flc, quency ] MAI _ /" __ Fimare 10 - Engine Loop With Multiplicative And 1o Disturbance Interaction Effects From The Airframe

g

Specifically, it can be shown that MA(S ) = GE-I {G_ _ - (Gin +G E _)K^ -IO 1o2 10-2 10-1 10o 10| [I+(GA+G _ _EA)KA] -I (G_ +GA _,_)1 Frequency in Rad/Sec (153 Figure 11 - Plot Of The Stability Robustness DA(S ) = (GEA +G E ___)K x [I+(GA+GAE _ )KA]-t Criterion Of Eq. (17) (Note that the engine affects the airframe in a dual manner, The sensitivity of the multiplicative interaction matrix and the dual expressions for these interaction terms can be found by interchanging all subscripts A and E in the above MA(S) tO uncertainties in the coupling dynamics shall now expressions.) be addressed. Sought were those coupling transfer Now the determinant of the return difference matrix for functions within GAE(S) and/or G_(s) that if varied would the system may be expanded as produce the largest variations in the magnitude of M^(jto).

Each coupling transfer function was varied (one at a time) det[I+GK] = det[-I+(GA+GAE _.A)KA] det[I+GE (I +MA)KE] by multiplying the nominal magnitude of the coupling (16) transfer function by the same magnitude variation, 8 m.

Then, at each frequency, the phase of the coupling transfer Given that K(s) stabilizes the system, the function was allowed to vary from nominal by an amount det[I+G(jto)K(jto)] is nonzero for all frequency, to.

5,, which ranged from-60degrees to +60degrees. The

over 180 degrees of phase variation before instability would occur.

"worst case" phase variation wasdefined asthe5, which

caused the largest difference (indB)in IMA(jto)I.

Table 3 - Variations In Coupling Transfer Functions

All 14coupling transfer functions (recall that GAE(S ) is

Required To Cause Instability

7xl andGEA(S) is IX7)werevaried, andtherespective

Coupling Magnitude Phase Frequency At IMA(jto)I foreach case is shown inFig.12.In thisfigure, Transfer Variation Variation lnstabihty

thenominal magnitude of MA(jto) is plotted in thesolid

Function (dB) (degrees) (rad/sec)

line.Each dashed lineisaplotofIMA(jto)I fora magnitude

q/wf 16 -115 1.3

variation 5m=3(--I0 dB) andthe "worstcase"phase

0 / w t 18 -85 0.8

variation at eachfrequency in oneparticular coupling

transfer function.It canbeseen fromthisfigure thatthe

Y/wt 10 -37 0.52

magnitude of themultiplicative interaction matrix is most

sensitive to magnitude andphase perturbations in four

V/wt 20 -120 0.38

couplingtransfer functions. Thesecoupling transfer

functions arefanspeed-to-ventral nozzle area, N2/ATs, fan N2/A78 20 -125 10

speed-to-aft nozzlearea,N2/A 8, fan speed-to-ejector

N2/All 19 -138 1.3

butterfly valveangle, N2/r I, andflightpathangle-to-fuel

b/2/r I 20 -132 0.3

flowrate, y/wf. Perturbations in the other ten coupling

transfer functions caused negligible variations in IMAfjto)I.

The flight path angle-to-fuel flow rate transfer It can be seen from Fig. 12 that, although perturbations in function, y/w t, is the most critical interaction, with both N2/A7s, N2/A 8 and N2_ caused variations in IMA(jto)I at all the smallest magnitude variation (10 dB) and the smallest frequencies, the perturbation in flight path angle-to-fuel phase variation (-37 degrees) required to cause instability.

flow rate caused the largest variation in IM^(jto)l, and For this perturbation, instability occurs at 0.52 rad/sec.

further, this occurred in the critical frequency range indicated Note that IMA(jto)I is greater than II+(K_:GE)-II for this in Fig. 11.

perturbation at 0.52 rad/sec, indicating the conservatism of 20 , . , . .... , . , ,,it, .... , , , ....... . ,,,.,, the stability robustness criterion of Eq. (17). However, for Variations _ Variations ,,- this perturbation, Omax(AV(jto))=10 dB for all to, indicating 10. inN:/_ i /inN2/A?8 ,//'¢_' that the structured singular value criterion shown in Fig. 8 is conservative as well.

Instability was determined by plotting the Nyquist plot o .......

of the determinant of I+G(jto)K(jto) with perturbations in the coupling transfer functions. Fig. 13 presents the plot of the det(I+G(jto)K(jto)) for both the nominal system, and -10 IM^I the system with the variations in the flight path angle-to- Variations _, Variations in N2/AB fuel flow rate transfer function that caused instability.

' in Y/wf -20 Although Fig. 13 only shows the Nyquist plot near the 10- 10-1 10o 101 102 origin, it can be shown that there are two ensuing clock- Frequency in Rad/Sec wise encirclements of the origin for the perturbed system.

This implies that a pair of closed-loop eigenvalues lies on Figure 12 - Sensitivity Of Multiplicative Interaction the jto-axis at + 0.52j.

Matrix To Perturbations In Coupling Transfer Functions The magnitudes and phases of all 14 coupling transfer functions were also varied until instability occurred. Listed in Table 3 are those transfer functions for which the -_ : /J smallest magnitude and phase variations, 5 m and 5,, would lead to instability. For each transfer function, the table lists the magnitude and phase variations required to cause instability, and the frequency at which the instability occurs. Note that the combination of magnitude and phase -5 ..... det(l+GK)=0 / .. Pe/_ _W, by: .......

variation required to cause instability is not unique. That at 0.52 rad/se._ S m= t0 dB is, more magnitude variation and less phase variation (or : S, = -37 deg vice-versa) can also cause instability, and Table 3 simply -10 lists example combinations. The first four transfer -10 -5 0 5 10 functions listed are engine-to-airframe (GAE(S)) interactions, Real Part and the last three listed are airframe-to-engine (G_^(s)) Figure 13 - Plot Of Airframe/Engine System's interactions. All coupling transfer functions not listed in Det[I+G(jto)K(jto)] With Variations In Flight Path Angle- this table required over 20 dB of magnitude variation and/or To-Fuel Flow Rate Transfer Function

The significance of the uncertainty leading to the

instability is depicted in Fig. 14. In the figure, the YE(s) = [I + G E 0+M^)K E]-I GE (I+M^)KE Yb: (s) nominal flight path angle-to-fuel flow rate transfer function + [I + GI:fI+M^)KEI -l D^ y^c(s) (19) is shown along with magnitude and phase variations of 10 dB and -37 degrees. It can be seen that the magnitude of It be can seen that disturbances to the engine responses uncertainty that causes instability is actually quite "'small" from airframe commands, y^¢(s) (for example, pilot stick in the physical units of deg/(lbm/hr). At the frequency of inputs), arise unless the disturbance interaction matrix, instability (0.52 rad/sec) the nominal magnitude is -56 dB D^(s), is zero. For this case study, D^(s) is a lx7 matrix (0.0016 deg/(lbm/hr)) and the perturbed magnitude is -46 dB and y^c(s) in Fig. 10 is (0.0050 deg/(lbm/hr)). This is a difference of only 0.0034 deg/(Ibm/hr).

Y^c(s) = [Qvc, q,, 0e, "I',, Vw, V¢, Vc] r (20) I -20; iiiiiii ....!! i ....:; I However, in [5] and [6] the actual airframe command vector "_ -4o : i ::::::: : • ::::::: 10dB ::: ! : ::::::1 consisted only of the commanded blended responses, Qvc and Vvc, and the commanded flight path angle, _'_. The -_ -60 other responses are regulated, or their commands are zero.

-so .i.i.iiii Define here the lx3 matrix DA(jo) as the "subset" of ._ool i i iiiiiii °__.".:.'.'.'_."_.' :i iiiil 0 D^(jo_) consisting of those elements corresponding to Qw, Vv¢ and _,,. Fig. 16 shows the magnitudes of the elements of D^(j_). These terms are seen to be approximately -10 dB in magnitude below 1.0 rad/sec.

.g -100 40 .... ,-, ............

"_, : ._::_.'X_ : : : Due to Variations In -/- 10.2 10d 10o 10t 102 Frequency in Rad/Sec _ 0 _ lO ..... _° _miml i t] _/-_i i i i' Figure 14 - Frequency Response Of Flight Path Angle-To- Fuel Flow Rate Transfer Function With Variations That Cause Instability 180 " /" . ..... :i Finally, Fig. 15 presents the Bode plot of the engine fan speed loop (with all other loops closed). It can be seen that this loop nominally has infinite gain margin and 80 o........ :, : ....... =:.

degrees of phase margin at a cross-over frequency of 5.5 rad/sec. However, the instability that occurs at 0.52 rad/sec due to the critical variations in the flight path angle-to-fuel -180 .........

flow rate coupling transfer function is also shown. It is lO-I 10o 10t clear that the classical phase margin defined at the gain Frequencyin Rad/Sec cross-over frequency does not indicate this critical Figure 1_ - Engine Fan Speed Loop Bode Plot With All frequency. The structured singular value of the closed-loop Other Loops Closed matrix Q22(jto) (Eq. (9)), shown in Fig. 8, also failed to indicate this critical frequency. However, as indicated by Fig. 11, the robustness criterion of Eq. (17) correctly indicated the frequency range in which instability would occur for the smallest variations in magnitude and phase of

ol 1

Multiplies Qv, one transfer function in G^E(s) or GEA(S ). Furthermore, this criterion is most sensitive to variations in ¥/wf within the critical frequency range, which is consistent with the results in Fig. 12.

__ __ultiplies % -20 Performance Analysis -40 Referring back to Fig. 10, the decoupled or non- interacting engine system's closed-loop responses are 10-2 10-1 I0O 10t 102 Frequency in Rad/Se.

(18) yE(s) = [I + G E KE] "l GE K E y_(s) - Frequency Response Magnitudes Of DA(jo_) However, with airframe/engine interactions, the engine system's responses are Theclosed-loop engine response (Eq. (19)) is the sum airframe's effects on the engine, the engine will also affect of two quantities: the complementary sensitivity function the airframe. Fig. 19 shows the magnitude of the operating on the engine commands, and the product of the airframe's closed-loop frequency response of flight path sensitivity function and DA(J¢O) operating on the airframe angle-to-commanded flight path angle, T/7,. This response commands. Fig. 17 presents the magnitude of the fan is presented because it was found to be the most sensitive speed response from commanded fan speed, N2c (the to variations in the flight path angle-to-fuel flow rate coupling transfer function, y/w r It can be seen that for the complementary sensitivity function), and the maximum singular value of the fan speed response from the nonzero nominal response, good command following performance is airframe commands (the product of the sensitivity function obtained out to a bandwidth of approximately 0.5 rad/sec.

However, responses are also shown that correspond to and DA(j¢o)). It can be seen from Fig. 17 that at all perturbations in the flight path angle-to-fuel flow rate frequencies the magnitude of the fan speed response from coupling transfer function. With a perturbation of 6 dB and the commanded fan speed is at least approximately 17 dB -22 degree (60% of 10 clB and 60% of -37 degrees), a peak greater than the maximum singular value of the fan speed magnitude of over 6 dB is seen in the flight path angle's response from the airframe commands. In other words, the closed loop response occurring at approximately 0.4 maximum singular value of the fan speed response from the rad/sec. This will clearly lead to unacceptable handling.

airframe commands is at most only approximately 14% of Hence, although the perturbations in this interaction are not the magnitude of the fan speed response from commanded "large" enough to cause instability, their effect on the fan speed. Unless more disturbance rejection performance closed-loop response is quite significant.

is required, it would seem that the fan speed loop should be able to adequately reject disturbances from airframe commands.

40 I [I+GE0+M_KEi.IGE(I+M_K El / (Complementary 20 _ Sensitivity) o -2 0 1 2 3 4 Time in Seconds lO-2 10-1 10o 101 1o2 Figur_ 18 - Fan Speed Response From A Maximum Frequency in Rad/Sec Allowable 4 Degree Step Flight Path Angle Command Figure 17 - Fan Speed Complementary Sensitivity ...... 60% of _m and i t _at'eau se'" Function And Disturbance Response From Airframe Interacfons 5 ,"" J instability ..,:_ .._. 4o%of_=_s, Fig. 18 presents the fan speed time response from a step flight path angle command, yc. The flight path angle was commanded to 4 degrees, which is its maximum allowable value, as given in Table 2(a). This constitutes a -10 Nominal \_, "worst case" fan speed disturbance response from -15 commanded flight path angle. It can be seen from this figure that the fan speed response has a peak magnitude of -20 ......

approximately 5% of the maximum allowable fan speed 10-1 10o 101 response of 120 rpms, also given in Table 2(a). Although Frequency in Rad/Sec not presented, the peak magnitudes of the fan speed response from the other airframe commands were even less Figure 19 - Closed Loop Flight Path Angle Response than that shown in Fig. 18. Therefore, the fan speed loop From Flight Path Angle Command seems to adequately reject disturbances from airframe commands, consistent with the results in Fig. 17. Conclusions Finally, recall that the system stability was sensitive For a particular airframe/engine system and integrated to magnitude and phase variations in the flight path angle- control law, a critical frequency range was identified along to-fuel flow rate coupling transfer function, y/wf, and that a with potentially poor stability robustness due to the interactions between the airframe and engine. It was found magnitude variation, 5=, of 10 dB, and a phase variation, that, within this critical frequency range, stability and _,, of -37 degrees in this coupling transfer function caused performance were sensitive to variations in the coupling instability. Therefore, performance robustness to between the airframe's flight path angle and the engine's uncertainties in the interactions should also be addressed.

fuel flow rate. A stability sensitivity study indicated that Although the focus of the analysis so far has been the the interactions between flight path angle and fuel flow rate Colu, ms 5 To 8

were,in fact,potentially the mostcritical. Instability

-1.418e-5 8.810e-3 4.259e-2 5.857e-6

occurred in thecriticalfrequency range indicated by a

-8.863e-2 3.009e-2 7.319e-2 -6.828e-5

stabilityrobustness criterion,whilethegaincross-over

-2.204e-2 -4.393e-3 -1.890e-3 3.523e-6 0.000e+0 O.O00e*O 0.000e¢0 0.000e+0

frequency for a classical single-loop analysis did not

1.733e-2 2.239e÷I 2.383e÷I 3.131e-2

correspond tothis critical frequency. Although the engine's

4.208e-2 -6.814e-I -7.588e-1 4.036e-2 -1.570e-3 -1.296e+0 -1.474e÷0 2.943e-2

fan speed loop seemed toadequately reject disturbances from

-I.059e-2 1.193e+1 1.256e+1 4.780e÷0

airframe commands, it wasshown thatuncertainties inthe

-7.510e-3 -1.477e÷I -1.573e+I 1.733e-2

coupling between flight path angle and fuel flow rate may

Columns I To 4 lead to unacceptable flight path angle command following 0.000e*0 0.000e+0 5.730e+I 1.719e÷I performance.

0.000e÷0 0.000e+0 5.730e¢I 0.000e¢0 0.000e÷0 0.000e÷0 0.000e+0 5.730e+I 7.348e-2 -4.176e-I 0.000e÷0 5.730e÷I Appendix A. State Space Description Of 1.661e-2 1.821e-2 7.367e-5 -3.213e+I System Dynamics And Control Law -8.188e-2 8.843e-4 7.367e-5 -3.213e+I 9.849e-I 1.733e-I 0.000e+0 1.526e-3 0.000e+0 0.000e+0 0.000e+0 0.000e+0 Control Law: Airframe/Engine System: Columns 5 To 9 x_ = A_xc + Bc(yc-y) x = Ax + Bu O.O00e+O O.O00e÷O O.O00e+O O.O00e÷O 0.000e+0 u = CA +D_(y_-y) O.O00e+O O.O00e+O O.O00e+O O.O00e+O 0.000e*0 y = Cx + Du 0.000e+0 0.000e÷0 0.000e+0 0.000e+0 0.000e÷0 -8.637e-8 -2.494e-8 -I.160e-9 -5.135e-8 1.851e-6 -1.465e-4 -3.587e-5 -1.849e-6 -7.480e-5 1. 693e-I State Vector:.

-1.460e-4 -3.573e-5 -1.842e-6 -7.449e-5 I. 693e-I x = [u, w, q, 0, N2, N25, T4h T3, P6] T -5.080e-6 ~1.477e-6 -6.915e-8 -3.047e-6 1.094e-4 l.O00e+O O.O00e+O O.O00e+O O.O00e+O O.O00e+O Definition Of States: Columns 1 To 4 O.O00e÷O O.O00e+O O.O00e÷O O.O00e+O u = axial velocity (ft/sec) O.O00e+O O.O00e+O O.O00e+O O.O00e+O w = vertical velocity (ft/sec) O.O00e+O O.O00e÷O O.O00e+O O.O00e+O q = pitch rate (rad/sec) -2.861e-6 O.O00e+O 1.885e-6 4.061e-6 -6.820e-2 O.O00e+O -2.080e-I -9.775e-2 0 = pitch attitude (rad) -6.820e-2 O.O00e+O -2.080e-I -9.776e-2 N 2 = fan SlXXXt (rpms) O.O00e+O O.O00e÷O 8.042e-5 9.331e-5 O.O00e+O O.O00e÷O O.OOOe+O O.O00e+O N2s = compressor speed (rpms) T41 = compressor turbine inlet temp. (degrees, R) Colun_ns 5 To 8 O.O00e+O O.O00e+O O.O00e+O O.O00e+O T 3 = combustor inlet temp. (degrees, R) O.O00e+O O.O00e+O O.O00e÷O O.O00e+O 0.000e+0 0.000e+0 0.000e+0 0.000e+0 P6 = tailpipe entrance total pressure (psi) 0.000e+0 -3.569e-7 8.772e-7 0.000e+0 -1.538e-2 1.389e-2 5.462e-2 -6.069e-6 Response And Control Vectors: (see Table 2) -1.538e-2 1.389e-2 5.462e-2 -6.064e-6 3.446e-8 -1.763e-6 5.181e-5 ~4.058e-8 O.O00e+O O.O00e÷O O.O00e+O O.O00e+O Y = [Qv, q, 0, y, V v, "_/, V, N2 ]T U = [%, Aq, L8, _, _79' ATS, As, wf ]T Ac Matrix; 14x14 Block Diagonal - Elements Not Listed = 0 Diagonal: {I,I) Through {5,5): -3.186e-3 -3.543e-3 -4.476e-3 -4.481e-3 -7.440e-3 Columns 1 To 4 Diagonal: (6,6) Through (I0,I0): -5.905e-2 7.184e-2 -2.283e+I -3.193e+I -8.954e-3 -1.817e-I -8.370e-I -1.348e+0 -1.348e+0 -1.369e-I -4.331e-I 1.297e+2 -3.922e+0 Diagonal: (II,II) Through (14,14) -1.240e-2 1.890e-2 -5.480e-I -1.402e-7 -2.360e÷0 -3.023e*0 -7.749e÷0 -7.749e÷0 0.000e+0 0.O00e+0 1.000e+0 0.000e+0 Other Nonzero Elements: -7.286e-I -1.282e-I 0.000e+0 -1.258e-3 Ac(9,10)=7.494e-2, Ac {10,9)=-7.494e-2 3.557e-1 6.258e-2 0.000e+0 6.142e-4 Ac(13,14)=5.629e÷0, Ac(14,13}=-5.629e÷0 1.523e-I 2.679e-2 0.000e+0 2.630e-4 4.853e+0 8.537e-I 0.000e+0 8.379e-3 Bc Matrix: Coltmans I To 4 -4.623e-1 -8.133e-2 0.000e+0 -7.983e-4 -3.522e-2 -8.032e-6 -3.826e-6 -I.142e÷I 1.673e-I 1.700e-4 1.399e-4 3.077e-I Col._ms 5 To 9 -8.150e-3 -1.847e-5 -_.588e-5 -4.455e-3 1.120e-3 3.348e-4 1.564e-5 6.916e-4 1.448e-I -1.919e-3 -1.497e-5 -1.074e-5 -1.810e-3 -7.208e-3 -2.109e-3 -9.950e-5 -4.360e-3 1.546e-I 7.453e-2 -4.129e-6 4.612e-6 9.322e-4 2.572e-4 7.677e-5 3.693e-6 1.596e-4 -3.305e-3 -1.458e+I -3.065e-5 -1.950e-5 9.773e-5 0.000e+0 0.000e+0 0.000e+0 0.000e+0 0.000e+0 2.107e+I -9.964e-I -7.692e-I -3.191e+0 -5.300e*0 5.580e+0 -1.659e-2 2.906e*0 5.050e*I -8.423e-I 3.781e÷0 2.923e÷0 -6.516e-I 7.035e-I -4.365e+0 7.495e-I 5.763e÷0 -1.442e¢0 1.198e+I -6.278e÷0 -4.850e÷0 -1.174e+I 3.419e-I 5.981e+0 -3.495e+1 1.720e÷0 -2.982e+0 -4.336e+0 1.726e+0 1.337e÷0 3.342e*0 -1.948e÷0 -1.529e+I 2.882e¢I -3.477e+I 2.701e¢I 1.506e+0 -1.180e+0 -9.109e-I -2.414e+0 1.528e+0 4.518e-I 2.064e-2 9.388e-I -1.992e+2 2.166e+0 -2.430e÷0 -1.878e÷0 -3.891e÷0 2.259e+0 1.453e÷0 1.123e+0 1.515e-I Columns I To 4 -7.202e.0 -6.871e÷0 -5.308e+0 -1.043e÷0 -3.178e-2 -I.030e-I -1.974e-1 -7.991e-2 -2.129e-I -8.065e-2 -7.841e-2 -1.100e-I -2.340e-2 2.629e-I -1.710e-2 2.988e-2 0.000e+0 0.000e+0 0.000e+0 0.000e+0 0.000e+0 -3.786e+2 -6.630e-5 4.304e+1 0.000e+0 -5.645e+2 -2.978e-3 -1.348e¢0 0.000e+0 -3.604e+2 -5.016e-3 -2.494e+0 0.000e+0 _.664e+3 1.217e-2 2.289e+I 0.000e+0 -1.401e+2 -5.077e-3 -2.838e*I I0 Coltunns 5 To 8 [2] Schierman, J., Schmidt, D., "Analysis Of Airframe/Engine -3.90Se-I 1,040e-5 -3.613e-5 2.995e-3 Interactions In Integrated Flight And Propulsion Control," 1.777e+I -9.835e-5 1.144e-4 -2.044e-2 2.467e-2 3.965e-6 2.092e-7 -I .451e-6 AIAA No. 91-2794, presented at the AIAA Guidance, I .833e-3 6.492e-6 3,166e-6 3.359e-5 Navigation and Control Conference, New Orleans, -2,056e-2 -2.179e-6 1.67Se-5 -1.200e+0 Louisiana, August, 1991.

1.880e-3 1.469e-5 4.748e-6 -6.802e-4 1,728e+1 3.871e-1 2,986e-1 -1.380e-I [31 Schierman, J., Schmidt, D., "Analysis Of Airframe And 4 . 657e-1 -I. 466e÷0 -I . 134e÷0 -6. 050e-3 I .303e÷0 2.436e÷0 I .881e÷0 -7.524e-2 Engine Control Interactions And Integrated -I. 097e÷0 -6. 671e-I -5. 184e-1 -3. 618e-2 Flight/Propulsion Control," September, 1991, to be -3. 617e-1 4,585e-1 3. 532e-I 8. 077e-3 published in the Journal Of Guidance, Control, And 3.378e-I 9.406e-I 7.284e-1 4.108e-2 Dynamics.

-1. 650e-4 -5. 650e-I -4. 357e-I -1. 211e-I 5.437e-I 2,666e+0 2. 059e*0 -I.079e-1 [4] Garg, S., Matterru D.L., and Bullard, R.E.. "Integrated Coltmuns 1 To 5 Flight/Propulsion Control System Design Based on a -I.033e-I -1.322e-I 1.611e-1 -5.057e-3 -4.509e-3 Centralized Approach," Journal of Guidance, Vol. 14, No.

-I. I18e-3 -I. 720e-3 -9,294e-4 -5. 827e-4 -5.309e-4 3. 429e-2 -8. 450e-2 -1. 243e-I -3,421e-2 -1.122e-2 1, pp. 107-116, Jan.-Feb., 1991. Originally presented as -I .315e-I -2.194e-1 1.221e-I -2.706e-2 -4.730e-2 AIAA Paper No. 89-3520, AIAA Guidance, Navigation and -I .343e-1 -2.199e-I 2.956e-I 1.504e-5 -3.630e-3 Control Conference, Boston, Massachusetts, August, 7.977e-I 9.973e-I -I.172e÷0 4.544e-2 -3.251e-I 1989.

4.042e-I 5.964e-1 -4.976e-I 4.357e-2 -6.710e-2 -1.892e+1 -2.098e+1 2.626e÷1 -7.981e-I -1.412e+I [5] Garg, S., Ouzts, P., "Integrated Flight/Propulsion Control Cc Matrix: Colt_ns 6 To 10 Design for a STOVL Aircraft Using H-Infinity Control 1.866e-I 1.504e-I 9.785e-2 -1.857e-I 4.699e-3 Design Techniques," presented at American Control -7.758e-4 -6.743e-5 7.813e-3 3.705e-3 1.039e-3 Conference, Boston, Massachusetts, June, 1991.

-4.394e-2 -9.341e-2 1.598e-I 1.773e-I 1.110e-2 1,316e-I 1.477e-I 3.251e-I -9,063e-2 3.817e-2 2.588e-I 2.826e-I 9.965e-2 -4.655e-I 3.603e-2 [6] Oarg, S., Mattern, D.L., "Application of an Integrated -9.835e-I -1.162e+0 -1.185e+0 1.266e÷0 -1.071e-1 Flight/Propulsion Control Design Methodology to a -5.213e-I -5.207e-I -5.652e-1 5.410e-1 -2.526e-2 2.246e+I 2.480e÷1 1.703e*1 -2.308e+I 7.521e-1 STOVL Aircraft," IAA Paper No. 91-2792, presented at the AIAA Guidance, Navigation and Control Conference, New Cc Matrix: Coltm_s II To 14 Orleans, Louisiana, August, 1991.

3.658e-I -5.758e-2 5.675e-2 1.330e-I -1.670e-2 1.529e-2 8.853e-3 -2.657e-2 I7] Shaw, P., et al., "Development and Evaluation of an -9.900e-1 2.647e-1 -6.822e-2 2.457e-I -1.322e-I 4.603e-I -1.151e-I -4.293e-1 Integrated Flight and Propulsion Control System," AIAA 1.471e*0 4.467e-I -2.130e-2 1.164e-1 Paper No. 85-1423, presented at the AIAA Joint Propulsion -9.120e-I -2.423e÷0 1.531e+0 1.402e+0 Conference, Monterey, California, July, 1985.

-7.606e-1 -1.544e-I 1.184e-1 3.201e-I 1.004e+I 1.295e÷I 2.233e÷I -I.169e+I [8] Rock, S.M., Emami-Naeini, A., Anex, R.P., "Propulsion IX: Matrix: Columns I To 4 Control Specifications in Integrated Flight/Propulsion -6.800e-3 -6.484e-2 -4.982e-2 -7.060e-2 Control Systems," AIAA Paper No. 88-3236, presented at 4.612e-4 4.161e-3 3.206e-3 4.498e-3 the AIAA Joint Propulsion Conference, Boston, 1.239e-3 1.225e-2 9,787e-3 1.300e-2 Massachusetts, 1988.

7.468e-3 6.278e-2 4,860e-2 6.805e-2 1.123e-3 1.555e-2 1.199e-2 1.754e-2 3.713e-2 3.194e-I 2.456e-I 3.468e-I [9] Doyle, J. "Analysis of Feedback Systems With Structured 1.911e-2 1.161e-I 8.783e-2 1.212e-1 Uncertainties" Proceedings of the Institute of Electrical 7.759e÷0 2.472e+I 1.911e÷I 2.549e+1 Engineers, Part D., 129, pp. 242-250, 1982.

Coltmuns 5 To 8 7.570e-4 2.518e-2 1.931e-2 -I.088e-2 [10] Maciejowski, J.M., Multivariable Feedback Design, -I.019e-4 -1.607e-3 -1.243e-3 8.586e-4 Addison-Wesley Publishing Company, New York, 1989.

-1.842e-3 -4.433e-3 -3.797e-3 4.780e-3 -1.377e-3 -2.395e-2 -1.884e-2 2.000e-2 2.039e-4 -6.021e-3 -4.647e-3 2.161e-3 [11] Kwakernaak, H., Sivan, R., Linear Optimal Control -2.155e-3 -1,229e-I -9.522e-2 9.407e-2 Systerns, Wiley-Interscience, New York, 1972.

1.093e-3 -4.552e-2 -3.403e-2 3.272e-2 1.276e+0 -9.525e+0 -7.411e¢0 4.660e÷0 [121 Doyle, J., Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Acknowledgements Transactions on Automatic Controls, Vol. AC-26, No. 1, This work was sponsored by the NASA Lewis Research pp. 4-16, Feb., 1981.

Center under Grant # NAG3-998. Dr. Sanjay Garg is the technical program manager.

References [1] Schmidt, D., Schierman, J., Garg, S., "Analysis of Airframe/Engine Interactions - An Integrated Control Perspective," AIAA Paper No. 90-1918, presented at the 26th Joint Propulsion Conference, Orlando, Florida., July, 1990.

A Comparative Study of Multivariable Robustness Analysis Methods as Applied to Integrated Flight and Propulsion Controit John D. Schierman¢, T. Alan Lovell* and David K. Schmidt** _, Aerospace Research Center Collegd_of Engineering and Applied Sciences Arizona Slate University Abstract allowable magnitude of airframe-to-engine interactions to Three multivariable robustness analysis methods will assure acceptable performance was recently developed and is presented as part of the IS methodology. The focus of the be compared and contrasted. The focus of the analysis will IS methodology is analysis of system stability and be on system stability and performance robustness to performance robustness to uncertainties in the dynamic uncertainty in the coupling dynamics between two cross-coupling between airframe and engine subsystems.

interacting subsystems. Of particular interest is interacting However, although this approach was developed for airframe and engine subsystems and an example analysis of interactions between the airframe and engine, airframe/engine vehicle configuration is utilized in the its application is not limited to these types of systems demonstration of these approaches. The Singular Value alone.

(SV) and Structured Singular Value (SSV) Analysis The STOVL configuration analyzed in Ref. [7] is methods will be compared to a method especially well considered representative of an advanced highly suited for analysis of robustness to uncertainties in maneuverable aircraft with integrated flight/propulsion subsystem interactions. This approach is referred to here control. This vehicle has the capabilities of re-directing as the Interacting Subsystem (IS) Analysis method. This engine thrust to generate forces and moments on the method has been used previously to analyze airframe, enhancing the lifting and maneuvering airframe/engine systems, emphasizing the study of capabilities. For this and similar configurations, the stability robustness. However, performance robustness is potential two-directional interactions between the airframe also investigated here, and a new measure of allowable and engine subsystems are of major concern. $ Engine uncertainty for acceptable performance robustness is introduced. The IS methodology does not require plant thrust can now directly influence the lift and attitude motion of the airframe, and in turn, the dynamic motion of uncertainty models to measure the robustness of the the airframe can affect the engine dynamics. Hypersonic system, and will be shown to yield valuable information single-stage-to-orbit vehicles, such as the X-30 aircraft regarding the effects of subsystem interactions. In contrast, the SV and SSV methods allow for the evaluation design concept, are also considered to possess significant of the robustness of the system to particular models of airframe/propulsion subsystem interactions, and will uncertainty, and do not directly indicate how the airframe require integrated airframe/engine control. 9,10 The (engine) subsystem interacts with the engine (airframe) dynamic interactions between airframe and engine subsystem.

subsystems are frequently difficult to model, and the uncertainties in these interactions can be potentially Introduction significant. Analysis methods are sought which can characterize effects of the interactions, such as critical The objective of this paper will be to compare and contrast aspects of three multivariable robustness analysis frequencies where robustness problems are most likely to methods when applied to interacting airframe/engine OCCur.

subsystems. These three approaches are denoted here as: The airframe/engine plant and control law used to demonstrate the three analysis techniques is presented in (1) Singular Value (SV) Analysis 1,2,3 the next section. The three sections following this will (2) Structured Singular Value (SSV) Analysis 3A and present the SV, SSV and IS analyses of this vehicle configuration, respectively. Each section presents ftrst the (3) Interacting Subsystem (IS) Analysis. 5-7 stability robustness analysis, then the performance robustness analysis. A brief review of the analysis theory This paper will focus on the analysis of both stability and is given in each section before presenting numerical performance robustness with all three methods.

results. Finally, f'mdings from this study are summarized The SV and SSV methods have been used for analysis and conclusions are drawn.

of multivariable systems in general. The development of the IS analysis method was motivated by the integrated System Description and Nomenclature flight/propulsion control problem. 5-7 A measure of the The vehicular system's input-output characteristics will be defined at one operating point by the matrix of transfer functions Jr Pres. at the AIAA GN&C Conf., Monterey, 1993.

:1: Doctoral Candidate, Student Member, AIAA.

* Graduate Fellow.

** Prof. of Aero. Eng., Now with the Dept. of Aero Eng., Univ. of Maryland, Assoc. Fellow, AIAA.

(1) Copyright © 1993 by D.K. Schmidt. Published by or y(s)=G(s)u(s) American Institute of Aeronautics and Astronautics, Inc.

with permission.

AIAA Paper No. 93-3809 can be more meaningfully compared. The normalized

Theairframe andengine response (y) and control (u)

frequency response magnitudes of the airframe's pitch vectors are denoted respectively by the subscripts "A" and attitude (0) to all control inputs listed in Table 1 are shown "E." Likewise, GA(S) and GE(S) represent the airframe and in Fig. 1. Likewise, the engine's fan speed (N 2) responses engine dynamics, respectively. Dynamic interactions are shown in Fig. 2. Although not shown here, the flight between the airframe and engine are reflected in the off- path angle (y) and forward velocity (V) frequency diagonal transfer function matrices, GAE(S) and GEA(S), responses are presented in Ref. [7]. It is evident from these referred to as the engine-to-airframe and the airframe-to- figures that the airframe/engine system is quite engine coupling or interaction matrices, respectively.

multivariable in nature in that each response is The control law is defined here as significantly influenced by several controls. The engine- to-airframe and airframe-to-engine coupling transfer functions in GAE(Jto) and GEA(Jto) are comparatively large K_ I_ J L y_ YE in magnitude. Fig. 1 shows that the fuel flow rate may

(2)

significantly affect the airframe's pitch attitude response.

or u(s) = K(s) [ y¢(s) - y(s) ]) Although not shown here, the fuel flow rate has an even more significant effect on the flight path angle response.

YAc(S) and yEc(S) are the vectors of commanded airframe In turn, Fig. 2 shows that the magnitudes of the engine's fan speed responses from the airframe controls are not and engine responses. KA(S) and KE(s) are the feedback insignificant.

control compensation matrices associated with the airframe and the engine control subsystems, respectively. The Table 1 - System Responses and Controls control cross-feeds are indicated by KAE(S) and KEA(S).

and Their Maximum Values The airframe/engine vehicle model analyzed in Ref. [7] will also be considered here. It is a delta wing supersonic System Responses Maximum Value aircraft with STOVL capabilities. The reference point about which the nonlinear system is linearized is the 0 - pitch attitude (deg) 21 deg steady-state wings-level decelerating transition, y- long. flight path angle (deg) 4.0 deg approaching hover. Note that the airframe's short period V - true airspeed fit/see) 76 ft/sec mode is unstable for this configuration and flight condition. At this reference point, the forces and moments N 2 - fan speed (rpm's) 120 rpm's controlling the aircraft are transitioning from those Maximum Value generated by the aerodynamic control surfaces to those System Control Inputs generated by the propulsion system.

Aq - pitch RCS area (in 2) 0.7 in 2 In this paper four responses and four controls (yielding a 4x4 compensation matrix) will be considered, and they - ejector butterfly valve angle (deg) 8.0 deg are listed in Table 1. The first three responses listed in A8 - aft nozzle throat area (ha2) 20 in 2 this table are airframe responses, while the fan speed, N 2, wf- fuel flow rate (lbm/hr) 1000 lbm/hr is a critical engine response. Therefore, the airframe and engine response vectors are (see Eq. (1)): YA(S) = [0, y, V] T and YE(S) = N 2 (3) "o -20 _ 0 The airframe and engine control vectors were selected as (see Eq. (1)): -40 UA(S ) = [Aq, 11, AS] T and UE(S) = wf (4) -60 A s _ "__ The Reaction Control System (RCS) draws bleed air from -80 the engine's compressor, and the Pitch RCS area controls 10-2 10-1 103 10t 102 the magnitude of RCS thrust. The ejector butterfly valve Frequency In Rad/Soc angle controls the amount of engine flow to the ejectors, thus the amount of ejector thrust. The magnitude of aft Fi_igllK¢_.l. - Pitch Attitude Frequency Response Magnitudes thrust is largely determined by the aft nozzle throat area.

With this selection, referring to Eq. (1), the airframe The feedback compensation for this system was transfer matrix, GA(S ), is 3x3, and the engine transfer designed using a standard Hoo control law synthesis function, GE(s), is a scalar. Thus, the engine-to-airframe formulation. 3,11 In this particular formulation, Fig. 3 coupling transfer matrix, GAE(S), is 3xl, and the airframe- shows the sensitivity transfer function matrix weighted by to-engine coupling transfer matrix, GEA(S), is Ix3.

Sd-l(s ) (where Sd(S) is the "desired" sensitivity matrix), Note that the responses and controls were normalized along with the control effort weighted by We(s). Sd(S) was by estimates of their respective maximum allowable chosen to equal the sensitivity matrix obtained by the perturbations from reference values, presented in Table 1.

system presented in Ref. [7] (in which eight responses and With this normalization, magnitudes of transfer functions margin of 90 degrees. Finally, the frequency response

controls were utilized). Wc(S) was chosen to weight the

magnitudes of the elements within K(s) were control effort greatest beyond specified actuation approximately the same order of magnitude as the bandwidths. Note that the purpose of this paper is neither corresponding elements of the compensator matrix to promote nor refute the I-I,0 control law synthesis presented in Ref. [7] (in which the control actuation was methodology. The elementary formulation shown in Fig.

not considered excessive).

3 was used simply to obtain a compensator in order to The compensator obtained by this synthesis procedure demonstrate the analysis methodologies presented in the was of 28th order, and was subsequently reduced to 14th next sections.

order by a frequency-weighted internally-balanced order reduction method presented in Ref. [12]. The partitioning of the control law matrix follows from the response vector (Eq. (3)) and the selection of airframe and engine controls (Eq. (4)). KA(S) is therefore 3x3, and KE(S) is a scalar.

r/_ "O The control cross-feed matrices, KAE(S)and KEA(S) are 3xl and lx3, respectively. The eigenvalues of the open-loop a airframe/engine plant, compensator and closed-loop system are presented in Table 2.

Table 2 - Open/Closed-Loop Eigenvalues Cl_ etl-l._op Eigenval ue_ Eigeawlues of the Open Loop Plant lO.2 tO.1 100 101 102 -1.9959e-02 -1.9970e+02 Frequency In Rad/Sec -2.1121e-01 t2.6096e*Oli -3.8212e+01 -3.7869e-01 -2.9395e÷01 - Fan Speed Frequency Response Magnitudes -3.1371_-01 t6.6357e-Oli -7.10876+00 -2.22556o01 -4.1220e+00 -1.5838e+01 Z5,47846+00i 1.29396+00 -9.2770e-00 -1.0629e-01 ±2.7932e-01i

[zt] -3.90466*00 ¢4.24116÷00J

-2.0918e+00 -3.1070e°00 _I.13476+00i Eigeavalues of the Compensator -2.4429eo00 ±I.0952e÷00i -2.2543e÷01 ±2.4624e÷01i -2.1509e*00 du __Z=e z2 -3.2040e÷01 -8.9366e-02 ±2.9916e-01i -2.33016÷01 -5.4206e-01 -1.6773e÷01 ±6.29156_00i -2.6680e-01 ±1.9755e-01i -4.9100e_00 ±4.5882e_00i -2.3179e-01 -4.2717e*00 -5.4666e-01 P(s) Zl -9.9883e-03 -8.3853e-03 -8.4735e-03 -].0029e-02 z2 The closed-loop responses from commands (Yc(S)) and disturbances (d(s)) for the systems are u e y(s) = T(s) yc(s) + S(s) d(s) (5) where T(s) and S(s) are the complementary sensitivity and sensitivity transfer function matrices, respectively. Fig. 4 presents the closed-loop pitch attitude (0) frequency response magnitude from a pitch attitude command, 0¢.

This figure also presents the "desired" performance (that The compensator synthesized by this procedure which was obtained by the feedback system presented in delivered tracking and disturbance rejection performance Ref. [7]) and the specified maximum allowable upper that approximately matched the performance obtained by bound. From the definition of the response vector given in the system presented in Ref. [7], and the closed-loop Eq. (3), the frequency response of 0/0¢ corresponds to the frequency response magnitudes did not exceed specified (1,1) element in T(s). Fig. 5 presents the closed-loop maximum allowable upper bounds. Further, the individual engine fan speed (N 2) frequency reslxmse magnitude from a loop transfers (with all other loops closed) exhibited fan speed command, N2c, along with its respective acceptable loop shapes and typically good classical gain and phase margins. The pitch attitude, flight path, forward "desired" performance and upper bound. This response speed and engine fan speed loops have cross-over corresponds to the (4A) element of T(s). Although not frequencies of 1.8, 1.5, 0.19 and 3.5 rad/sec, respectively. shown, similar disturbance rejection performances were The pitch attitude and flight path angle loops both have seen for these loops. Further, both the tracking and approximately 60 degrees of phase margin, and 16 and -10 disturbance rejection performances for the flight path angle dB of gain margin, respectively. The forward speed loop (y) and forward velocity (V) responses were likewise has gain and phase margins of 55 dB and 75 degrees, while acceptable.

the fan speed loop has infinite gain margin and a phase

Note,however, thattheclosed-loop system is not

0 Maximum decoupled, and each command can elicit responses in the _, -_.. AAllowable Upper other channels. Fig. 6 presents the pitch attitude response

o Bo0nd

from flight path, velocity, and engine fan speed commands.

These responses correspond to the (1,2), (1,3) and (1,4) elements in TOm) (as well as in S(jto)). It can be seen - that a flight path angle command can produce a pitch '_ -40 attitude response greater than -20 dB between approximately 0.05 and 10 rad/sec. Although not shown, •-_ -20 -60 a pitch attitude command can, in turn, produce a significant flight path angle response within this frequency range.

-80 Both pitch RCS jets and ejector thrust produce airframe 10-2 10.1 lO0 IO1 102 pitching moments, and it would be difficult to decouple Frequency in Rad/Sec pitch attitude and flight path angle responses. (Note that even larger magnitudes were seen in the corresponding off- - Pitch Attitude From Other Commands diagonal elements of TOm) and S(jto) for the system Singular Value (SV) Analysis presented in Ref. [7].) In general, it was found that TOm) The integrated airframe/engine system with was not decoupled above approximately 0.05 tad/see, and unstructured output multiplicative uncertainty is shown in S0to) was not decoupled below approximately 10 rad/sec.

Fig. 7. In this figure, the response, control and command However, the responses did not exceed their respective vectors, y(s), u(s) and yc(s), and the plant and control law allowable upper bounds, and therefore the over-all closed- transfer function matrices, G*(s) and K(s), are defined as in loop performance for this system was deemed acceptable.

Eqs. (1) and (2). Note that G*(s) denotes the "nominal" plant with no uncertainty (M(s)=0). Again, d(s) in Fig. 7 .tab ,,, 2 radlsec is a vector of exogenous disturbances corrupting the responses of the system.

t_ "O Achieved / Performance " " y4s)_ _s) Maximum Allowable -60 Upper Bound F_igllL¢_2 - Feedback System With Uncertainty, M(s) -80 Stability Robustness Analysis 10-2 10-1 100 101 102 It is shown in Refs. [1] and [2] that for a system with Frequency in Rad/Sec unstructured output multiplicative plant uncertainty, M(s), system stability is assured if K(s) stabilizes the nominal - Pitch Attitude-From-Pitch Attitude Command system (M(s)=0), if MOt,) does not alter the encirclement requirement (for stability) of the Nyquist plot (plot of i toe - 3 tad/see det[l+0+M)GK]), and if 0 i .......

rn Omax(M(j¢o)) < Omm(l+[G(jco)K(jc0)] "1) for all ¢o>0 (6) -20- Maximum / "_.

o Allowable _- I Upper Bound "Desired "_ 1,> where Omax() and Omin( ) are the maximum and t- e_ -40 F Performan_/ minimum singular values, respectively. This inequality may be used as a stability robustness criterion. If this 450 - Achieved Performance criterion is not met, stability can no longer be assured.

Although the analysis presented in this paper will focus on .801 10-2 lO-t tOo lOt 102 uncertainty at the plant output, a complete analysis should also address robustness to multiplicative uncertainty at the Frequency in Rad/Scc plant input, which may be analyzed by a similar criterion.

- Fan Speed From Fan Speed Command The uncertainty matrix, M(jto), in Eq. (6) can be of any general structure. However, since the focus of this Note that the upper bounds shown in Figs. 4-6 will be study is robustness to uncertainties in the airframe/engine utilized in the analyses discussed next. The matrix of interactions (GAE(S) and GEA(S)), consider that the airframe maximum allowable upper bounds on the complementary and engine.plants, GA(S) and GE(S), are reasonably well sensitivity matrix is denoted Tu(jto), and likewise the modeled, but the interactions contribute the most matrix of maximum allowable upper bounds on the significant sources of uncertainty in the model of the sensitivity matrix is denoted Su(Jm).

system's dynamics. The airframe/engine plant description with additive uncertainty in the coupling dynamics is G(s) = G*(s) + A(s), where respectively. The nominal (M=0) complementary sensitivity and sensitivity transfer function matrices shall

G*(s)=[ GA G_landA(s)=I 0 AAE]

be denoted as T*(s) and S (s), respecuvely.

Consider, as an example, the following constant uncertainty matrix for the airframe/engine system under study (recall, AAE is 3xl and AEA is Ix3), 2O ee_ Omin(l+(GK)'l)_ .=_.

(8) o o l A =8oA1, AI-- e o o 1

[i°°1]

1 l I o -20 where 5 o is a scalar. Using the following relationship (M = 8.5e-4 M1) between additive and output multiplicative uncertainty, -40 10-2 10t 10o 101 10z G(s) = G*(s)+A(s) = (I+M(s))G*(s) (9) Frequency in Rad/Sec F_igC.._r¢_- Singular Value Stability Robustness Criterion the equivalent multiplicative uncertainty for this example is Often, aspects of aircraft flying or handling qualities M(s) = A(G*(s)) q = _ioMl(s) (10) are considered acceptable if frequency response magnitudes where (such as pitch rate-to -pilot stick input) lie within defined upper and/or lower allowable bounds. Allowable bounds Ml(S ) = AI(G*(s)) -1 (I 1) may also be utilized to determine acceptable tracking and disturbance rejection performance of the engine loops.

Fig. 8 presents the plot of Eq. (6) for the feedback Recall that upper bounds on the elements of TOm) and system discussed in the previous section along with the SOu)) were presented in the last section, and the matrices example uncertainty above. It can be seen that the of these allowable upper bounds were denoted Tu(jo)) and minimum value of Omin(I+[G(jo))K(jo))] -1) is Su(Jo)). It is stated in Ref. [3] that multivariable tracking approximately -7 dB and occurs in the frequency range and disturbance rejection performance may be defined between 0.4 rad/sec and 0.8 rad/sec. However, it is also acceptable if shown in this figure that when _io = 8.5e-4, Eq. (6) is no Omax(Tu-l(jo)) TOo))) _<I for all to longer satisfied for frequencies greater than 30 rad/sec. Yet, Omax(Su'l(jo)) S(jo))) < 1 for all O) (14) the system is stable for this uncertainty. Note that the criterion of Eq. (6) is known to be a conservative measure of stability robustness. From the Nyquist plot, it was These inequalities constitute multivariable performance found that when 8 0 = -0.0665 the system becomes robustness criteria Acceptable performance is assured if these criteria are met for all frequencies.

unstable at a frequency of 0.36 rad/sec. This is an increase from 8.5e-4 by approximately 38 dB (a factor of 78). Fig. 9 presents the complementary sensitivity Further note that the frequency at which the criterion fails performance robustness criterion of Eq. (14) for the feedback system under study. It can be seen that this in Fig. 8 does not correspond to the frequency of criterion is not met for the nominal system (M--0), even instability.

though the magnitudes of all closed-loop frequency Again, the multiplicative uncertainty defined in Eq.

(10) is just an example. Different uncertainty matrices responses lie below their upper bounds (see Figs. 4-6).

with smaller maximum singular values may exist which Thus, the criterion of Eq. (14) is conservative in this case.

cause the system to become unstable. Finding the Recall that TOm) is not decoupled (not diagonally particular critical multiplicative uncertainty matrix with dominant) beyond approximately 0.05 rad/sec, and smallest maximum singular value (thus minimizing the o=_(T,l(jo)) T*(jc0)) begins to grow larger than 0 dB conservatism of the criterion of Eq. (6)) can be a difficult around this frequency. Recall as well that S(jc_) is not task, and it may represent variations in the plant that are decoupled below approximately 10 rad/sec, and, although physically unrealistic.

not shown, Omaa(su'l(jo)) S*(jo))) is greater than 0 dB until approximately this frequency. The maximum singular Performance Robustness Analysis value of a matrix is only an accurate measure of the The closed-loop responses of the system shown in magnitude of the element with largest magnitude when the Fig. 7 are matrix is diagonally dominant. Holding the diagonal y(s) = (I + (I+M)GK)-I(I+M)GK yc(s) elements constant, as the off-diagonal elements increase in magnitude, the maximum singular value will also increase + (I + fl+M)GK)-ld(s) (12) in size. This property adds to the conservatism of the or, criterion of Eq. (14).

y(s) = T(s) yc(s) + S(s) d(s) (13) Fig. 9 also shows the criterion for the system with an example MOo) ) = -0.016 MI(jo)) (=25% of the uncertainty where again T(s) and S(s) are defined as the complementary which causes instability), and Fig. 10 presents the pitch sensitivity and sensitivity transfer function matrices,

attitude (0) response from fan speed command (N2c) for the

[,]:[oll ol21Eyc]

system with this value of uncertainty. It can be seen that u' Q21 Q_ y' this response increased beyond its maximum allowable (16) where, upper bound for frequencies above 0.1 rad/sec. Although Qll = G*K(I+G*K) 1, QI2 = (I+G*K) 1 not shown, the increases in magnitudes of the flight path Q21 = P ( I+KG*)IK, Q22 = "P (I+KG*) IK angle and velocity responses from fan speed command were just as large. Although an increase in o,_(Tu't(jto) T(jto)) Note that P relates the off-diagonal uncertainty matrix, from the nominal value o,,.,_(T,_ltjto)T'(jto)) is noted in A(s), to the block-diagonal uncertainty matrix, AD(S).

Fig. 9, the performance degradations in the airframe That is, responses from engine commands were discovered only after investigating all closed-loop responses from all A(s)=AD(s)P, p=[0 I] (17) commands.

I 0 I0 yc(s) y(s) v y, _ u' \ -5 (M = 25% of M Which Causes Instability) __ Figure 11 - System With Block-Diagonal Uncertainty -i0 Omax.(T_l (jm) T* (jm)) _; Stability Robustness Analysis (M = O) -15 From Refs. [3] and [4], the system shown in Fig. 11 10-2 10-1 10O 101 102 remains stable if and only if Frequency in Rad/Sec - SV Performance Robusmess Criterion IIADII, < 1 , where (18) IIQ2211p llQ2211_ = sup [I_(Q22(j_))], & IIADll. = sup [am_,(Ao(j0_))].

Response With Maxunum 25% of Uncertainty Here _t(Q22(Jto)) is the structured singular value of Which Causes Instability Allowable Upper Bound Q22(Jto).

-8 _= Fig. 12 presents the inverse of g(Q22(Jto)) for the

\

"2 feedback system under consideration. It can be seen that the minimum value of 1/l_(Q22(Jt_)) is -31 dB at a frequency of approximately 0.36 rad/sec. The structured singular value theory states that at each frequency an 10-2 10-1 10o 101 102 uncertainty matrix ADcrit(Jto) exists that causes the system Frequency in P-_l/See to become unstable and Omax(Ai)crit(Jto))=l]l.t(Q22(Jto)).

Therefore, at 0.36 rad/sec an uncertainty matrix Acrit(Jto) Figure 10 - Pitch Attitude From Fan Speed Command exists that causes instability and has a corresponding block-diagonal matrix, ADcrit(Jo), with a maximum Structured Singular Value (SSV) Analysis When more structure can be given to the uncertainty singular value equal to -31 dB. Recall that for the in the system, the SSV analysis method takes advantage of uncertainty matrix A = -0.0665 A_, where A t is defined in this knowledge to give a less conservative measure of Eq. (8), instability occurred at 0.36 rad/sec. Unlike the SV stability robustness. For this method, the feedback loop analysis method, here the frequency of instability is for the overall interacting system is represented as shown consistent with the critical frequency indicated in Fig. 12.

in Fig. 11. Utilizing the specific structure of uncertainty However, as also shown in Fig. 12, for the specific as defined in Eq. (7), AD(S) in Fig. 11 reflects the structure of uncertainty defined in Eq. (8), Omax(A D) = -31 uncertainties in the coupling dynamics into the following dB when 8 o = 0.016, hence, the criterion of Eq.(18) is no block-diagonal form: longer satisfied. Yet, recall that this is only approximately 25% of the value of uncertainty that causes instability.

AD(S) =[ AAE 0 1 (15) Again, the additive uncertainty def'med in F..q. (8) is just an 0 AEA example, and is certainly not the critical uncertainty matrix, Acrit(J_). Acrit(Jm), may have different Q(s) in Fig. 11 represents the nominal closed-loop magnitudes and phases for each element.

system, and it can be shown that can be seen that the performance robustness criterion of Eq.

(23) simply "combines" the singular value tracking and disturbance rejection performance criteria of the last section o (see Eq. (14)).

Fig. 13 presents the criterion of Eq. (23) for the

.E

-to feedback system under study. Just as with the SV

-8

_=,=(AD) _(-----__ analysis, it can be seen that the criterion is not met even (A =0.016 A1)

'E

-2o for the nominal system since Omax(Qll) > 1 throughout the frequency range shown. Again, this is due to the fact -30 that the closed-loop system is not diagonally dominant.

Fig. 13 also shows the criterion of Eq. (23) for the -4o system with A = -0.016A 1 (25% of uncertainty that causes lO-_ 10-t 10o 101 102 instability). Although an increase from the nominal value Frequency in Rad/Sec (Omax(Qll)) is noted in Fig. 13, as with the SV analysis, - SSV Stability Robustness Criterion this criterion does not directly indicate which elements of TOm) are increasing in magnitude.

Finally, although the structured singular value stability criterion in Fig. 12 has only just failed for A = -0.016 Al, recall from Fig. 10 that for this value of uncertainty the pitch attitude response from engine fan er_ speed command violates its maximum allowable upper .S bound. As expected, uncertainty in the system will cause U performance requirements to fail before stability robusmess "2 0| o=,x(Qt]+Qlz(I-fDQ22) ADQ2fl ] requirements.

-5 _ (A = 25% of Uncertainty ] / Which Causes Instability) .. -- • Performance Robustness Analysis -10_ o=,x(Qlt)=Om_//Su'tS 0to)I_ The structured singular value performance robustness -ts[ (LT__ 1"0o) ]1 criterion, inlroduced in Ref. [13], is also presented in Ref.

102 l(_t 100 10t 10z [3]. Note that in Fig. 11, Frequency in Rad/Sec y'(s) = AD(S) u'(s) (19) Figure 13 - Performance Robustness Criterion of Eq. (23) Finally, as discussed in Ref. [3], note that both robust Substituting Eq. (19) into Eq. (16) , the closed-loop stability and performance can be assured by one structured responses of the system shown in Fig. 11, with the singular value criterion. That is, the stability criterion of uncertainty matrix AD(S ), are Eq. (18) and the performance criterion of Eq. (23) are assured to be met ff and only if y(s) = {Qll + Q12(I'ADQ22)'lADQ21 } Yc(s) (20) 1 < 1 (24) II Q I1_ In Ref. [3], the system outputs are then redefined to be Although further manipulations on the system and block- diagonal uncertainty matrix are required in the development (21) of this criterion, the matrix Q in this inequality is %](s) y(s) essentially that defined by Eqs. (16) and (22). The criterion of Eq. (24) can also be used as an objective in the control Again, Tu(s) and S,(s) are the matrices of maximum law synthesis. If it is met, robust stability and allowable upper bounds on the closed-loop frequency performance are assured for uncertainty in the interacfons response magnitudes. With this selection of outputs, between the airframe and engine. Although not shown, this criterion is not met for the feedback system analyzed here, since the criterion of Eq. (23) is not met even for the Qn = [ S_l(s)Si(s)], Q] 2 = "-S_'l(s) S_(s)] (22) nominal system (see Fig. 13).

L Td(s) T (s) J '_J(S) S (S)] Interacting Subsystem (IS) Analysis, Note that Q21 and Q22 remain the same as in Eq. (16).

The main objective of this analysis methodology is to Using Eq. (20) with these new definitions for Q11 and reveal how the interactions between the airframe and engine QI2, performance robustness of the system may be are manifested, and to assess their significance. This considered acceptable ff method is presented in Refs. [5]-[7]. It is shown in these references that through block-diagram manipulation, the °max(Qll + Q12(I'ADQ22)-lADQ21) < 1 for all to (23) airframe/engine plant (Eq. (1)) and the control law of Eq.

(2) may be described as shown in Fig. 14. The effects of Note that the nominal (AD(jto)=0) performance robustness the airframe on the engine loop due to the dynamic criterion is Omax(Q 11) < 1 for all to. From Eq. (22), it coupling between these subsystems is represented by the between the frequencies of 0.2 and 0.5 rad/sec, and that the multiplicative and disturbance interaction matrices, MA(S) "robustness margin" is seen to be approximately 6 dB.

and DA(S). It can be shown that MA(S)=IGEAg_AE- (GEA + GEXEA)RIGE "1 1 + i • TheallowablesizeoflMa(j_)l where, 6O R = KA [I+(GA+GAE XEA)KA]" 1(GAE+GA _:) ] _1 to assure l + (I+M/O_E¢ 0/ 4O (25)

".7/

o DA(S ) = (GEA+GEXEA)KA[I+(GA-_3AEXEA)KA] 1 2O .2 g_ where, with reference to Eq. (2), note that 0 -20 KAE(S) = XAE(S)KE(S), KEA(S ) = XEA(S)KA(S) (26) _- x. Critical Frequency Range -40 10-2 10-I l_ 101 lOZ Also, note that the engine affects the airframe in a dual manner, and the dual expressions for these interaction Frequency in Rad/Sec terms can be found by interchanging all subscripts A and E Figure 15 - IS Stability Robustness Criterion in the above expressions.

Fig. 16 presents the criterion of Eq. (28) with the uncertainty matrix A=-0.058A 1, where A 1 is defined in Eq.

(8). Recall from the previous section that instability occurs when A =-0.0665 A1. Also recall that the structured singular value stability robustness criterion failed for A=-0.016A 1. Hence, for this particular structure of uncertainty, this analysis method gives the least conservative measure of stability robustness. Note too, as Figure 14 - Engine Loop With Effects From Airframe indicated in Fig. 16, the criterion fast fails at 0.36 rad/sec, which is precisely the frequency at which instability occurs Stability Robustness Analysis for this structure of uncertainty. Finally, the dual of the The determinant of the return difference matrix for the criterion of Eq. (28) (for analysis of the engine's effects on airframe/engine system (Eqs. (1),(2)) may be expressed as the airframe loops) was also seen to indicate the frequency of 0.36 rad/sec as most critical.

det[I+GK] = det[I+(GA+GAEXEA)KA]det[I+(I+MA)GEKE] Fig. 17 presents the Bode plot of the airframe's pitch (27) attitude loop (with all other loops closed). It can be seen that this loop nominally has a minimum gain margin of Therefore a necessary condition guaranteeing 16 dB at a phase cross-over frequency of 6.5 rad/sec, and a det[I+G(jto)K(.jto)]_3 is that the det[I+(I+MA)GEK E] is phase margin of 60 degrees at a gain cross-over frequency nonzero for all frequency, and this is assured if the engine of 1.8 rad/sec. However, the instability that occurs at 0.36 control law KE(S) stabilizes the (non-interacting) engine rad/sec due to the uncertainty (A=-0.0665A 1) in the loop (in which case the det[I+GEKE]_)), and if 1 interactions between the airframe and engine is also shown.

The gain and phase cross-over frequencies in the classical Omax(MA(jO))< Omin(I+[GE(jtO)KE(jo_)] -1) for all to > 0 single-loop analysis do not correspond to this critical frequency. However, the SV, SSV and IS stability (28) robustness criteria all correctly indicated frequency ranges This inequality may be considered a stability robustness around 0.36 rad/sec as being critical.

criterion. In order to assure that the det[I+(I+MA)GEK E] is nonzero at each frequency, Omin(I+(GE(jtO)KE(jCo)) "1) is the maximum allowable size of Omax(MA(JCo)). The tO smallest difference between Omax(MA(J0_)) and .,.q Omin(l+[GE(jt0)KE(jto)] -1) may therefore be considered a o -10 "robustness margin," which indicates the "size" of allowable uncertainty in MA(jto). Note that Eq. (25) -20 shows that MA(S) is an explicit function of the coupling "_ I MA(io) " I MAtio) -30 matrices GAE(S) and GEA(S), and uncertainty associated for A=-0.058 AI with the coupling dynamics is therefore reflected in the -40 uncertainty in MA(jto). 10.2 10.1 102 lOt 102 Fig. 15 shows the plot of the stability robustness Frequency in Rad/Sec criterion of Eq. (28) for the airframe/engine system under Figure 16 - IS Robustness Criterion With Uncertainty study. This figure indicates that the smallest difference between ]MA(jo)I and II+[GE(jO)KE(jo_)]-II occurs magnitude of MA(jto) is smaller than MA(JCo), then 4O acceptable tracking performance is assured.

_ta 2O For scalar engine systems, 9dA(j00) can be directly ,, calculated. MA(jo ) is determined by solving a static

/ oc -- 1.8

minimization problem at each frequency. The loss g_ Loop Transfer _ t "'x_-_""" GM - 16 function to be minimized is With Uncertainty Fretluencv _. - dB -20 Which Causes of Instability _".

J = IMA(jCo)l (31) Instability: = 0.36 rad/sec .... X ._.....

..40 with the constraint A = -0.0665 A 1 ITu(jO_)l = I(I+(I+MA)GEKE) -1 (I+MA)GEKE I or, (32) i o ITu(jO_)I = ITE(jto)I

100 ,,: \ -....._

The augmented loss function was therefore defined as PM = 60 dog 10-2 lOd I00 101 102 g = IMA(jO0)l Frequency in Rad/Sec + _,{ ITu(jto)l 2 - I(I+(I+MA)GEKE) -1 (I+MA)GEKE 12 } (33) Figure 17 - Pitch Attitude Loop Transfer Function where _. is the Lagrange multiplier. (Note that the square of the magnitudes was utilized in order to simplify the Engine Performance Robustness Analysis problem.) The following are the necessary conditions for From Fig. 14, the engine response from engine finding the minimum magnitude of MA(jto): command for the integrated system is o: _ o, o: - o, _=0 (34) YE(S) = [1 + (I+MA)GEKE]-I(I+MA)GEKE YEc(S), or 01M^I 0Z(M^) 0_.

(29) YE(S) = TE(S) YEt(s) Expanding these necessary conditions and solving for Z(MA) gives the phase angle for MA(jCo ) as When MA(S)--O, the "non-interacting" closed-loop engine response is defined as Z(_tA) -- tandt -ITul2 sin(_) (35) YE(S) = [1 + GEKE] -1 GEK E YEc(s) ITol2(m+cos(qb))-m or, (30) where m and ¢ are definedas themagnitude and phase of YE(S) = TE'(S ) yEt(S) the "non-interacting" engine loop transfer function.That is, The tracking performance may be considered acceptable if the magnitude of the engine response for the interacting GEK E = m ej¢ (36) system (MA(jto)#0) lies below the magnitude of the upper Once the phase of 9,(A(jo) is determined, the magnitude of bound defined as ITu(jto)l. Fig. 18 shows the magnitude of 9¢t'A(jt_) the root of the following quadratic with minimum TE'(jo) along with the magnitude of the specified upper magnitude: bound, ITu(j¢o)l, as already shown in Fig. 5.

C 11MAI2+[C2cos(Z(9,fA))+C3sin(Z(gd'A))] IMAI+C 4 = 0 (37) rr_0to)l C 1 = m2(ITul 2- 1), C2 = 2(C 1 + m ITul2COS(0)) _ta C 3 = -2m ITul2sin(¢), C a = C 1 + 2m ITul2cos(0) + ITu 12 i -20 Non-Interacting Response: _ _ "-.

o YE = (I+GBKE)'1GEKE y_: _ - For the airframe/engine system considered here, Fig.

-40 19 shows IMA(jto)I, the actual IMA(jto)l , and the allowable Interacting system's response, \ IMA(JO)I that assures that the I+(I+MA)GEKE¢0 (the YE = (1+(1 +MA)GL*K E)'I(1 +M^)GEKE Y_ \ -60 stability robustness metric - see Fig. 15). It can be seen must lie below upper bound for acceptable tracking performance that the magnitude of 9,fA(jto) is lower than that which -80 10-2 10-1 103 101 102 indicates stability robustness (as expected). A "performance robustness margin" may be defined in a Frequency in Rad/Sec manner analogous to the "stability robustness margin," as Fimn'e 18 - Non-lnteracting Fan Speed Response the minimum difference between 19,4A(jto)l and IMA(jto)l.

In Fig. 19, it can be seen that this "robustness margin" is 9,(A(jto) is defined here as the maximum allowable approximately only 1 dB less than the "stability robustness magnitude of MA(Jto) that assures the interacting engine's margin." In fact, as shown in the figure, the engine's closed-loop frequency response magnitude is less than its closed-loop fan speed response will not exceed its upper maximum allowable upper bound. That is, if the actual bounds until A=-0.055A 1. This is a comparatively large uncertainty and indicates that the engine system's tracking and indicates frequencies centering around 0.3 and 30 performance is robust to uncertainties in the rad/sec as critical. Fig. 21 presents the pitch attitude airframe/engine interactions. Although not shown, this response from pitch attitude command for an uncertainty result is consistent with closed-loop fan speed frequency A =-0.058 At (that just causes the stability robustness responses with uncertainty in the system.

criterion of Eq. (28) to fail - see Fig. 16), and shows that Finally, note that the maximum allowable magnitude the frequencies at which this response deviates greatest of MA(jo) such that the engine's sensitivity function lies from the nominal are consistent with the critical frequency below its upper bound can be solved in the manner just ranges indicated by the plot in Fig. 20.

descn'bed.

6O 1 +._1.._ - The allowable size oflMA(J_)[ / 40 I GL:KE[ to assure l + (I+MA)GI_K a _,0 / Allowable size of l M^(jo) _ y / .s to assure acceptable _ / / .8 -8

2 p rfo-, c

/_. Nominal(_o).

t-m _o -20 TAO )o -(I_A)'I(I+ME)G^K^ 10-2 10-1 tO0 lOt 102 10-2 10-1 10o 101 102 Frequency in Rad_e,c Frequency in Rad/Sec Figure 20 - Performance Robustness Measure - '/(jo_) Figure 19 - MA(jO) = Allowable IMAI Airframe Performance Robusmess Analysis From Eq. (39) it can be seen that the effect of the engine commands on the airframe responses may be It can be shown that the integrated system's airframe indicated by comparing the "sizes" of the airframe's responses are complementary sensitivity matrix, TA(jto), with the product of the airframe's sensitivity matrix, SA(J_ ), and YA(S) = [I + (I+ME)GAKA] "I(I+ME)GAK A YAc(S) the disturbance interaction matrix, DE(jta ). Fig. 22 shows + [I + (I+ME)GAKA] "ID E yEc(s) (38) the minimum singular value of TA(jto) along with the or, maximum singular value of (SA(jtO)DE(jto)) (a "worst YA(S) = TA(S) YAc(S) + SA(S) DE(S) YEc(S) (39) case" study). It is seen that beyond 1 rad/sec the "size" of the responses due to engine commands becomes greater than 10 percent of the "size" of the responses due to where ME(S ) is the dual of MA(S). When ME(S)=0, the airframe commands. Uncertainty in the system can "non-interacting" closed-loop airframe responses from increase the "sizes" of both the airframe's sensitivity airframe commands is defined as matrix and the disturbance interaction matrix (both are functions of the interactions GAE(JO) and GEA(jto)), YA(S)= [I+ GAKA] -IGAK A YAc(S) further increasing the magnitudes of the responses from or, (40) engine commands, and this is consistent with the result t shown in Fig. 10.

YA(S) = T A (s) YAc(S) Now define the maximum singular value of the "ratio" of _.. Maximum I TA'(jt0) and TA(JO) as Allowable Upper[ \-__._- _ Bound ) . -] .

_gjo) = Omax((T A OC0)) TA(jO)) (41) -20 o Response With "_x "i

]

-40 If no interactions are present (MEOtO)--0) then _jto) =1 for A=-0.058 As __t all frequency. Therefore, with interactions (ME(JO))_0), '/(jo) can indicate those frequencies where the effects of the

Nodal j\ 1

uncertainties in the interactions will be most prominent for R_pon_- _.l -80 the closed-loop airframe responses only from airframe 10.2 10.1 I0O lOa 102 commands. Unlike the performance robustness criteria of Frequency in Rad/'Scc the SV and SSV analyses (see Figs. 9 and 13), this measure will not be "clouded" by the effects of the engine Figure 21 - Pitch Attitude From Pitch Attitude Command commands on the airframe responses. Fig. 20 presents the plot of '/(jto) for the airframe/engine system under study, References [1] Doyle, J., Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis," IEEE Transactions on Automatic Controls, Vol. AC-26, No.

-.j 1, pp. 4-16, Feb., 1981.

-20 .g .2.

12] Ridgely, D., Banda, S., "Introduction to Robust r.

Multivariable Control," AFWAL-TR-85-3102, Flight Dynamics Lab, Air Force Wright Aeronautical Labs, -60 Dayton, February, 1986.

[3] Maciejowski, J.M., Multivariable Feedback Design, -80 10-2 IO-I tOO 10t 102 Addison-Wesley Publishing Company, New York, 1989.

Frequency in Rad/Sec 141 Doyle, J. "Analysis of Feedback Systems With Structured Figure 22 - A "Worst Case" Measure of the Effects of Uncertainties" Proceedings of the Institute of Electrical Engineers, Part D., 129, pp. 242-250, 1982.

Engine Commands On Airframe Responses Schierman, J., Schmidt, D., "Analysis Of Airframe And Conclusions [5] Engine Control Interactions and Integrated The Interacting Subsystem (IS) analysis method Flight]Propulsion Control," Journal Of Guidance, specifically addresses effects of the interactions between the Control, And Dynamics, Vol. 15, No. 6, Nov.-Dec., airframe and engine. The Singular Value (SV) and 1992, pp. 1388-1396.

Structured Singular Value (SSV) methods provide criteria that, if met, assure robust stability and performance. [6] Schierman, J., Schmidt, D., "Analysis Of However, if these criteria are not met, the causes of the Airframe/Engine Interactions In Integrated Flight And Propulsion Control," AIAA No. 91-2794, proceedings problems are not apparent.

It was seen that the stability robustness criterion of of the AIAA Guidance, Navigation and Control Conference, New Orleans, August, 1991.

the SV analysis method can be a conservative measure.

Further, for the case study, the critical frequency range Schierman, J., Schmidt, D., Lovell, T., "Analysis Of indicated by the stability robustness criterion of the SV [7] Airframe/Engine Interactions For A STOVL Aircraft With analysis method did not correspond to the frequency of Integrated Flight/Propulsion Control," AIAA No. 92- instability. Uncertainty was considered to be significant 4623, proceedings of the AIAA Guidance, Navigation only in the interactions between the airframe and engine.

and Control Conference, Hilton Head, August, 1992.

Utilizing this structured uncertainty, the stability robustness criterion of the SSV analysis method indicated a [81 Garg, S., Mattern, D.L., "Application of an Integrated critical frequency range that was consistent with the Flight/Propulsion Control Design Methodology to a STOVL Aircraft," AIAA Paper No. 91-2792, proceedings frequency of instability. This critical frequency of of the AIAA Guidance, Navigation and Control instability was also accurately indicated by the IS analysis Conference, New Orleans, Louisiana, August, 1991.

method. Further, for this case study, the IS method gave the least conservative measure of stability robustness.

SchmidL D., "Dynamics and Control of Hypersonic Although the performance of the nominal [9] Aeropropulsive/Aeroelastic Hypersonic Vehicles," airframe/engine system was considered acceptable, the AIAA Paper No. 92-4326, proceedings of the AIAA performance robustness criteria of both the SV and SSV Guidance, Navigation and Control Conference, pp. 161- analysis methods were not met for the nominal system.

171, Hilton Head, South Carolina, August, 1992.

These criteria were conservative because the closed-loop system was not diagonally dominant. Further, little [10l Schmidt, D., "Integrated Control of Hypersonic Vehicles - A Necessity Not Just a Possibility," AIAA Paper No.

insight into the effects of uncertainty on the closed-loop 93-3761, to be presented at the Guidance, Navigation performance was gained by these criteria. However, the IS and Control Conference, Monterey, August, 1993.

analysis method was able to indicate an accurate "performance robustness margin" that measured the Doyle, J., Glover, K., et. al, "State-Space Solutions to allowable magnitude of the interactions from the airframe I 11 l Standard H 2 and H** Control Problems," proceedings of such that acceptable tracking performance in the engine the American Control Conference, Atlanta, June, 1988, was still assured. The IS analysis method also indicated pp. 1691-1696.

critical frequency ranges where the airframe tracking performance would be most affected by uncertainty in the It 21 Bacon, B., Schmidt, D., "Multivariable Frequency- interactions. Finally, this analysis method also correctly Weighted Order Reduction," Journal of Guidance, Voi.

indicated that disturbances from the engine's fan speed 12, No. 1, Jan.-Feb., 1989, pp. 97-107.

command could be significant on the closed-loop airframe Doyle, J., Wall, J., Stein, G., "Performance and responses. [ 131 Robustness Analysis for Structured Uncertainty," proceedings of the IEEE Conference on Decision and Acknowledgments Control, Orlando, 1982, pp. 629-636.

This work is sponsored by the NASA Lewis Research Center under Grant # NAG3-998. Dr. Sanjay Garg is the technical program manager.