LIDS-P-1932 NOVEMBER 1989 Application of Modern Control Theory on a Model of the X29 Aircraft Munther A. Dahleh David M. Richards October 22, 1989 Abstract In this report, two different control strategies for a model of the X29 are presented. The first is based on the 11 design methodology and the second is based on the HE design methodology. The purpose here is to show that these methodologies make the design process quite systematic, and that the resulting controllers conform with our intiuition. 1 X29 Model The model for the X29 was taken directly out of [8]. A diagram of a forward swept wing aircraft is shown in Fig. 1. The model describes the use of the canard and flaperon to control the pitch angle and angle of attack. The pitch angle 8 is the angle of the nose of the aircraft with respect to horizontal, and the angle of attack a is the angle of the nose with respect to the direction of the aircraft's velocity. The flight path angle -7 is 0.- a and is the angle of the velocity with respect to horizontal. These quantities are all shown in Fig. 2. Three different systems are presented in [8]. The first has the canard deflection as input and the angle of attack as output. The second has pitch attitude as output, and the third system uses both the canard and the flaperon to control the pitch attitude and the angle of attack. In this work, only the first case is treated. -- -- I -----------~--~ - ----------- 1 ~"I~-~-~~--- ~ ---- t/ 1 _____ canard flaperon stFake - on a Figure 1: Diagram of the X29 Control Surfaces 2 plane axis ity 0 r horizon Figure 2: Pitch, Flight Path, Angle of Attack 2 11 Control Design The challenge of the control design in the case of the X29 is to meet the various design specifications. For the first design case with the canard controlling the angle of attack, the specifications are that 10% error or less is required in command following for 0.01 < w < 1 rad/sec, disturbances must be attenuated by at least a factor of 1.5 for w < 1 rad/sec, and the system must be robust in the face of multiplicative errors. Multiplicative errors are introduced from ignoring the high frequency wing torsion mode and from the scaling. A final objective is to avoid exciting the wing bending mode at about 60 rad/sec. To accomplish this, it is desired to have the system bandwidth less than 60 rad/sec. This will cause attenuation of any excitation of the bending mode. In the design, this spec is approximated by requiring the crossover frequency of GK to be below 60 rad/sec. All together, these specs require that the sensitivity be small and that the closed loop transfer function rolls off sufficiently at high frequency to guarantee robust performance. We will employ the l1 methodology to achieve this. For details on the 11 problem consult (1,3,4,5,6,9,101. To meet these requirements, a good strategy is to minimize the sensitivity function, particularly in the range w < 1. With no weighting of the sensitivity, the design specifications were not quite met. The results are shown in Figs. 3-4. The sensitivity was sufficiently small in the appropriate range at low frequencies, but the bandwidth of the system was too high and did not satisfy the 3 20 -c -0 L ..LJ.. I.. j II. . 0' E -20--I 1··· Il l 4 · I!Il· I···· l II · I II ,11 .I !I I 1141 !111 - .L4U...j.I .Ll O o c..ooo iT!!i···-·· · i !] ~·1001000 i ] :: ~' ····r~··iiriiii · ·r,.,ii~ I Ii1!i!--'i.. LI·UII-~~·· III· 1Ii··I·Ii!11ii',.,, iiii~···~ i0 .1-~.·I.-iii· .001i,.01 ii'10 I·· i tll · I ·· I1m (· · I· · I IIIII~l IIIIIIII eeelll I ll! I···r · ·I I 1t 1 11·· 1 I 1Ii Il· eIl tl It t lle Ill' -40- ® ~·~W·· q:D~~~~~~~~~~~~~~~~~~~~~ I· i··iiiii~ i·Iiii: ii··iiiii~ ii: ; ;! '-iLiiiii_. ··r i iii --'· iii '1.. i.1 i =_ .... i ii.- ii I~i' 20·· E~~~~~·· for Unweighted Optimization Figure 3: Sensitivity ~~~~~~~~~qe, · rl 40 · · · · .···· q'l' ".' t · r t,~.l · iiii'i]]''iiF-i i-Iii [ili··i' .·~......... [iiii'··i'!iiii:__ !r ]]i ]]] ~ ·ill~L- ]]]]i]] ]]]]]!] illr ·~~~~~~~~~~~~~~~~~ · :::001.01 .~ 11111 1011l 1111l 001 00 1II:I )·(Z-pimzr F~~~~gure ~ C~) ~ - ~ fo· I· naghe 20 :::..e. -0~ C7 for 1II - GK) IFigure 4: GK(I .001.01 0 EF 20 · M ZI.TI I ilIItI i -40 tI 11111i i iIi i 1~ I i·· 10 1001000o 1IIIIUnweighted Optimization I Ii iI I.IIIII. I I IIi i t i I 1 1. i I 1. 21110 !I· ~ 60 30 II-11 .- III: I II L- II: _0 -3 CD 60 0 I9i i E i 4 I eINl I ebill . III· ...... -120 ·001.0 !I .IIIt II.IIll 1 iee1h I lii ....................... 10 11110 1001000 Figure 5: Optimal Sensitivity With Weight 1+ robustness requirements. Infact the requirements were not met even with several weights which were used, an example of which is depicted in Fig. 5. The weighting was the discretized equivalent of 1 Compared to Fig. 3, Fig. 5 demonstrates the effect of the higher weighting at low frequency. The low frequency sensitivity has been decreased at the expense of the high frequency sensitivity as expected. After using several weighting functions, it was determined that the requirements could not be met (or would be difficult to meet) using only the sensitivity function. The next step was to use the complementary sensitivity GK(I - GK)- 1 in the objective function. The sum of the sensitivity and the complementary sensitivity wras used as the objective. The results of the new optimization are shown in Figs. 6-8. The low frequency sensitivity has increased slightly and GC crosses the 0db point at about 40 rad/sec. This new system meets the design specs. Various weights were used to try to tune the system better, 5 20 ::I .'-. :::.:: ::':: ::::" :: :::: :: 1. 10, C -' 10~iii !iiii iii:iiii -20 IJ Figre 0 .001.0 ~~ ~ iiiiiii !i ii""'i i 10 1 001000 7:.G'r.C ..-. ~~~~.~.-.~ . 11 11 -. ,,...... . -,,,, .0-i'- ,-""";:... 'E C20 Figure 6: New Sensitivity "~- - £ ..- 0 40:(···· 0 1.1 . 11 10111 ...1111I 001 -~0 ·. ~.. 0'E fi -20 ~~ ,. .001 ; ;.=: ~; : ;: 00·C ......... .. r ::.~::= : ;; : ~ : ... = ....... : .-..-.. ........ ...-..-....... ... "::: : : - ::::::::: ...........:= ..-. ... ~.- 1111l11101 00100 0111 .11111 · ~~~ ~ 1111II1(6 1~1 :.:::::::::...:! ::: Figur 7:" CX··IIII1·1·)~ I :::::::::.. 11.0 00 7 ..... I' ii7ii- . I"a . ;: 2Q C ,llll i,··rr11ijj) toliC 7 Ileele ··· eeleelI i j4,iw· Lt ! Ie Ill·il r.1111 II'naaI ~ elee ·· r· ,i · iit ,·· · Iel eeieel 1~1 fgf) , ell , Ji ilil I·· -50 "'".001 .00· 01'1.01·~.l··1 ::1 1 1110101 01000l II 1001000: Fgre butnone· 8: Complelentary Sensitivity·~r~· ~ prde uc mpoedrs lts wereII found11 wich 3~ ~o·~~~~~~·· C· ontrol Des1·ign To~~~~··· the·· meetIIaboveH specif·icain cnieeie estv problem~1··· directly. Bas··icly wereuie hesnstvy to be small low~~~··· < w frequency· (0··_< 1),adtecmlmnaysniii ob small~~~···~···I at··· hih frequency Afesevera iteains h wih o 2~~~~~~00 W=?I.~·~rf~?B-f?!V1.. ~~rR? Figure 8: --~~~·· 0"'···-"" ]···· · 11111 I···· Complelentary r~l( z 1111 l)2 'Sensitivity I Lt· To-us ~ereetI algorihm(·ha~ involvesoving ~w icai q t none i ns[],i necessary which I(·o· produced include 11he·· control ~rnseresults. fncin n but wereisfound improved objective.final Hence obetieis~ "he muchinmze~e~nomo ·I· · 1··· V · · I·~n· Control Design ~· IIIII· ··I 1··¢K$ I ~·I 1 3 H,, ¢1111 a1··where~~~~1·· i s s m a1·· l l · I nubr hersingcoe lornse uc CTinspecifications· ar hw nFgrs -01 eo.I i paet~a are me~ ·· .· · · ~ theI· mixed To meet the above specification, we )I·I considered sensitivity problem directly. Basically, we require the sensitivity to be small at tionsfrequency are shown Figures 9-10-11 below. It is apparent thatto the are esensitivity low (0 in <met. uw <~~·specifications 1), and the complementary be small at high frequency. After several iterations, the weight on the sensitivity function was choosen as 100 ( + 1)2' To use the recent algorithms that involve solving two Riccati equations (71, it is necessary to include the control transfer function in the objective. Hence the final objective is to minimize the H,, norm of pKS~ where p is a small number. The resulting closed loop transfer func- m a 10 t U -IO r-------------T-~--1-'~I~ii -----Figure 10: GK e a · --- --- ,__4~. 0 I . , , , 11 ( 1 III , ............. (, 1,I ,I II,~~~~~~~~~~~~~t'- -- I Figure 9: Sensitivity 4~~~~~~~~~0~~~ frequency I in rad/sec I i 10 io a g 0 e ri,, fI~~ r i~~IiequencyI~ . , -20 - ----------....... Figure ,,,, . . 1 i d~ ,, . i .. -e i e i ,vi S U 10 i i0 0 10ii i i i Ifrequency in rad/sec References (11 M.A. Dahleh and Y. Ohta. "A necessary and sufficient condition for robust BBO stability," Systems and Control Letters, Vol. 11, pp. 271-275, 1988. (21 M.A. Dahleh and J.B. Pearson. a'I optimal feedback controllers for discrete-time systems, 1986. [31 Proceedings A CC, Seattle, WA, June M.A. Dahleh and J.B. Pearson. 'I' optimal feedback controllers for MMO discrete-time systems," IEEE Trans. A-C, Vol AC-32, April 1987. (41 M.A. Dahleh and J.B. Pearson. "Ll optimal feedback compensators for continuous-time systems," IEEE Trans. A-C, Vol AC32, October 1987. [51 M.A. Dahleh and J.B. Pearson. "Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization," IEEE Trans. Automat. Contr., Vol AC-33, pp. 722-731, August 1988. [61 M.A. Dahleh and J.B. Pearson. "Minimization of a regulated response to a fixed input," IEEE Trans. Automat. Contr., Vol d AC-33, pp. 924-930, October 1988. [71 J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis. "State space solutions to standard H 2 and H,o control problems," IEEE Trans. A-C, Vol AC-34, pp. 831-847, August 1989. [81 W. Quinn. 'Multivariable control of a forward swept wing aircraft,' Laboratory for Information and Decision Systems Report LIDS-TH-1530, MIT, 1986. [9] D.M. Richards. '1I-optimal control: Solution software and design ezamples,' M.S. Thesis, LIDS-TH-1906, MIT, Cambridge, August 1989. [101 J.S. Shamma and M.A. Dahleh. 'Time varying vs. time invariant compensation for rejection of persistent bounded disturbances and robust stability," to appear in IEEE Trans. A-C. 10