Chapter t h r e e d e t a i l s t h e s t e p s used i n l i n e a r i z i n g t h e inner-loop
The r e p o r t is organized i n t o s i x chapters. Chapter two p r e s e n t s nonlinear block diagrams of t h e inner-loop c o n t r o l laws and t h e corre- sponding l i n e a r time-invariant equations of t h e inner-loop models.
Chapter t h r e e d e t a i l s t h e s t e p s used i n l i n e a r i z i n g t h e inner-loop c o n t r o l laws. Chapter four combines t h e l i n e a r inner-loop models i n t o t h e continuous-time p l a n t r e p r e s e n t a t i o n shown i n Eq. 1. The p r o p e r t i e s of p r o p o r t i o n a l p l u s i n t e g r a l feedback of a c c e l e r a t i o n e r r o r i n t h e inner-loop c o n t r o l l a w s a r e a l s o discussed i n Chapter four. Chapter f i v e p r e s e n t s simulations comparing l i n e a r model time h i s t o r i e s with nonlinear model time h i s t o r i e s . The r e p o r t is summarized i n Chapter s i x . Appendix A compares nonlinear ACSL simulations with a r e c e n t l y developed ATOPS B-737 FORTRAN simulation.
11. INNER-LOOP MODELS A. AUTOTHROTTLE The autothrottle inner-loop is the most complicated of the control designs. Two error signals are computed and compared to threshold values. Control feedback paths are switched on and off as the error signals cross the threshold values. A windshear estimator is used to produce feedback for windshear compensation. The versine of the roll angle is fed back to advance the throttle just as the airplane rolls into a turn. The roll angle signal is washed out to prevent throttle advance in a steady turn. A block diagram of the autothrottle inner- loop control system is shown in Fig. 1 . The gains in the block diagram are defined in Table 1.
A detailed explanation of the windshear estimator is given in Ref.
3. Basically, true airspeed, TAS and longitudinal acceleration, irGs are employed in a complementary filter to produce $ where CF VW is the wind velocity making fr a washed-out estimate of wind.
CF +CF is next passed through a wind turbulence filter to filter out high frequency components in v CF ' h k12 +kll A 2 - - -
Bound - (icF - vcF)
ircF s -kll
The negative wind shear estimate,cCp, is subtracted from tGS to form
the acceleration feedback sisnal.
The acceleration feedback signal is to form the command error.
subtracted from the acceleration command, % C ' The integral of the command error becomes one of the feedback paths to the incremental throttle command 8 TC " (EPR) approaches the maximum safe If the engine pressure ratio value forEPR(lZXEPR), special logic, (switch B as shown in Fig. 1 ) switches The effect is to the EPR error signal to input the throttle integrator.
cause a decrease in the incremental throttle command. Switch A causes a throttle down error command to immediately decrease the rate of change of the throttle integrator.
A linear model for autothrottle is as follows
[ : ; I
" w o
kl*k7 sin -kl/k2 sin The state A c i s t h e i n t e g r a l of p e r t u r b a t i o n command e r r o r . The s t a t e X A$wo i s t h e p e r t u r b a t i o n washed o u t v e r s i n e r o l l c o r r e c t i o n f a c t o r . The a c c e l e r a t i o n measurement, is t h e p e r t u r b a t i o n r a t e of change of ground speed. AVTAS i s t h e measured p e r t u r b a t i o n t r u e a i r s p e e d . A 4 is t h e measured p e r t u r b a t i o n r o l l a n g l e and A% is t h e p e r t u r b a t i o n o u t e r - C loop command. I n s t r a i g h t and l e v e l f l i g h t , t h e v e r s i n e r o l l c o r r e c t i o n s t a t e can be e l i m i n a t e d . The v a r i a b l e , Y, i n Eq. 7 i s an o p t i o n t o account f o r switch A a s follows
e = MXEPR - 0.1 - EPR
EPR 0
k4 e~~~ ' k3
0.0 < eEpR < k3 (k4 /k3 *eEpR 0.0 e < 0 . 0 EPR OPTION 1, A > 0, Y = Z (11) OPTION 2 , A < 0, Y = 1.0 (12) For a given t r i m f l i g h t c o n d i t i o n s , outer-loop g a i n s could be determined f i r s t f o r OPTION 1 t h e n f o r OPTION 2 . I f n o t i c a b l e changes a r e a p p a r a n t , t h e e f f e c t of A may have t o be accommodated i n t h e outer-loop c o n t r o l l a w . The n o n l i n e a r bounds caused by t h e g a i n s kg, kll, k13 and k a r e n e g l e c t e d when t h e l i n e a r model is c o n s t r u c t e d i n Eq. 7. An approach u s i n g d e s c r i b i n g f u n c t i o n s shown l a t e r i n Eq. 1 8 could be used t o model The v a r i a b l e s AV n o n l i n e a r bounds i n t h e l i n e a r p l a n t r e p r e s e n t a t i o n .
GS ' AvTAs and Av a r e measurement n o i s e .
cp B. VERTICAL PATH A block diagram of t h e v e r t i c a l p a t h inner-loop f o r t h e e l e v a t o r is shown i n Fig. 2. The g a i n s a r e given i n Table 2. N e i t h e r t h e e l e v a t o r inner-loop, n o r t h e a u t o t h r o t t l e inner-loop employ p i t c h a t t i t u d e feedback f o r s t a b i l i t y d u r i n g p a t h t r a c k i n g . Reference 3 e x p l a i n s t h a t r c p l a c i n g a p i t c h a t t i t u d e command system w i t h a v e r t i c a l a c c e l e r a t i o n command system allows t h e a i r c r a f t t o weather cock v e r t i c a l l y upon encountering v e r t i c a l g u s t s and s h e a r s s o t h a t g l i d e s l o p e beam t r a c k i n g performance can be enhanced. Other t y p e s of c o n t r o l d e s i g n procedures can produce an e l e v a t o r c o n t r o l system t h a t weather cocks, employs p i t c h a t t i t u d e feedback, and has good g l i d e s l o p e t r a c k i n g performance a s d i s c u s s e d i n Ref. 9.
The e l e v a t o r inner-loop f i l t e r s t h e v e r t i c a l a c c e l e r a t i o n measure- ment, Forms t h e a c c e l e r a t i o n command e r r o r , then f e e d s t h e e r r o r s i g n a l i n p r o p o r t i o n a l - i n t e g r a l form t o t h e e l e v a t o r . The i n t e g r a t o r o u t p u t is p o s i t i o n l i m i t e d . Washed-out p i t c h r a t e is fed back f o r improved s t a b i l i t y . The v e r t i c a l a c c e l e r a t i o n command is p o s i t i o n l i m i t e d t o i n s u r e passenger comfort. The feedback s i g n a l t o e l e v a t o r is m u l t i p l i e d by a g a i n t h a t d e c r c a s c s w i t h i n c r e a s i n g c a l i b r a t e d a i r s p e e d . The l i n e a r model f o r e l e v a t o r is The perturbation states are the integral of command error, Ach, filtered h
vertical acceleration, AG, and washed out pitch rate, . The measure-
A % o
ments are pitch rate, A g , and INS (Invertial Naviation System) vertical acceleration, A ; . The outer-loop control input is the vertical acceler- ation command, A$. The limit on hhc could be handled similiar to Eq. 18.
C. HORIZONTAL PATH The horizontal path inner-loop feeds back roll rate for stability augmentation and the error between the outer-loop roll angle command and roll angle. A block diagram of the control law is shown in Fig. 3. The gain values are defined in Table 3 . The feedback signal which forms the aileron actuator command is multiplied by the gain k which is a function v of calibrated airspeed, CAS, as shown in Fig. 3. The roll command signal is limited.
A linear model of the horizontal path inner-loop which accounts for the roll command rate limit is The perturbation states are filtered roll rate, A$, and filtered roll h command, A . The measurementas are roll rate, Ap, and roll angle A $ .
The outer-loop control input is the roll command, A $ , Using describing functions, if A$C changes abruptly then The advantage of allowing J to be variable is that outer-loop guidance gains can be designed for different values of A.
D. RUDDER The Boeing inner-loop control system for rudder is the yaw damper shown in Fig. 4. The gains are shown in Table 4 . Body axis yaw rate is filtered to suppress measurement noise. The filtered yaw rate is multiplied by a gain which is a function of CAS as shown in Table 5.
The filtered yaw to decrease the gain as airspeed increases.
rate signal is washed out for turn coordination then position limited to reduce control authority. Outer-loop commands directly actuate the ruddqr surface. The Boeing outer-loop control design comands rudder for decrab during landings.
The linear model of the inner-loop is
-1'0/k42 -I* Olk4 [ : ] + [ o
Avr) (19) 0.0 -1.0/k4 K u ~ * k ~ ~ / k ~ ,
= [1.0 1 . 0 1 + [O.O][Ar + Avr] + [1.0 (20)
A 6 ~ The position limits in Fig. 4 could similarily be incorporated in the linear model using Eq. 18 if this is deemed desirable. The perturbation states in the linear model are filtered yaw rate, A?, and washed out yaw rate, Arm.
The measurement is yaw rate, Ar. The outer-loop control is Au RC' 111. LINEAR MODEL ANALYSIS A . LINEARIZATION OF FEEDBACK ELEMENTS The Boeing inner-loop system uses the derivative of ground speed, ..
true airspeed, and vertical acceleration, h as feedback ' G S ' ' T A S elements. Linear analyisis requires that these elements be linearized and expressed in terms of the perturbed states and controls of the air- craft.
This section derives the perturbation relationships.
Figure 5 shows the relationships between the accelerations of the vehicle in a local-level north pointing frame and the along track and cross track accelerations. The north pointing frame is denoted as the geographic coordinate system. In the figure, V is the ground speed, a G
is the vehicle acceleration in the horizontal plane, and 5 is the ground
track angle.
From the figure it follows that
tan = f V , = (t + 92)4
ir '
The along track, a and cross track, a accelerations are related ATK' CTK3 to 2 and through the transformation,
Expressing cosc and sin5 in terms of i and 9 the following occurs
+ +y
- -
a ATK V~
ky - $%
a = CTK
v~
Note that taking the derivative of V in Eq. Zlb, shows that V and aATR G G ,. 24 produces Eq. 2 3 are equal as required. Perturbing which is rewritten in matrix form as
6 , = H Aic + H . . ls
x - x - (26) B Let HG($,8,$) be the transformation from body axes to geographic axes where $, 0, and I/J are the Euler angles (platform axes and geographic axes are assumed to coincide).
It follows that rn where and The vector wG represents the body axis angular rates in radians, (a flat B nonrotating earth is assumed) - G w is the matrix representation of the vector cross product and is given B by Perturbing Eq. 28 produces where and Perturbing Eq. 30 produces
A expression for AkB can be determined from the linear aerodynamic model
of the aircraft,
Extracting the equation for AiB from Eq. 37 produces
The states, Aw - are gust disturbances. Substituting Eq. 37 into Eq. 36 results in Substituting Eq. 33 and Eq. 39 into Eq. 36 determines the desired equation for fl in terms of the aircraft perturbation states and controls, G G
~i~ = H AV + H d B + H AV + D AU + D Aw
w - B w - B vu - V W -
where A similiar expression for ~h is obtained by noting that where
Hi;= [ 0 0 - 11
Substituting Eq. 39 into Eq. 51 determines the desired perturbation ..
expression for Ah in terms of the perturbation aircraft states and controls.
True airspeed is the velocity of the vehicle relative to the atmosphere, The vector w . + represents the 3-axis steady state wind above the earth's surface and is not modeled in this analysis. From Eq. 54, the perturbed value for AVTAS is where
Substituting the perturbed value for A I A obtained from Eq. 56 produces
the final expression for AVTAS, (59)
AVTAS = % A A x B - % A AX
a.
The final expressions relating the perturbed values for A $ G ' Ah, and AVTAS and the linear aircraft model are determined by distributing the elements in Eqs. 59, 53, and 40 into Eq. 77 for the chosen order of A x and A x in Eqs. 78 and 79.
B . GUST MODEL The gust terms in the model are of a random nature and can be Modeled using the well-known Dyrden spectrum, Ref. 10. The modeling effort consists of using spectral factorization methods to obtain a dynamical system which generates a random process having the specified power spectral density when driven by a white noise process, Ref. 11.
The transfer functions Rotational gusts around the aircraft are ignored.
for the gusts are as follows, The airspeed, V is defined in Eq. 54, LU, Lv, and L are the scales TAS ' W of turbulence, and o and o are the variance of the gust. The up OV' W scales and gust variance are shown in Table 6.
A state-space realization of Eqs. 6 0 to 62 is q in Eq. 64 is a 3-vector of independent Gaussian white noise processes - with unit variance.
IV. A LINEAR MULTIVARIABLE INNER-LOOP MODEL USEFUL FOR OUTER-LOOP DESIGN The block diagrams for the inner-loops in Figs. 1, 2, 3, and 4 are relatively straight forward and easily transformed into a nonlinear simulation using the ACSL programming system. Formulating a linear multivariable model for the inner-loop system which is useful for linear simulations and linear analysis is the purpose of this section. The acceleration feedback in the inner-loop complicates the derivation.
Each inner-loop linear model discussed in Chapter I1 can be placed in the following form A & = ElAx + E2[AyIL + AxIL] + E 3 A .
- E The elevator inner-loop model is used as an example. The other inner- loop linear models representations use A, R, or T in place of E in Eqs. 68 and 69 for aileron, rudder, or throttle inner-loops. The states of the inner-loop dyanmic models shown in Chapter I11 are as follows The vector Au represents the commands to the inner-loop system -C These commands become the new controls when the inner-loop control system closes the loop around the aircraft dynamics. The vector A y represents the inner-loop measurements used for feedback, A~' = [Ap Aq A A AVGs
A v ~ ~ s ~h I
The last three inner-loop measurements, As, are expressed in terms of the aircraft states, controls, and gust in Chapter 111, The vector A x represents the states of aircraft, in body axes,
AX^ - = [Au Aw Aq A0 Av Ap A A A Ax Ay Az] (78)
The vector, AX, represents the aircraft controls, The vector Av in Eqs. 68 and 69 are white zero-mean Gaussian noise -1L states representing the measurement noises for the sensors used in the inner-loops. The vector has the following covariance T 2 2 2 2 2 a 2 2 E{AV AV I = DIAGONAL [GAP 'JAq Gnr OA4 0 -1 = VIL (80) Ah -1L -1L "TAS The expressiona is the standard deviation for the measurement noise A P The other variables in Eq. 78 are standard of the roll rate gyro.
2 1 deviations of the measurement noises for the pitch ratio gyro, yaw rate gyro, roll angle from the INS (Inertial Navigation System), inertial , along track acceleration from the INS, airspeed sensor, and vertical acceleration from the INS. The actuator command vector, A s , is composed of elements from each of the inner-loop control systems discussed in Chapter 11, The model for the linearized actuator dynamics is discussed in Ref. 13, and has the following form, In Eq. 79, AsTH is thrust while A6TC in Eq. 80 is the throttle command.
If all the inner-loop models are combined, the result is as follows, Table 7 shows how the matrices in Eqs. 84 to 86 are constructed.
In the rest of the derivation, the inner-loop control system is closed around the aircraft perturbation dyanmics. Substituting Eq. 86 into Eq. 85 and the result into Eq. 81, one obtains w h e r e T T T
A x = [nxT A % < AzIL]
- - P K1l = [ G U H ~ ~ C ~ ~ G u H ~ ~ D ~ w Fu G u H ~ ~ l - K21 - G u H ~ ~ K31 = G u H ~ ~ K41 = G H D u I Y I L D e f i n i n g -1
Z = (I - K41)
u Eq. 87 reduces t o
A s = K A x + K A u + K3AxIL
1 - p 2-2 w h e r e - - - K1-ZuKll, K 2 - Z U K 2 1 ' K 3 - - Z u K j l R e w r i t i n g Eq. 86 as
AxIL = K A x + K A u
4~ --P 5 Y w h e r e K = [CIL 4Y O D~~ K5y = D~~ and s u b s t i t u t i n g f o r A u - f r o m Eq. 94 r e s u l t s i n
AxIL = K A x + K A u + K6AvIL
4 7 . 3 5-c where K 4 = K + K K (100) 4 Y 5~ 1 = K K (101) 5 y 2 K 6 = K K (102) 5 Y 3 Substituting Eq. 99 into Eq. 83 and regrouping produces
A i I L = [ A + B K ] A x + [BIy + BIyK6 1 A I L + [BIG + B1yK5 1 A s (103)
1 I Y 4 -p where Substituting Eq. 99 into Eq. 85 produces (105)
A s - c = [H1 + HIyK41 AX + [HIy + HI$6] AxIL + [HIC + HIyK5 1 A +
7' where 82 one obtains Substituting the above into Eq.
where A2 = [0 0 A 01 U The aircraft dynamics satisfy the equation
Ak - = AAx - + BAu
- Substituting Eq. 94 into Eq. 109 produces
Ak - = [A, + BK ]Ax
+ BK3AxIL + BK2Azc
1 --P where Combining Eqs. 110, 6 4 , 107, and 103 determines the desired closed-loop model using the inner-loop control system for the linear aircraft dynamics Equation 112 is the expanded version of Eq. 1 discussed in the introduction.
A. FEATURES OF ACCELERATION FEEDBACK The Boeing inner-loop control system feeds back the integral of the acceleration error qunatities to improve stability and tracking. The properties of this type of feed- back can be studied by investigating a simple scalar system with the control law The closed-loop system for the scalar plant is The two rows in the closed-loop plant matrix in Eq. 118 are linearly dependent implying that one of the closed-loop eigenvalues is always zero. Two of the four zero eigenvalues in Table 9 are caused by the way integral feedback is used in the Boeing inner-loop control law.
The steady-state tracking ability of the control law can be investigated using the model when 2 i s a constant, an expression f o r 6 can be obtained from B q . 116.
C 119 using 6 i s The closed-loop system f o r Eq.
The steady s t a t e value f o r it is given by Unless a is zero, t h e r e is no value f o r k which makes 2 equal t o f c i n steady s t a t e . A s t e p command f o r ~h i n Chapter I V shows t h a t p e r f e c t C steady s t a t e t r a c k i n g is not obtained i n simulation even though i n t e g r a l feedback is employed i n t h e Boeing inner-loop c o n t r o l system.
V . INNER-LOOP MODEL VERFICATION The purpose of this chapter is to compare time histories between the linear inner-loop closed-loop models and the nonlinear ACSL simu- lation model which includes nonlinear models of the inner-loops control systems. A number of options are available when the linear model is constructed. Table 8 shows the options and the recommended settings currently employed.
The effect of varying the engine dynamics EPR time constant is investigated in the longitudinal dynamics verification. The effects of the lead/lag filter and spoiler aerodynamics are investigaged in the lateral-directional dyanmics verification. The choice of the first two options shown in Table 8, remain unresolved.
A. LONGITUDINAL The simulation comparison has two purposes. The first purpose is computer coding verification. The second purpose is to identify non- linearities which cause a significant descrepancy between linear and nonlinear dynamics.
The autothrottle has a significant nonlinearity in the upper and lower saturation limits for the rate of change of EPR (engine pressure ratio) as discussed in Ref. 12. The effect of the saturation limit is to decrease the EPR dynamics time constant in a manner similiar to Eq. 18.
Unsaturated, the EPR dynamics time constant is -0.2 sec. A time constant of -2.0 sec is used in Ref. 12. Figure 6 shows the effect of rEpR for -0.2, -0.5, and -2.0. The simulation comparisons in Fig. 6 are in good agreement. The recommended T is -0.2, i.e., no saturation effect EPR is needed in the linear model.
Originally there were significant mismatch between linear and non- linear autothrottle simulations. The mismatch was traced to a subtle error in the calculation of the upper saturation limit of EPR in the ACSL nonlinear simulation, which was subsequently corrected. The error in the ACSL program existed, but apparantly did not affect the results in Ref. 12.
The time history comparisons for a ~h command is shown in Fig. 7 C and has excellent agreement. The Ah response is also shown in Fig. 7 and as discussed in Chapter 111, the steady state error is not zero even though the inner-loop control system feeds back the integral of the command error.
The inner-loop closed-loop eigenvalues for the longitudinal system are shown in Table 9. The phugoid mode and the short period are stable and well damped. The effect of changing T has almost no effect on EPR the eigenvalues. Residualizing the elevator actuator dynamics primarily
affects the eigenvalue for the ~i filter by further stabilizing its
value. The two zero eigenvalues for ASx and AS are a feature of feeding
h back the integral of acceleration as discussed in Chapter 111. The states associated with the gust model (Awgl, Awg2, and Aw ) and the wind shear A A g3 estimator A and A are uncontrollable and have eigenvalues which remain fixed for variations in the aircraft dynamics.
B. LATERAL-DIRECTIONAL The largest discrepancy between linear and nonlinear models occurs for the aileron inner-loop control system. The discrepancy is caused by the highly nonlinear aileron/spoiler actuator system discussed in Ref. 12.
The linear model for the inner-loop control system and actuator is con- structed so that any subset of the nonlinearities can be represented in the model.
Three linear models are simulated for step A$c commands of 2.0 deg and are shown in Fig. 8 . The roll command is large enough so that spoiler is activated when A+c = 2.0 deg, but when A$ returns to zero, little C spoiler is used. The assumptions in constructing the three linear models are discussed in Table 10, which shows the closed-loop eigenvalues.
The closed-loop eigenvalues indicate that the Dutch Roll mode, spiral mode, and roll mode are very stable. In CASE 1, the slow actuator eigen- value, -0.975, is deceptive since it can be shown that the eigenvalue is almost cancelled by a zero of nearly equal value. The aileron actuator in CASE 1 responds almsot immediately to the commanded aileron value as long as the value has a finite rate. The simulations in Ref. 12 for step (% infinite rate) commands in the aileron actuator can be misleading in this regard.
CASE 2 models the aileron actuator as it existed before many of the mechanical nonlinearities were added. The fast aileron actuator mode forms a complex pair with the yaw rate filter mode. The roll mode becomes more stable.
CASE 3 includes the effect of spoiler. In the nonlinear model, spoiler is activated after the aileron actuator surface commands exceeds certain values (Qj 2.2 d e g ) . Including the effect of spoiler when A$ is C small produces incorrect results. Not including spoiler when A$ is of C moderate value may not produce correct results.
The simulations in Fig. 8 show that CASE 1 and 2 match the non- linear simulation for small A$ commands. Using a A$ of -25 degs and C C Eq. 18, Fig. 9 shows the descrepancies that occur as A$ is increased.
C The effect of the saturation limit in reducing the roll response rate limit for large roll commands is clearly evident in Fig. 9. The roll command saturation limits are apparantly not used in the simulations in Ref. 6 where -25 deg roll commands are used to test for limit cycles.
Rudder commands of 2.0 deg are simulated in Fig. 10 for two cases.
The rudder linear and nonlinear response match well. Continued problems with the linear model for the aileron/spoiler actuator system is evident in the aileron response in Fig. 10. Including spoiler feedback increases the aileron command control effectiveness.
VI. S U M M A R Y The l i n e a r and nonlinear inner-loop c o n t r o l system models t o be used i n a 3-D R/NAV outer-loop c o n t r o l s y n t h e s i s problem a r e presented i n t h i s report. Most of t h e modes with f a s t eigenvalues a r e r e t a i n e d i n t h e l i n e a r model s i n c e they do not pose a problem t o t h e outer-loop (limited s t a t e feedback) c o n t r o l s y n t h e s i s procedure. Closed-loop eigenvalues f o r t h e inner-loop c o n t r o l system discussed i n t h e r e p o r t show t h a t a l l inner- loop complex modes a r e w e l l damped and inner-loop r e a l modes a r e acceptably s t a b l e .
Small s t e p commands i n each of t h e outer-loop c o n t r o l v a r i a b l e s show good agreement between l i n e a r and nonlinear model time h i s t o r i e s .
The r o l l inner-loop c o n t r o l system is i d e n t i f i e d a s t h e model with t h e most descrepancy primarily because of t h e highly nonlinear a i l e r o n / s p o i l e r a c t u a t o r .
REFERENCES Halyo, Nesim, "Development of an Advanced 3D Guidance and Control Law for Curved Path Trackingq', ICS Technical Proposal, September 1978.
BoeingDocument D6-32669,"Flight Critical Control Laws for the NASA Terminal Controlled Vehicle," September 1975.
Boeing Document D6-41565, !"Guidance Algorithms and Non-Critical Control Laws for ADEDS and the AGCS", May 1974.
Boeing Document D6-32686-1 "Terminal Configured Vehicle (TCV) B-737 Navigation Computer Software Description", NASA Langley, Hampton, VA August, 1978.
Boeing Document D6-34279, "NASA 515 Flight Control System Description- RSFS ~ircraft", September 1976.
" NASA 515 Roll Attitude Loop Stability
Boeing Document D6.-42695, Investigations", February 1976.
Margolis, S. P., Wolverton, D. A., H a m , R . W., "Flight Control Computer Software Descriptionsq',CSC Number 33509, Computer Sciences Corporation, Hampton, VA, July 1979.
Broussard, J. R, and Glasson, D. P., "Optimal Multirate Flight Control System Design", JACC, San Francisco, August 1980.
Broussard, J. R., "Design, Implementation and Flight Testing of PIF Autopilot Designs for General Aviation ~ircraft", Information & Control Systems, Inc., Report No. 681102, October 1981.
Martin, D. J., "~eal-Time Simulation of Atmospheric Turbulenceq1, Masters Thesis, The George Washington University, May 1979.
Halyo, Nesim, "~evelopment of a Digital Guidance and Control Law for Steep Approach Automatic Landings Using Modem Control Tech- niques", NASA CR-3074, February 1979.
Broussard, J. R., and Stallman, S. T., "Modification and Verification of an ACSL Simulation of the ATOPS B-737 Research Acrcraft", NASA CR-166049, February 1983.
APPENDIX A
APPENDIX A Recently, an alternative FORTRAN simulation of the ATOPS B-737 aircraft was made available to ICS by NASA. As part of the verifica- tion of the new simulation, inner-loop step commands time histories computed using the ACSL simulation were similarily computed using the FORTRAN simulation. The descrepancies between the simulations were identified and partially eliminated.
Figures 11 to 14 show the comparisons for the same step commands made in Figs. 6, 7, 8 and 10.
The most notable difference is for the autothrottle control system in Fig. 11. The aileron response difference in Fig. 13 is caused by the fact that the FORTRAN simulation at the time the simulation was per- formed did not include the leadllag compensator and did have spoiler feedback. In Fig. 14, the FORTRAN simulation included the lead/lag compensator and spoiler feedback and the aileron responses are in reasonable agreement.
AUTOTHROTTLE BLOCK D I A G R M G A I N VALUES TABLE 1.
-
G A I N VALUE 5.0 kl 16.0 k2 0.3 1.0 k4 10.0 2.0 1.5 60.0 1.2 5.0 k10 1.0 kll TABLE 2.
ELEVATOR B L O C K DIAGRAM GAIN VALUES GAIN VALUE 296.5 k14 1 . 0 k15 0.275 k16 120.0 k17 360.0 k18 20.0 k19 0 . 1 k20 5 . 0 k21 4 . 0 k22 80.0 k23 0.25 k2 4 0.25 k25 0.004 k26 2.16 k2 7 16 . O k28 10.0 k29 62.4 k30 TABLE 3.
AILERON/SPOILER BLOCK DIAGRAM GAIN VALUES P GAIN VALUE 20.0 k31 0.05 k32 1 . 4 k33 50.0 k34 25.0 k35 5 . 0 k36 4 . 0 k37 2 . 0 k38 97.66 k39 TABLE 4 .
RUDDER BLOCK DIAGRAM GAIN VALUES t GAIN VALUE 2 . 3 1 k40 0.143 k41 3 . 3 3 k42 1 . 0 k43 4 . 0 k44 TABLE 5 BREAKPOINTS FOR YAW DAMPER GAIN VERSUS AIRSPEED INPUT CAS 100.00 122.4 150.0 206.0 450.0 (kts) OUTPUT KYD 1.0 0.765 0.61 0.395 0.31 TABLE 6 SCALES AND VARIANCE FOR GUST MODELS ALTITUDE h (ft) 0-60 60-328 328-1750 >I750 a (ft/sec) 16.0 60.72h- u a (ft/sec) 12.7 26.50h- v a (ft/sec) W 560.0 560.0 LU (ft) 320.0 102.17h Lv (ft) Lw (ft) 174.0 12.65h F T 1 O 0 0 0 E l 0 0 0 O A l O O O O R 1 T 4 0 0 0 0 E 4 0 0 0 0 A 4 0 O O O R 4
H~~ = 1 . 1
0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 'a aw INNER-LOOP CONTROL SYSTEM MATRICES TABLE 7.
I OPTIONS RECOMMENDED VALUE 1
INCLUDE :SPOILER FEEDBACK CURRENTLY 0PTIMA.L CURRENTLY OPTIMAL ENGINE EPR TIME CONSTANT -0.2 SEC RESIDUALIZE SPOILER ACTUATOR STATE TRUE INCLUDE EFFECTS OF PCU RATE LIMITS FALSE INCLUDE EFFECTS OF BACKLASH IN AILERON ACTUATOR MODEL FALSE RESIDUALIZE RUDDER ACTUATOR STATE TRUE RESIDUALIZE ELEVATOR ACTUATOR STATE FALSE INCLUDE EFFECT OF EPR REDUCTION IN CHOOSING Y IN EQ. 11 FALSE, Y = 1.0 RESIDUALIZE A 5 STATE FALSE RESIDUALIZE A : : STATE FALSE RESIDUALIZE A? STATE FALSE INCLUDE'SATURATION DESCRIBING FUNCTION FACTOR IN A+c FALSE INCLUDE SATURATION DESCRIBING FUNCTION FACTOR IN A g C FALSE INCLUDE SATURATZON DESCRIBING FUNCTION FACTOR IN Ahc FALSE TABLE 8 OPTIONS IN CONSTRUCTING THE INNER-LOOP MODEL I I
T EpR=-O. 2 T EpR=-O. 5 EPR=-0.5 ,C6e -
CASE L EpR=-2. 0 ACTUATOR STATE RESIDUALIZED A x 0.0 0.0 0.0 0.0 AZ 0.0 0.0 0.0 0.0 i 0.0 0.0 0.0 0.0 * t X 0.0 0.0 0.0 0.0 PHUGOID -0.0572j0.03 -0.057kj0.03 -0.057kj0.03 -0.058Cj0.03 MODE ~ ~ = 0 . 0 6 5 , [ = 0 . 8 7 8 ~ ~ = 0 . 0 6 5 , 5 = 0 . 8 7 8 ~ = 0 . 0 6 5 , 5 = 0 . & 7 8 ~ ~ = 0 . 0 6 6 , 5 = 0 . 8 7 8
t
-0.138 -0.138 -0.138 -0.138 &,l -0.149 -0.149 -0.149 -0.149 &g2 -0.149 -0.149 -0.149 -0.149 &g3 , .
-0.20 -0.20 -0.20 -0.20 A ~ C f -0.242 -0.254 -0.254 -0.335 j0.155* 4 0 -5.53 -2.11 -2.11 ~,=0.369, =0.907* *%PR SHORT PERIOD -1.772j1.94 -1.772j1.93 -1.782j1.86 -1.76kj1.94 MODE ~,,=2.62,5=0.675 %=2.63,5=0.673 y,-2=2.6O,~=O9693 a,,=2.62,5=0675 -5.0 -5.0 -5 .O -5.0 A%, ?: A h -7.39 -7.39 -8.49 -7.39
I
A 6 -23.5 -23.5 --- -23.5
e
I
* Two Reals Formed a Complex P a i r
INNER-LOOP LOGTITUDINAL CLOSED-LOOP EIGENVALUES TABLE 9 CASE 1 2 3 0.0 0.0 0.0 AY CASE 1 0.0 0.0 0.0 -0.149 -0.149 -0.149 NO SPOILBR AERODYNAMIC EFBECT . A w , l IN B MATRIX, LEADLAG COMPENSATOR -0.149 -0.149 -0.149 DYNAMICS INCLUDED IN A MATRIX AWg2 -0.273 -0.275 -0.397 " W O CASE 2 -0.975 -6.09+j2.13* -0.975 a NO SPOILER AERODYNAMIC EFFECT DUTCH -0.875+j0.605 -0.902+j0.574 -1.01fj0.24 IN B MATRIX, LEADLAG COMPENSATOR ROLL DYNAMICS NOT INCLUDED IN A MATRIX MODE w =1.06,5=0.822 w =1.07,5-0.843 u =1.04,5=0.973 n n n SPRIAL -1.6 -1.6 -1.6 CASE 3 MODE ROLL -2.73 -3.56 -4.58 SPOILER AERODYNAMIC EFFECT IN B MODE MATRIX, SPOILER ACTUATOR STATE IS ROLL -5.0 -5.00 -5.0 RESIDUALIZED, LEADUG COMPENSATOR COMMAND DYNAMICS INCLUDED IN A MATRIX -5.85 w =6.45,5=0.943*
Ai
n -10.5+j5.0* -17.4 -22.5 w n =11.6,5=0.903*
A 6
* Reals Combined to Form a Complex Pair
TABLE 10
INNER-LOOP LATERALIDIRECTIONAL CLOSED-LOOP EIGENVALUES
PI: W A u 6 ' ; P ( r O .
\ Z PI: W A ..
X a ATK F I G U R E 5 LOCAL L E V E L VECTOR DIAGRAM
-
20.0 FIGURE 6 SIMULATION RESPONSE F O R A AXc C O ~ A N D O F 2.0 FPS'.
.0.00 4.00 8.00 12.0 16;. 0 20.0 TIME (SEC) FIGURE 6 (CONTINUED) SIMULATION RESPONSE F O R A A% COMMAND O F 2.0 F P S ~ . c I J A C S L Ln rl I 0.00 4.00- 8.00 12.0 -- 16.0 20.0 TIME (SEC) b -' ....
I - -
0.00 4.00; 8 10d- 12: o 1 c 0 20.0
TIME (SEC) FIGURE 7 SIMULATION RESPONSE F O R A ~h~ C O M H R N D OF 2.0 FPS'.
LINEAR 4 . 0 0 6 100 8 r00"- 10.0 T I M E (SEC) LINEAR 9 LEAD/LAG NO SPOILER FEEDBAC T I M E (SEC) FIGURE.8 SIMULATION RESPONSE FOR A A$, COMMAND OF 2.0 DEG.
, d.
-a I FIGURE 9 SIMULATION RESPONSE FOR A A$, COMMAND OF -25.0 DEG.
TIME (SEC) , ACSL 8.00 12.0 TTMF I C r r \ w m o =A I ACSL m I V) I 0.00 4.00. 8.00. 12.0 TIME (SEC) T I M E (SEC) FIGURE 12 SIMULATION RESPONSE FOR A ~h~ COMMAND OF 2 . 0 FPS'.
0~1aWfiL PAGE @
OF. pOOR w a ~ \ f l
TIME (SEC) FIGURE 1 2 (CONTINUED) SIMULATION RESPONSE FOR A ahc COMMAND OF 2 . 0 F P S ~ .
o Q ~ Q ~ H L ~ Wi2E E % '
Q F ptmR 3 , v j A ~ m ACSL-RESPONSE 0.00 2.00 4.00 6.00 8.00 10.0 TIME (SEC) TIME (SEC) FIGURE 13.SIMULATION COMPARISON F O R A OC C O M M A N D O F 2.0 DEG.
a ~ R ~ @ N A L
pWR Q U A L ~ I 0.00 2.00 4.00 6.00 8.00 10.0 TIME (SEC) 0 .
Tf 0 b.
CU -0
E?
n -0 Z ACSL-RESPONSE 0.00 2.00 .4.00 6.00 % 8.00 10.0 TIME (SEC) FIGURE 1 4 SIMULATION COMPARISON F O R A U , C O M M A N D O F 2.0 DEG.
ATOPS B - 7 3 7 INNER-LOOP CONTROL SYSTEM LINEAR MODEL CONSTRUCTION A N D VERIFICATION JOHN R. BROUSSARD SYSTEMS, INC.
28 RESEARCH DRIVE HAMPTON, VA 23666 6 SPACE ADMINISTRATION WASHINGTON, DC 20546 LANGLEY TECHNICAL MONITOR: RICHARD M. HUESCHEN INTERIM REPORT Nonlinear models and b l o c k diagrams of an i n n e r - l o o p c o n t r o l s y s t e f o r t h e ATOPS B-737 Research A i r c r a f t a r e p r e s e n t e d . Continuous- time l i n e a r model r e p r e s e n t a t i o n s of t h e n o n l i n e a r i n n e r - l o o p c o n t r o l system a r e d e r i v e d . C l o s e d - l o o p a i r c r a f t s i m u l a t i o n s comparing n o n l i n e a r and l i n e a r dynamic r e s p o n s e s t o s t e p i n p u t s a r e used t o v e r i f y t h e i n n e r - loop c o n t r o l system models.
UNCLASSIFIED - UNLIMITED AIRCRAFT SIMULATION SUBJECT CATEGORY 08 For sale by the National Technical Informalion Service, Springfield, Virginia 22161