Sampled-data H 1 state-feedback control of systems with state delays

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1 INT. J. CONTROL, 2, VOL. 73, NO. 12, 1115± 1128 Sampleddata H 1 statefeedback control of systems with state delays EMILIA FRIDMAN{{ and URI SHAKED{ The nite horizon piecewiseconstant statefeedback H 1 control of timeinvariant linear systems with a nite number of point and distributed timedelays is considered. For the controller, coupled Riccati type partial di erential equations between sampling and updating the terminal conditions are derived. For small time delays the solutions and the resulting controllers are approimated by series epansions in powers of the largest delay. Unlike the in nite horizon case, these approimations possess both regular and boundary layer terms. It is shown that the controller obtained by highorder approimations improves the performance of the system. The performance of the closedloop system under the memoryless zeroapproimation controller is analysed. Bounded Real Lemmas for statedelay systems with jumps are obtained. 1. Introduction Continuoustime H 1 control problem of statedelay systems has been studied by Bensoussan et al. (12), van Keulen et al. (13), van Keulen (13), McMillan and Triggiani (13), Lee et al. (14), Ge et al. (16), Fridman and Shaked (18, 2). Bensoussan et al. (12), van Keulen et al. (13), van Keulen (13) and McMillan and Triggiani (13) have obtained the controller by solving Riccati operator equations. Lee et al. (14) and Ge et al. (16) have designed a delayindependent controller. Fridman and Shaked (18, 2) have derived the controller from Riccati type partial di erential equations (RPDEs) or inequalities, and the solution of the RPDEs has been approimated by epansions in the powers of the delay. In Shaked et al. (18) a bounded real lemma has been obtained in terms of di erential linear matri inequalities. Sampleddata H 1 control of systems without delay has been studied by Bamieh and Pearson (12), Toivonen (12), Khargonekar et al. (13), Sivashankar and Khargonekar (14), Basar and Bernard (15), Sagfors and Toivonen (17). Two main approaches have been used. The rst one is based on the lifting technique in which the problem is transformed into equivalent nitedimensional discrete H 1 problem (Bamieh and Pearson 12, Toivonen 12). The second, more direct, approach is based on the representation of the system in the form of hybrid discrete/continuous one and the solution is obtained in terms of Riccati equations with jumps (Khargonekar et al. 13, Sivashankar and Khargonekar 14, Basar and Bernard 15, Sagfors and Toivonen 17). Received 1 March 1. Revised 1 January 2. Communicated by Professor V. Kucera. { Department of Electrical EngineeringSystems, Tel Aviv University, Ramat Aviv 678, Tel Aviv, Israel. { Author for correspondence. emiliaeng.tau.ac.il In the present paper, we adopt the second approach to the nite horizon sampleddata statedelay H 1 control problem. Note that in the case of statedelays the lifting technique leads to complicated discrete system with in nitedimensional state variable. We obtain the piecewiseconstant statefeedback controllers by solving coupled RPDEs between sampling and updating the terminal conditions at the sampling instants. We derive an asymptotic approimation to the solution of these RPDEs by epanding it in the powers of the largest delay. The resulting approimation is obtained by solving uncoupled loworder partial di erential equations. The performance of the system with the controller that has been obtained using the zero approimation (the one that corresponds to zero delay) is analysed when the openloop system possesses a nonzero delay. Bounded real lemmas for systems with state delays and jumps are obtained. 2. Problem formulation Throughout this paper we denote by j j the Euclidean norm of a vector or the appropriate norm of a matri. Given t f >, let L 2 ;t f Š be the space of the square integrable functions with the norm jj jj L 2 and let C a;bš be the space of the continuous functions on a;bš with the norm j j c. We denote t ˆ t ; y t ˆ y t ; 2 ;Š and t ˆ lim <s! t s, and t ˆ t. Prime denotes the transpose of a matri and col f;yg denotes a column vector with components and y. Consider the system _ t ˆ L t Bu t Dw t = 1 z t ˆ col fc t ;u t g ; t 2 R n is the state vector, u t 2 R l is the control signal, w t 2 R q is the eogenous disturbance, and z t 2 R p is the observation vector, B; C and D are constant matrices of appropriate dimensions. The R n valued International Journal of Control ISSN 2± 717 print/issn 1366± 582 online # 2 Taylor & Francis Ltd

2 1116 E. Fridman and U. Shaked function L which carries R n valued functions on ;Š into R n is de ned as L t ˆ Xr iˆ t i A 1 s t s ds 2 ˆ r < r1 < < 1 < ˆ, A ;A 1 ;... ;A r are constant matrices and A 1 s is a smooth enough matri function. Consider the `structural operator F : L 2 ;Š! L 2 ;Š given by F t ¹ ˆ Xr ¹ t i ¹ À i ¹ A 1 p t p ¹ dp 3 À i is the indicator function for the set i ;Š, i.e. À i ¹ ˆ 1 if ¹ 2 i ;Š and À i ¹ ˆ otherwise. This operator was introduced by Delfour (186) and the use of this operator in the Lyapunov± Krasovskii functional leads to simpli ed RPDEs for H 2 and H 1 design (Delfour 186; Fridman and Shaked 2). Given >, and assuming that w 2 L 2 ;t f Š, we consider the following performance inde J ˆ jjzjj 2 L 2 2 jjwjj 2 L 2 E tf 4 E tf ˆ t f P f t f 2 t f Q f ¹ F tf ¹ d¹ F tf s R f s;¹ F tf ¹ ds d¹ 5 and P f ˆ P f. The form of E tf stems from the form of Lyapunov± Krasovskii functional. We assume that the matrifunctions Q f and R f are continuous and piecewise continuously di erentiable functions of their arguments, that satisfy the relations: P f ˆ Q f ; Q f ¹ ˆ R f ;¹ 6 We are looking for the piecewiseconstant statefeedback controller of the form u t ˆ tk ; ; t1 ; ; t k µ t < t k1 ; k ˆ ; 1 ; ;K 1 7 < t 1 < < t K ˆ t f are the sampling instants. This type of control is encountered in many practical problems a zeroorderhold is used to provide an analog input signal at the output of a digital controller. The problem is to nd a statefeedback controller of (7) which ensures that J µ for all w 2 L 2 ;t f Š and for the zero initial conditions ½ ˆ ; ½ µ. This means that the H 1 norm of (1), which is de ned by the supremum over w 2 L 2 ;t f Š of the ratio between fjjzjj 2 L 2 E tf g 1=2 and jjwjj L 2, is not greater than. 3. Piecewiseconstant controller design Following Basar and Bernard (15) and Toivonen and Sagfors (17) we denote A ˆ A B ; ˆ Ai ; A 1 ˆ A1 " D ˆ D # " ; C ˆ C # 8 I l " A d ˆ In ; B d ˆ # The system of (1) attains then the form t _ ˆ Xr iˆ t h i I l A 1 s t s ds Dw t ; z t ˆ C t ; t k µ t < t k1 t k ˆ A d t k B d u k 1 t ˆ col f t ;u t g. Similarly to (3) we denote F t ¹ ˆ Xr ¹ t i ¹ À i ¹ A 1 p t p ¹ dp 11 clearly F t ¹ ˆ col ff t ¹ ; g. The Lyapunov± Krasovskii functional for the system of () is given by Delfour (186) V t; t ˆ t P t t 2 t Q t;¹ F t ¹ d¹ F t s R t;s;¹ F t ¹ ds d¹ 12 P ˆ P M M U ; Q ˆ Q N ; R ˆ R 13 In the latter formulas the zeroblocks stem from the zero delay in u. Then E tf ˆ E tf ; E tf ˆ t f P f tf 2 t f Q f ¹ F tf ¹ d¹ F tf s R f s;¹ F tf ¹ ds d¹ 14

3 Sampleddata H 1 statefeedback control 1117 and P f ˆ Pf ; Denote Q f ˆ Qf ; S ˆ 2 DD ; S ˆ 2 D R f ˆ Rf 15 Consider the following RPDEs with respect to the n l n l matrices P t ; Q t;¹ and R t;¹;s P t _ A P t P t A Xr Xr Ai D Q t; i Q t; i A i P t S P t C C Q t; A 1 d A1 Q t; d ˆ Q t;¹ Q t;¹ ˆ A P SŠ Q t; ¹ t ¹ Xr Ai R t; i ;¹ A1 s R t;s;¹ ds R t;¹;s R t;¹;s R t;¹;s t ¹ s ˆ Q t;¹ S Q t;s P t ˆ Q t; ; Q t;¹ ˆ R t;;¹ ; M t ˆ N t; R t;¹;s ˆ R t;s;¹ P t f ˆ P f ; U t f ˆ ; M t f ˆ ) N t f ;s ˆ ; Q t f ;¹ ˆ Q f ¹ ; R t f ;¹;s ˆ R f ¹;s 2 A solution of (16)± (2) on t K1 ;t f Š is a triple of n l n l matrices fp t ; Q t;¹ ; R t;¹;s g t 2 t K1 ;t f Š;¹ 2 ;Š;s 2 ;Š, P t ; Q t;¹ and R t; ¹; s are continuous and piecewise continuously differentiable functions of their arguments that have a form of (13), satisfy initial and terminal conditions (1), (2) for every t; ¹ and s and satisfy partial di erential equations (16)± (18) for almost every t; ¹ and s. Lemma 1: T he supremum value with respect to w of ) J K ˆ E tf jzj 2 2 jwj 2 Š dt t K1 for the system of is given by sup J K ˆ J K ˆ V t K1 ; tk1 21 w the matrifunctions P; Q and R of the form 13 satisfy (16)± (2). The proof of Lemma 1 is given in the Appendi. Writing (16)± (18) for the various blocks of (13), we obtain and _P t A P t P t A Xr Xr A i Q t; i Q t; i P t SP t C C Q t; A 1 d A1 Q t; d ˆ t Q t;¹ ¹ Q t;¹ ˆ A PSŠQ t;¹ Xr A i R t; i ;¹ A 1 s R t;s;¹ ds t R t;¹;s ¹ R t;¹;s s R t;¹;s _M t A P t SŠM t P t B Xr A i N t; i ˆ Q t;¹ SQ t;s A1 N t; d ˆ t N t;¹ ¹ N t;¹ ˆ Q t;¹ B SM t Š _U t B M t M t B M t SM t I l ˆ 27 We obtain the following result. Theorem 1: Assume that for every k ˆ K;... ;1 there eists a solution to 16 ± 1 on t k1 ;t k Š with the following terminal conditions: 2 for k ˆ K and for k < K

4 1118 E. Fridman and U. Shaked P t k ˆ P t k M t k U 1 t k M t k U t k ˆ ; N t k ;¹ ˆ ; M t k ˆ 28 Q t k ;¹ ˆ Q t k ;¹ M t k U 1 t k N t k ;¹ R t k ;s;¹ ˆ R t k ;s;¹ N t k ;s U 1 t k N t k ;¹ T hen, the controller u t ˆ U 1 t k M t k t k N t k ;s F tk s ds ; t k µ t < t k1 ; k ˆ K 1;... ; 2 solves the sampleddata statefeedback H 1 control problem. Proof: We shall apply dynamic programming arguments, by adapting the dynamic programming equation to the sampling intervals: inf u sup J ˆ inf sup w u w 1 jzj 2 2 jwj 2 Š dt inf u J k ˆ V t k1 ; tk1 t k1 M t k1 U 1 t k1 M t k1 t k1 F tk1 ¹ N t k1 ;¹ d¹u 1 t k1 N t k1 ;¹ F tk1 ¹ d¹ 2M t k1 t k1 3 k ˆ K. Therefore at the stage K 1 we obtain the performance cost J K1 ˆ inf u J K K1 t K2 jzj 2 2 jwj 2 Š dt By the same arguments and due to (28) for k ˆ K 1 we see that inf u J K1 is given by (3), k ˆ K 1. Similarly it can be shown that (3) holds for all k ˆ K 1;... ;1. Then inf u J µ inf u J 1 ˆ since ˆ : & inf u sup w inf u sup w jzj 2 2 jwj 2 Š dt t K1 E tf We rst consider stage K. By Lemma 1 the supremum value of J K ˆ E tf jzj 2 2 jwj 2 Š dt t K1 with respect to w is given by sup J K ˆ J K ˆ V t K1 ; tk1 w ˆ V t K1 ; tk1 u t K1 U t K1 u t K1 2 t K1 M t K1 u t K1 2u t K1 N t K1 ;¹ F tk1 ¹ d¹ V t; t ˆ t P t t 2 t Q t;¹ F t ¹ d¹ F t s R t;s;¹ F t ¹ ds d¹ P; Q and R of the form and the matrifunctions (13) satisfy (16)± (18) and (1). The unique minimizing u for J K is given by (2), k ˆ K 1, and the corresponding minimum value of J K is given by 4. Asymptotic approimation of the H 1 controller 4.1. Asymptotic solutions to the RPDEs As we have seen the H 1 controller has been found by solving a set of coupled RPDEs. Finding a solution to the latter is not an easy task and we are, therefore, looking for a solution to the RPDEs on t k ;t k1 Š; k ˆ K 1;... ; in a form of asymptotic epansion in the powers of the delay h P t ˆ P t h P 1 t P 1P ½ Š h 2 P 2 t P 2P ½ Š Q t;h± ˆ Q t;± h Q 1 t;± P 1Q ½; ± Š h 2 Q 2 t;± P 2Q ½;± Š R t;h±;h ˆ R t;±; h R 1 t;±; 31 P 1R ½;±; Š h 2 R 2 t;±; P 2R ½;±; Š t 2 t k ;t k1 Š; k ˆ K 1;... ; ½ ˆ tk1 t h ; ± 2 1;Š; 2 1;Š Note that we consider the asymptotic epansion of R instead of R since R is the only nonzero block of R. The `outer epansion terms fp i ; Q i ;R i g; i ˆ ;1;... constitute the major part of the solution that satis es (16)± (1) for t 2 ;t f Š; t 6ˆ t k1 ; 2 1;Š; ± 2 1; Š. The boundarylayer correction terms

5 Sampleddata H 1 statefeedback control 111 P ip; P iq and P ir will be chosen such that (31) satis es the terminal conditions of (2) and that jp ip ½ j sup jp iq ½; ± j ±2 1;Š sup ±; 2 1;Š jp ir ½;±; j! as ½! 1 32 Since ½ is a stretchedtime variable around t ˆ t k1, (32) asserts that P ip;p iq and P ir are essential only around t ˆ t k1 and they thus provide a correction to the outer epansion at the terminal point t ˆ t k1. In the sampleddata case we approimate the solution to the RPDEs on each sampling segment: t K1 ;t f Š;... ; ;t 1 Š. We assume that the terminal values P f ;Q f and R f are approimated as: P f ˆ P f Xm1 h i P fi Q f h± ˆ Q fi ˆ P fi Q f Xm1 h i Q fi ± R f h±;h ˆ P f Xm1 h i R i ±; R fi ; ˆ Q fi 33 Q fi and R fi are continuous and piecewisecontinuously di erentiable functions of ± and. The remainders P f ;m1 ; Q f ;m1 and R f ;m1 may depend on h and are uniformly bounded functions. We realize that, in practical cases there is a little sense in taking delaydependent P f. We nevertheless consider the general case of nonzero P fi since this will be required in our analysis on the sampling segments t k1 ;t k Š; k < K. For simplicity we assume that A 1 ˆ. Note that A 1 appears in the integral terms of RPDEs and the latter terms are of the order of O h. That is why in the case of A 1 6ˆ the asymptotic approimation is similar. We substitute (31) in (16)± (1) and equate, separately, outer epansion and boundarylayer correction terms with the same powers of h. We notice that for t ˆ t k1 h½; ¹ ˆ h± and s ˆ h we have t ˆ 1 ½ ; ¹ ˆ h1 ± ; s ˆ h1 Thus, for the zeroorder terms we obtain from (23), (24) and (1) Q t;± ˆ P ; R M t;±; ˆ P t 34 P ˆ P M M U Then, from (22), (25) and (27), we obtain the di erential equations _P t Xr iˆ A i P t Xr iˆ P t P t SP t C C ˆ 35 _M t Xr Ai P t S M t P t B ˆ iˆ _U t B M t M t B M t SM t I l ˆ 36 with the terminal conditions P t K ˆ P f P t k ˆ P t k M t k U 1 t k M t k k ˆ K 1; ; 1 M t k ˆ ; U t k ˆ ; k ˆ K;... ;1 37 These equations correspond to the sampleddata H 1 control of systems without delay (Basar and Bernard 15). We make here the following assumption: Assumption 1: For a speci ed value of, the DREs 35, 36 and 37) possess a bounded solution on ;t f Š. Assumption 1 means that the H 1 statefeedback sampleddata control problem for (1) without delay has a solution. If this were not the case, even P, the zeroorder term in (31), would not eist. To determine the rstorder terms we start with t K1 ;t f Š and with the equations for Q 1 : Q 1 t;± ˆ M t Q ± t Q _ t " Q 1 t; ˆ P 1 t I # n 38 M ˆ Xr iˆ S Then Q 1 t;± ˆ Q 1 t; M t Q t Q _ t Š± Substituting this epression into the equation for P 1, we obtain P_ 1 M P 1 P 1 M Xr Xr P g i A i Q M _ Q g i M Q _ Q ˆ P 1 t f P 1 P ˆ P f 1 ; g i ˆ h i =h 3

6 P P 112 E. Fridman and U. Shaked P P P P P P P P It follows from (22) that _ 1P ½ ˆ. Since 1P vanishes for ½! 1, we have 1P ½ ² ; ½. Hence, P 1 t f ˆ P f 1, and P 1 is a solution to the linear di erential equation (3) with the latter terminal condition. For 1Q ; R 1 and 1R we obtain from (17), (18) and (34) 1 ½ Q ½;± 1 ± Q ½;± ˆ Q 1 t f ;± 1Q ;± ˆ Q f 1 ± 1Q ½; ˆ 1P ½ ˆ ± R 1 t;±; R 1 t;±; ˆ P t SP t _P t R 1 ½;; ˆ Q 1 ½; 4 P P P and ½ 1R ½;±; ± 1R ½;±; 1R ½;±; ˆ R 1 t f ;±; P 1R ;±; ˆ R f 1 ±; P 1R ½;; ˆ P 1Q ½; " Q 1 ˆ Q1 ; P N1 Q ˆ P Q # 1 1 N 1 41 Then, for ½ and t 2 t K1 ;t f Š, we nd successively ( Q f 1 ± ½ Q 1 t f ;± ½ ; if ½ µ ± P 1Q ½;± ˆ ; if ½ > ± R 1 t;±; ˆ R 1 t; ;± ˆ ± P t SP t _P t Š Q 1 t; ± ; P 1R ;±; ˆ R f 1 ±; ± P f SP f _P t f 1R ½;±; ˆ P Therefore Q 1 t f ; ± ± 8 P 1R ;± ½; ½ > if ½ µ ±; µ ± 1R ½; ;± ˆ P 1Q ½ ±; ± <>: if ½ > ±; µ ± P 1Q ½;± ˆ ; ½ ± > P 1R ½;±; ˆ P 1Q ½ ±; ± ˆ ½ > ; µ ± The rstorder and the higher order terms of the epansions can be similarly found on all sampling segments Nearoptimal piecewiseconstant H 1 control For small enough values of h the approimations of the controller contain the outer epansion terms only, i.e. all the boundarylayer terms P i vanish. This is due to the fact that we need an approimation of the solution to RPDEs only on the left end of the sampling segment. In the Appendi we prove the following. Theorem 2: Under Assumption 1 the following holds for all small enough delays h: (1) T he system of (16)± (2), (22)± (28) has a solution. T his solution is approimated for any integer m by P t ˆ P t Xm h i P i t P ip ½ Š O h m1 Q t;h± ˆ Q t Xm R t;h±;h ˆ P t Xm h i Q i t;± P i Q ½;± Š O h m1 h i R i t;±; P ir ½;±; Š O h m1 ½ ˆ tf t h ; ± 2 1;Š; 2 1;Š the boundarylayer terms satisfy 42 P ip ½ ˆ ; ½ > i 1; P iq ½;± ˆ ; ½ ± > i 1 P ir ½;±; ˆ ; ½ > i 1; µ ± and jo h m1 j µ ch m1, c is a positive scalar which is independent of h;t;± and. The matrices P i and Q i are taken with bars and have the structure of (13), all the components are taken with inde i. (2) Denote by Y i ; i ˆ ;... ;m the terms of the epansions of U 1 in the powers of h, i.e. X m h i U i t 1ˆ Xm iˆ iˆ Y t ˆ U 1 t h i Y i t O h m1 ;

7 T hen the controller of (2) is approimated by u t ˆ u m t O h m1 u m t ˆ Xm iˆ X mi jˆ h ij Y i t k M j t k t k 1 t k µ t < t k1 ; k ˆ K 1; ; N j1 t k ;s F tk hs ds 43 N 1 ˆ and N ˆ M. T he approimate controller u m guarantees an attenuation level of O h m1. 5. The zeroorder controller performance We study the performance of the system under the zeroorder controller u t ˆ U 1 t k M t k t k which solves the H 1 control problem for (1) without delay L 2 gain of statedelay systems with jumps In order to study the performance under the zeroorder controller in the sampleddata case we note that the closedloop system of (1), u ˆ u t, has the form of () with the jumps condition t k ˆ G k t k ; k ˆ 1;... ;K 1 44 G k ˆ I U 1 t k M t k 45 We derive rst the relevant bounded real lemma for () with (44), for any given A i ; A 1 ; D ; C and G k of the appropriate dimensions. Consider J ˆ jj zjj 2 L 2 2 jjwjj 2 L 2 E tf 46 E is given by (14), F is de ned by (11) and P f ; Q f and R f are any continuous and piecewise continuously di erentiable functions of their arguments. We then apply the lemma to the special matrices of (8) and (45). Lemma 2: Assume that for every k ˆ K;... ;1 there eists a solution on t k1 ;t k Š to (16)± (1) such that P t ˆ Q t; ; Q t;¹ ˆ R t; ; ¹ 47 with the following terminal conditions Sampleddata H 1 statefeedback control 1121 P t K ˆ P f ; R t K ;s;¹ ˆ R f s;¹ ; Q t k ;¹ ˆ Gk Q t k ;¹ ; R t k ;s;¹ ˆ R t k ;s;¹ ; Q t K ;¹ ˆ Q f ¹ ; and P t k ˆ Gk P t k G k and k < K 48 T hen the performance inde J of (46) for the system of () with the jumps condition (44) is nonnegative for all w 2 L 2 ;t f Š and ˆ. Proof: Similarly to Theorem 1 we use the dynamic programming argument sup J ˆ sup w w sup w 1 _ jzj 2 2 jwj 2 Š dt sup w E tf jzj 2 2 jwj 2 Š dt t K1 We rst consider stage K. By Lemma 1 the supremum value of J K ˆ E tf jzj 2 2 jwj 2 Š dt t K1 with respect to w is given by sup J k ˆ V t k1 ; tk1 w ˆ t k1 P t k1 t k1 2 t k1 Q t k1 ;¹ F tk1 ¹ d¹ F tk1 s R t;s;¹ F tk1 ¹ ds d¹ 4 The matrifunctions P; Q and R satisfy (16)± (18) and (47). Substituting now (44) into (4) we obtain the following performance cost for the stage K 1 J K1 ˆ t K1 GK1 P t K1 G K1 t K1 2 t K1 GK1 Q t K1 ;¹ F tk1 ¹ d¹ F tk1 s R t;s;¹ F tk1 ¹ dsd¹ K1 jzj 2 jwj 2 Š dt t K2 By Lemma 1 this cost has a supremum value given by (4), k ˆ K 1, since the corresponding RPDEs with terminal conditions (48) have no escape point.

8 1122 E. Fridman and U. Shaked Similarly it can be shown that (4) holds for all k ˆ K 1;... ;1. Then sup w J ˆ sup w J 1 µ due to ˆ : & Note that for h! ; P! P and Q! Q (cf. (42)). Therefore the latter lemma coincides with the corresponding results in Sivashankar and Khargonekar (14) Robustness of the performance under u Theorem 3: Under Assumption 1 the controller u for all small enough h leads to a performance level of. Proof: Applying u to (1), we obtain () and (44). By Lemma 2 this closedloop system has an induced L 2 gain less or equal to if the corresponding RPDEs of (16)± (1) with the terminal conditions of (48) have a bounded solution. The eistence of a solution to these RPDEs, approimated by (42) with m ˆ, can be proved similarly to (i) of Theorem 2. & Given > and h, one should verify that the corresponding RPDEs have a solution in order to make certain that u leads to a performance level of. This is not an easy task. That is why one may resort to more conservative, but computationally simpler, conditions in terms of di erential linear matri inequalities (DLMI) or Riccati di erential inequalities (RDI) that were formulated for the case of one delay in Shaked et al. (18). Generalization of these lemmas to the case of r delays and jumps in the system is given below in Lemmas 3 and Bounded real lemmas for statedelay systems with jumps using DL MI Consider () with (44), A i ; D ; C and G k are any given matrices of appropriate dimensions. For simplicity we assume that A 1 ˆ. Consider J ˆ jjzjj 2 L 2 2 jjwjj 2 L 2 t f P f t f 5 P f is any matri. Lemma 3: T he performance inde J of (5) for the system of () with the jumps condition of (44) is nonpositive for all w 2 L 2 ;t f Š, and ˆ, if there eist square integrable matrices Q i t ˆ Qi t ; i ˆ 1;... ;r that are positive de nite on t f h i ;t f and allow an absolutely continuous solution P t ˆ P t on t 2 t k1 ;t k ; k ˆ 1;... ;K to the following DL MI 2 C t P t A 1 P t A 2 P t A r P t D 3 A1 P t Q 1 t h 1 N t ˆ 6 7 µ 64 ArP t Q r t h r 75 D P t 2 I 51 C t 7 _P t A P t P t A C C Xr with the terminal conditions Proof: P t K ˆ P f and P t k ˆ G kp t k G k ; k ˆ K 1;... ;1 We consider the function V t; t 7 t P t t Xr t i ) Q i t 52 ½ Q i ½ ½ d½ Using (), it can be easily established that for t 6ˆ t k d dt V t; t ˆ t _P t A P t P t A C C Xr Q i t Š t 2 t P Xr t C C t Xr 2 t P t Dw t y i t y i t Q i t h i y i t y i t 7 t h i : Denoting ¹ 7 y1 yr w Š it follows from (54) that dv dt ˆ ¹ N ¹ 2 jwj 2 j zj 2 ; t 2 t k ;t k1 55 From (52), (53) and the fact that t ˆ ; 8 t µ, it is readily obtained that dv d½ d½ ˆ V t f ; tf V t K1 ; tk1 V t K1 ; t K1 V t 1 ; t 1 ˆ V t f ; tf ˆ E tf Xr t f i By considering (55), this implies that J ˆ ¹ N ¹d½ Xr ½ Q i ½ ½ d½ Q i d½ µ t f i 8w 2 L 2 ;t f Š Lemma 3 provides a bounded real criterion in terms of the DLMI of (51). Note that if Q t > ; t 2 ;t f Š, we obtain using Schur complements, that (51) is equivalent to the RDI &

9 F _P t A P t P t A P t 2 DD Xr Q i 1 t h i A i P t C C Xr Q i t µ 56 The result of Lemma 3 is most powerful, in the sense that it establishes a condition for the performance inde J to be nonpositive almost independently of the delay length. It is thus too conservative. A less conservative condition which eplicitly depends on the length of the delay is derived in the net lemma. Sampleddata H 1 statefeedback control 1123 _P t Xr iˆ Xr A i P t P t Xr iˆ P 2i h i AP 1i A Xr A X r j A j 2 P t 4 2 D I Xr h i P 3i Xr Ai Š 1 D # G i 2 H i P t µ t 2 t k ;t k1 C C 58 Lemma 4: Assume that h µ min kˆ1;...;k t k t k1. The performance inde J of (5) for the system of () with the jumps condition of (44) is nonpositive for all w 2 L 2 ;t f Š if for every k ˆ K 1;... ; there eist constant symmetric positive de nite matrices P 1i ;P 2i ;P 3i and P 4i ; i ˆ 1;... ;r with I P r h ip 3i >, that allow a solution P t ˆ P t, that is absolutely continuous for t 6ˆ t k, to the following RDI for k ˆ K 1;... ;1 _P t Xr iˆ Xr A i P t P t Xr iˆ P 2i h i AP 1i A Xr A X r j A j " P t 2 D I Xr Xr h i P 3i 1 D # G i 2 H i Ai P 4i 1 À ik t Š P t µ and to the following RDI for k ˆ t 2 t k ;t k1 C C 57 with the terminal conditions P t K ˆ P f and P t k ˆ GkP t k G k I Gk Xr h i Ai P 4i I G k H i t 7 h i DP 3i 1 k ˆ K 1;... ;1 D ; G i 7 h i P 1i 1 P 2i 1 A i 5 6 and À ik t 7 À i t t k h i, i.e. À ik t ˆ 1 if t 2 t k ;t k h i Š and À ik t ˆ otherwise. The proof is given in the Appendi. Using Schur complements we can rewrite (57) in the form of equivalent DLMI (see (61) below) ˆ _P t Xr Xr iˆ A i P t P t Xr iˆ P 2i h i AP 1i A Xr A X r j A j C C The RDI (58) can also be brought into the form of (61), the rows and the columns containing P 4i ;i ˆ 1;... ;r should be deleted. 2 F PD h 1 PA 1 h 1 PA 1 h 1 PA 1 D PÀ 1k h r PA r h r PA r h r PA 3 r D PÀ rk D P 2 P r I ip 3i h 1 A1P 1 P 11 h 1 A1P 1 P 21 h 1 D A1P 2 h 1 P 31 PÀ 1k P 41 µ 61 h r ArP r P 1r 6 h r ArP r P 2r 7 64 h r D 75 ArP 2 h r P 3r PÀ rk P 4r

10 1124 E. Fridman and U. Shaked Remark 1: In the case when A i I G k ˆ for some i > we set P 4i ˆ P 4i 1 ˆ in (57) and (5) and we delete the row and the column containing P 4i ;i ˆ 1;... ;r in (61). The above result is obtained for any matrices in (), (44). In the special case of matrices of the form (8), (45) we obtain that I G k ˆ for all i ˆ 1;... ;r and hence for all k ˆ K 1;... ; we solve the RDI of (58) with the terminal conditions given by (52). We obtain the following corollary. Corollary 1: Assume that for every k ˆ K 1;... ; there eist constant symmetric positive de nite matrices P 1i ;P 2i and P 3i i ˆ 1;... ;r with I P r h ip 3i >, that allow a solution P t ˆ P t, that is absolutely continuous for t 2 t k ;t k1, to the RDI of (58) (to the DL MI of (61), the row and the column containing P 4i ;i ˆ 1;... ;r is deleted), with the terminal conditions of (52), ; D ; C;G k are given by (8), (45) and P f ˆ. Then the performance inde J of (4) with E ˆ for the closedloop system of (1), u ˆ u t, is nonpositive for all w 2 L 2 ;t f Š and ˆ. Remark 2: Note that the case of A 1 6ˆ requires modi cation of technique used in lemmas 3 and 4 and leads to more complicated DLMIs and RDIs. Eample: Consider the system _ t ˆ t t h u w; z ˆ col f;ug 62 and P f ˆ Q f ˆ R f ˆ. For h ˆ this eample coincides with the one in Basar and Bernard (15, p. 135). We chose t f ˆ 1 s, K ˆ 2, t 1 ˆ :5 s and ˆ 1. We veri ed that (35) (and, hence, linear equations (36)) with terminal conditions (37) had bounded solutions on ; 1Š: P t ˆ U t ˆ tan 1 t ; M t ˆ cos 1 1 t 1 P t ˆ tan :721 t ; µ t < :5 Therefore, Assumption 1 holds and :5 µ t µ 1 u t ˆ ; µ t < :5; u t ˆ :2553 :5 From (3) and (43) we obtained P 1 :5 ˆ 1:155; M 1 :5 ˆ :3113 U 1 :5 ˆ :12 u 1 t ˆ ; µ t < :5 u 1 t ˆ u t h 3:253 :5 :5 µ t < 1 :2553 :5 hs ds ; :5 µ t µ 1 1 Considering the system of (62), u ˆ u, and applying the RDI of Lemma 3, we were not able to nd a timeinvariant Q > for which the RDI of (56) holds. If there eisted such a Q, then u would lead to a closedloop performance level of ˆ 1 for all h. Applying therefore Lemma 4 and choosing P 11 ˆ P 21 ˆ P 31 ˆ P 41 ˆ I we nd that the RDI of (58) with terminal condition of (5) has a solution for all µ h µ :17 s. For h ˆ :18 s the solution of (58), with the equality sign and (5), has an escape point (see Figure 1). The control law of u ˆ u thus guarantees ˆ 1 for all µ h µ :17s, as for h ˆ :18 our theory, which only provides a su cient condition cannot guarantee the performance level of ˆ h= h= Values of P Values of P Time t in sec Time t in sec Figure 1. The (1, 2) component of the solution P to (57) for h ˆ :17 s and h ˆ :18 s.

11 Sampleddata H 1 statefeedback control 1125 w 1 sin 2ºt cos 2ºt sin 4ºt cos 4ºt sin6ºt cos6ºt J :571 :4446 :4827 :4874 :424 :446 :486 The behaviour of (62) for h ˆ :17 under the controller u ˆ u and the disturbances w ˆ 1; w ˆ sin 2ºnt; w ˆ cos 2ºnt; n ˆ 1;2;... (that constitute the orthogonal basis in L 2 ;1Š) has been simulated. We see from Table 1 that indeed the resulting J µ for all values of w under consideration. 6. Conclusions Table 1. The values of J for h ˆ :17 s. The paper presents a solution to the sampleddata statefeedback H 1 control of linear timeinvariant systems with state timedelays in the nite horizon case. Similarly to sampled data H 1 control of systems without delay (Khargonekar et al. 13, Basar and Bernard 15, Sagfors and Toivonen 17), our solution includes two steps: one of solving the continuoustime problem between sampling and the second is an updating the terminal conditions at the sampling instants. One additional outcome that stems from the theory of this paper is the derivation of new bounded real lemmas for invarianttime systems with statedelays and jumps. The theory that has been developed here shows that for small time delays, similarly to the case of singularly perturbed systems (Pan and Basar 13, Fridman, 16), our controllers are a ected by the boundarylayer phenomenon. This fact requires evaluation of both, outer epansion and boundarylayer corrections. One can adopt the Riccati operator equations approach of Bensoussan et al. (12), van Keulen et al. (13) and McMillan and Triggiani (13) to solve sampleddata statedelay case. This leads to di erential Riccati operator equations with jumps. The performance that is obtained in this case can be analysed, similarly to Ndiaye and Sorine (2), only for the zeroorder approimation. Appendi Proof of Lemma 1: Let t be a solution of (). Then, di erentiating V t; t with respect to t we obtain d V t; t ˆ 2 L t Dw Š dt P t t Q t; ¹ F t ¹ d¹ t t _ P t t 2 2 t Q t;¹ d dt F t Š ¹ d¹ Q t;¹ F t ¹ d¹ t d dt F t s R t;s;¹ F t ¹ Š ds d¹ 63 L t is de ned by (2) with all matrices taken with bars. Denote by w ˆ 2 D P t Then, integrating by parts in (63) e.g. Q t;¹ d F t ¹ d¹ ˆ dt Q t;¹ F t ¹ d¹ d Q t; ¹ d¹ F t ¹ Š ) A 1 ¹ t d¹ ˆ Q t; L t A t Š Xr Q t; i A i t Q t;¹ F t ¹ d¹ ¹ Q t;¹ A 1 ¹ d¹ t due to (1) Q t; L t A t Š ˆ P t L t A t Š and applying (16)± (1) we get (64) dv dt ˆ t P t _ A P P A Xr Ai A1 2 t Q t; i Xk Q t; d Q t;¹ Xr A Q t; i A i Q t;¹ Q t; ¹ t ¹ Ai A1 R t; ;¹ d R t; i ;¹ Q t; A 1 d Š t F t ¹ d¹

12 1126 E. Fridman and U. Shaked R t;¹;s t F t s R t;¹;s R t;¹;s ¹ s F t ¹ d¹ ds 2 2 w w ˆ t C C P S P t 2 2 w w 2 t PS Q t;¹ F t ¹ d¹ F t s Q t;¹ S Q t;s F t ¹ d¹ ds ˆ t C C t 2 w w w w 2 w w It follows from (64) that E tf V t K1 ; tk1 The latter relation implies (21). t K1 jzj 2 2 jwj 2 Š dt ˆ 2 jjw w jj L 2 Proof of Theorem 2: The proof of (i) is similar to Fridman and Shaked (18). Equation (43) follows from (42). We prove only that u m leads to an attenuation level of O h m1. We apply u and u m on () and (1). We obtain correspondingly and m that satisfy () and the jump conditions t k ˆ W kt ; mt k k ˆ W k O h m1 Š mt k W k t ˆ col t ;U 1 t k M t k t N t k ;s F t s ds Thus, y t ˆ t mt satis es _y t ˆ Xr iˆ y t h i A 1 s y t s ds z t ˆ C t ; t k µ t < t k1 65 y t k ˆ W k y t O h m1 k mt k We nd that jjz m zjj 2 L 2 µ c jy tf j 2 c h m1 j tf j 2 c c 1 jy t j 2 c h m1 j t j 2 cš dt µ c jy tf j 2 c hm1 j tf j 2 c 66 c 1 jjyjj 2 L 2 h m1 jj t jj 2 L 2 Š; c 1 > 67 jj t jj L 2 ˆ jj ma 2 ;Š t jj L 2. Evidently, jj t jj L 2 µ jj t jj L 2. Let X t be a transition matri for (), X ˆ I and X t ˆ for t <. Denote T t X ˆ X t. By the variation of constants formula (Hale 177) t ˆ T t t k W k t T t s X k Dw s ds t k t k < t µ t k1 68 From the latter equation and the condition ˆ we have j t 1 j c µ c 1 jjwjj L 2. By induction it is easy to obtain from (68) that j t k j c µ c jjwjj L 2 ; k ˆ 1;... ;K; c > 6 Then from (68) and (6) we derive jj t jj 2 ˆ XK tk1 L 2 j s j 2 c ds µ c X K 3 j t j 2 k c jjwjj2 L 2 kˆ t k kˆ µ cjjwjj 2 L 2 Similarly to the latter inequality, one can derive jj mt jj 2 L 2 µ cjjwjj 2 L 2 ; and jjy t jj 2 L 2 µ ch m1 jj t jj 2 L 2 µ c 2 h m1 jjwjj 2 L 2. The latter inequalities, together with (67), imply jjz m jj 2 L ˆ 2 jjzjj 2 L 2 O h m1 jjwjj 2 L 2 : Since jjzjj 2 L 2 µ 2 jjwjj 2 L 2, we derive jjz m jj 2 L 2 µ 2 O h m1 Šjjwjj 2 L ˆ 2 O h m1 Š 2 jjwjj 2 L 2 : Proof of Lemma 4: Since t ˆ and w t ˆ for t < and h ˆ we nd that for t 2 t k ;t k h i t ˆ t k t k Xr A j ½ hj jˆ d½ Dwd½ t k and t k ˆ t tk h i Xr A j ½ h j d½ t i jˆ k Dw d½ t i Summing the latter two equations and using (44) we obtain that for all t 2 ;t f Š t h i ˆ t t Xr A j ½ h j d½ t i jˆ Dw d½ I G k t k À ik t t i Thus, for t 2 ;t f Š &

13 t _ ˆ Xr iˆ Xr t X r Xr " t X # r A j ½ h j t i jˆ t D w d½ Dw t t i I G k t k À ik t De ning the function V t; t 7 t P t t, we obtain that for almost all t d dt V t; t ˆ t _P t Xr Aj P t P t Xr A j ² 1 t 7 Xr ² 2 t 7 Xr ² 3 t 7 Xr t i t i t i ² 4 7 t P t Xr jˆ t 2 ² 1 ² 2 ² 3 ² 4 t P t A i A ½ d½ jˆ " t P t X # r A j ½ h j d½ d½ 7 t P t A i Dw ½ d½ t P t Dw t I G k t k À ik t Š Since for any z; y i 2 R n ; i ˆ 1;... ;r and for any symmetric positive de nite matrices X i 2 R n n 2 Xr y i z µ Xr yi X i 1 y i Xr z X i z we nd that for any n l n l constant symmetric matrices P 1i > ;P 2i > ;P 3i ;P 4i ; i ˆ ;... ;r, and for any q q symmetric constant matri P 5 > 2² 1 µ Xr Xr 2² 2 µ Xr h i PA i P 1 Xr t i 1i A h i PA i P 1 t i i P AP 1i A d½ 2i A i P Xr ½ h j A X r P 2i A j ½ h j d½ Sampleddata H 1 statefeedback control 1127 j 71 2² 3 µ 2 Xr Xr 2 t i w P 3i w d½ PA i H i Ai P 2 w P 5 w 2 PDP 5 1 D P 2² 4 µ t P t Xr P 1 4i À ik t P t t t k I Gk Xr Ai P 4i À ik t I G k t k H i is de ned in (6). Since 71 d dt V t; t dt ˆ E tf XK1 V t k ; tk V t k ; t k Š we nd, using (7)± (71), that J ˆ d dt V t; t dt XK1 kˆ1 kˆ1 V t k ; tk V t k ; t k jjzjj 2 L 2 2 jjwjj 2 L 2 ( t k P t k t k µ XK1 kˆ1 t k P t k t k t k I Gk ) Xr h i Ai P 4i I G k t k S 1 t dt 2 X r Xr t i ½ h j Aj w P 5 I w dt ( AP 1i A ) X r P 2i A j ½ hj 2 w P 3i w d½ dt S 1 t 7 _P Xr iˆ " P Xr 2 A i P P Xr iˆ G i 2 H i Xr DP 5 1 # D P C C P 1 4i À ik t 72 73

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