Robust Stability and Stabilization for Singular Systems With State Delay and Parameter Uncertainty

Transcription

1 11 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY Now, we choose 8= 1; ; 1 9= 19; 5 5; 7 he corresponding Riccai equaion (13) admis he maximal soluion 5= 1; ; 7 : By heorem 2, all opimal conrols are given as follows: u() = ; ; 7 x() +v() wih v(1) 2 L 2 F (< m ). Moreover, one feedback law is u() = ; ; 7 x(): Nex, we would like o see how he choice of 8 and 9 migh affec he form of he opimal conrols. ake he following marices: 8 " = 1+"; ; 1+" 9 k = : 19k; 5k 5k; 7k parameerized by " and k wih j"j < 1=2 and <k<18. Boh 8 " and 9 k are posiive definie and he corresponding Riccai equaion (13) admis he maximal soluion 5 ";k = k; ; (1 + ") " In his case, i follows from heorem 2 ha all opimal conrols of he original LQ problem can also be given as follows: u() = ; ; " x() +v() wih v(1) 2 L 2 F (< m ). Hence, a differen opimal feedback law is u() = ; ; " x(): I is ineresing o noe ha he aforemenioned opimal conrols do no depend on he parameer k. REFERENCES [1] M. A. Rami and X. Y. Zhou, Linear marix inequaliies, Riccai equaions, and indefinie sochasic linear quadraic conrols, IEEE rans. Auoma. Conr., vol. AC-45, pp , June 2. [2] M. A. Rami, X. Y. Zhou, and J. B. Moore, Well-posedness and aainabiliy of indefinie sochasic linear quadraic conrol in infinie ime horizon, Sys. Conrol Le., vol. 41, pp , 2. [3] S. Chen, X. Li, and X. Y. Zhou, Sochasic linear quadraic regulaors wih indefinie conrol weigh coss, SIAM J. Conrol Opimiz., vol. 36, pp , [4] M. Kohlmann and X. Y. Zhou, Relaionship beween backward sochasic differenial equaions and sochasic conrols: A linear-quadraic approach, SIAM J. Conrol Opimiz., vol. 38, pp , 2. [5] R. Penrose, A generalized inverse of marices, in Proc. Cambridge Philos. Soc., vol. 52, 1955, pp [6] H. Wu and X. Y. Zhou, Sochasic frequency characerisic, SIAM J. Conrol Opimiz., vol. 4, pp , 21. [7] J. Yong and X. Y. Zhou, Sochasic Conrols: Hamilonian Sysems and HJB Equaions. New York: Springer-Verlag, [8] X. Y. Zhou and D. Li, Coninuous-ime mean-variance porfolio selecion: A sochasic LQ framework, Appl. Mah. Opimiz., vol. 42, pp , 2. : Robus Sabiliy and Sabilizaion for Singular Sysems Wih Sae Delay and Parameer Uncerainy Shengyuan Xu, Paul Van Dooren, Radu Şefan, and James Lam Absrac his noe considers he problems of robus sabiliy and sabilizaion for uncerain coninuous singular sysems wih sae delay. he parameric uncerainy is assumed o be norm bounded. he purpose of he robus sabiliy problem is o give condiions such ha he uncerain singular sysem is regular, impulse free, and sable for all admissible uncerainies, while he purpose of robus sabilizaion is o design a sae feedback conrol law such ha he resuling closed-loop sysem is robusly sable. hese problems are solved via he noions of generalized quadraic sabiliy and generalized quadraic sabilizaion, respecively. Necessary and sufficien condiions for generalized quadraic sabiliy and generalized quadraic sabilizaion are derived. A sric linear marix inequaliy (LMI) design approach is developed. An explici expression for he desired robus sae feedback conrol law is also given. Finally, a numerical example is provided o demonsrae he applicaion of he proposed mehod. Index erms Coninuous singular sysems, delay sysems, linear marix inequaliy (LMI), robus sabiliy, robus sabilizaion. NOAION hroughou his noe, for real symmeric marices X and Y, he noaion X Y (respecively, X>Y) means ha he marix X Y is posiive semidefinie (respecively, posiive definie). I is he ideniy marix wih appropriae dimension, he superscrip represens he ranspose, kxk is he Euclidean norm of he vecor x, while (M ) denoes he specral radius of he marix M. I. INRODUCION Conrol of delay sysems has been a opic of recurring ineres over he pas decades since ime delays are ofen he main causes for insabiliy and poor performance of sysems and encounered in various engineering sysems such as chemical processes, long ransmission lines in pneumaic sysems, and so on [8]. Recenly, he problems of robus sabiliy analysis and robus sabilizaion for uncerain delay sysems have been sudied. Like in he case of uncerain sysems wihou delay, he mehod based on he conceps of quadraic sabiliy and quadraic sabilizabiliy has been shown o be effecive in dealing wih hese problems in boh coninuous and discree conexs [12], [18]. On he oher hand, conrol of singular sysems has been exensively sudied in he pas years due o he fac ha singular sysems beer describe physical sysems han regular ones. Singular sysems are also referred o as descripor sysems, implici sysems, generalized saespace sysems, differenial-algebraic sysems, or semisae sysems [4], [11]. A grea number of resuls based on he heory of regular sysems (or sae-space sysems) have been exended o he area of singular sysems [4], [11]. Recenly, robus sabiliy and robus sabiliza- Manuscrip received June 2, 21; revised November 13, 21. Recommended by Associae Edior L. Dai. his work was suppored in par by he Belgium Programme on Iner-Universiy Poles of Aracion, iniiaed by he Belgian Sae, Prime Miniser s Office for Science, echnology, and Culure, and in par by he Universiy of Hong Kong under Gran 1233/1987. he work of R. Şefan was suppored in par by NAO. he scienific responsibiliy ress wih he auhors. S. Xu, P. Van Dooren, and R. Şefan are wih he Cener for Sysems Engineering and Applied Mechanics (CESAME), Universié caholique de Louvain, B-1348 Louvain-la-Neuve, Belgium. J. Lam is wih he Deparmen of Mechanical Engineering, Universiy of Hong Kong, Hong Kong. Publisher Iem Idenifier 1.119/AC /2$17. IEEE

2 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 1123 ion for uncerain singular sysems have been considered. he noions of quadraic sabiliy and quadraic sabilizabiliy for regular sysems have been exended [2], []. I should be poined ou ha he robus sabiliy problem for singular sysems is much more complicaed han ha for regular sysems because i requires o consider no only sabiliy robusness, bu also regulariy and absence of impulses (for coninuous singular sysems) and causaliy (for discree-singular sysems) a he same ime [6], [7], and he laer wo need no be considered in regular sysems. Very recenly, much aenion has been paid o singular sysems wih ime delay. For he discree-ime case, when srucured uncerainy appears, some resuls on robus sabiliy were given in [19] by using properies of modulus marix. When unsrucured uncerainy appears, he resuls on robus sabiliy and robus sabilizaion were repored in [17], a linear marix inequaliy (LMI) design mehod was developed. For he coninuous-ime case, numerical mehods for such sysems were discussed in [1] and [3], while [23] sudied he sabiliy problem by analyzing he sysem s characerisic equaion and some frequency domain condiions for sabiliy were given. I is worh poining ou ha no parameer uncerainy was considered in [23]. o he bes of our knowledge, when parameer uncerainy appears, here are no resuls on he problems of robus sabiliy and sabilizaion for coninuous singular delay sysems in he lieraure. In his noe, we address he problems of robus sabiliy and sabilizaion for uncerain coninuous singular sysems wih sae delay. he parameer uncerainies are ime invarian and unknown, bu norm bounded. he purpose of he robus sabiliy problem is o develop condiions such ha he uncerain singular sysem is regular, impulse free and sable for all admissible uncerainies. Following he same idea as in dealing wih he robus sabiliy problem for uncerain singular sysems wihou delay [2], [], we inroduce he concep of generalized quadraic sabiliy. I is shown ha generalized quadraic sabiliy implies robus sabiliy. A necessary and sufficien condiion for generalized quadraic sabiliy is obained in erms of a sric LMI. Similarly, he concep of generalized quadraic sabilizaion is proposed when dealing wih he robus sabilizaion problem, he purpose of which is he design of memoryless sae feedback conrol laws such ha he resulan closed-loop sysem is regular, impulse free and sable for all admissible uncerainies. A sric LMI design approach is proposed and an explici expression for he desired robus sae feedback conrol law is given. I is worh poining ou ha mos LMI-ype condiions for singular sysems in he lieraure conain equaliy consrains [13], [21], [], which will resul in numerical problems when checking such nonsric LMI condiions since equaliy consrains are fragile and usually no me perfecly [15]. herefore, he sric LMI design approach proposed in his noe is much more reliable in numerical compuaion. II. PRELIMINARIES AND PROBLEM FORMULAION Consider a linear singular sysem wih sae delay and parameer uncerainies described by (6) : E _x() =(A +1A)x() +(A d +1A d )x( ) +(B +1B)u() (1) x() =(); 2 [; ] (2) x() 2 n is he sae, u() 2 m is he conrol inpu. he marix E 2 n2n may be singular, we shall assume ha rank E = r n. A, A d and B are known real consan marices wih appropriae dimensions. >is a consan ime delay, () is a compaible vecor valued coninuous funcion. 1A, 1A d and 1B are ime-invarian marices represening norm-bounded parameer uncerainies, and are assumed o be of he following form: [1A 1A d 1B ]=MF()[N A N d N B ] (3) M, N A, N d and N B are known real consan marices wih appropriae dimensions. he uncerain marix F () saisfies F ()F () I (4) and 2 2, 2 is a compac se in. Furhermore, i is assumed ha given any marix F : FF I, here exiss a 2 2 such ha F = F (). 1A, 1A d and 1B are said o be admissible if boh (3) and (4) hold. Remark 1: I should be poined ou ha he srucure of he uncerainy wih he form (3) and (4) has been used in oher papers dealing wih he problem of robus sabilizaion for regular and singular uncerain sysems in boh coninuous and discree ime conexs; see, e.g., [16] and []. he nominal unforced singular delay sysem of (1) can be wrien as E _x() =Ax() +A d x( ): (5) Definiion 1: [4], [11]: 1) he pair (E;A) is said o be regular if de(se A) is no idenically zero. 2) he pair (E;A) is said o be impulse free if deg(de(se A)) = rank E. he singular delay sysem (5) may have an impulsive soluion, however, he regulariy and he absence of impulses of he pair (E;A) ensure he exisence and uniqueness of an impulse free soluion o his sysem, which is shown in he following lemma. Lemma 1: Suppose he pair (E;A) is regular and impulse free, hen he soluion o (5) exiss and is impulse free and unique on [; 1). Proof: Noing he regulariy and he absence of impulses of he pair (E;A) and using he decomposiion as in [4], he desired resul follows immediaely. In view of his, we inroduce he following definiion for singular delay sysem (5). Definiion 2: 1) he singular delay sysem (5) is said o be regular and impulse free if he pair (E;A) is regular and impulse free. 2) he singular delay sysem (5) is said o be sable if for any " > here exiss a scalar (") > such ha, for any compaible iniial condiions () saisfying sup k()k ("), he soluion x() of sysem (5) saisfies kx()k" for. Furhermore x()!!1: hroughou his noe, we shall use he following noion of robus sabiliy and robus sabilizaion for uncerain singular delay sysem (6). Definiion 3: he uncerain singular delay sysem (6) is said o be robusly sable if he sysem (6) wih u() is regular, impulse free and sable for all admissible uncerainies 1A, and 1A d. Definiion 4: he uncerain singular delay sysem (6) is said o be robusly sabilizable if here exiss a linear sae feedback conrol law u() =Kx(), K 2 m2n such ha he resulan closed-loop sysem is robusly sable in he sense of Definiion 3. In his case, u() = Kx() is said o be a robus sae feedback conrol law for sysem (6). he problem o be addressed in his noe is he developmen of condiions for robus sabiliy and robus sabilizabiliy for he uncerain singular delay sysem (6) given in (1) and (2). III. MAIN RESULS In his secion, we give a soluion o he robus sabiliy analysis and robus sabilizaion problems formulaed previously, by using a sric LMI approach. Firs, we presen he following resul for singular delay sysem (5), which will play a key role in solving he aforemenioned problems.

3 1124 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY heorem 1: he singular delay sysem (5) is regular, impulse free and sable if here exis a marix Q> and a marix P such ha EP = PE (6) AP + PA + A d P Q 1 PA d + Q<: (7) For he proof of heorem 1, we need he following resuls. Lemma 2 [13]: he singular sysem E _x() =Ax() (8) is regular, impulse free and sable if and only if here exiss a marix P such ha EP = PE AP + PA <: Lemma 3: Consider he funcion ': +!.If _' is bounded on [; 1), ha is, here exiss a scalar >such ha j _'()j for all 2 [; 1), hen ' is uniformly coninuous on [; 1). Lemma 4 (Barbala s Lemma) [9]: Consider he funcion ': +!.If' is uniformly coninuous and 1 '(s)ds < 1, hen lim '() =:!1 Proof of heorem 1: Suppose boh (6) and (7) hold for Q>, hen from (7) i is easy o see ha AP + PA < : (9) By Lemma 2, i follows from (6) and (9) ha he pair (E;A) is regular and impulse free. Nex, we shall show he sabiliy of he singular delay sysem (5). o his end, we noe ha he regulariy and he absence of impulses of he pair (E;A) implies ha here exis wo inverible marices G and H 2 n2n such ha [4] E := GEH = I r A := GAH = A 1 I nr (1) I r 2 r2r and I nr 2 (nr)2(nr) are ideniy marices, A 1 2 r2r. According o (1), le A d :=GA d H = A d11 A d12 A d21 A d ; P :=GP H = Q :=GQG = hen, from (6) and (7), we have P 11 P12 P 21 P ; Q 11 Q 12 Q12 Q : (11) E P = P E (12) A P + P A + A dp Q 1 P A d + Q<: (13) By using a Schur complemen argumen, i follows from (13) ha A P + P A + Q A dp P A d Q < : (14) Noing he expression of E in (1) and using (12), we can deduce ha P 11 = P 11 and P 21 =; herefore P reduces o P 11 P12 P = P : (15) Subsiuing (1), (11) and (15) ino (14), one evenually ges (16), as shown a he boom of he page. hus P + P + Q A dp P A d Q < : (17) Since Q > and he inequaliy (17) holds, we have ha P is inverible. herefore, i follows from (17) ha ~ Q A d P 1 P A d P 1 + P + ~ Q < ~Q = P 1 Q P > : (18) By [5, h. 1], we have ha (18) implies A ~ d Q A d Q ~ < : (19) herefore Now, le () = (A d ) < 1: (2) 1() 2 () = H 1 x() 1() 2 r, 2() 2 nr. Using he expressions in (1) and (11), he singular delay sysem (5) can be decomposed as (6 D): 1() _ =A 1 1() +A d11 1( ) + A d12 2 ( ) (21) = 2 ()+A d21 1 ( ) + A d 2( ): () I is easy o see ha he sabiliy of he singular delay sysem (5) is equivalen o ha of he sysem (6 D). In view of his, nex we shall prove ha he sysem (6 D ) is sable. Since P 11 = P 11 and P 11 A 1 + A 1 P11 + Q 11 < as (16) shows, i follows ha P 11 >. Define V ( )= 1 () P ()+ = ( + ); 2 [; ]: (s) 1 P Q P (s)ds Recall ha for any marices K 1, K 2 and K 3 of appropriae dimensions wih K 2 > K 1 K 3 + K 3 K 1 K 1 K 2 K 1 + K 3 K 1 2 K 3 : hen, he ime-derivaive of V ( ) along he soluion of (21) and () is given by _V ( )= d d (() P 1 E()) + () P 1 Q P () ( ) P 1 Q P ( ) = () P 1 E ()+ _ () _ E P 1 () + () P 1 Q P () ( ) P 1 Q P ( ) =2() P 1 A() +() P 1 Q P () +2() P 1 Ad ( ) ( ) P 1 Q P ( ) () P 1 A P + P A + Ad P Q 1 P A d + Q 2 P (): P 11A 1 + A 1 P11 + Q 11 P12 + Q 12 A d11 P11 + A d12 P 12 A d12 P P 12 + Q 12 P 11 A d11 + P 12 A d12 P A d12 P + P + Q A d21p11 + A dp 12 A dp P 11 A d21 + P 12 A d Q 11 Q < : (16) 12 P A d Q 12 Q

4 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 1125 I follows from his inequaliy and (13) ha _ V ( ) < and 1 k 1 ()k 2 V ( ) 1 () P () V ( ) 1 () P () = + V ( ) 2 2 (s) P 1 Q P (s)ds _V ( s )ds k(s)k 2 ds k 1 (s)k 2 ds < (23) 1 = min P 1 11 > ; 2 = max[ P 1 ( A P + P A + A d P Q 1 P A d + Q) 2 P ] > : aking ino accoun (23), we can deduce ha herefore and 1 k 1 ()k k 1 (s)k 2 ds V ( ): k 1()k 2 m 1 (24) k 1 (s)k 2 ds m 2 (25) m 1 = 1 1 V ( ) > ; m 2 = 1 2 V ( ) > : hus, k 1 ()k is bounded. Considering his and (2), i can be deduced from () ha k 2 ()k is bounded and, hence, i follows from (21) ha k _ 1()k is bounded. herefore, (d=d)k 1()k 2 is bounded oo. By Lemma 3, we have ha k 1 ()k 2 is uniformly coninuous. herefore, noing (25) and using Lemma 4, we obain lim k1()k =: (26)!1 Now, noing ha for any >, here exiss a posiive ineger k such ha k <k, and considering () we have 2 () =(A d ) k 2 ( k) k i=1 his, ogeher wih (2) and (26), implies ha (A d ) i1 A d21 1 ( i ): lim!1 k 2()k =: hus, (6 D) is sable. his complees he proof. Remark 2: heorem 1 provides a sufficien condiion for he singular delay sysem (6) o be regular, impulse free and sable. When E = I, he singular delay sysem (6) reduces o a sae-space delay sysem and i is easy o show ha heorem 1 coincides wih [1, Lemma 1]. herefore, heorem 1 can be viewed as an exension of exising resuls on sae-space delay sysems o singular delay sysems. Furhermore, by comparing heorem 1 wih [13, Lemma 2], we can regard heorem 1 as an exension of exising resuls on singular sysems wihou delay o singular delay sysems. Following he same philosophy as in dealing wih he problems of robus sabiliy and robus sabilizaion for uncerain singular sysems wihou delay [2], [], and aking ino accoun heorem 1, we inroduce he following definiions. Definiion 5: he uncerain singular delay sysem (6) is said o be generalized quadraically sable if here exis marices Q> and P such ha EP = PE (27) (A +1A)P + P (A +1A) +(A d +1A d )P Q 1 P (A d +1A d ) + Q< (28) for all admissible uncerainies 1A and 1A d. Definiion 6: he uncerain singular delay sysem (6) is said o be generalized quadraically sabilizable if here exiss a linear sae feedback conrol law u() =Kx(), K 2 m2n, marices Q> and P such ha EP = PE (29) (A K +1A K )P + P (A K +1A K ) +(A d +1A d )P Q 1 P (A d +1A d ) + Q< (3) for all admissible uncerainies 1A, 1A d and 1B, A K = A + BK; 1A K =1A +1BK: (31) he following lemma shows ha generalized quadraic sabiliy and generalized quadraic sabilizaion imply robus sabiliy and robus sabilizaion, respecively. Lemma 5: Consider he uncerain singular delay sysem (6).Ifiis generalized quadraically sable, hen i is robusly sable. If i is generalized quadraically sabilizable, hen i is robusly sabilizable. Proof: From heorem 1, he desired resuls follow immediaely. In view of his, necessary and sufficien condiions for generalized quadraic sabiliy and generalized quadraic sabilizabiliy for he uncerain singular delay sysem (6) are derived. In order o obain hese resuls, he following lemma is needed. Lemma 6 [14]: Given marices, and 4 of appropriae dimensions and wih symmerical, hen +F ()4 + (F ()4) < for all F () saisfying F ()F () I, if and only if here exiss a scalar > such ha < : For simpliciy we inroduce he marix 8 2 n2(nr) saisfying E8 =and rank 8 = n r. Now, we are in a posiion o give he quadraic sabiliy resul. heorem 2: he uncerain singular delay sysem (6) is generalized quadraically sable if and only if here exis a scalar >, marices X>, Q>and Y such ha he LMI (32) holds, as shown a he boom of he page. Proof: (Sufficiency) Assume ha here exis a scalar >, marices X>, Q>and Y saisfying (32). By seing P = EX + Y 8,i is easy o see ha EP = PE : (33) A(EX + Y 8 ) +(EX + Y 8 )A + MM + Q A d (EX + Y 8 ) (EX + Y 8 )NA (EX + Y 8 )A d Q (EX + Y 8 )Nd N A (EX + Y 8 ) N d (EX + Y 8 ) I < : (32)

5 1126 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY Observe ha for any F () saisfying (4) and any scalar > 1AP + P 1A herefore 1A d P P 1A d = M F () 2 [ N A P N d P ] + PN A F () 2 [ M ] M M + 1 PN A 2 [ N A P N d P ] : (A +1A)P + P (A +1A) + Q (A d +1A d )P P (A d +1A d ) AP + PA + M M + Q A d P PA d Q + 1 PN A [ N AP N d P ] : Q By using a Schur complemen argumen, i follows from his inequaliy and (32) ha: (A +1A)P + P (A +1A) + Q (A d +1A d )P P (A d +1A d ) < Q or, equivalenly (A +1A)P + P (A +1A) +(A d +1A d )P Q 1 P (A d +1A d ) + Q<: his inequaliy and (33) are precisely (27) and (28) in Definiion 5. Hence, he uncerain singular delay sysem (6) is generalized quadraically sable. (Necessiy) Assume ha he uncerain singular delay sysem (6) is generalized quadraically sable. I follows from Definiion 5 ha here exis marices Q>and P such ha (27) and (28) hold. hus, for all F () saisfying (3) and (4), he following inequaliy holds: (A +1A)P + P (A +1A) + Q (A d +1A d )P P (A d +1A d ) < Q which can be rewrien as AP + PA + Q + M PA d + PN A A d P Q F ()[N A P N d P ] F () [ M ] < : By Lemma 6, i follows ha here exiss a scalar > such ha: AP + PA + Q A d P PA d Q + MM + 1 PN A [ N A P N d P ] < : Invoking again a Schur complemen argumen, one obains AP + PA + M M + Q A d P PN A PA d Q N AP N d P I < : (34) From Lemma 2, i can be shown ha (34) implies ha he pair (E; A) is regular and impulse free. herefore, i follows from [4] ha here exis wo inverible marices U and V 2 n2n such ha: E := UEV = I r A := UAV = A 1 I nr (35) I r 2 r2r and I nr 2 (nr)2(nr) are ideniy marices, A 1 2 r2r. Le P := UPV, hen from he proof of heorem 1, we have ha P akes he form P = P 11 P 12 P (36) P 11 >, P 12 2 r2(nr) and P 2 (nr)2(nr). On he oher hand, from E8 = and rank 8 = n r, we can show ha here exiss an inverible marix 3 2 (nr)2(nr) such ha Hence 8=V P =U 1 P 11 P 12 P V I nr = U 1 I r V 1 V 3: (37) P 11 I nr + U 1 P 12 P 3 3 [ I nr ] V =EX + Y 8 X = V P 11 I nr V V ; Y = U 1 P 12 P 3 : Finally, since X > and by replacing P ino (34), he desired resul follows immediaely. he generalized quadraic sabilizabiliy resul is presened in he following heorem. heorem 3: he uncerain singular delay sysem (6) is generalized quadraically sabilizable if and only if here exis a scalar >, marices X>, Q>, Y and Z such ha he LMI (38) holds, as shown a he boom of he page, W =A7(X; Y ) +7(X; Y )A + BZ + Z B + M M + Q 7(X; Y )=EX + Y 8 wih 7(X; Y ) inverible. In his case, a robusly sabilizing sae feedback conrol law is given by u(k) =Z7(X; Y ) x(): (39) W A d 7(X; Y ) 7(X; Y )N A + Z N B 7(X; Y )A d Q 7(X; Y )N d N A 7(X; Y ) + N B Z N d 7(X; Y ) I < (38)

6 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 1127 Proof: According o Definiion 6, he sysem (6) is generalized quadraically sabilizable wih respec o he uncerainy srucure (3) if and only if here exiss K 2 m2n such ha he resulan closed-loop sysem wih E _x() =(A c +1A c )x() +(A d +1A d )x( ) (4) A c = A + BK; 1A c =1A +1BK is quadraically sable wih respec o he uncerainy srucure [1A c 1A d ]=MF()[N A + N BK N d ] : By invoking now heorem 2 for he closed-loop sysem (4), one deduces ha (6) is generalized quadraically sabilizable if and only if here exiss K 2 m2n and a scalar >, marices X>, Q> and Y such ha he LMI holds, as shown in (41) a he boom of he page, wih 2=(A + BK)(EX + Y 8 ) Define +(EX + Y 8 )(A + BK) + M M + Q: Z = K(EX + Y 8 ) and observe ha he LMI (41) is precisely inequaliy (38) in he saemen of heorem 3. Hence, necessiy is proved. Now, wihou loss of generaliy, we can assume ha 7(X; Y ) = EX + Y 8 is inverible, oherwise we can choose a sufficienly small scalar > such ha ^7(X; Y ) = 7(X; Y )+I also saisfies (38) wih ^7(X; Y ) inverible. If (38) holds, hen (41) is saisfied for K = Z7(X; Y ). aking ino accoun he aforemenioned consideraions, i follows ha (6) is generalized quadraically sabilizable. his proves sufficiency. Remark 3: heorem 3 presens a necessary and sufficien condiion for generalized quadraic sabilizabiliy. he desired robusly sabilizing sae feedback for uncerain singular sysem (6) can be obained by solving he sric LMI (38), which can be solved numerically very efficienly by using inerior-poin algorihm, and no uning of parameers is involved [2]. I is worh poining ou ha sric LMI (38) is expressed by using he sysem marices of (6). he design procedure involves no decomposing of he sysem, which can ge around cerain numerical problems arising from decomposiion of marices and hus makes he design procedure relaively simple and reliable. IV. NUMERICAL EXAMPLE In his secion, we give an example o demonsrae he effeciveness of he proposed mehod. Consider he linear uncerain singular delay sysem (6) wih parameers as follows: E = A d = :5 :3 :5 1 A = B = 1:5 : : M = :5 :2 :1 N A =[:2 :1 :3] N d =[:1 :2 :5] N B =[:1 :1]: In his example, we assume ha he ime delay = 1:5 and he uncerain marix F () = sin(). he purpose is he design of a sae feedback conrol law such ha, for all admissible uncerainies, he resulan closed-loop sysem is regular, impulse free and sable. o his end, we choose 8=[1 1 2] : Using Malab LMI Conrol oolbox o solve he LMI (38), we obain he soluion as follows: X = Q = Y = Z = :2682 :167 :312 :167 :2976 :3568 :312 :3568 :6443 :9575 :475 :475 :475 :9538 :53 :475 :53 :9538 :2467 :1484 :213 :9452 :416 :5859 :7454 :5912 :7552 =1:21: herefore, by heorem 3, a robusly sabilizing sae feedback conrol law can be obained as u() = 13: : :6474 6: :7 11:8584 V. CONCLUSION x(): he problems of robus sabiliy and sabilizaion for uncerain coninuous singular sysems wih sae delay and parameer uncerainy have been sudied. Based on he noions of generalized quadraic sabiliy and generalized quadraic sabilizaion, hese problems have been solved. Necessary and sufficien condiions for generalized quadraic sabiliy and generalized quadraic sabilizaion are presened in erms of a sric LMI, respecively. he proposed sae feedback conrol law guaranees ha he resulan closed-loop sysem is regular, impulse free as well as sable for all admissible uncerainies. 2 A d (EX + Y 8 ) (EX + Y 8 )(N A + N BK) (EX + Y 8 )A d Q (EX + Y 8 )Nd (N A + N B K)(EX + Y 8 ) N d (EX + Y 8 ) I < (41)

7 1128 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY REFERENCES [1] U. Ascher and L. R. Pezold, he numerical soluion of delay-differenial-algebraic equaions of rearded and neural ype, SIAM J. Numer. Anal., vol. 32, pp , [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Marix Inequaliies in Sysems and Conrol heory. Philadelphia, PA: SIAM, [3] S. L. Campbell, Singular linear sysems of differenial equaions wih delays, Appl. Anal., vol. 2, pp , 198. [4] L. Dai, Singular Conrol Sysems. Berlin, Germany: Springer-Verlag, [5] M. C. de Oliveira, J. Bernussou, and J. C. Geromel, A new discree-ime robus sabiliy condiion, Sys. Conrol Le., vol. 37, pp , [6] C.-H. Fang and F.-R. Chang, Analysis of sabiliy robusness for generalized sae-space sysems wih srucured perurbaions, Sys. Conrol Le., vol. 21, pp , [7] C.-H. Fang, L. Lee, and F.-R. Chang, Robus conrol analysis and design for discree-ime singular sysems, Auomaica, vol. 3, pp , [8] J. K. Hale, heory of Funcional Differenial Equaions. New York: Springer-Verlag, [9] M. Krsić and H. Deng, Sabilizaion of Nonlinear Uncerain Sysems. London, U.K.: Springer-Verlag, [1] J. H. Lee, S. W. Kim, and W. H. Kwon, Memoryless H conrollers for sae delayed sysems, IEEE rans. Auoma. Conr., vol. 39, pp , Jan [11] F. L. Lewis, A survey of linear singular sysems, Circuis, Sys. Signal Processing, vol. 5, pp. 3 36, [12] M. S. Mahmoud and N. F. Al-Muhairi, Quadraic sabilizaion of coninuous ime sysems wih sae-delay and norm-bounded ime-varying uncerainies, IEEE rans. Auoma. Conr., vol. 39, pp , Oc [13] I. Masubuchi, Y. Kamiane, A. Ohara, and N. Suda, H conrol for descripor sysems: A marix inequaliies approach, Auomaica, vol. 33, pp , [14] I. R. Peersen, A sabilizaion algorihm for a class of uncerain linear sysems, Sys. Conrol Le., vol. 8, pp , [15] E. Uezao and M. Ikeda, Sric LMI condiions for sabiliy, robus sabilizaion, and H conrol of descripor sysems, in Proc. 38h IEEE Conf. Decision Conrol, Phoenix, AZ, USA, Dec. 1999, pp [16] L. Xie and C. E. De Souza, Robus H conrol for linear ime-invarian sysems wih norm-bounded uncerainy in he inpu marix, Sys. Conrol Le., vol. 14, pp , 199. [17] S. Xu, J. Lam, and C. Yang, Robus H conrol for discree singular sysems wih sae delay and parameer uncerainy, Dyna. Coninuous, Discree, Impul. Sys., o be published. [18], Quadraic sabiliy and sabilizaion of uncerain linear discree-ime sysems wih sae delay, Sys. Conrol Le., vol. 43, pp , 21. [19] S. Xu, J. Lam, and L. Zhang, Robus D-Sabiliy analysis for uncerain discree singular sysems wih sae delay, IEEE rans. Circuis Sys. I, vol. 49, pp , Apr.. [2] S. Xu and C. Yang, An algebraic approach o he robus sabiliy analysis and robus sabilizaion of uncerain singular sysems, In. J. Sys. Sci., vol. 31, pp , 2. [21], H sae feedback conrol for discree singular sysems, IEEE rans. Auoma. Conr., vol. 45, pp , July 2. [] S. Xu, C. Yang, Y. Niu, and J. Lam, Robus sabilizaion for uncerain discree singular sysems, Auomaica, vol. 37, pp , 21. [23] W. Zhu and L. Pezold, Asympoic sabiliy of linear delay differenialalgebraic equaions and numerical mehods, Appl. Numer. Mah., vol. 24, pp , Singular LQ Problem for Nonregular Descripor Sysems Jiandong Zhu, Shuping Ma, and Zhaolin Cheng Absrac In his noe, a singular linear quadraic (LQ) problem for nonregular descripor sysems is invesigaed. Under some general condiions, he opimal conrol and he opimal sae of he LQ problem are given. he opimal conrol is synhesized as sae feedback. All he finie eigenvalues of he closed-loop sysem are locaed on he lef-half complex plane. he sae of he closed-loop sysem has he leas free enries. Index erms Nonregular descripor sysems, singular linear quadraic (LQ) problem, sae feedback. I. INRODUCION Descripor sysems have comprehensive pracical background, such as power sysems [11], social economic sysems [18], circui sysems [21], and so on. Grea progress [1], [3], [8] has been made in he heory and is applicaions since 197s. Linear quadraic (LQ) opimal conrol problem (LQ problem) is imporan in conrol heory and has been used in pracice widely. here have been a lo of excellen resuls [2], [5], [7], [16], [23] abou LQ problem for descripor sysems. In he case of he weighing marix R in he linear quadraic cos being posiive definie, he heory has maured. Cobb [7] considered he problem wih geomeric mehod. Bender and Laub [2] reduced he problem o solving a generalized Riccai equaion. Cheng e al. [5] ransformed he nonsingular LQ problem for descripor sysems ino a nonsingular LQ problem for sandard sae space sysems. hey gave sufficien condiions for he solvabiliy of he nonsingular LQ problem. However, wih R being semidefinie posiive, here was no much work unil now. Chen e al. [4] discussed he problem for a special kind of descripor sysems based on Weiersrass canonical form and he assumpion ha he conrol is sufficienly smooh. Zhu e al. [24], who ransformed he singular LQ problem for descripor sysems ino a nonsingular LQ problem for sandard sysems, gave a new mehod of dealing wih he problem. In recen years, nonregular descripor sysems were discussed [9], [12], [13], wih many open problems unsolved. In [9], Geers discussed he LQ problem via linear marix inequaliies (LMIs). In his noe, we exend he mehod used in [24] o nonregular descripor sysems and obain some new resuls. Using elemenary linear algebra and he equivalence principle for opimal conrol problem, we derive he relaionship beween he singular LQ problem for nonregular descripor sysems and he singular or nonsingular LQ problem for sandard sae space sysems. Under some general condiions, he opimal conrol-sae pair is derived. he opimal conrol is expressed as sae feedback. he sae of he closed-loop sysem has he leas free enries and all he finie eigenvalues are locaed on he lef half complex plane. he resricion imposed on sysems in his noe is weaker han ha in [2], [5], [7], and [24]. his noe is organized as follows. Secion II is a saemen and ransformaion of he problem. Secion III considers he soluion of he problem. Secion IV is a brief conclusion /2$17. IEEE Manuscrip received November 13, 21. Recommended by Associae Edior L. Dai. his work was suppored by he Projec 973 of China, he Universiy Docoral Foundaion of he Educaion Minisry of China, and he Naural Science Foundaion of Shandong Province under Grans G199823, 98412, and Q99A7. he auhors are wih he Deparmen of Mahemaics, Shandong Universiy, Jinan 251, P. R. China. Publisher Iem Idenifier 1.119/AC