Robust Control of the Aircraft Attitude

Transcription

1 Robus Conrol of h Airraf Aiu F X Wu 1, an W J Zhang Darmn of Mhanial Enginring Univrsiy of Sasahan, Sasaoon, SK S7N 5A9, Canaa Chris_Zhang@EngrUsasCa 1 On h sial laving from Norhsrn Ployhnial Univrsiy, Xi'an, China, Wuf@EngrUsasCa Ky Wors: airraf, aiu-raing onrol, moifi Rorigus aramr, robus, nonlinar H onrol Absra h aiu onrol of airrafs lays an imoran rol in h airraf moion his roblm may b absra as ha of h aiu onrol of rigi boy, hih is a omlx nonlinar onrol roblm Firsly, h aiu onrol of airrafs is insr ino h fram of nonlinar H onrol Unr his fram, sign a H onrollr ihou solving any nonlinar Bllman-Jaobi-Raai uaion Our onrollr osssss h robusnss no only agains h unrainy of h airraf aramrs bu also agains h sir signal Unr our onsiraion, h aiu sa is aramriz by moifi Rorigus aramrs h valiiy of our rsuls is illusra in simulaion 1 Inrouion h aiu onrol of airrafs suh as airlan, sa saion an salli) is on of h imoran roblms in h fligh onrol For xaml, if h laning aiu of h airlan is no onroll o a r-sifi sa, ossibly i ill aus srious fligh ain Whhr h aius of sa saion an salli ar aura or no is a ruial faor ha hy run normally h roblm of h aiu onrol of airrafs may b absra as ha of h aiu onrol of rigi boy for insan [1,3]) h aiu sa of rigi boy is no a sa, bu a oma manifol SO 3), an so is h aiu rror sa [-4] Hovr, hr xis many R n ays o aramriz h manifol SO 3) [4], suh as uni uarnion, Rorigus aramr, an moifi Rorigus aramr h uni uarnion an globally aramriz h manifol SO 3) hil Rorigus aramrs, an moifi Rorigus aramrs only an loally aramriz h manifol SO 3) Hovr, Rorigus aramrs or moifi Rorigus aramrs only onain hr sals hil h uni uarnion onains four sals ih a onsrain uaion Sin moifi Rorigus aramrs may aramriz a lagr omain han Rorigus aramrs, mloy moifi Rorigus aramrs o srib h aiu sa in his ar h aiu onrol roblm may b srib as a lass of nonlinar onrol roblm Wn an Krnuz-Dlgao rsn a sris of h onrol las o h aiu onrol roblm bas on Lyaunov aroah in [], hr h aiu sa is aramriz by uni uarnion Sloin an Bn [1] rsn an aaiv onrol la o h aiu onrol of rigi saraf similar o h aaiv robo maniulaor onrol algorihm of Sloin an Li [7] Diffrn from hos xising rsuls for xaml [1,]), in his ar suy h aiu raing onrol roblm unr h fram of nonlinar H hory his ar is organiz in h folloing Sion sribs h ynamis an inmais uaions of h aiu moion by mloying moifi Rorigus aramrs an formulas h raing onrol roblm of airraf In Sion 3, sign h onrollr o h aiu raing roblm, an isuss h robusnss of his onrollr agains h aramrs of airrafs Furhrmor, suy h robusnss of his onrollr agains h sir raing signals by mloying Barbala Lmma W o som simulaion rsarhs o sho h valiiy of our onrollr in sion 4 A onlusion is ran in 1

2 sion 5 Euaions an Problm Aoring o Eulr ynamis uaions [,8], h angular vloiy rrors ar govrn by h folloing iffrnial uaions: Jω& + ω Jω u 1) hr J is h inria of airraf, u is h onrol oru, an ω ω ω is h angular vloiy rror an ω an ω xrss h angular vloiy of airraf an h angular vloiy of h sir angular vloiy, rsivly h J ω + ω Jω inu isurban, an is rgar as h h is h angular momn of h sir aiu All variabls ar xrss in h boy fram h oraor nos h vor rou oraion In orr o suy h aiu raing roblm, h inmais uaions of h aiu rror n b sulmn Bas on h inmais uaion of h aiu rror aramriz by uni uarnion [3] an rlaionshi bn uni uarnion an moifi Rorigus aramrs [3], h inmais uaions of h aiu rror aramriz by moifi Rorigus aramrs may b srib by h folloing uaions & ) hr G ) ω 1 1+ G ) I + + I) 3) Aoring o h finiion of G ) in uaion 3), h folloing roris hol 1 G ) 1 + ), 4 1 G ) 1 + ) 4) 4 If hr r no informaion abou h inu isurban, i is imossibl o sign h aiu raing onrollr Hovr, i is also imossibl o obain h rf informaion of h inu isurban in h raial aliaions hrfor, h rasonabl goal of h aiu raing onrol is o sign h onrollr u, ω ) suh ha h los-loo sysm saisfy h folloing rformans P1) h los-loo sysm is asymoially sabl hn ; P) For som virual ouu y, ω ) h L gain of h los-loo sysm is no grar han a givn onsan > ), i y γ γ γ L [, +) iniiaing from ) an ω ) [ ] 5) [ ] h sriion of h aiu raing onrol of airraf is aually h roblm of h nonlinar H In his ar, onsir h folloing virual ouu y ω + 6) 3 Conrollr sign an is robusnss 31 Prvious Knolg In his subsion, inrou som funamnal nolg, hih is ruir in orr o rov our main rsuls Consir h nonlinar sysm blo: x& f x), z h x) x) 7) Assum ha x is is balan oin, ha is, f ) W rall [5] ha h sysm 7) is globally zro-sa abl if z ) imlis lim x ) funion ah On h ohr han, a nonngaiv V : R n R is sai o b ror if for 1 a >, h s V [, a]) { R n : V x) a} x is oma Lmma 1 [5] If h sysm 7) is globally zro-sa abl an V x) is ror an h rivaiv V x) ih rs o im along h rajory of h sysm 7) saisfis V x )) z h x) h x), > hn h sysm 7) is globally asymoially sabl Lmma Barbala Lmma [6] ) If h funion

3 g) saisfis g ), g ) L, an g ) L, [1, ) hn lim g ) Lmma 3 h aiu raing sysms srib by uaions 1) an ) ih h ouu srib by uaions 6) ar globally zro-sa abl 3 Conrollr sign For h aiu raing sysms srib by uaions 1) an ), onsir h sa ransformaion as follos: I z J 1 I ω, > 8) Obviously, i is a iffomorhism From uaions 1), ) an 8), h n sa variabls z, ) ar govrn by h folloing uaions J z+ ω Jz u + G ) ω + ω 1 ) z J ) 9) G 1) horm 1 Consiring h folloing onrol la hr u ω + ) G ) ω ω an 11) ar o osiiv sal, h los-loo sysm omos of uaions 1), ), 6) an 11) saisfis h rformans P1) an P), hr γ Whr J) + λ J ) 1 λ J ) 1) λ an J ) λ no h imum an imum ignvalu of h inrial marix J, rsivly Proof: subsiuing h onrol lo 11) ino uaion 9) yils o: & 13) Jz + ω Jz ω + ) Dfin h folloing Lyaunov funion V ω, ) z J z + 4 ln1 + ) 14) h rivaiv of V x) ih rs o im along h rajory of h sysm 9) an 1) yils o: V& ω, ) z z J J ω Jz ω Jω ) + ω 8 & z& + 1+ ω + ) ω J ω J 15) W hav us h inial uaion z J ω Jz) inualiis blo: hr By mloying h 1 ω J ω Jω + J 16) ) Euaion 15) may b ru o V& ω, ) ω Jω y λ + Jy + J ) y y + J + λ I + ) J ) + λ ω J J ) 18) Euaion 6) has bn ali hr For, ingraing uaion 18) from o yils o V ) V ) + + λ J ) λ J ) y y 19) On h on han, V ) hn h iniial valus ar ) [ ] an ω ) [ ] On h ohr han, V ), for hrfor hn h inualiy 1) hols h los-loo sysm omos of uaions 1), ), 6) an 11) saisfis h rforman inx P) from uaion 19) Fuhrmor, hn, inualiy 18) is 3

4 ru o V ω, ) λ J ) y y ) Aoring o h finiion in uaion 14), h Lyaunov funion V ω, ) is ror hrfor, lim ) [ ] an lim ω ) [ ] hol by mloying lmma 1 an lmma, ha is, h los-loo sysm omos of uaions 1), ), 6) an 11) also saisfis h rforman inx P1) Rmar: From uaion 11), h onrollr in horm 1 is innn of h sruur aramrs of airrafs; hrfor i osssss h robusnss agains h unrainy of h airraf aramrs 33 Analysis of Robusnss of h onrollr In his subsion, furhr isuss robusnss of h onrollr 11) agains h sir signal By mloying h Barbal Lmma, obain h folloing rsul: horm Consir h aiu raing sysms srib by uaions 1) an ), if h inualiy 4 < 1) λ J ) hols, hn unr h onrol la 11), h aiu rror an h angular vloiy rror ar globally onvrgn for any inu isurban, ha is, L [, +) L[, +) lim ) [ ] lim ω ) [ ] Proof: x Qx x from Euaion 15), may obain V& z) ω Jω ω J hr ω x J Q J b F ) λ F a x ω J + b x J J I, F, I J ) + > an ) Aoring o inualiy 1), h symmrial marix Q is osiiv, so a λ Q) >, ingraing uaion ) from o yils o: V ) V ) a x s + b x s For 3) No ha V +), hrfor, by mloying Sharz inualiy, an hav V ) a x + b x L L L 4) hr h oniion L [, +) has bn us Solving ou inualiy 4) yils o b + b L L a L + 4aV ) x 5) so x L [, +) From uaion 3), 5) an 14), hav V ) a b b x + b x x L x s + V ) s + V ) L s + + V ) hrfor, h sa x is uniformly boun, ha is, x L [, +) Furhrmor, mloying h oniion L [, +) an uaion 1), ) an + 11) yils o x L [, ) hrfor, aoring Barbala Lmma, onlu ha lim x ), ha is, lim ) lim ω ) [ ] 4 Simulaion [ ] an In orr o xam our rsuls, his sion givs a simulaion rsarh h airraf inrial unr onsiraion is J iag[1 31] In h simulaion, h airraf as ruir o ra h sir angular vloiy ω from 4

5 h iniial aiu ih ω ), hr h vor [ ] an φ 4648ra141 ) orrson o h Eular axis an Eular angl of h iniial aiu rror marix, rsivly hus h iniial rrors ar an ) [ ] ω ) In h simulaion, h onrollr gains r sl 6, 8, 6 h simulaion rsuls ar i in Fig1, hr soli lins xrss h rsonss hn airraf aramrs ar auraly non, an ash lins xrss h rsonss hn hr xiss h aiion isurban inrial follos as: J As assr in horm, Fig 1 shos ha hn h inrial marix is xaly non, our onrollr guaran ha h los-loo sysm is asymoially sabl Furhrmor, hn hr xiss h unrainy in h inrial marix, again as assr in horm 1 an horm, Fig1 also shos ha h los-loo sysm is sill asymoially sabl ) ) a) f) b) Fig 1 Fig1 a)-): Profils of angular vloiis raing rrors Fig1)-f): Profils of moifi aramrs raing rrors 5

6 In a or, h simulaions hav shon ha our onrollrs guaran h robusnss no only agains h unrainy of h airraf aramrs bu also agains h sir signal 5 Conlusion I has shon from h ring isussion ha h aiu onrol roblm of airraf may b solv in h fram of nonlinar H onrol hory his an furhr la o h sign mhoology of h robus onrollr for h aiu onrol roblm of airraf Boh h hori rsuls an h simulaiv suis hav shon ha our onrollr for h aiu raing roblm of airraf osssss h robusnss no only agains h airraf aramrs, bu also agains h ra signals Rfrns [1] J J Sloin an M D Di Bno, Hamilonian aaiv onrol of saraf, IEEE ransaion on Auomai Conrol ): 848~85 [] J Wn an K Krnuz-Dlgao h aiu onrol roblm IEEE ransaion on Auomai Conrol ) 1148~116 [3] H Wiss Quarnion-bas ra/aiu raing ih aliaion o gimbal aiu onrol J of Guian, Conrol, an Dynami, 1993, 161):1345~1349 [4] M D Shusr A survy of aiu rrsnaions J Of h Asronauial Sins, 1993, 433): 439~514 [5] C I Byns, AIsiori, an J C Willms Passiviy, fba uivaln, an h global sabilizaion of imum has nonlinar sysms IEEE rans on Auo Conr 1991, 3611): [6] S Sasry an MBoson, Aaiv Conrol, N Jrsy: Prni-Hall, 1989 [7] J J Sloin an W Li, Aaiv maniulaor onrol: A as suy, IEEE ransaion on Auomai Conrol ): 955~13 [8] A J van r Shaf L Gain an Passiviy hnius in Nonlinar Conrol, Sringr-Vrlag, Lonon, 6