Reliable and Robust H∞ Control for a Class of Uncertain Descriptor Systems

This paper discusses the reliable and robust H∞ control problem for a class of descriptor systems with parametric uncertainties and external disturbances in the case of actuator failures. By solving a set of matrix inequalities, a sufficient condition and a controller are obtained for the reliable and robust H∞ control problem of the descriptor systems. Introduction Descriptor systems are generalized and natural representations for many practical control plants such as robots, electric circuits, complex chemical engineering process, economical systems and so on. Compared with the standard state-space models, the form of descriptor system model can indicate the equality constraint conditions about dynamic expressions. In recent years, control problems of descriptor system have attracted much attention [1-4]. Robust control mainly focus on controller design to deal with uncertainties which are widely existed in practical control systems, for example, model parametric uncertainty and external disturbance. As an important approach of robust control, ∞ H control for uncertain systems has been intensively investigated in the last two decades [1, 2, 4, 5]. A control system designed to tolerate failures of sensors or actuators, while maintaining an acceptable level of the closed-loop system stability and performance, is called a reliable control system. For safety-critical systems, such as air vehicles, chemical plants and nuclear power plants, safety and reliability are even more important than good performance. So the reliability should be an important consideration when designing the controller. Recently, reliable control problem has also received considerable attention and some research results have been obtained for the reliable control in the case of sensor and actuator failures [6-10]. This paper deals with the theoretical aspects of the ∞ H control problem for a class of descriptor systems with parametric uncertainties and external disturbances in the case of actuator failures. Based on a set of matrix inequalities, a sufficient condition is obtained to solve the reliable and robust ∞ H control problem. By solving the matrix inequalities, the controller can be obtained. Proceedings of The 2008 IAJC-IJME International Conference ISBN 978-1-60643-379-9 Problem Statement Consider a class of descriptor systems with parametric uncertainties and external disturbances described by Gw Bu t x A A t x E + + ∆ + = ) ( ] [ ) ( & (1a) Cx y = (1b) where x X R ∈ ⊆ , R t∈ , m R u∈ , l y R ∈ and w denote the descriptor state, time, control input, measurement output and external disturbance, respectively. A, B, C and G are smooth matrix-functions, n n R A × ∈ , m n R B × ∈ , C l × ∈ n R . m, n and l are positive integers. A ∆ is the parametric uncertainty, and satisfies the assumption A1: A1: F D A ∆ = ∆ (2) p n R D × ∈ , q p R × ∈ ∆ , n q R F × ∈ , E and F are the known matrix-functions, ∆ is known matrix-function, and satisfies : I T ≤ ∆ ∆ (3) q q R I × ∈ is an identity matrix. The uncertain descriptor system under consideration is depicted in Figure 1. Figure 1 Descriptor system block diagram In the following, the definition of the reliable and robust ∞ H control problem to be discussed is addressed. Definition 1 [5]: System (1) is said to be tolerant and have ∞ H performance γ , if for given positive number γ , the system (1) is canonical, stable and pulseless, and the 2 L -gain from the disturbance w to the control output z is finite, i.e., ∫ ∫ ∞ ∞ ≤ 0 2 2 2 0 2 2 w dt z γ P (A+∆A)