[06ac]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.

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