Copyrgt IFAC 1t Trnnal World Congrss, Barclona, Span ROBUST H-INFINITY REDUCED ORDER FILTERING F OR UNCER TAIN BILINEAR SYSTEMS H Souly Al, M Zasadznsk, H Rafarala y and M Darouac CRAN, IUT d Longy, Unvrst Hnr Poncar ; Nancy I 1, ru d Lorran, Cosns t Roman, FRANCE -mal : {souly,mzasad}@ut-longyup-nancyfr Abstract: Ts papr nvstgats H1 rducd ordr unbasd ltrng problms for a nomnal blnar systm and a blnar systm actd b y norm-boundd structurd uncrtants n all t systm matrcs Frst, t un basdnss condton s drvd Scond, a cang of varabl s ntroducd on t nputs of t systm to rduc t consrvatsm nrnt to t ltr stablty rqurmnt and to trat t product of t nputs by t dsturbancs Tn t soluton s xprssd n trms of LMI b y transformng t problm n to a robust stat fdback n t nomnal cas and a robust statc output fdback n t uncrtan cas Copyrgt c IFAC Kyords: Blnar systms, Robust stmaton, H-nnt y optmzaton, Uncrtanty 1 INTRODUCTION T functonal ltrng purpos s to stmat a lnar combnaton of t stats of a systm usng t masurmn ts In ts papr, a rducd ordr H1 ltrng mtod s proposd to rconstruct a lnar combnaton of t stats of blnar systms by xplotng t nonlnarts n t nomnal and t robust cass Ts s acvd troug t dsgn of an obsrvr c dynamcs as t sam dmnson tan ts lnar com bnaton In addton to t stablty and L gan attnuaton rqurmnts, t ltr must also b unbasd, t stmaton rror dos not dpnd on t stats of t systm Ts condton s xprssd n trms of as many Sylvstr quatons as tr ar nputs, t an addtonal Sylvstr quaton du to t lnar part of t blnar systm Our approac s basd on t rsoluton of a systm of quatons to nd condtons for t xstnc of t unbasd rducd ordr ltr tn solv t xponntal convrgnc and L gan attnuaton problms c ar rducd to a robust stat fdback n t nomnal cas It s son tat t robust functonal unbasd ltrng problm for uncrtan blnar systms subjctd to tmvaryng norm-boundd uncrtants can b sn as a partcular cas of a statc output fdback on undr som condtons Ts problm rqurs to solv Lnar Matrx Inqualts (LMI) t an addtonal non convx Blnar Matrx Inqualty (BMI) constrant T papr s organzd as follos T condtons for t unbasdnss, xponntal convrgnc and L gan attnuaton of a rducd ordr H1 functonal ltr for contnuous-tm nomnal blnar systm ar studd n scton It s son troug scton tat t concrnd robust ltrng problm for blnar systm actd by structurd norm-boundd tm-v aryng uncrtants can b solvd as a statc output fdback problm Tn, scton concluds t papr REDUCED ORDER UNBIASED H1 FILTERING IN THE NOMINAL CASE 1 Problm Formulaton T nomnal blnar systm consdrd n ts scton s gvn by _x A x 1 y Cx D z Lx A u x B (1a) (1b) (1c)
r x(t) IR n s t stat vctor, y(t) IR p s t masurd output and z(t) IR r s t vctor of varabls to b stmatd r r n T vctor (t) IR q rprsnts t dsturbanc vctor Wtout loss of gnralty, t s assumd tat rank L r T problm s to stmat t vctor z(t) from t masurmnts y(t) As n t most cass for pyscal procsss, t blnar systm (1) as knon boundd control nputs, u(t) IR m, r ( 1 ::: m) : n u(t) IR m j u mn u (t) u max o : () T ROUF (Rducd Ordr Unbasd Fltr) s dscrbd by _ H 1 ^z Ey H u J y 1 J u y (a) (b) r ^z(t) IR r s t stmat of z(t) In ordr to avod tat som lnar combnatons of componnts of vctor z b drctly stmatd from t masurmnt y tout usng t ltr stat vctor, ts assumpton s mad n t squl Assumpton 1 Matrcs C and L vrfy rank C T T L rankc rankl: () T stmaton rror s gvn by z ; ^z Lx ; ^z ; ED () r (a) L ; EC: (b) x ; T problm of t ROUF dsgn s to dtrmn H, H, J, J and E suc tat () t ltr () s unbasd f, t stmaton rror s ndpndnt of x, () t ROUF () s xponntally convrgnt for u(t), () t mappng from to as L gan lss tan a gvn scalar for u(t) (van dr Scaft, 199) Unbasdnss condton fulllmnt From (), notc tat t tm drvatv of t rror s functon of t tm drvatv of t dsturbancs To avod t us of _ n t dynamcs of t rror, consdr as a n stat vctor Sttng u 1, tn t L gan from to as t follong stat spac ralzaton _ ; ED u H ( B ; ( A ; H ; J C)u x J Du ) () and t unbasdnss of t ltr s acvd t follong Sylvstr quatons A ; H ; J C ::: m () old As matrx L s of full ro rank, t rlatons n () ar quvalnt to( ::: m) ( A ; H ; J C) L y y I n ; L L (9) r L y s a gnralzd nvrs of matrx L satsfyng L LL y L (snc rank L r, av LL y I r) Usng t dnton of, (9) s quvalnt to( ::: m) A L y ; H L y ; J CL y (1a) A H EC ; J C (1b) r A A (I n ; L y L) and C C(I n ; L y L) Usng (a), rlaton (1) can b rrttn as H A ;K C ::: m (11) CA L y CL and y r A LA L y, C K E K t K J ; H E: (1) Tn rlaton (1a) can b xprssd n t follong compact form K LA (1) r A A ::: A m and K CA CA 1 ::: CA m C ::: C ::: m C E K ::: K (1a) (1b) and a gnral soluton to quaton (1) f t xsts, s gvn by K LA y Z(I (m)p ; y ) (1) r Z Z E Z ::: Z m s arbtrary Lmma T unbasdnss of t ltr () s acvd f and only f t follong rank condton olds rank rank (LA) T T (1) Proof Usng t prvous dvlopmnts, ltr () s unbasd, rlaton () olds, tr xsts a soluton K to (1), tn t rank condton (1) s satsd Unbasdnss condton undr ED Usng t prvous stps, t mappng from to, gvn by (), bcoms _ ; u A ; LA y 1(u) ; Z 1(u) LB ; LA y (u) ; Z (u) ; ED u A ; LA y 1(u) ; Z 1(u)!! ED (1)
r Z Z(I (m)p ; y ) and 1(u) P m u CA L y CL y u 1 CL y u m CL y (u) CB D u 1 D u m D : (1) Du to t product Z(I (m)p ; y ) 1(u)E, t rror s blnar n t gan paramtr Z n systm (1) Ts blnarty sntrnscally lnkd to t unbasdnss condton () Indd, t blnarty H n () ylds a gan K (s (1)) contanng t product H E In ordr to avod ts blnarty, consdr ED n t squl ts allos us to av LMI tractabl formulaton for t problm nstad of ntractabl BMI on Addng t constrant ED, rlatons (1), (1) and (1) bcom K LA r D (19) and a soluton to (19), f t xsts, s gvn by K LA y Z(I (m)p ; y ): () So, gv t follong lmma c s drvd from lmma Lmma T unbasdnss of t ltr () s acvd undr ED t follong rank condton LA rank rank (1) olds No, assum tat t condton (1) n lmma olds Tn rlaton () s vrd t K gvn by () and av (t) (t) n (1), _ u A ; LA y 1(u) ; Z 1(u) P {z } m u H LB ; LA y (u) ; Z (u)! () r Z Z(I (m)p ; y )No, as t tm () of t dsgn objctvs as bn solvd, t rmans to trat t ponts () and () of ts objctvs Exponntal stablty and L gan attnuaton No, ntroducacang of varabls by consdrng ac u (t) n () as a structurd uncrtanty Notc tat t dnton of t uncrtanty st n () can lads to som consrvatsm snc, n t gnral cas, ju mnj ju maxj t ju mnj 1and ju maxj 1To rduc ts consrvatsm, ac u (t) can b rrttn as follos r ( 1 ::: m) u (t) (t) () 1 (u mn u max), 1 (u max ; u mn) () and 1, T n uncrtan varabl s (t) IR m r s dnd as : (t) IR m j mn ;1 (t) max 1 : () By usng rlatons ()-(), t dynamcs of t rror (t) n () can b rrttn as follos _ t A A ; ZC B ; ZG A ; Z C ()H A ; LA y B ; Z G ()H () Mor, () IR mrmr, () IR mqmq, H IR mrr and H IR mqq ar dnd by (a) C (I (m)p ; y ) (b) A LA y ; (c) 1 A 1 ::: m A m ; C (I (m)p ; y ); (d) B LB;[ LA ] y G (I (m)p ; y ) () B ; [ LA ] y D G (I(m)p ; y )D (f) and ; D 1 CA 1 L y ::: m CA m L y ::: 1 CL y ::: m CL y ::: ::: 1 D ::: m D () bdag( 1 I r ::: m Ir) () bdag( 1 I q ::: m Iq) CB D m D P m CA L y CL y m CL y H [I r ::: Ir ] T H [ Iq ::: Iq ] T (a) : (b) (9a) (9b) (9c) r bdag() dnots a block-dagonal matrx From (), t uncrtan matrcs () and () ar boundd as k()k 1 and k()k 1: () Accordng to t prvous dvlopmnts, () can b rrttn as t follong systm _ (A ; ZC ) q q H I r A B B ; Z H p C G G p p p (1) connctd t p () q : () p () q At ts stp, can s tat t H1 ROUF can b solvd as a partcular cas of a dual robust stat fdback problm t structurd uncrtants So, cangv t torm c nsurs t xponntal stablty of t ROUF () and t L gan attnuaton from to
Torm Suppos tat condton (1) s vr- d If tr xst matrcs P P T >, S >, S >, Y and a scalar > suc tat ( s t transpos of t o-dagonal part) (a) H T S A T P ;C T Y ;S B T P ;G T Y ;S B T P ;G T Y ; Iq H T S SH ;S SH ;S < : () t (a) A T P P A ; C T Y ; Y T C (1)I r, Y Z T P and r S and S ar dagonal matrcs t t sam structur as () and (t) rspctvly, tn t ROUF () for t systm (1) s xponntally stabl and as a L gan from to lss tan or qual to Proof By consdrng systm (1)-() as a dagonal norm-bound lnar drntal ncluson (Boyd t al, 199), t follong auxlary systm drvd from (1) (s (L and Fu, 199) for dtals, omttd bcaus of lack of plac) _ (A ; ZC ) A ;1 S ;1 B S ;Z C ;1 S ;1 G S q q S1 H I r ;1 B ;1 B p p ;1 S 1 H p p () s ntroducd, r S and S av bn dnd n t torm Lt Y Z T PTnby usng t boundd ral lmma, systm (1)-() s xponntally stabl and as a L gan from to lss tan or qual to f tr xst P P T >, S >, S >, Y and a scalar > suc tat matrcs n systm () satsfy ts nqualty (a) Ir S ;1 T A P ;S ;1 T C Y ;Imr S ;1 T B P ;S ;1 T G Y ;Imq ;1 T B P ; ;1 T G Y ;Iq S 1 H ;Imr (b) ;Imq Ir ;Ir < t (b) ;1 S 1 H Pr- and post-multply ts 1 nqualty by bdag(i n S S 1 1 I q S S 1 I r), and us t Scur lmma to dlt t last r ros and t last r columns, t LMI () s obtand Assum tat (1) s vrd Tn usng Z P ;1 Y T and (), t matrcs of ROUF () ar gvn by (11) and (1) ROBUST REDUCED ORDER H1 FILTER In ts scton, ll mak rfrnc to t follong uncrtan blnar systm _x (A A (t))u x (B B(t)) (a) y (C C (t))x (D D(t)) z Lx (b) (c) r x(t), y(t), z(t), (t) and u(t) av bn dnd n scton T matrcs A, A, B, C, D and L ar knon constant matrcs tat dscrb t nomnal systm of () gvn by (1) T uncrtan matrcs A (t), B(t), C (t), D(t) and A (t) can b rttn as A (t) B(t) Mx C(t) D(t) My (t) A 1 (t) ::: A m (t) {z } (t) bdag( (t)) Ex E (a) M 1 x ::: M m x E 1 xt ::: E m x T T (b) r M x IR n`, M y IR p`, E x IR `n and E IR `q ( ::: m) ar knon constant matrcs c spcfy o t lmnts of t matrcs of t nomnal systm ar actd by t uncrtan paramtrs n (t), ( ::: m) T tm-varyng uncrtants n (a) and (b) ar assumd to b structurd and boundd, (t) ar dagonal matrcs satsfyng (t) I` For t sam rasons xpland n scton, t constrant E D () M y s usd nstad of ED n t squl of scton Tn (19)-(1) must b rplacd by Kb LA t b M y (a) K LA b y Z(I(m)p ;b b y ) (b) LA rank b rank b : (c) Introducng t augmntd stat vctor gvn by T x T T (9) usng () and t cang of varabls n ()-(), t systm obtand by t concatnaton of () and ()-(b) s gvn by _ (t) A 1 ; J My ; M x H Mx M x M x ;P m (t)[e x ] J M (t)[ex y ] M H x Mx (t)[ex ] P B m B; J D A (t)[ex ] Mx ;P Mx m 1 I r J D J M y J M y (t)e (t)e! () r matrx satss t unbasdnss rlaton () Usng (11), (1), ()-(9) and (a), t uncrtan systm () s quvalnt to
(t) _ P m A A;ZC p p p M x M BM ;ZGM B M ;ZG M M B M ;ZG M B M ;ZG M p B M ;ZG M B B;ZG q x q q q q q q q q I r connctd t H x H E x E x E x E x A A ;ZC B;ZG p H E E p T x p T p T p T T p p T T T p p p T T {z } M x BM ;ZGM bp qx T q T q T q T T q qt T T q q q T T {z } bq p x p p p (1) b ( t) () t b ( t) bdag( x() () () (t) (t) (t) ( t) ( t) ( t)) and r t matrcs c av not yt bn dnd ar gvn by (t ` `1 ::: `m) A 1 A 1 m A m (a) M 1 Mx 1 m Mx m M 1 Mx 1 m Mx m B M LM x ; (b) (c) LA b y M (d) GM (I (m)p ; b b y ) M () B M LM ; LA b y M (f) G M (I (m)p ; b b y )M (g) B M LM ; LA b y M (a) G M (I (m)p ; b b y )M (b) B M ; LA b y M (c) G M (I (m)p ; b b y )M: (d) H x I n ::: In T IR mnn E x E x Ex 1 T E E T () E xt ::: E x T T IR m`n (f) ::: E m x T T ::: E T T IR m`q x() bdag( 1 I n ::: m In) ( t) bdag( 1 (t) ::: m (t)) ( t) bdag( 1 1 (t) ::: m m (t)) t IR `n (g) () () (j) (k) M CM x M y m M y M M CM M CM ::: ::: 1 M y ::: m M y (a) (b) Notc tat, from () and t dnton of (t), av t follong bounds kx()k 1 ( t) 1 ( t) 1: () Not tat t unbasdnss condton () for t ltr () s vrd for t nomnal cas, for M x,( ::: m)andm y Dn a block-dagonal matrx S > t t sam structur as t uncrtan matrx b ( t) gvn n () S bdag(sx S S S S S S S S ): () r all t submatrcs ar dagonal Tn, n t systm (1), t dtrmnaton of gan Z can b transformd nto ts robust statc output fdback control problm (s (L and Fu, 199) for dtals, omttd bcaus of lack of plac) _ A B bp B uu bq C z Dz bp y C y Dy bp u ;Z y (a) (b) r Z s t statc output fdback controllr to b dsgnd n ordr to acv stablty and prformancs lk attnuaton from t augmntd prturbaton [bp T T ] T to t augmntd controlld output [bq T T ] T T vctors u and y play trolofcontrol nput and masurd output, rspctvly T matrcs and t vctors ntroducd n () ar gvn by S B B ps ;1 ;1 1 C q B C z C S 1 D qps ;1 ;1 S 1 D q D z D ps ;1 ;1 D D y D yps ;1 ;1 D y r A A A B p B B u I r C y C A M x M M M x B C q C T (9a) (9b) (9c) A B BM B M B M B M B M B M B HT x ExT T Ex E T T x Ex H T I r
D qp D q D p D D yp D y H E E C G GM G M G M G M GM G M G : Tn t follong torm s ddcatd to t dsgn of t gan Z Torm Assum tat t rank condton n () olds, tr xsts a robust ROUF () for t uncrtan systm () f tr xst matrcs P P T >, Q Q T > and dagonal matrcs S >, S >, suc tat (t ` `1 ::: `m, s mn mr mq ` ` m`) K T y K T u A T PPA PB PBp C T C T q S B T P ; Iq D T DT q S B T p P ;S DT p DT qp S C D Dp ;Ir SCq SDq SDqp ;S QA T AQ QC T QCT q B BpS CQ ;Ir D DpS CqQ ;S Dq DqpS B T D T DT q ; Iq SB T p SD T p SDT qp ;S I nr PQ Ky< (a) Ku< (b) (c) r S S ;1, K y bdag(ky Irs ) and K u bdag(k u Iqs ) suc tat K y and K u ar bass of t null spacs of [ Cy Dyp Dy ] and [ B T u ], rspctvly All gans Z ar gvn by Z B y RKC y L Z ; B y RB RZC LC y L (1) t KR ;1 1 V1 R ;CRV 1C T R V 1 V R 1 ; B T L B Q C B L R ;1 V 1 ;V 1 C T R ;1 ; ;R;1 1 BT L V1CT R CRV T 1CR ;1 1 By L ; Q ;1 > C R V 1 C T R ;1 C R V 1 ;Bu QA T AQ QC T QCT q B BpS CQ ;Ir D DpS Cq Q ;S Dq DqpS B T D T DT q ; Iq SB T p SD T p SDT qp ;S CyQ SDyp Dy B L and R 1, R and Z ar arbtrary matrcs satsfyng R 1 R T 1 > and kr k < 1 Matrcs B L, B R, C L and C R ar any full rank factors suc tat B B LB R and C C LC R Proof By usng t boundd ral lmma, t torm can b provn usng t projcton lmma and formulas gvn n (Iasak and Sklton, 199) Tn t robust ROUF () s nally obtand by usng rlatons (11), (1) and () Rmark Unlk t robust ROUF cas c as bn solvd abov as a statc output fdback, notc tat t full ordr robust ltrng problm can b transformd nto a full ordr dynamc output fdback problm and s tn solvabl va LMI only (L and Fu, 199 Bttant and Cuzzola, ) Tn, n full ordr robust ltrng, > Tatst BMI (1c) bcoms P In In Q man drnc t t rducd ordr ltrng r tr s an addtonal non convx constrant (BMI) Tr s no cnt algortm to solv ts non convx problm c can only av local solutons by mans of urstcs suc as t con complmntary lnarzaton (El Gaou t al, 199) CONCLUSION Ts papr as prsntd a smpl soluton to t H1 ROUF problm va LMI mtods for blnar systms Aftr gvng condtons for t xstnc of t ROUF, t s son tat t ROUF dsgn s rducd to a robust stat fdback problm n t nomnal cas and to a statc robust output fdback on n t blnar systm s actd by t structurd norm-boundd tmvaryng uncrtants tn tr ar an addtonal non convx rlaton to solv REFERENCES Bttant, S and FA Cuzzola () Unbasd robust H1 ltrng by mans of LMI optmsaton In: Proc IFAC Symposum on Robust Control Dsgn Pragu, Czc Rpublc Boyd, SP, L El Gaou, E F ron and V Balakrsnan (199) Lnar Matrx Inqualty n Systms and Control Tory SIAM Pladlpa El Gaou, L, F Oustry and M A t Ram (199) A con complmntary lnarzaton algortm for statc output-fdback and rlatd problms IEEE Trans Aut Contr, 111 11 Iasak, T and RE Sklton (199) All controllrs for t gnral H1 control problms : LMI xstnc condtons and stat spac formulas Automatca, 111 L, H and M Fu (199) A lnar matrx nqualty approac to robust H1 ltrng IEEE Trans Sgn Proc, van dr Scaft, AJ (199) L-gan analyss of nonlnar systms and nonlnar statfdback H1 control IEEE Trans Aut Contr,