Robust stabilization of uncertain input-delayed systems using reduction method

doi:10.1016/S0005-1098(00)00145-X

Automatica

Volume 37, Issue 2, February 2001, Pages 307-312

https://doi.org/10.1016/S0005-1098(00)00145-X Get rights and content

Abstract

This paper concerns a robust stabilization problem that arises when applying the so-called reduction method to multiple input-delayed systems with parametric uncertainties. A robust stabilizing controller that can be constructed easily by solving convex problems in terms of linear matrix inequalities is presented. Numerical examples are provided to show that in many cases the proposed controller is less conservative and easier to obtain than the existing controllers.

Introduction

Time delays in control inputs are often encountered in many industrial processes. The presence of input delays, if not considered in a controller design, may cause instability or serious deterioration in the performance of the resulting control systems (Marshall, 1979).

An easy way of dealing with input-delayed systems is to reduce them to delay-free ordinary systems by using a certain state transformation (Kwon & Pearson, 1980; Arstein, 1982). This control strategy, the so-called reduction method, has been shown to overcome some of the inherent problems of the conventional Smith predictor method (Watanabe & Ito, 1981; Furukawa & Shimemura, 1983). For example, unstable systems can be stabilized and the effects of the initial conditions are taken into consideration. The reduction method, however, suffers from a weakness that the complete reduction to a delay-free system is only possible with an exact model of the system. Hence, uncertainties existing in real systems may sometimes cause serious problems in using the reduction method.

The robust stabilization problem in applying the reduction method to uncertain input-delayed systems has not yet been studied in depth (Pandiscio & Pearson, 1993; Cheres, Palmor & Gutman, 1990). In Pandiscio Jr. and Pearson (1993), a robust stability condition is suggested using the structured singular value method for pure input-delayed systems. However, their result cannot be applied to multiple input-delayed systems that are affected by both delayed and current inputs, since the properties of these two delayed systems are intrinsically different from each other. For multiple input-delayed systems, a stabilizing robust controller was proposed by Cheres et al. (1990), based on the reduction scheme. However, the suggested controller is not satisfactory in that the procedure for the design of the controller is quite complex and the stabilizability condition is too strong.

Another approach to robust stabilization of uncertain input-delayed systems is to obtain a stabilizing controller directly rather than using the reduction method (Choi & Chung, 1995; Kim, Jeung & Park, 1996). In this approach, the so-called memoryless controllers, which have feedback of the current state only, are designed to guarantee the delay-independent stability of the closed-loop systems. Although these delay-independent, memoryless controllers have the merit that they are easy to implement, they tend to be unduly conservative when the actual size of the delay is small. In fact, information on the size of the delay is often available in many processes. Hence, by using this information and employing a feedback of the past control history as well as the current state, controllers using the reduction method may provide better performance than memoryless controllers.

In this paper, we present a new robust stabilizing controller for multiple input-delayed systems with parametric uncertainties using the reduction method. The proposed controller is formulated in terms of linear matrix inequalities (LMIs) that can be solved efficiently using recently developed convex optimization algorithms (Boyd, Ghaoui, Feron & Balakrishnan, 1994). Compared with the previous result by Cheres et al. (1990), the suggested controller is easier to obtain and the corresponding stabilizability condition is less conservative. In addition, the proposed controller can stabilize a larger class of systems than existing memoryless controllers (Choi & Chung, 1995; Kim et al., 1996) for the case where information on the size of the delay is available. These facts will be illustrated using numerical examples.

Section snippets

Overview of the reduction method

Let us consider an input-delayed system $x ̇ (t)=Ax(t)+B_{0} u(t)+B_{1} u(t−h)$ with initial conditions x(0)=x₀ and u(t)=φ(t) for $t∈[−h, 0].$

The reduction method for the input-delayed system (1) was suggested by Kwon and Pearson (1980). An extension to time-varying systems with distributed delays can be found in Arstein (1982). The outline of the method is as follows.

Consider a linear transformation from x(t) and $u(α), α∈[t−h,t]$ to z(t) $z(t)≜x(t)+ ∫ t−h t e^{A(t−α−h)} B_{1} u(α) d α.$ Then, system (1) is transformed to $z ̇$

Numerical examples

In this section, we present numerical examples comparing the controller proposed in this paper with existing results.

Example 1

Consider the following input-delayed system with a state of uncertainty (Cheres et al., 1990) $x ̇ (t)=(A+ Δ A)x(t)+B_{1} u(t−0.2),$ where $A= 01 −1.25 −3, Δ A= 00 v 0, |v|≤ρ_{v},$ $B_{1} = 01 .$ In Cheres et al. (1990), it was shown that their suggested controller can stabilize the system (22) when ρ_v is less than 1.7. The maximum value achievable from the memoryless controllers of Choi and Chung (1995) and Kim et

Conclusions

This paper presented a robust stabilizing controller for uncertain multiple input-delayed systems using the reduction method. Compared with the existing results on robust control using the reduction method, it was shown that the controller proposed in this paper is less conservative and easier to construct. For the case where information on the size of the delay is available, it was also shown that the proposed controller can stabilize a larger class of uncertain systems than existing

Acknowledgements

The second author's work was supported by Korea Research Foundation under the Grant No. 1998-001-E01231.

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^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Middleton under the direction of Editor Paul Van den Hof.

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Technical CommuniqueRobust stabilization of uncertain input-delayed systems using reduction method☆

Abstract

Introduction

Section snippets

Overview of the reduction method

Numerical examples

Conclusions

Acknowledgements

Automatica

Automatica

Linear systems with delayed control: A reduction

IEEE Transactions on Automatic Control

Min-max predictor control for uncertain systems with input delays

IEEE Transactions on Automatic Control

Predictive control for systems with time delay

International Journal of Control

Technical Communique
Robust stabilization of uncertain input-delayed systems using reduction method☆