Delay-dependent Robust Passive Control for Uncertain Discrete-time Systems with Time Delays

Copyright © 2013 Jufang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This paper considers the problem of robust passive control for uncertain discrete systems with time-varying delays. We pay attention to designing a state feedback controller which guarantees the passivity of the closed-loop system for all admissible uncertainties. In terms of a linear matrix inequality, a sufficient condition for the solvability of this problem is presented and the explicit expression of the desired state feedback controller is given.


Introduction
In the past several years, much attention has been paid to the study of stability of systems with control input delay.Much of them is focused on the passivity analysis for classes of time-delay systems.Using classical definitions of passivity and positive realness, the conditions for a nonlinear system can be rendered passive via smooth state feedback, see [1,2].The robust passive control problem for time-delay systems was dealt with in [3,4] via various approaches.The robust passivity synthesis problem for discrete-time-delay systems is investigated in [5,6], but all these time delays are constant.To the best knowledge of authors, the problem of robust passive control for discrete-time systems with time-varying delays has not been fully investigated, which is more complex.
In this paper, we deal with the problem of robust passive feedback control for discrete systems with parameter uncertainties and time-varying delays.The parameter uncertainties are assumed to be time-varying but normbounded.The purpose is to construct a state feedback controller such that the closed-loop system is strictly passive and obtain a delay-dependent condition for the solvability of the problem.

Statement of the Problem
Consider the following uncertain discrete-time system with time-varying delays:  is a positive integer representing the time-varying delay of the system, which satisfies the following assumption: and 2 are unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the form , then the following inequality holds for any matrices R, S 1 , 3) is called passive if there exists a scalar 0 where  is some constant which depends on the initial condition of the system.In addition, the systems (2.1)-( 2.3) is said to be strictly passive if it is passive and .In the sequel, we provide conditions under which a class of discrete- time linear dynamical systems with time-varying parameter uncertainties can be guaranteed to be strictly passive.First, we have the following result pertaining to the system (2.1)-(2.3).

Proof of Main Results
Theorem 3.1 The discrete-time systems with time delay (2.3) is strictly passive if there exist symmetric positive definite matrices P, R, Q and , such that the following LMI holds: where Proof.Choose a Lyapunov function candidate for the system (2.1) -(2.3) as follows: where Now, by some calculations, we can get that We define that From the Lemma 2.3, for , we can have that 1 2 We have (3.3) and (3.4) into (3.2),after some manipulation, then obtain the following inequality: where

If
, then , and from which it follows that . Application of Lemma 2.1 to the above inequality, it puts into the following form: Substituting the uncertainty structure (2.5) into (3.9) and rearranging, we get the following inequality Then by Lemma 2.2, the inequality (3.10) holds if and only if for some 0 for all admissible uncertainties satisfying (2.4).On using Lemma 2.1 in (3.11), it becomes that in (3.1).This completes the proof.0  

Robust Passive State Feedback Controller
We now build on the foregoing results by considering the passive control problem, that is, designing a state feedback controller to render the closed-loop time-delay system passive.Extending the system (2.1)-(2.3),we consider a class of time-delay systems of the form: where   is the control input, , 2 , are known real constant matrices; 1 and 2 are unknown matrices representing time-varying parametre uncertainties, and are assumed to be of the form: Then the transformed system becomes then we observe that Copyright © 2013 SciRes.OJDM   The following theorem establishes the main result.Theorem 4.1 Consider the uncertain discrete-time delay system (4.4),(4.5).If there exists a positive scalar 0   , a real matrix Y, three symmetric positive definite matrices X , , such that the following inequality holds:

 1 . 4 . 1
then the systems (4.4),(4.5)are strictly passive, and the state-feedback gain matrix is given by 1 K YX   .Proof.Similar to Theorem 3.Remark It is noted that the matrix inequalities conditions in Theorem 4.1 are not LMIs.In order to solve the matrix inequalities conditions in Theorem 4.1, we can follow a similar line as in Lee et al. (2004) and Moon et al. (2001) to provide a nonlinear minimization problem subject to LMIs.