Robust sampled-data control of linear singularly pertu

470IEEE TRANSACTIONS ON AUTOMA TIC CONTROL,VOL.51,NO.3,MARCH 2006

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

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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)

472IEEE TRANSACTIONS ON AUTOMA TIC CONTROL,VOL.51,NO.3,MARCH 2006

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)

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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)

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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)

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475

6162hB 2Y 2

r H

B 19T F 0+Y T F T 209T

C T +Y T

D T

12

30 (9+9T

)h B 2Y 2

r H B 10h R 20

330h R

200hY T 20hY T 2D T

123330 r I n 000033330 2I q

F T

100333330 r I n 00

3333330h R 20

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 2Y =[Y 1Y 2]

(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 f 9;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.

Theorem 4.1: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 L 2-gainless than for all small enough " 0,if for some prescribed scalar =0there exist n 12n 1matrices P

1>0;91;n 22n 2matrices 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 (37)with 9and P

givenby (28)is f easible.The state-feedback "-independent gain is given by K =Y 901.ii)The gain K =[K 1K 2]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 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 (33)is feasible and E "P ">0,where P "is givenby (18).

Example 4.1:Consider the uncertain system (11),(29)with H =I 2;F 0=0:11I 2;F 1=F 2=[0:10:1]T .We ?nd by Theorem 4.1,where =00:1,that the "-independent fast-rate controller (10),where K =[03:004900:5954],leads to L 2-gainless than 2:6(which is less than 2:8achieved 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 L 2-gainless than2.6f or all the samplin gs 0 t k +10t k 0:1and for all 0<" 0:49.

V .C ONCLUSION

Sampled-data state-feedback H 1control 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.

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