Robust
Sampled-Data Control of Linear Singularly
Perturbed Systems
Emilia Fridman
Abstract—State-feedback
H control problem for linear singularly per-turbed systems with norm-bounded uncertainties is studied.The fast vari-ables are sampled with fast rates,while for the slow variables both cases of slow and of fast sampling are considered.The recent “input delay”ap-proach to sampled-data control is applied,where the closed-loop system is represented as a continuous one with time-varying input delay.Linear ma-trix inequalities (LMIs)for solution of
H control problem are derived via input-output approach to stability and L -gain analysis of time-delay sys-tems.A numerical example illustrates the ef?ciency of the method.Index Terms—
H control,linear matrix inequality (LMI),sampled-data control,singularly perturbed systems,time-delay.
I.I NTRODUCTION
Singular perturbations in control systems often occur due to the presence of small “parasitic”parameters,such as small masses,small time-delays.The mainobjective o f sin gular perturbationmethods is to alleviate the dif?culties caused by the high dimensionality and the ill-conditioning that results from the interaction of slow and fast dynamical modes.Decomposition of the full-order problem to the "-independent reduced-order slow and fast subproblems was started with the classical Tikhonov theorem on the asymptotic behavior of the solution to the initial value problem [19]and developed further to composite controller design [2],[15](see a survey [17]for recent references).A LMI approach to linear singularly perturbed systems was introduced in [6],[9].
Two mainapproaches have beenused to the sampled-data robust control.The ?rst one is based on the lifting technique [1],[21]in which the problem is transformed to equivalent ?nite-dimensional discrete problem.This approach was applied to sampled-data nonlinear singu-larly perturbed systems,where the composite controller with the fast sampling in the fast variables was suggested [4].The second approach is based onthe represen tationof the system inthe orm of hybrid dis-crete/continuous model.This approach leads to necessary and suf?-cient conditions for stability and L 2-gainan
alysis inthe orm of dif -ferential equations (or inequalities)with jumps and it was applied to sampled-data H 1control of linear singularly perturbed systems [18],where the slow sampled-data controller was designed.The above ap-proaches do not work in the cases with uncertain sampling times or uncertain system matrices.
A new “input delay”approach to sampled-data control has been sug-gested recently in [7].By this approach,a digital control law is repre-sented as a delayed control as follows:
u (t )=u d (t k )=u d (t 0(t 0t k ))=u d (t 0 (t ))
t k t (t )=t 0t k (1) where u d is a discrete-time control signal and the time-varying delay (t )=t 0t k is piecewise linear with derivative _ (t )=1for t =t k . Manuscript received March 14,2005;revised August 2,2005.Recommended by Associate Editor L.Glielmo.This work was supported by the Kamea Fund of Israel. The author is with the Department of Electrical Engineering-Systems,Tel Aviv University,Tel-Aviv 69978,Israel (e-mail:emilia@eng.tau.ac.il).Digital Object Identi?er 10.1109/TAC.2005.864194 Moreover, t k +10t k .The solutionto the problem is f oun d thenby solving the problem for a continuous-time system with uncertain but bounded (by the maximum sampling interval)time-varying delay in the control input via Lyapunov technique.Given h >0,the conditions obtained are robust with respect to different samplings with the only requirement that the maximum sampling interval is not greater than h .Stability of singularly perturbed systems with a constant delay h has beenstudied intwo cases:1)h is proportional to "(small delay),and 2)"and h are independent.The ?rst case,being less general than the second one,is encountered in many publications (see,e.g.,[11],[10],and the references therein).The second case has been studied in the frequency domain [16].A Lyapunov-based approach to the problem leading to LMIs has been introduced in [6]for the general case of in-dependent delay and ".In the case of constant delay,it was shown [6],that the necessary condition for robust stability of singularly perturbed system for all small enough values of singular perturbation parameter ">0is the delay-independent stability of the fast subsystem,which is rather restrictive.The same is true for systems with uncertain and bounded time-varying delays,where constant delay is just a particular case of delay.Therefore,it is natural to design a delayed state-feedback controller with a small delay in the fast variable " (t ).This corresponds to the fast sampling of fast variables considered in [4].Inthis n ote,we solve the state-f eedback sampled-data H 1-control problem by applying the input delay approach to sampled-data con-trol and by developing the input-output approach to singularly per-turbed time-delay systems.The input-output approach was introduced for regular systems with constant delays in [13]and further developed in [12](see also references therein),where it was generalized to the time-varying delays with the delay derivative less than q <1.Recently, the input–output approach has been developed to L 2-gainan alysis of regular systems with time-varying bounded delays without any con-straints on the delay derivative [8].It is the objective of the present note to develop this approach to singularly perturbed systems with time-varying delay.Two controller designs are considered:1)With the fast sampling in the fast variables and the slow one in the slow vari-ables,and 2)with the fast sampling in both variables. Notation:Throughout this note,the superscript “T ”stands for ma-trix transposition,R n denotes the n -dimensional Euclidean space with vector norm k 1k ;R n 2m is the set of all n 2m real matrices,and the notation P >0,for P 2R n 2n means that P is symmetric and positive de?nite.The symmetric elements of the symmetric ma-trix will be denoted by 3.L 2is the space of square integrable functions v :[0;1)!C n with the norm k v k L = [1 0k v (t )k 2dt ]1=2. II.P ROBLEM F ORMULATION Giventhe ollowin g system: E "_x (t )=(A +H 1 F 0)x (t )+(B 1+H 1F 1)w (t ) +(B 2+H 1F 2)u (t )(2)z (t )=Cx (t )+D 12u (t ) (3) where x (t )=col f x 1(t );x 2(t )g ;x 1(t )2R n ;x 2(t )2R n is the system state vector,u (t )2R `is the control input,w (t )2R q is the exogenous disturbance signal,and z (t )2R p is the state combination (objective function signal)to be attenuated.The matrix E "is givenby E "= I n 00 "I n (4) where ">0is a small parameter. 0018-9286/$20.00?2006IEEE Denote n1=n1+n2.The matrices A;B1;B2;F0;F1;F2;H;C and D12are constant matrices of appropriate dimensions.The matrices in (2)and(3)have the following structures: A=A1A2 A3A4 H= H1H2 H3H4 F0= F01F02 F03F04 B i=B i1 B i2 C=[C1C2]F i= F i1 F i2 ;i=1;2:(5) We do not require A4to be nonsingular.Such a system is a non-standard singularly perturbed system[14].In the case of singular A4 open-loop system(2)with"=0has index more than one and pos- sesses animpulse solution[3]. The uncertain time-varying matrix1(t)=[11(t)12(t) 13(t)14(t) ]satis- ?es the inequality 1T(t)1(t) I n;t 0:(6) We are looking for a piecewise-constant control law of two forms. 1)A multiple(slow/fast)rate state-feedback u(t)=u s(t)+u f(t)u s(t)=K1x1(t k);t k t u f(t)=K2x2("t k)"t k t<"t k+1(7) where0=t0 2)A single(fast)rate state-feedback u(t)= Kx("t k);"t k t< "t k+1,where0="t0<"t1<111<"t k<111are the fast sampling instants and lim k!1t k=1. Given >0our objective is to?nd a piecewise constant controller which internally stabilizes the system and leads to L2-gainless than . The latter means that the following inequality J=k z k2L0 2k w k2L<0(8) holds for x(0)=0and for all nonzero w2L2. We represent a piecewise-constant control law as a continuous-time control with a time-varying piecewise-continuous(continuous from the right)delay (t)=t0t k as given in(1),corresponding to the slow sampling,and with small delay" (t)="(t0t k),corresponding to the fast sampling.We will thus look for state-feedback controllers of two forms u(t)=K x1(t0 (t)) x2(t0" (t)) K=[K1K2](9) and u(t)=Kx(t0" (t)):(10) We assume that A1)t k+10t k h8k 0. From A1it follows that (t) h since (t) t k+10t k. To guarantee that for all small enough">0the full-order system is stabilizable-detectable we assume[20]. A2)Both pencils[sE00A;B2]and[sE00A T;C T]are of full row rank for all s with nonnegative real parts,where E0is givenby(4)with"=0. A3)The triple f A4;B22;C2g is stabilizable-detectable. III.M ULTIPLE R ATE H1C ONTROL A.Input–Output Model Substituting(9)into(2),we obtain the following closed-loop system: E"_x(t)=(A+H1F0)x(t)+(B2+H1F2)K 2 x1(t0 (t)) x2(t0" (t)) +(B1+H1F1)w(t) z(t)=Cx(t)+D12K x1(t0 (t)) x2(t0" (t)) :(11) We will further consider(11)as the system with uncertain and bounded delay (t)2[0;h]. We represent(11)in the form E"_x(t) =(A+B2K+H1(F0+F2K))x(t)0(B2+H1F2)K 2 0 (t) _x1(t+s) ds 0" (t) _x2(t+s)ds +(B1+H1F1)w(t) z(t) =(C+D12K)x(t)0D12 K 0 (t) _x1(t+s) ds 0" (t) _x2(t+s)ds :(12) We follow the idea of[13]and[12]to embed the perturbed system(12) into a class of systems with additional inputs and outputs,the stability of which guarantees the stability of(12).Consider the following for-ward system: E"_x(t)=(A+B2K)x(t)+hB2Kv(t)+B1w(t)+Hv3(t) z(t)=(C+D12K)x(t)+hD12Kv(t) y(t)=E"_x(t)=(A+B2K)x(t)+hB2Kv(t) +B1w(t)+Hv3(t) y3(t)=(F0+F2K)x(t)+hF2Kv(t)+F1w(t)(13a-d) where v(t)= v1(t) v2(t) y(t)= y1(t) y2(t) with feedback v1(t)=01 h 0 (t) y1(t+s)ds v2(t)=01 "h 0" (t) y2(t+s)ds v3(t)=1y3(t):(14) Note that for h!0the above model(13),(14)corresponds to the closed-loop system(2)with the continuous state-feedback u(t)=Kx(t). Assume that y i(t)=0;8t 0;i=1;2;3.The following holds for n i2n i-matrices R i>0;i=1;2and a scalar r>0[12] k p R i v i k L k p R i y i k L;i=1;2k p rv3k L k p ry3k L: (15) For"!0inequality(15)is valid and y2givenby(13c)van ishes. Thus,for"!0(13),(14)is the input–output model,which corre-sponds to the descriptor system without delay in x2 E0_x(t)=(A+H1F0)x(t)+(B1+H1F1)w(t) +(B2+H1F2)u(t) u(t)=K1x1(t k)+K2x2(t)t2[t k;t k+1) 0 t k+10t k h:(16) Remark 3.1:Descriptor system canbe destabilized by arbitrary f ast sampling in the fast variable of the state-feedback even if the system is stable under continuous-time state-feedback.Consider the following simple example: E 0_x (t )= 01011x (t )+0 1 u (t ); x (t )2R 2: (17) It is clear that the closed-loop system is stable with the continuous state-feedback u (t )=02x 2(t ),while it is unstable with u (t )=02x (t k );t 2[t k ;t k +1),for any sampling t k .Really,the resulting closed-loop triangular system is stable if equation x 2(t )+u (t )=0is stable.However,this equationinthe sampled-data case x 2(t )=2x 2(t k );t 2[t k ;t k +1)is unstable.B.L 2-Gain Analysis Consider the Lyapunov function V (t )=x T (t )E "P "x (t ),where P "has the structure of P "= P 1 "P T 2P 2 P 3 ;P 1>0;P 3>0:(18) Note that P "is chosento be o the orm o (18)(as,e.g.,in[20]),such that for "=0,the function V with E "=E 0and P "=P 0,corresponds to the descriptor case). Given ">0,from (15)it follows that the following condition along (13a): W 1 =_V (t )+h 2i =1 y T i (t )R i y i (t )+r k y 3(t )k 2 0h 2 i =1 v T i (t )R i v i (t )0r k v 3(t )k 2+k z (t )k 20 2k w (t )k 2 <0 (k x (t )k 2+k u (t )k 2+k w (t )k 2); >0(19) guarantees the internal stability of (11)and that L 2-gainof (11)less than .Moreover,since y (t )depends on _x (t ),we consider the deriva-tive condition _V (t ) 0 (k x (t )k 2+k _x (t )k 2); >0.Such deriva-tive condition corresponds to the descriptor model transformation in-troduced in[5]. Given n 2n -matrices 8j = 8j 10 8j 2 8j 3 ;j =2;3; 8j 12R n 2n 8j 32R n 2n (20) denote P "= P " 82 83 :(21) We have,similarly to [5],the ?rst equationshownat the bottom of the page. Thus,along the trajectories of (13)we obtain W T (t ) 0 (t )+h 2i =1 y T i (t )R i y i (t )+r k y 3(t )k 2+k z (t )k 2(22) where (t )=col f x (t );E "_x (t );v (t );v 3(t );w (t )g and 0 =0" h P T " 0B 2K P T "0H P T " 0B 1 30hR 100hR 2 00330rI n 03330 2I q 0"=P T "0I n A +B 2K 0I n + 0A T +K T B T 2I n 0I n P ": (23a,b) By applying Schur complements to the term h 2i =1y T i (t )R i y (t )+r k y 3(t )k 2+k z (t )k 2we conclude that (19)is satis?ed if (24),as shown at the bottom of the page,holds. Denote by 4";" 0the matrix inthe le t-han d sid e o (24).I 40<0,i.e.,(24)is feasible for "=0,then f or the same values of P 1;P 2;P 3;R;82and 83the full-order LMI (24)is feasible for small enough values of ",since 4"=40+"M ,where M is some constant matrix.Hence,40<0implies (19)for small enough ".We thus proved the following.Lemma 3.1: i)Given >0;h >0and m 2n -matrix K ,(11)is internally stable and has L 2-gainless than for all small enough ">0and 0 (t ) h ,if there exist n 12n 1matrices P 1>0;R 1>0;821;831;n 22n 2matrices P 3>0;R 2>0;823;833;n 12n 2-matrices P 2;822;832and a scalar r >0such that LMI (24)is feasible for "=0,where P 0and 00are givenby (18),(20),(21),and (23b). ii)Given ">0; >0;h >0and m 2n -matrix K ,(11)is inter-nally stable and has L 2-gainless than for all 0 (t ) h ,if there exist n 12n 1matrices P 1>0;R 1>0;821; _V (t )=2x T (t )P T "E "_x (t )=2 x (t ) E "_x (t )T P T " E "_x (t ) (A +B 2K )x (t )+hB 2Kv (t )+B 1w (t )+Hv 3(t )0E "_x (t )0"h P T " 0B 2K P T " 0H P T " 0B 1r (F 0+F 2K )T 0hR C T +K T D T 12 03 0hR 00hrK T F T 2 0hK T D T 12 330rI n 00003330 2I q rF T 10033330rI n 00333330hR 0333 3 3 3 0I p <0 R = R 100 R 2 (24) 831;n 22n 2matrices P 3>0;R 2>0;823;833;n 12n 2-ma-trices P 2;822;832and a scalar r >0such that LMI (24)is fea-sible and E "P ">0,where P "and 0"are givenby (18),(20),(21),and (23b). If (24)is feasible for "=0,thenthe slow (descriptor)system (16)is internally stable and has L 2-gainless than .Moreover,the fast LMI,shownin(25)at the bottom o the page,is easible.The latter LMI guarantees that the fast _x 2(t )=(A 4+H 414F 04)x 2(t )+(B 12+H 414F 12)w (t ) +(B 22+H 414F 22)u (t )u (t )=K 2x 2(t k ); t 2[t k ;t k +1); 0 t k +10t k h (26) system is internally stable and has L 2-gainless than .Thus the fea-sibility of "-independent LMI (24),where "=0,implies that the fast subproblem is solvable by a sampled-data controller,while the slow subproblem is solvable by a mixed controller (continuous in the fast variable and sampled-data in the slow one).C.State-Feedback Design Our objective now is to ?nd K .Inorder to obtainanLMI in (24)we have to restrict ourselves to the case of block-diagonal 82=diag f 821;823g and to 83= 82,where =0is a scalar parameter.Note that 82is nonsingular due to the fact that the only matrix which can be negative de?nite in the second block on the diagonal of (24)is 0 (82+8T 2).De?ning 9=8012= diag 80121801 23 P =9T P 09 R =9T R 9 r =r 01and Y =K 9;multiplying LMI (24)by diag f 9;9;9;I n ;I q ;I n ;9;I p g and its transpose,from the right and the left,respectively,we obtain the LMI with a tuning parameter ,as shownin(27)at the bottom of the page.Note that P and R have the same,block-triangular and block-diagonal structures,as P 0and R correspondingly. Theorem 3.1:Given >0,consider the system of (2)and the multirate state-feedback law of (9).Assume A1–A3. i)The state-feedback (9)internally stabilizes (2)and guarantees L 2-gainless than for all small enough " 0,if for some prescribed scalar =0there exist n 12n 1-matrices P 1>0; R 1>0;91;n 22n 2-matrices P 3>0; R 2>0;93,an n 12n 2-matrix P 2,a p 2n -matrix Y and a scalar r >0such that LMI (27)with 9=910093 P = P 10 P 2 P 3 R = R 100 R 2(28)is feasible.The state-feedback "-independent gain is given by K =Y 901. ii)The gain K =[K 1K 2]obtained in i)solves the slow (16)and the fast (26)subproblems. iii)Given ">0the gain obtained in i)internally stabilizes (2)and guarantees L 2-gainless than if there exist n 12n 1matrices P 1>0;R 1>0;821;831;n 22n 2matrices P 3>0;R 2>0;823;833;n 12n 2-matrices P 2;822;832and a scalar r >0such that LMI (24)is feasible and E "P ">0,where P "is given by (18). Example 3.1:[18]Consider (11)with A 0=210102 B 2= 22 B 1= 13 C = 211 30 D 12 = 00 1(29) where H =0.Given =3and the uniform sampling t k +10t k =0:1,it was shownin[18]that the slow state-f eed-back u (t )=01:1618x 1(t k );t 2[t k ;t k +1)solves the H 1-control problem for the full-order system for all small enough ">0.More-over,the slow controller can not achieve <2:85 . 0f h P T f 0B 22K 2P T f 0H 4P T f 0B 12 r (F 04+F 22K 2)T 00hR 2C T 2+K T 2D T 12 030hR 200hrK T 2F T 220hK T 2D T 12 330rI n 00 003330 2I q rF T 120033330rI n 00333330hR 2 033333 3 0I p <0 0f =P T f 0I n A 4+B 22K 20I n +0A T 4+K T 2B T 22I n 0I n P f P f = P 30 823833 (25) 61 62hB 2Y r H B 19T F T 0+Y T F T 209T C T +Y T D T 12 30 (9+9T )h B 2Y r H B 10h R 0 330h R 00hY T F T 2 0hY T D T 123330 r I n 000033330 2I q F T 100333330 r I n 003333330h R 0 3333 3330I p <0 61=A 9+9T A T +B 2Y +Y T B T 2;62= P T 09+ 9T A T + Y T B T 2(27) Consider the uncertain system(11),(29)with H=I2;F0=0:11 I2;F1=F2=[0:10:1]T.Applying Theorem3.1with the smaller =2:8,the same h=0:1and choosing =00:1,we?nd that the multirate controller(9)with"-independent gain K=[02:44070 0:5788]leads to L2-gainless than2:8for all small enough">0 and all the samplings with t k+10t k 0:1.By applying Lemma 3.1to the resulting closed-loop system for h=0:1and for different values of">0we verif y that this gainleads the ull-order system to L2-gainless than2:8for all0<" 0:49and for all the samplings 0 t k+10t k 0:1.The possibility to treat the uncertain system,as well as to check the solvability of the H1control problem for given values of h and",are the advantages of the LMI approach. IV.F AST S AMPLE-R ATE H1C ONTROL Substituting(10)into(2),we obtain the following closed-loop system: E"_x(t)=(A+H1F0)x(t)+(B2+H1F2)Kx(t0" (t)) +(B1+H1F1)w(t) z(t)=Cx(t)+D12Kx(t0" (t)):(30) Similarly to the previous sectionwe in troduce the f orward system: E"_x(t)=(A+B2K)x(t)+hB2K v(t)+B1w(t)+Hv3(t) z(t)=(C+D12K)x(t)+hD12K v(t) y(t)=E"_x(t)=(A+B2K)x(t)+hB2K v(t) +B1w(t)+Hv3(t) y3(t)=(F0+F2K)x(t)+hF2K v(t)+F1w(t)(31a-e)where v(t)= "v1(t) v2(t) y(t)= y1(t) y2(t) with feedback v1(t)=01 "h 0" (t) y1(t+s)ds v2(t)=01 "h 0" (t) y2(t+s)ds;v3(t)=1y3(t):(32) Inequalities(15)are valid here.The only difference with the previous case that v1in(31)is multiplied by",which leads to the full-order LMI for stability and L2-gainshownin(33)at the bottom of the page. Setting"=0into(33)and applying the Schur complements to the row and the column with the only nonzero(diagonal)element hR1(and thus deleting also the row and the column containing hR1)we obtain the"-independent LMI shown in(34)at the bottom of the page. Feasibility of(34)implies feasibility of(33)for small enough">0 and R1>0.LMI(34)implies the same fast LMI(25)and the fast problem(26),while the slow LMI has a orm shownin(35)at the bottom of the page,and corresponds to the slow problem with a con-tinuous-time state-feedback E0_x(t)=(A+H1F)x(t)+(B1+H1F1)w(t) +(B2+H1F2)u(t)u(t)=Kx(t): (36) 0"h P T" "B2K1B2K2 P T" H P T" B1 r(F0+F2K)T hR C T+K T D T12 30hR00hr "K T1 K T2 F T20h "K T1 K T2 D T12 330rI n0000 3330 2I q rF T100 33330rI n00 333330hR0 3333330I p <0 R=diag f R1;R2g (33) 00h P T0 B2K2 P T0 H P T0 B1 r(F0+F2K)T hR2 C T+K T D T12 30hR200hrK T2F T20hK T2D T12 330rI n0000 3330 2I q rF T100 33330rI n00 333330hR20 3333330I p <0 (34) 00P T0 H P T0 B1 r(F0+F2K)T C T+K T D T12 30rI n000 330 2I q rF T10 3330rI n0 33330I p <0(35) 6162hB2Y2 r H B19T F0+Y T F T209T C T+Y T D T12 30 (9+9T)h B2Y2 r H B10h R20 330h R200hY T20hY T2D T12 3330 r I n0000 33330 2I q F T100 333330 r I n00 3333330h R20 33333330I p <0 61=A9+9T A T+B2Y+Y T B T262= P T09+ 9T A T+ Y T B T2Y=[Y1Y2](37a,b) Note that singularly perturbed systems with small delay are usually decomposed into the nondelayed slow subsystem and the delayed fast one(see,e.g.,[10]). Multiplying(34)by diag f9;93;9; r I n;I q; r I n;93;I p g and its transpose,on the left and on the right,respectively,we obtain the fol- lowing"-independent LMI with a tuning parameter ;see(37a,b),as shownat the top of the page. Theorem4.1:Given >0,consider the system of(2)and the fast-rate state-feedback law of(10).Assume A1–A3. i)The state-feedback(10)internally stabilizes(2)and guaran- tees L2-gainless than for all small enough" 0,if for some prescribed scalar =0there exist n12n1matrices P 1>0;91;n22n2matrices P 3>0; R 2>0;93,an n12n2-matrix P2,a p2n matrix Y and a scalar r>0 such that LMI(37)with9and P givenby(28)is f easible.The state-feedback"-independent gain is given by K=Y901. ii)The gain K=[K1K2]obtained in i)solves the slow(36)and the fast(26)subproblems. iii)Given">0the state-feedback(10)with K from i)internally stabilizes(2)and guarantees L2-gainless than if there exist n12n1matrices P1>0;R1>0;821;831;n22n2matrices P3>0;R2>0;823;833;n12n2-matrices P2;822;832and a scalar r>0such that LMI(33)is feasible and E"P">0, where P"is givenby(18). Example4.1:Consider the uncertain system(11),(29)with H= I2;F0=0:11I2;F1=F2=[0:10:1]T.We?nd by Theorem4.1, where =00:1,that the"-independent fast-rate controller(10),where K=[03:004900:5954],leads to L2-gainless than2:6(which is less than2:8achieved by the multi-rate controller(9))for all small enough ">0and all the samplings with t k+10t k 0:1.Moreover,this gain leads the full-order system to L2-gainless than2.6f or all the samplin gs 0 t k+10t k 0:1and for all0<" 0:49. V.C ONCLUSION Sampled-data state-feedback H1control problem for singularly perturbed system with norm-bounded uncertainties has been solved via input delay approach to sampled-data control.The only assump- tion on the sampling that the distance between the sequel sampling times is not greater than some h>0.Two kinds of controllers have been designed(both with the fast sampling in the fast variables): the multirate state-f eedback(slow rate inthe slow variables)an d the fast-rate state-feedback.The"-independent gains of the controllers are found from"-independent LMIs."-dependent LMIs are derived which give suf?cient conditions for the solvability of the full-order system. Anillustrative example shows that the f ast-rate con troller leads to better performance,than the multirate one.The tradeoff is in the fast sampling of the slow variables. R EFERENCES [1] B.Bamieh,J.Pearson,B.Francis,and A.Tannenbaum,“A lifting tech- nique for linear periodic systems,”Syst.Control Lett.,vol.17,pp.79–88, 1991. [2]J.Chow and P.Kokotovic,“A decomposition of near-optimum regula- tors for systems with slow and fast modes,”IEEE Trans.Autom.Control, vol.AC-21,no.5,pp.701–705,Oct.1976. [3]L.Dai,Singular Control Systems.Berlin,Germany:Springer-Verlag, 1989. [4]M.Djemai,J.-P.Barbot,and H.Khalil,“Digital multirate control for a class of nonlinear singualrly perturbed systems,”Int.J.Control,vol.72, no.10,pp.851–865,1999. [5] E.Fridman,“New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,”Syst.Control Lett.,vol.43, pp.309–319,2001. [6],“Ef ects of small delays onstability of sin gularly perturbed sys- tems,”Automatica,vol.38,no.5,pp.897–902,2002. [7] E.Fridman,A.Seuret,and J.-P.Richard,“Robust sampled-data stabi- lization of linear systems:An input delay approach,”Automatica,vol. 40,pp.1441–1446,2004. [8] E.Fridman and U.Shaked,“Input-output approach to stability and L-gain analysis of systems with time-varying delays,”in Proc.44th Conf.on Decision and Control,Seville,Spain,2005,pp.7175–7180. [9]G.Garcia,J.Daafouz,and J.Bernussou,“The in?nite time near op- timal decentralized regulator problem for singularly perturbed systems: A convex optimization approach,”Automatica,vol.38,pp.1397–1406, 2002. [10]V.Glizer and E.Fridman,“ H control of linear singularly perturbed systems with small state delay,”J.Math.Anal.Appl.,vol.250,pp.49–85, 2000. [11] A.Halanay,“An invariant surface for some linear singularly perturbed systems with time lag,”J.Diff.Equat.,vol.2,pp.33–46,1966. [12]K.Gu,V.Kharitonov,and J.Chen,Stability ofTime-De lay Syste ms. Boston,MA:Birkh?user,2003. [13]Y.-P.Huang and K.Zhou,“Robust stability of uncertain time-delay sys- tems,”IEEE Trans.Autom.Control,vol.45,no.11,pp.2169–2173,Nov. 2000. [14]H.K.Khalil,“Feedback control of nonstandard singularly perturbed sys- tems,”IEEE Trans.Autom.Control,vol.34,no.10,pp.1052–1060,Oct. 1989. [15]P.Kokotovic,H.Khalil,and J.O’Reilly,Singular Perturbation Methods in Control:Analysis and Design.New York:Academic,1986. [16] D.W.Luse,“Multivariable singularly perturbed feedback systems with time delay,”IEEE Trans.Autom.Control,vol.AC-32,no.11,pp. 990–994,Nov.1987. [17] D.S.Naidu,“Singular perturbations and time scales in control theory and applications:An overview,”Dyna.Contin.,Discrete Impul.Syst. (DCDIS)Series B:Appl.Algorithms,vol.9,no.2,pp.233–278,2002. [18]Z.Pan and T.Basar,“H-in?nity optimal control for singularly perturbed systems with sampled state measurements,”in Advances in Dynamic Games and Applications,T.Basar and A.Haurie,Eds.Boston,MA: Birkh?user,1994,vol.1,pp.23–55. [19] A.Tikhonov,“Systems of differential equations containing small param- eters multiplying some of derivatives,”Mathematica Sborniki,vol.31, pp.575–586,1952. [20]H.Xu and K.Mizukami,“In?nite-horizon differential games of singu- larly perturbed systems:A uni?ed approach,”Automatica,vol.33,pp. 273–276,1997. [21]Y.Yamamoto,“New approach to sampled-data control systems—A function space method,”in Proc.29th Conf.Decision and Control, Honolulu,HI,1990,pp.1882–1887.