Gain scheduled observer state feedback controller for rational LPV systems

Transcription

1 Proceedigs of the 7th World Cogress he Iteratioal Federatio of Atomatic Cotrol Seol Korea Jly Gai schedled observer state feedback cotroller for ratioal LPV systems Boali A Yagobi M Chevrel P Istitt de Recherche e Commicatios et Cyberétiqe de Nates Re de la Noë BP 90 F-4430 Nates Frace Ecole des Mies de Nates 4 Re Alfred Kasler BP 07 F Nates Frace AisBoali@emfr Abstract: his paper addresses the desig of gai schedled observer-based cotrollers for ratioal liear parameter varyig systems (LPV Sch systems are eqivaletly recast as affie descriptor LPV systems Based o this ew realizatio a descriptor observer-based cotroller is desiged by meas of some ew sfficiet coditios give as LMIs he stability of the descriptor closed-loop system is proved A state space ratioal cotroller with a observer-based strctre is the derived he stability of the obtaied ratioal closed-loop is also proved A merical eample is preseted to illstrate the efficiecy of the method INRODUCION Aalysis ad cotrol of liear parameter varyig systems have bee a very poplar topic drig the last two decades sice this class of systems covers a large scale of practical systems icldig some o liear systems as illstrated for istace i (Apkaria et al 994 (Leith et al 000 ad may other papers Several techiqes ca be fod i the literatre abot gai schedlig cotrol of LPV systems Oe of these approaches cosists i iterpolatig several ivariat cotrollers ted for differet operatig poits his classical gai schedlig desig is obtaied i three steps: defie the operatig poits for the LPV system desig a liear time ivariat cotroller for each oe of them ad fially bild the gai schedled cotroller he last step of the desig is based o iterpolatio techiqes Nmeros iterpolatio schemes ca be sed as for istace the iterpolatio of the state space matrices the poles zeros ad gais of the cotrollers(see for eample (Stillwell et al 999 his simple scheme has the drawback that i most cases o theoretical proof of the stability of the closed-loop is give However i (Stilwell et al 000 the stability of the closed-loop is esred thaks to some additioal costraits Frthermore iterpolatio of observer ad state feedback gais has bee tackled with i several papers as for istace i (Raharijoaa et al 006 ad (Berriri et al 006 Aother way to desig gai schedled cotrollers is based o the Lyapov s theory ad the etesio of the LMI coditios kow i the LI case he mai advatage of this approach is that the stability of the closed-loop system is esred Recetly sigificat progress has bee made i this area by sig some special represetatios for LPV systems: LF (liear fractioal trasformatio affie polytopic represetatios I the case of polytopic systems the desig of a observer-based cotroller was addressed by (Bara et al 00 (Bara et al 00 Or objective is to eted these reslts to the case of ratioal LPV systems Nmeros reslts o the cotrol of LPV ratioal system ca be fod i the literatre Most of them are based o the se of a LF ad provide sfficiet coditios for the desig of stabilizig cotrollers (see for eample (Scherer et al 00 I this paper the desig of observer-based cotrollers for ratioal LPV systems is tackled with via a eqivalet descriptor realizatio with a affie depedecy o the varyig parameter his idea was sed i (Boali et al 006 ad (Boali et al 007a i the state feedback sythesis case his paper is orgaized as fallows We preset the problem der cosideratio i Sectio Some prelimiary reslts cocerig the stability of LPV descriptor systems are preseted i Sectio 3 I sectio 4 we cosider the desig of a observer-based cotroller for the eqivalet LPV descriptor system he obtaied descriptor cotroller is sed i Sectio 5 to derive a state space ratioal cotroller with a observer-based strctre Fially a merical eample is give i Sectio 6 to illstrate the efficiecy of the proposed method before cocldig Notatio: he otatio A > 0 (respectively A 0 stads for A defiite positive (respectively semi-defiite positive he otatio Bdiag( A A deotes for a bloc diagoal matri with A ad A o the pricipal diagoal He { A } stads for A + A ad for terms that are idced by symmetry PROBLEM FORMULAION he cosidered class of LPV systems is described by /08/$ IFAC / KR-00303

2 7th IFAC World Cogress (IFAC'08 Seol Korea Jly ( t Ar ( θ( t ( t + Br ( θ( t ( t ( Σr : ( y( t Cr ( θ( t ( t + Dr ( θ( t ( t where the state matrices are assmed to be ratioal fctios of the time varyig vector θ ( t he state vector is give by t he ipt vector is t ad y y( t is the otpt vector he vector ( t ( t ( t q ( t θ θ θ θ is assmed to be real cotios time varyig ad satisfyig the followig costrais: -Each parameter θ i ( t is real time measrable ad rages betwee kow etremal vales θi ( t θ i θ i -he variatio rate of each parameter θ i ( t is limited by kow pper ad lower bods that is θ i ( t τ i τ i As a coseqece the varyig parameter ad its rate evolve both i hyper rectagles with vertices defied by Ξ {( ω ω q \ ωi { θi θi} } ad {( τ τ q \ τi { τi τi} } Ω he set of possible trajectories θ ( is oted Θ he mai objective of this paper is to desig a observer state feedback cotroller which is ratioally depedet o the time varyig parameter he soght cotroller ca be give by ˆ Ar ˆ + Br + Lr ( yˆ y ( Kr : yˆ Cr ˆ + Dr ( Fr ˆ + Gr ( yˆ y where Lr ( θ ad Fr ( θ are respectively the parameter varyig observer ad state feedback gais which are ratioally depedat o the varyig parameter I order to desig a stabilizig cotroller as give by ( we propose to se a descriptor realizatio Ideed it has bee demostrated i (Boali et al 006 that the ratioal LPV system give by ( ca be eqivaletly recast ito a descriptor realizatio as follows I 0 A θ A θ B w 0 0 A3 A4 + B ( d : Σ y( t ( C C (3 where ( A i ( θ i { 3 4} are affie fctios of the parameter θ ( t ad matrices B B C ad C are all costat Matri A4 ( θ is o siglar for all trajectories θ( he followig eqatios describe the relatio betwee realizatio ( ad (3 Ar A A A4 A3 (4a Br B A A4 B (4b Cr C A4 A3 C (4c D C A B (4d r 4 he desig of the cotroller give by (3 ca be doe thaks to the sythesis of a observer state feedback cotroller for the eqivalet affie descriptor realizatio (3 his cotroller ca be give by Eˆ A ˆ+ B + L ( yˆ y ( Kd : yˆ Cˆ (5 F( θ ˆ where the geeralized state vector is ˆ ˆ ˆ the state matrices are partitioed as follows I 0 E 0 0 A θ A θ A θ A3 A4 B B B L θ L θ L ( θ ( C C C F F F ( ad he desig of the cotroller ( thaks to classical LMIs as those proposed i (Bara et al 00 is iterestig whe state matrices are polytopic De to the ratioal depedecy of the state matrices i ( we ca o loger se a fiite set of LMIs as doe i (Bara et al 00 he se of realizatio (3 is a itermediary step I fact we propose here ew sfficiet coditios allowig to desig a descriptor observer-based cotroller by meas of a fiite set of LMIs Sice (3 is affiely depedat o the varyig parameter it seems atral to choose the same strctre for the descriptor cotroller A ratioal cotroller of the form ( ca the be easily etracted 3 PRELIMINARY RESULS 3 Admissibility coditios of LPV descriptor systems I this paper the admissibility of LPV descriptor systems as defied i (Masbchi et al 003 is cosidered I fact cosider the followig LPV descriptor system E A( θ (6 with t ad rak ( E We assme withot loss of geerality that I 0 E 0 0 sice it is always possible to cosider a siglar vale decompositio (SVD of matri E Moreover this particlar strctre is atral whe cosiderig the LPV descriptor system ( Σ d heorem : (Masbchi et al 003 he descriptor system give by (6 is admissible if there eists a cotiosly differetiable fctio P : Θ sch that for all θ( E P P E 0 (7a d He { A P } + E ( P < 0 (7b A eqivalet strict LMI coditio has bee proposed i (Boali et al 006a ad is remided et 493

3 7th IFAC World Cogress (IFAC'08 Seol Korea Jly heorem : (Boali et al 007a he descriptor system give by (6 is admissible if there eist cotiosly differetiable fctios P : Θ ad ( S : Θ sch that for all θ( P P > 0 (8a d He { A ( P E US V } E ( P E 0 ( VU + + < (8b where are matrices of fll colm rak ad composed of bases of ker E ad ker E respectively Remark : Coditios give by (7 or (8 are hardly eploitable for the desig of a observer state feedback cotroller withot avoidig the se of griddig techiqes For this reaso a ew coditio is proposed et i terms of a eteded LMI heorem 3: he descriptor system (6 is admissible if there eist cotiosly differetiable fctios P : Θ ( S : Θ ad a matri W sch that for all θ( P P > 0 ad S S > 0 (9a He { W } W A W + Φ dp ( θ Φ ( θ + Bdiag ( 0 0 < 0(9b Φ ( θ with Φ : Bdiag ( P S Proof For brevity reasos oly the mai lies of the proof are preseted here Let assme that coditios (9 hold Cosiderig that ll spaces of ( I 0 0 ad 0 I 0 ( I A( θ I are A( θ I I ad I 0 I we ca drop matri W by applyig projectio lemma ad Schr complemet his leads to coditios (8 As a coseqece the descriptor system give by (6 is admissible Remark : Whe applyig coditios preseted i heorem 3 i (Boali et al 006a for a state feedback desig it appears that a additioal strctre costrait o the matri W is ecessary to solve the LMIs his is obviosly ot the case whe sig heorem 3 Oe of the advatages of this eteded LMI coditio is that there is o loger prodcts betwee the state matrices ad the kow fctios P ( ad S ( he parameterized LMIs proposed i heorem 3 ca be solved thaks to a fiite set of LMIs whe dealig with polytopic descriptor systems A dal form of the previos heorem ca easily be derived 3 Strog eqivalece ad admissibility Whe two LI realizatios are eqivalet the stability of the first oe implies the stability of the secod his simple reslt is o loger tre whe cosiderig time varyig systems as illstrated i (Cobb 006 I this paper we itrodce a ew characterizatio of eqivalece of two realizatios i the descriptor LPV case I fact cosider two LPV descriptor realizatios give by the pairs ( EAθ EA ( θ (0 Defiitio : he two realizatios (0 are said strogly eqivalet if there eist two cotiosly differetiable fctios M : Θ N : Θ sch that for all θ( i M ( θ ad N ( θ are o siglar matrices ii M ( θ N ( θ are cotiosly differetiable ad the followig eqatios hold M EN( θ E (a M A N A (b d M E ( N( θ 0 (c his property is sed i the et reslt heorem 4: (Boali et al 007b Cosider the two strogly eqivalet descriptor realizatios ( EAθ ad ( EAθ he followig statemets are eqivalet i here eists a cotiosly differetiable fctio P : Θ sch that for all θ( E P P E 0 d He { A P } + E ( P < 0 ii here eists a cotiosly differetiable fctio P : Θ sch that for all θ( E P P E 0 d He { A P } + E ( P < 0 33 Admissibility of descriptor systems with a special strctre I this sectio we cosider two descriptor polytopic pairs give by ( EA ( θ ad ( EA ( θ Lemma : If there eist affie fctios : X Θ ad X : Θ sch that for all θ( coditios hold the followig E X i X i E 0 (a d He { Ai Xi } + E ( Xi < 0 (b 494

4 7th IFAC World Cogress (IFAC'08 Seol Korea Jly for i { } the for ay boded matri A3 ( θ the E 0 A ( θ 0 descriptor pair give by 0 E is A3 A admissible Proof Assme that coditios ( hold for the pairs ( EA ( θ ad ( EA ( θ hs there eist positive real mbers r ad r sch that He A X + E X θ r I { } { } ri He A X + E X < 0 Sice X ( θ ad A3 ( θ are boded for all θ( there eists a real positive mber r 3 sch that X A3 r3i Let s cosider ow the matri X Bdiag ( X λx with λ > 0 First it is easy to see that X ( θ is a cotiosly differetiable fctio sch that for all θ( Bdiag( E E X X Bdiag( E E 0 holds We show et that the ieqality iclded i coditios (7 holds for the pair E 0 A ( θ 0 0 E A3 A Ideed by otig that He { A X } + E X λa3 X He{ A X } + E X X θ A3 θ ri + λra 3 X X A3 ri + λr( r3 I ad by choosig 0 < λ < rr ( r3 it comes that He { A X } + E X λa3 X ( He{ A X } + E X X A3 < 0 ad λ( He { A X} + E X < 0 Applyig Schr complemet leads fially to A ( θ 0 E 0 He X + X < 0 A3 A 0 E E 0 A ( θ 0 which implies that the pair 0 E A3 A is admissible accordig to heorem 4 DESCRIPOR OBESERVER-BASED CONROLLER DESIGN I this sectio the state feedback ad the observer problems are addressed separately Sfficiet coditios for the sythesis of a stabilizig state feedback gai are proposed Similarly dal coditios will be give for the sythesis of the observer gai Fially the stability of the closed-loop with the descriptor observer-based cotroller is proved 4 A ew LMI-based coditio for state feedback sythesis for descriptor systems Let s cosider the followig descriptor LPV system give by E A( θ + B y C rak( E (3 heorem 5: here eists a descriptor state feedback cotroller sch that the closed-loop pair give by ( EA + BF is admissible if the there eist cotiosly differetiable fctios P : θ P S : θ S R : θ R of appropriate dimesios ad a matri W sch that for all θ( P P > 0 ad S S > 0 (4a He { W } W A R B W + +Φ dp ( θ ( θ Bdiag ( 0 0 Φ < 0 ( θ Φ with Φ : Bdiag ( P S he state feedback gai is the give by F( θ R W (4b Proof By settig R ( F( θw this reslt ca easily be proved sig heorem 3 I the case of a polytopic descriptor system previos coditio ca be epressed thaks to a fiite set of LMIs as follows Corollary : here eists a descriptor state feedback cotroller sch that the closed-loop pair give by ( EA + BF is admissible if the there eist affiely depedat fctios P : θ P S : θ S R : θ R of appropriate dimesios ad a matri W sch that ( ωi τj Ξ Ω P( ωi P( ωi > 0 ad S ωi S ωi > 0 He { W } W A( ωi R( ωi B ( ωi W + +Φ ( ωi Bdiag ( P ( τj P ( Φ + < 0 Φ( ωi with Φ ( ωi : Bdiag ( P ( ωi S ( ωi he state feedback gai is the give by F( θ R W 4 A ew LMI-based coditio for observer sythesis of descriptor systems New sfficiet coditios similar to those proposed for the state feedback desig are proposed et heorem 6: here eist a descriptor observer gai sch that the closed-loop pair give by ( EA + L C is 495

5 7th IFAC World Cogress (IFAC'08 Seol Korea Jly admissible if the there eist cotiosly differetiable fctios P : θ P S : θ S H : θ H of appropriate dimesios ad a matri W sch that for all θ( P P > 0 ad S S > 0 (5a He { W } W A H C W + + Φ dp ( θ Φ ( θ + Bdiag ( 0 0 < 0 (5b Φ ( θ with Φ : Bdiag ( P S he observer gai is the give by L( θ W H As i Corollary those coditios ca be tred ito a fiite set of LMIs i the case of affie LPV descriptor systems 43 Stability of the closed-loop LPV system with the observer state feedback cotroller We cosider the closed-loop descriptor system ( with the cotroller (3 We assme that the state feedback F( θ ad the observer L( θ have bee sythesized thaks to heorem 5 ad heorem 6 he closed-loop ca be described by E 0 A BF 0 E ˆ L( θc A( θ + BF( θ + L( θc ˆ Let s defie e ˆ he closed-loop system is eqivalet to E 0 e A + L C 0 e 0 E (6 ˆ L( θc A( θ + BF( θ ˆ Accordig to the desig of the state feedback ad the observer the pairs ( EA + BF ad ( EA + L C are both admissible As a coseqece applyig Lemma lead to the admissibility of the closedloop give by (6 5 OBESERVER SAE FEEDBACK DESIGN FOR RAIONAL LPV SYSEMS I this sectio we etract the ratioal observer state feedback cotroller thaks to the descriptor cotroller (5 desiged i previos sectio Oce the state space ratioal LPV cotroller obtaied we still eed to prove the stability of the state space closed-loop hese reslts are formlated i et theorem heorem 8: here eists a ratioal observer-based cotroller ( stabilizig the ratioal LPV system give by ( if there eists a descriptor observer-based cotroller give by (5 stabilizig the descriptor LPV system (3 he ratioal state feedback ad observer gais are the give by Fr ( I F ( A4 + BF B FR (7a Lr L A A4 L (7b with FR F F A4 A3 (7c ad the matri Gr ( θ is give by G F ( A + B F L (7d r 4 Proof: Realizatios ( ad (3 are strogly eqivalet (see Defiitio Ideed if we cosider the two followig cotiosly differetiable ad o siglar fctios I A A4 I 0 M N 0 A4 A4 A3 I it appears that d M EN E M E ( N 0 A A3 A r ( θ 0 M N A A4 0 I ad that M N are o siglar ad cotiosly differetiable fctios Based o this a strogly eqivalet system to the descriptor cotroller (5 ca be give by I 0 ˆ A ˆ r θ ˆ 0 I ˆ X X Br Lr ( θ + ( yˆ y + A 4 θ B A4 θ L θ (8 ˆ ( FR F Xˆ yˆ Cr ( θ ˆ + CX where ˆ X A4 A3 ˆ + ˆ L L A A L r 4 ad FR F F A4 A3 Elimiatig ˆX from (8 leads to the followig state space realizatio for the cotroller ˆ Ar ˆ + Br + Lr ( yˆ y ( Kr : yˆ Cr ˆ + Dr Fr ˆ + Gr ( yˆ y with Fr ( I F ( A4 + BF B FR ad Gr F ( A4 + BF L Frthermore the admissibility of the descriptor LPV system (6 proves that ( ˆ ˆ 0 which implies that ( ˆ ˆ 0 his meas that the ratioal LPV closed-loop system is stable Fially the ratioal cotroller is etirely described by eqatios (7 ad stabilizes the state space ratioal LPV system ( 6 NUMERICAL EXAMPLE Let s cosider the followig ratioal LPV system θ θ 3θ θ θ y + with θ 5 5 ad θ 496

6 7th IFAC World Cogress (IFAC'08 Seol Korea Jly he simple chage of variables 3 θ ( θ + leads to the followig affie LPV descriptor realizatio 0 0 θ θ θ ( θ y ( 0 3 Applyig heorem 5 ad heorem 6 leads to the followig gais F θ θ θ L θ 0 06θ θ he ratioal LPV cotroller is the derived thaks to eqatios (7 We obtai the followig ratioal matrices 0387θ 095θ θ 086θ 446 F r ( θ θ + θ + L r 04θ 394θ θ θ + 000θ 0558θ 057 G r ( θ θ + o see the iflece of the chose parameter variatio the eigevales domai of the estimatio state error ad the state feedback is give i Figre Figre : he eigevales domai for matrices A ( θ + L ( θc ad A ( θ + B F r r r 7 CONCLUSION r r r I this paper a method for the desig of gai schedled observer-based cotrollers for ratioal LPV systems is preseted Based o a eqivalet descriptor affie LPV realizatio a descriptor observer-based cotroller is desiged by meas of some ew sfficiet coditios give as LMIs A ratioal LPV cotroller with a observer-based strctre is the derived he stability of the obtaied ratioal closed-loop is proved he effectiveess of the proposed method has bee tested o a merical eample REFERENCES Apkaria P Gahiet P ad Becker G (994 Self schedled H cotrol of liear parameter varyig systems Proceedig of the America Cotrol Coferece o Cotrol pp Bara G I Daafoz J Ragot J (00 Gai Schedlig techiqes for the desig of observer state feedback cotrollers 5 th IFAC World Cogress Barceloa Spai Bara G I Daafoz J Kratz F (00 Advaced gai schedlig techiqes for the desig of parameterdepedet observers Proceedig of the 40th IEEE Coferece o Decisio ad Cotrol Orlado Florida USA Berriri M Chevrel P Yagobi M Clavea F (006 Observer-based gai schedlig H cotroller for a 3 motors web trasport system 5 th IFAC Symposim o Robst Cotrol Desig olose Frace Boali A Chevrel P Yagobi M (006 Abot gai schedled state feedback cotrollers for ratioal LPV systems 9 th Iteratioal Coferece o Atomatio Robotics Cotrol ad Visio Sigapore Boali A Yagobi M Chevrel P (007a New LMI-based coditios for stability H performace aalysis ad state-feedback cotrol of ratioal LPV systems Eropea Coferece o Cotrol Kos Greece Boali A Yagobi M Chevrel P (007b Gai schedled state feedback desig for a class of LPV descriptor systems Coferece o Systems ad Cotrol Marrakech Morocco Cobb J D (006 State Feedback Implse Elimiatio for Siglar Systems over a Hermite Domai SIAM Joral o Cotrol ad Optimizatio Vol 44 N 6 pp Leith D Leithead W (000 Srvey of gai schedlig aalysis ad desig Iteratioal Joral of Cotrol Vol 73 N pp Masbchi I Kato J Saeki M ad Ohara A (003 Sythesis of Otpt Feedback Gai Schedlig Cotrollers Based o Descriptor System Represetatio IEEE Coferece o Decisio ad Cotrol pp Hawaii USA Raharijoaa Chevrel P Dc G (006 Redced order ad observer-based LPV cotrol for lateral drivig assistace 5 th IFAC Symposim o Robst Cotrol Desig olose Frace Scherer C W (00 LPV cotrol ad fll block mltipliers Atomatica (37 pp Stilwell D J Rgh W J (999 Iterpolatio of observer state feedback cotrollers for gai schedlig IEEE rasactio o Atomatic ad Cotrol 44 N 6 pp 5-9 Stilwell D J Rgh W J (000 Stability preservig iterpolatio methods for the sythesis of gai schedled cotrollers Atomatica (36 pp