Robust stabilization of uncertain systems with unknown input delay

doi:10.1016/j.automatica.2003.10.005

Automatica

Volume 40, Issue 2, February 2004, Pages 331-336

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Abstract

This paper is concerned with the robust controller design for uncertain input-delayed systems. The time delay is assumed to be an unknown constant. A controller with delay feedback for the robust stabilization of the system is proposed. The stability criterion of the closed-loop system is derived in terms of linear matrix inequalities (LMIs). Examples show that in many cases our method can give less conservative results than those by the existing methods. Moreover, for the same cases, our controllers have lower feedback gains than the existing ones.

Introduction

In real control systems, input delays are often encountered because of transmission of the measurement information. The existence of these delays may be the source of instability or serious deterioration in the performance of the closed-loop system.

Recently, the control design problem of input delayed systems has attracted considerable attention. In Kolmanovskii and Myshkis (1992), Choi and Chung (1996), Kim, Jeung, and Park (1996) and Li, Huay, Huang, and Grant Fisber (1999), the memoryless controllers were proposed and the stability criteria were independent of the size of the time delay. Moreover, these stability criteria are expressed in the form of Riccati matrix equations. Although the memoryless controllers in Kim et al. (1996) and Choi and Chung (1996) are easy to implement, it was pointed out (Moon, Park, & Kwon, 2001) that they tend to be more conservative when the time delay is small. Based on the reduction method (Kwon & Pearson, 1980), Moon et al. (2001) proposed a robust controller for the uncertain input-delayed systems, which has a feedback of the current state and the past input history. It was shown by examples that the controller with delay compensation can have more robustness than the memoryless controllers. However, the shortcoming of the method is that the exact value of the time delay must be known.

In this paper, we investigate the controller design for the uncertain input-delayed systems. Different from Moon et al. (2001), it is not required to know the exact value of the time delay in this paper. Moreover, comparing with the memoryless controllers (Choi & Chung, 1996; Kim et al., 1996; Li et al., 1999), more information on the system is employed to implement the given controller, therefore, a less conservative result can be obtained for the system by using the controller given in this paper. The stability criterion is given in the form of LMIs, which can be effectively solved by various convex optimization algorithms (Boyd, Ghaoui, Feron, & Balakrishnan, 1994). For comparison, we give some examples.

Notation: Rⁿ denotes the n-dimensional Euclidean space, R^n×m is the set of n×m real matrices, I is the identity matrix, ||·|| stands for the induced matrix 2-norm. λ_min(.), λ_max(.) denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. The notation X>0 (respectively, X⩾0), for X∈R^n×n means that the matrix X is real symmetric positive definite (respectively, positive semi-definite). C₀ denotes the set of all continuous functions from [−H,0] to Rⁿ.

Section snippets

Preliminaries and main result

Consider the following uncertain system with unknown input delay: $x ̇ (t)=[A + Δ A (t)] x (t)+[B_{0} + Δ B_{0} (t)] u (t)+ [B_{1} + Δ B_{1} (t)] u (t −τ),$ $x (0)= x_{0}, u (s)=φ(s), s ∈[−τ,0],$ where x(t)∈Rⁿ and u(t)∈R^m denote the state vector and input vector, respectively. $A, B_{0}$ and B₁ are known parameter matrices of appropriate dimensions. ΔA(t), ΔB₀(t) and ΔB₁(t) are time-varying parameter uncertainties which satisfy the following conditions: $Δ A (t)= DF (t) E, Δ B_{i} (t)= D_{i} F_{i} (t) E_{i}, i=0,1,$ where $D, E, D_{i}$ , and E_i are real constant matrices with

Examples

Example 1

Consider the system with state uncertainty and input delay $x ̇ (t)=(A + Δ A) x (t)+ B_{1} u (t −τ)$ with $A = 01 −1.25 −3, Δ A = 00 v 0, | v |⩽γ,$ $B_{1} = 01 .$ When τ=0.2, the controller design and robustness to the uncertainty of system (27) were studied in Moon et al. (2001), and Cheres, Palmor, and Gutman (1990). It was found that the maximum allowable value of γ is 11.6895. For the case of γ=10, the controller gain designed in Moon et al. (2001) is $[−551.63 −137.19]$ . However, applying Theorem 1 with ρ=4, we found that under the

Conclusions

This paper addressed the delay-dependent robust controller design for uncertain input-delayed systems. To design the controller, it is not required to know the exact value of the time delay. Therefore, it is more realistic than the controllers in Moon et al. (2001) and Cheres et al. (1990). The design method of the controller is applicable to the cases where the time delay is known or unknown. The method of this paper can also be extended to the system with time-varying input delay, which will

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^☆: This paper was not presented in the IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Geir E. Dullerud under the direction of Editor Paul van den Hof. This work is supported by the National Natural Science Foundation of China (69874042) and the Foundation for University Key Teachers by the Ministry of Education and the Teaching and Research Award Program for Outstanding Young Teachers in Nanjing Normal University.

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Technical communiqueRobust stabilization of uncertain systems with unknown input delay☆

Abstract

Introduction

Section snippets

Preliminaries and main result

Examples

Conclusions

Automatica

Automatica

Automatica

Linear matrix inequalities in systems and control theory

Delay dependent robust H∞ control for uncertain systems with time varying delays

IEE Proceedings: Control Theory and Applications

Technical communique
Robust stabilization of uncertain systems with unknown input delay☆

Delay dependent robust H_∞ control for uncertain systems with time varying delays