Document
NASA Technical Memorandum 86801 NASA- TM-8680 1 t 9860016867
Decoupling Control Synthesis
for an Oblique-Wing Aircraft
Gurbu)( s. Alag, Robert W. Kempel, and Joseph W. Pahle
June 1986 I ., '. q '<'\U' ..- _ . (_ .... v .LANGLEY RES!::ARCH CENTER LIBRARY, NASA HA'.~PTOUt VIRGIt/IA
NI\5/\
National Aeronautics and Space Administration 111111111111111111111111111111111111111111111 NF00052 3 1176 01308 6724 NASA Technical Memorandum 86801
Decoupling Control Synthesis
for an Oblique-Wing Aircraft
Gurbux S Alag Western Michigan University, Kalamazoo, Michigan Robert W Kempel and Joseph W Pahle Ames Research Center, Dryden Flight Research FacIlity, Edwards, California
NI\S/\
National Aeronautics and Space Administration Ames Research Center Dryden Flight Research FacIlity Edwards. California 93523 DECOUPLING CONTROL SYNTHESIS FOR AN OBLIQUE-WING AIRCRAFT Gurbux S. Alag* Western M1chigan University Kalamazoo, Michigan and Robert W. Kempel** and Joseph W. Pahle** NASA Ames Research Center Dryden Fl1ght Research Fac1lity Edwards, Californ1a Abstract s complex frequency Interest 1n obl1que-wing a1rcraft has surfaced u input vector periodically since the 1940's. Th1S concept offers some substant1al aerodynam1c performance specified components of eigenvector v advantages but also has slgnificant aerodynam1c and 1nert1al cross-coupling between the a1rcraft w vector m-d1mensional 10ngitud1nal and lateral-direct10nal axes. This paper presents a techn1que for synthes1zing a x state vector decoupling controller wh1le prov1ding the des1red stabil1ty augmentation. y output vector The proposed synthes1s procedure uses the e1genvector achievable for specif1ed Z concept of a real model-followlng control sys- components tem. Feedforward gains are selected on the assumpt10n that perfect model-following cond1- a angle of attack, deg tions are sat1sfied. The feedback ga1ns are obtained by uS1ng eigensystem ass1gnment, and sldesl1p angle, deg the a1rcraft lS stab1l1zed by uS1ng part1al state feedback. The effectlveness of the control laws control surface deflection developed in achieving the des1red decoupl1ng lS e p1tch angle, deg lllustrated by appllcat10n to llnear1zed equat10ns of mot10n of an obl1que-wing a1rcraft for a glven flight cond1t10n. eigenvalue bank angle, deg 4> Nomenclature ( ••• )R1 reorder1ng operat10n A,B,C system matrlces Subscr1pts d e1genvector of components, unspec1f1ed aL left aileron e error aR ri ght ail eron ident1ty matrix 1 eft horlZonta 1 hL J r1ght hor1zontal hR K feedback ga 1n 1th value L,M matrlces of appropr1ate d1mens1on m model OWRA obl1que-w1ng research a1rcraft plant (aircraft) P roll rate, deg/sec p set of real numbers R p1tch rate, deg/sec q u 1nput vector RMF real model-follow1ng x state vector yaw rate, deg/sec r *Associate Professor, Electrical Eng1neering d desired value Department.
**Aerospace Eng1neer.
number of outputs m number of Inputs In thIS paper, the development of control laws for OWRA by integration of the two above-mentioned n number of states technIques lS demonstrated. The results show the effectIveness of the controller In obtaInIng the t pseudo-Inverse decoupled response for a gIven flIght condltlon and wing skew.
IntroductlOn Problem DefInItion The advantages of an oblIque wIng were fIrst noted In the 1940's. However, only recently has The concept of model-followlng IS useful when the Interest, technology, and mIssIon of an an ldeal set of plant equatIons of motIon can be oblIque-WIng desIgn evolved Into a full-scale speCIfIed. The ldeal obJective of model-follOWIng flIght research program. The NASA Ames Research flIght control lS to force the aircraft to respond Center and the U.S. Navy are developIng an as the model would to a gIven pIlot command. It is often deSIrable to SImulate the model dynamICs oblIque-WIng research aIrcraft (OWRA). Gregory1 has outlIned a number of potentIal advantages and In the flIght computer and to generate the aIr- dIsadvantages of thIS type of aIrplane. Theo- craft control SIgnal uSIng the aIrcraft outputs, retIcal and WInd-tunnel studIes have shown that a the pIlot Input commands, and the model states.
varIable skew oblIque wIng offers a substantIal This sItuatIon IS sometImes referred to as the aerodynamIC performance advantage for aIrcraft pIlot flYIng the computer, whIle the computer IS mIssIons that requIre both hIgh effICIency In flYIng the aIrcraft.
subsonIC flIght and supersonIc dash or cruIse.
The most obVIOUS dIsadvantage of the oblIque-WIng More preCIsely, the model-follOWIng problem concept IS the asymmetry assocIated WIth WIng-skew can be stated as follows. The lInearIzed dynamICs angle. ThIS asymmetry results In sIgnIfIcant are gwen as aerodynamIC and InertIal cross-couplIng between .
the aIrcraft longItudInal and lateral-dIrectIonal (1)
xp = Apxp + Bpu
p axes. Current tYPIcal deSIgn procedures synthe- sIze aIrcraft controllers based on 2- or at most (2)
YP = Cxp
3-degree-of-freedom solutIons. However, the OWRA stabIlIzatIon and decoupllng must consIder at where xp ERn, up E Rm, and YP E Ri, and Ap. B p' least 5 degrees of freedom SImultaneously.
and Care matrlces of approprIate dimenSIons. The control up must be determIned such that the plant The baSIS for OWRA WIll be NASA's F-8 dIgItal fly-by-wlre aIrcraft. ThIS aIrcraft WIll be output YP apprOXImates, reasonably well, some modIfIed by the removal of the current hIgh wIng model output vector Ym defIned by the equatIons: and lnstallatlon of a wIng PIVOt and a composlte wIng. A maJor part of the OWRA program WIll be (3) the syntheSIS of a control system that WIll pro- VIde acceptable stabIlIzatIon and decoupllng (4) Ym = CXm across the Mach, angle-of-attack, and wlng-skew envelope. The alrcraft thus offers an opportunlty where xm ERn, um E Rm, and Ym E Ri, and Am. B , m to apply modern control theory technIques to the and C are matrIces of approprIate dImenSIons.
solutIon of problems assocIated WIth OWRA.
For OWRA, the state, Input, and output vectors Model-followlng has been a popular method for are gl ven by the deSIgn of multlvarlable control systems over the last two decades. In thlS method, the deSIred v veloclty, m/sec behaVIor of the plant lS prOVIded by an ldeal a angle of attack, deg model, and the problem IS one of deSIgnIng a sldesllp angle, deg fl sUltable controller for the plant so that ItS .,.
bank angle, deg response follows that of the model.
X =
e pItch angle, deg roll rate, deg/sec p Yore IndIcated the use of thlS method for pItch rate, deg/sec q slmultaneous stablllty augmentatIon and mode yaw rate, deg/sec r decoupllng. H1S syntheSIS procedure consisted of constructIng an Ideal model, deSIgnIng feedback left horlzontal tall deflectIon. deg gaIns by quadratIC optImIzatIon, and deSIgnIng rIght horIzontal tall deflectIon. deg feedforward galns. A dIsadvantage of thlS method IS that selectIon of feedback gaIn IS an Iterative left aIleron deflectIon. dpg and time-consumIng process. The determination of rIght aIleron deflectIon. deg this gaIn becomes a more complex problem when all rudder deflectIon. deg states are not avaIlable and therefore output feedback lS used.
roll rate. deg/sec pItch rate, deg/sec Another technIque for decoupled flIght control yaw rate. deg/sec deSIgn IS the elgenstructure asslgnment. 3 In bank angle, deg thIS technlque, the performance speclflcatlons angle of attack. deg can be Interpreted In terms of the eigenvalues SIdeslIp angle, deg and elgenvectors of the closed-loop system.
Broussard and Berry4 have established the eqUIvalence of thlS technlque to the deSIgn uSIng model-followlng systems.
The des1rerl model of the a1rcraft, def1ned hy method of e1genstructure ass1gnment w1ll be matr1ces Am and 13 as well as the a1rcraft matr1- used to select the ga1n K. ThlS enables the m des1red e1genvectors and e1genvalues to be ces Ap and Bp are glven 1n Table 1. The a1rcraft selected to ensure sat1sfactory tran1S1ent matr1ces correspond to a fl1ght cond1t10n of 0.8 response of the a1rcraft.
Mach number and an alt1tude of 6096 m at 45' w1ng ske~. The model used 1n th1S study lS a mod1f1ca- tlOn of the zero-Illng-skew conf1gurat10n at the same fl1ght cond1t10n. Am and 8 are m elements Two w1dely used synthes1s techn1ques of modern mod1f1ed to 1ncrease damp1ng and to el1m1nate control theory are the llnear quadrat1c regulator zero-w1ng-skew coupl1ng terms. Th1S model lS pre- des1gn and the modal control theory 1nvolv1ng pole llm1nary and may not represent 1deal dynamlcs but placement or e1genvalue-e1genvector ass1gnment. 7 does 1ncorporate the des1red a1rcraft decoupl1ng.
One of the purposes of feedback control of alr- craft lS to 1mprove or enhance the flY1ng quall- .!i.o_d_e_l_-ro_l_l_ow_1_n_g __ C_o_n_t:_o_l __ Sy_s~_e_m.
t1es of an a1rcraft. The dlff1culty In lncor- porat1ng spec1f1cat10ns such as damp1ng, natural There are two conf1gurat10ns of model- frequency, and decoupl1ng w1th1n a quadrat1c per- follow1ng, one lS 1mpl1c1t model-follow1ng, and formance 1ndex makes the e1gensystem synthesls the other 1S real model-follow1ng (RIIF). In procedure a prom1s1ng des1gn alternat1ve. The 1mpl1c1t model-follow1ng, the model lS not part of performance spec1f1cat10ns can be 1nterpreted the system. In Rt1F, however, the model lS part of 1n terms of des1red closed-loop e1genvalues and the systel'l as control 1 aw requ1 res the states of e1genvectors. MooreR and others have shown how the model. The techn1que of RMF has been shown to feedback can be used to place closed-loop e1gen- he amenable to the Solut10n of many a1rcraft value and shape closed-loop e1genvectors. Cun- control problems.
9, n1ngham and Andry, Shap1ro, and Chung,10 and Frzberger estahllshed cond1t10ns for perfect 11 Sohel and Shap1ro have successfully demon- model-follow1ng that pnable an 1deal match of the strated the use of e1genstructure ass1gnment dynam1cs of the compensated plant w1th those of procedure for a1rcraft control system des1gn.
the model. However, the cond1t10ns for perfect model-follow1ng are never atta1nahle 1n the real The handl1ng qual1t1es data base may be used I~orld. An asymptot1c Rt1F control law was der1ved to obtaln deSlred pole locatlons dlrectly. The by Chan for the class of plants and models whose add1tlonal des1gn obJect1ve of obta1nlng augmented output vectors are 1dentlcally thelr state vec- dynam1cs slmllar to those ohtalned 1n fl1ght leads tors. Chan showed that, even 1f the cond1t10nS to spec1f1cat10ns on e1genvectors or deS1red mode for perfect model-follow1ng are not sat1sfled, use shapes. For example, pltch att1tude must he dom1- of perfect model-follow1ng ga1ns can Y1eld a nant for the short-per10d mode, and speed must be control capable of keep1ng error between the model dom1nant for the phUg01d mode.
and plant to a "sJ'lall" reg10n of state space.
Chan chose up as Deta1led d1Scuss10ns on elgenspace may be found 1n Ref. 11. However, some bas1c results for (5 ) controllahle and observable systems are summarlzed up ul + u2 1n the follow1ng d1Scuss10n.
where Cons1der the system (6) ul Ke .
x Ax + Bu t {3t (Am - Ap)xm u2 + BpBmuJ'l p y Cx (7) Kxmxm + Kumum where x eRn, u £ Rm, and y £ R~, and A, B, and C are matr1ces of appropr1ate d1mens10ns. If the t and Bp lS the pseudo-1nverse of 8p, and Kxm and system 1S controllable and observable, and the Kum are the feedforward ga1ns uS1ng model states matr1ces Band C are full rank, the followlng results hold and command 1nput. Also, 1. The pos1t10ns of maX1mum (m,~) closed-loop (8) e = xm - xp elgenvalues can be ass1gned arb1trar11y w1th the st1pulat10n that 1f A1 1S a complex closed-loop The control up w1ll ensure perfect model- * elgenvalue, ltS complex conjugate A1 must also be follow1ng, 1f 1t lS poss1ble. If perfect model- a closed-loop elgenvalue.
follow1ng lS not poss1ble, the error sett1ng rates would depend on elgenvalues of the closed- loop system and can thus be controlled 1n RMF.
2. The shape of maX1 mum (m rv) el genvectors
Also, 1f only partlal state feedback lS poss1ble can be altered. If the shape of a complex elgen- In the plant, perfect real model-followlng lS * vector V1 lS altered, 1tS complex conjugate v st1ll poss1ble.
l must be altered 1n the same way.
For OWRA, because part1al state feedback lS to be used, the feedback galn K must be selected 3. For each e1genvector whose shape 1S to ensure stab1l1ty of the closed-loop a1rcraft altered, m1n1mum (m,r) e1genvector elements and placement of ltS closed-loop elgenvalues can be chosen arbltrarlly.
at the des1red 10cat10n 1n the s-plane. The 4. Attalnable elgenvectors must lle In the values and deslred elgenvalues, the feedback galn matr1x K, and the feedforward ga1n matrlces subspace spanned by the columns of (All - A)-I B Kxm and Kum.
of dlmenSlon m that lS the number of lndependent F1gures 3(a) to 3(c) and 4 111ustrate the control varlables. A deslred elgenvector v~ wlll, closed-loop system response to the same elevator In general, not reslde In the prescrlbed subspace command 1nput. The pltch rate In Flg. 3(a) lS and cannot be achleved. The achlevable elgen- attenuated as compared w1th the open-loop response. However, the system 1S very closely vector v~ 1S obta1ned by orthogonal proJect10n follow1ng the des1red model response as lllus- d -1 of v onto the subspace spanned by (~I - A) B. trated In Flg. 4. The yaw rate and bank angle are v1rtually nonexlstent as lllustrated In FlgS. 3(b) It w111 generally be true that only a few of the and 3(c), thus ach1ev1ng the des1red decoupllng.
components 1n V1 are actually spec1f1ed. The rema1nder can be arb1trary. To account for th1S, F1gures 5(a) to 5(c) 111ustrate the open-loop vl lS reordered and part1t10ned as follows system response to a one-degree alleron command 1nput. Pltch and yaw angular rate and bank angle are agaln shown. The relatlve pltch coupl1ng lS (g) not as severe for the alleron command case as the roll coupllng lS for the elevator command, how- ever, coupllng lS stlll present.
where F1gures 6(a) to 6(c), 7(a), and 7(b) 111us- trate the closed-loop system response to the same alleron command. Pltch rate lS vlrtually v; the spec1f1ed subvector nonexlstent, and the deslred yaw rate and bank angle are achleved, glvlng the deslred decoupllng.
d1 the vector of unspec1f1ed components ConcluslOns (. the reorder1ng operat10n A method lS presented to obtaln a decoupled If we let control for a hlghly coupled asymmetrlc alrcraft.
The method utll1zes a real model-follow1ng control law 1n WhlCh ga1ns for perfect model-followlng are used even when the condltlons for perfect model- followlng are not satlsfled. The feedback galn, uS1ng output feedback, lS computed by uSlng then, as shown 1n Ref. 13, Zl may be selected to e1genstructure asslgnment. The results 1nd1cate best approX1mate v~ w1th v~. By the method of that the method does obtaln the decoupllng lncor- porated 1n the 1deal model for the fllght con- orthogona I project 1ons, Zl 1 S obta1ned d1t10n cons1dered.
Future 1nvestlgatlons wlll be conducted to
Zl = (L'L)-l L'v; (11)
evaluate the control algorlthm under nonllnear 6-degree-of-freedom fllght condltlons. These 8, the feedback As shown by Moore ga1n K lS 1nvestlgatlons w111 cons1der such factors as gl yen by nonllnear aerodynamlc data, control system surface rate and posltlon constralnts, and (12) system hysteresls.
where w1 1S obta1ned from the relat10n References (13) (All - A)V1 = BW1 IGregory, T., "Ob 11 que Wl ng Ready for Research Alrcraft," Aerospace Amenca, June 1985, Results pp. 78-81.
To 11lustrate the degree of coupl1ng 1n the 2Yore, E., "Opt1mal Decoupl1ng Control," 9th open-loop system and decoupl1ng 1n the closed- loop system, a one-degree control command was J01nt Automat1c Control Conference of the 1nput for 2 sec as shown 1n F1g. 1. Th1S command Amer1can Automat1c Control Counc11, New York, 1nput was e1ther elevator or a1leron and was 1968, pp. 327-336.
reduced to zero after 2 sec. F1gures 2(a) to 2(c) 11lustrate the open-loop system response to an 3S r1nathkumar, S., "Modal Control Theory and elevator command 1nput for p1tch rate, yaw rate, Appl1cat10n to A1rcraft Lateral Handl1ng Qual1t1es and bank angle, respect1vely. Slgn1f1cant yaw Des1gn," NASA TP-1234, 1978.
rate and bank angle are generated as a result of the p1tch command, and of part1cular lnterest lS 4Broussard, J.R., and Berry, P.W., "The Rela- the very large change 1n bank angle 11lustrat1ng tlonsh1p Between Impl1c1t Model Follow1ny and the slgn1f1cant cross-coupl1ng.
E1genvalue E1genvector Placement," Proceed1ngs IEEE Conference on Dec1s10n and Control, Table 2 shows the des1red e1yenvector ass1yn- Oct. 1979, pp. 38-42.
ment spec1t1cat10n, open- and closed-loop e1gen- Modal Control," Proceedings of the IEEE Conference 5Erzberger, H., "On the Use of Algebraic Methods on DeC1Sl0n and Control, Dec. 1980, pp.178-186.
1n the Analys1s and Des1gn of Model-Follow1ng Control Systems," NASA TN D-4663, 1968.
10Andry, A.N., Shap1ro, E.Y., and Chung, J.C., "On Elgenstructure Asslgnment for llnear Systems," 6Chan, Y.T., "Perfect Model Follow1ng w1th a Real IEEE Transact10ns Aerospace and Electronlc Model," 14th J01nt Automat1c Control Conference of Systems, Sept. 1983, pp. 711-729.
the Amer1can Automat1c Control Counc1l, Columbus, Oh10, Paper 10-5, 1973, pp. 287-293.
11S o bel, K.M., and Shap1ro, E.Y., "Appl1catlon of E1gensystem Ass1gnment to lateral Translatlon and 7Alag, G.S., and Duke, E.l., "Development of Yaw P01nt1ng Fl1ght Control," 23rd IEEE Conference Control laws for a Fl1ght Test Maneuver AutOP1lot on Declslon and Control, las Vegas, Nev., Dec.
for an F-1~ A1rcraft," NASA TM-86736, 1985.
1984, pp. 1423-1428.
8Moore, B.C., "On the FleX1b1l1ty Offered by Full 12l1ebst, B.S., Garrard, W.L., and Adams, W.M., State Feedback 1n Mult1var1able Systems Beyond "Elgenspace Des1gn of an Actlve Flutter Sup- Closed loop E1genvalue Ass1gnment," IEEE Trans.
presslon System," AIAA Paper 84-1867, Aug. 1984.
Auto. Control, Vol. 21, Oct. 1976, pp. 689-692.
13Harvey, C.A., Steln, G., and Doyle, J.C., '"Opt1- 9Cunn1ngham, T.B., "E1genspace Select10n mal Llnear Control' Characterlzatl0n of Multl- Procedures for Close Loop Response Shap1ng w1th Input Systems," ONR-CR-215-238-2, TR-2, 1977.
TABLE 1. - AIRCRAFT AND MODEL MATRICES Plant (a1rcraft) matr1ces 0.0094 22.0707 10.5479 -0.1341 -32.1127 0.0057 -0.0005 -0.0265 -0.0001 -0.7826 0.0958 0.0000 0.0000 0.0030 0.9926 -0.0003 0.0000 -0.0592 -0.2908 0.0387 -0.0002 0.0259 0.0001 -0.9920 0.0000 0.0000 0.0000 0.0000 1.0000 0.0247 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 33.1432 -53.6933 0.0000 0.0000 -3.1250 2.0552 1. 7210 -0.0002 -8.6816 0.7975 0.0000 0.0000 0.1679 -1.0352 0.1810 0.0000 -1.0092 10.7521 0.0000 0.0000 -0.0213 0.0080 -0.7129
---------------------------
0.0844 -0.0309 -0.2210 -1.2572 3.6598 -0.0974 -0.0974 -0.0198 -0.0302 0.0000 -0.0166 0.0166 0.0008 -0.0005 0.0647 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.9804 15.8467 13.2774 -22.2654 -11.8422 0.5694 -9.4073 -10.8655 -1.2311 0.8797 1.9854 -2.2579 0.5262 -0.3276 -6.2499 Mode 1 mat rl ces 0.0000 0.0000 0.0000 -0.0077 23.5966 0.0000 0.0000 -32.1129 -0.0001 -1.1062 0.0000 0.0000 0.0000 0.0000 0.9909 0.0000 0.0000 0.0000 -0.6000 0.0387 0.0000 -0.0148 0.0000 -0.9919 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 -0.0133
Am =
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -44.3777 0.0000 -10.0000 -12.1514 0.0000 0.0000 0.0000 0.0000 - .0000 0.0000 0.0001 0.0000 12.1943 0.0000 0.0000 0.0000 0.0000 -2.0000 0.0000 0.0000 -2.2032 -2.2032 -0.8354 -0.8354 -0.0848 -0.0848 -0.0494 -0.0494 0.0000 0.0647 -0.0166 0.0166 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Bm =
0.0000 0.0000 0.0000 0.0000 0.0000 !
11.2379 -11.2379 29.0513 -29.0513 9.6847 0.0000 -7.8229 -7.8229 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -6.6502 TABLE 2. - EIGENSYSTEM ASSIGNMENT AND GAINS a DeSlred elgenvectors Short perlod Dutch roll Splra 1 Roll subsldence 0 0 0 0 0 0 0 1 x 0 0 x x 1 0 0 x 0 0 0 0 1 x x x x x 0 0 0 0 0 0 x 1 x 1 0 0 x 0 0 x 1 x 0 ------------------------ Elgenva1ues Des 1 red Achleved Condltl0n Open loop closed loop closed loop Short perlod -1.0433 ± J2.8269 -2 ± J3.5 -2 ± J3.5 Dutch ro 11 -0.5463 ± J3.3816 -3 ± J4.0 -3 ± J4.0 SPl ra1 -0.0118 -0.1 -0.1 -2.7544 -7.0 -7.0 Roll subSldences PhugOld -0.0053 ± JO.0455 -0.0047 t JO.0455 Feedback galn p q r cp a ~ -0.3471 -1.4507 0.1476 -0.5902
.0.9587]
[ 0.1454
0.4180 -0.0580 1.0127 -0.0915 1.0377 0.1151 K = -0.3710 0.9639 2.2955 -0.2967 -5.0019 6.9733 0.1133 0.1199 -0.0982 0.0194 -7.2635 1.3083 0.0114 -0.1653 0.1540 0.0210 0.6898 -1.9884 Feed forward galns t
Kxm = Bp(Am - Apl
---------------------------------------------------------------------------.---------- a cp a p v ~ q r 6.5743 -3.6200 0.0729 -0.0003 0.1698 -0.2893
[ 0.0010 .0.3591J
-0.0006 -3.6600 2.2241 -0.0452 0.0002 -0.0560 0.4205 0.2696 9.4682 -0.1429 0.0004 -0.5446 0.3861 0.5095
Kxm = -0.0019 -9.0867
0.0002 7.0692 2.0716 0.0037 -0.0002 0.1608 -0.6155 -0.0284 0.0004 2.1139 -1.4962 0.0272 -0.0001 0.0164 -0.1778 0.0388 -0.8204 -0.5391 -0.0271 0.7842 -2.1203] 1.0516 0.9978 -0.1411 -0.3323 1. 3850 Kum = 3.1869 1.6393 2.1149 -1. 9684 3.9478 0.0239 0.2235 0.8121 1.4435 -0.1933 [ -0.3735 -0.4054 0.1779 0.127R 0.2326 ax lS arbltrary.
Elevator deflection, deg o 2 3 4 5 6 7 8 9 10 Time, sec Fig. 1 Command input to system.
O~-+-------------------------- -5 o~--~---r~~~~---------- Yaw -10 Pitch rate, rate, deg/sec -1 5 deg/sec -2 -20 -4 -25 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 Time, sec Time, sec (b) !az.1 l'ate.
(a) Pitoh l'ate.
-10 -20 Bank angle, -30 deg -40
-so
-60 0 2 3 4 5 6 7 8 9 Time, sec (0) Bank angLe.
Fig. 8 Open-Loop system ~esponse to eLevato~ oommand input.
o~--~~~------------------ Pitch rate, deg/sec -2 _4~-L __ ~~ __ ~~ __ -L __ L--L __ ~-J
o 2 3 4 5 6 7 8 9 10
Time, sec (a) Pitoh l'ate.
Bank
05~
Yaw
2~
rate, 0 .. "---.......:~_--------- deg/sec -
'~~"=.::=
-2-' , o 1 234 5 6 7 6 9 10 o 1 234 567 8 9 10 Time, sec Time, sec (b) Iaz.1 l'ate. (0) Bank angLe.
Fig. 3 CLosed-Loop system ~esponse to eLevato~ command input.
-- Plant (aircraft) --- Model Pitch rate, deg/sec -2 -4 0 2 3 4 5 6 7 8 9 10 Time, sec Fig. 4 Model-following response to elevator command input.
Pitch 0 I----,I-\l--\--,I----...:: ...... -===---~- Yaw rate, rate, 4 deg/sec deg/sec - 1 -2 -3 2 3 4 5 6 7 8 9 10 o 2 3 4 5 6 7 8 9 10 Time, sec Time, sec (b) YaLJ rote.
(a) pitch rote.
Bank angle, deg 2 3 4 5 6 7 8 9 10 Time, sec (c) Bank angle.
Fig. 5 Open-loop system response to aileron corrrnand input.
Pitch
·1 ~
rate, 01-===------------- deg/sec _.1~_L __ ~_L __ ~_L __ L__L __ L_~~ o 2 3 4 5 6 7 8 9 10 Time, sec (a) Pitoh rate •
•
..
Yaw Bank rate, angle, 6 deg/ •• c deg .2 O~----------------------------- 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 0 Time, sec Time, sec (0) Bank angle.
(b) latJ rate.
Fig. 6 Closed-loop system ~esponse to aile~on command input.
-- Plant (aircraft) -- Plant (aircraft) --- Model --- Model I I
--------
----- I
----
I Yaw I Bank I rate, I angle, I deg/sec I deg I I I I I I - 1 Q 2 3 4 5 6 7 8 9 10 0 2 3 4 5 6 7 8 9 10 Time, sec Time, sec (a) !(l!J rate.
(b) Bank angle.
Fig. 7 Mod8l-foll~ing ~espon8e to aile~on command input.
Report No Recipient's Catalog No
1 I 2 Government Accession No 3
NASA TM-86801 4 Title and Subtitle 5 Report Date June 1986 DECOUPLING CONTROL SYNTHESIS FOR 6 Performing Organization Code AN OBLIQUE-WING AIRCRAFT Author(s) 8 Performing Organization Report No Gurbux S. A1ag, Robert W. Kempel, and Joseph W. Pah1e H-1339 10 Work Unit No Performing Organization Name and Address 9 RTOP 533-06-01 NASA Ames Research Center 11 Contract or Grant No Dryden F11ght Research Fac111ty P.O. 80x 273 Edwards, CA 93523-5000 Type of Report and Period Covered 12 Sponsoring Agency Name and Address Techn1ca1 Memorandum Nat10na1 Aeronaut1cs and Space Adm1n1strat10n 14 Sponsorl ng Agency Code Wash1ngton, D.C. 20546 15 Supplementary Notes Prepared as Amer1can Automat1c Control Counc11 paper for presentat10n at Amer1can Control Wash1ngton, June 18-20, 1986. Dr. A1ag lS aff111ated w1th Western Conference, Seattle, Messrs. Kempel and Pah1e are aff111ated w1th NASA Ames-Dryden.
M1ch1gan Un1vers1ty, 16 Abstract Interest 1n ob11que-w1ng a1rcraft has surfaced per10d1ca11y Slnce the 1940's. Th1S concept offers some substant1a1 aero- dynam1c performance advantages but also has slgn1f1cant aerodynam1c and 1nert1a1 cross-coup11ng between the a1rcraft 10ng1tudlna1 and 1atera1-d1rect10na1 axes. Th1S paper presents a techn1que for synthes1z1ng a decoup11ng controller wh11e prov1d1ng the des1red stab111ty augmentat10n.
The proposed synthesls procedure uses the concept of a real mode1-fo110w1ng control system. Feedforward ga1ns are selected on the assumpt10n that perfect mode1-fo110w1ng cond1t10ns are satlsf1ed.
.
The feedback ga1ns are obta1ned by uS1ng e1gensystem asslgnment, and the a1rcraft lS stab111zed by uS1ng part1a1 state feedback. The effect1veness of the control laws developed 1n ach1ev1ng the des1red decoup11ng lS 111ustrated by app11cat10n to 11near1zed equat10ns of mot10n of an ob11que-w1ng alrcraft for a glven f11ght cond1t10n.
, 17 Kev Words (Suggested by Author(s)) 18 Distribution Statement Asymmetrlc a1rcraft control Unc1ass1f1ed - Un11m1ted E1gensystpm synthesls F11 ght control systems Mu1t1var1ab1e control systems Perfect mode1-fo110w1ng STAR category 08 Price" Security Classlf (of thiS report) 20 Security CI~sslf (of thiS page) 21 No of Pages 19 22 Unc1ass1f1ed Unc1ass1f1ed 10 A02 ~For sale by the National Technical Information Service. Springfield. Virg~nia 22161.