Optimal guaranteed cost control of neutral systems with interval time - varying delayed state and control = Bài toán điều khiển giá trị tối ưu cho hệ phương trình vi phân trung tính có trễ trên trạng thái và điều khiển

Mai Viet Thuan va dug Tap chi KHOA HOC & CONG NGHE 83(07): 95 - 102

)ptimal guaranteed cost control of neutral systems with

interval time-varying delayed state and control

MAI VIET THU.AN AND HOANG THI HA

College of Sciences, Thainguyen university, Thainguyen, Vietnam

E-mail: maithuankl'Sgmail.com

Dstract. This paper deals with the problem of op￾lal guaranteed cost control for neutral systems with

erval time-varying delayed state and control. The

le delay is assumed to be a continuous function be￾iging to a given interval, but not necessary to be dif￾entiable. A linear-quadratic cost function is consid-

'd as a performance measure for the closed-loop sys￾II. By constructing a set of time-varying Lyapunov￾asovskii functional combined with Newton-Leibniz

mula, a guaranteed cost controller design is pre￾ited and sufficient conditions for the existence of a

aranteed cost state-feedback for the system are given

terms of linear matrix inequalities (LMIs). A numer￾1 example is included to illustrate the effectiveness

our results.

-Ij words. Guaranteed cost control. Stability, Stabi￾ition. Interval delay. Neutral system, Linear matrix

quality.

i Introduction

ibility analysis of dynamical control time-delay sys￾ns is fundamental to many practical problems and

>: received considerable attention, see e.g [8, 9] and

! references therein. Various stability techniques

re been applied to derive new conditions for asymp-

•ic stability and stabilization of the systems by many

earchei's [10, 11, 13). On the other hand, in many

ictical system, it is desirable to design the control

Item which is not only asymptotically stable but also

irantee an adequate level of performance.

The linear quadratic stabilization of systems was

isidered in the context of the guaranteed control cost

)l)lem, where the approach concerns finding upper

iinds on the quadratic cost for the closed-loop linear

item. This approach has the advantage of providing

upper bound on a given performance index and thus

the system performance degradation incurred by the

uncertainties is guaranteed to be less than this bound.

Based on this idea, many significant results have been

proposed for the continuous-time case [5, 12] and for

the discrete-time case [4, 14, 15]. However, the interval

time-varying delaj-ed control input was not considered

there. In addition, although the disign problem of a

robust delay-dependent guaranteed cost control of a

class of uncertain nonlinear neutral systems with both

norm-bounded uncertainties and nonlinear parameter

perturbations was considered in [1, 2], the approach

used there can not be applied to the systems with non￾differentiable time-varying delays in state and control.

Moreover, in all that papers, the time delay is aissumed

to be either a constant or a differentiable function.

In this paper, we consider the problem of guar￾anteed cost control for a more general neutral de￾layed system, where the control input coiitaiiis inter￾val time-varying delays. By constructing a set of time￾varying Lyapunov-Krasovskii functionals combined

with Newton-Leibniz formula, new delay-dependent

criteria for existence of the guaranteed cost controller

are derived in terms of LMIs. Compared to the exist￾ing results, our result has its own advantages. First, the

time delay system is time-varying delayed in state and

control. Since the controller input also contains contin￾uously distributed time-varying delays, the techniques

used in the previous papers cannot be applied directly

to solve the stabilization problem for the system. Sec￾ond, the time delay is assumed to be any continuous

function belonging to a given interval, which means

that the lower and upper bounds for the time-varying

delay are available, but the delay function is bounded

but not necessary to be differentiable. This allows the

time-delay to be a fast time-varying function and the

lower bound is not restricted to being zero.

The outline of the paper is as follows. Section 2

presents definitions and some well-known technical

propositions needed for the proof of the main result.

Optimal guaranteed cost control of neutral systems

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