Robust H_∞ filter design for discrete-time switched systems with interval time-varying delay and linear fractional perturbations: LMI optimization approach

Abstract

In this paper, the robust H_∞ filter scheme for the discrete-time switched systems with interval time-varying delay and linear fractional perturbations is proposed. A convex optimization problem with some LMI constraints is formulated to design the stable switched filter which minimizes the H_∞ norm bound for filtering error system. Some nonnegative inequalities and the free weighting matrix technique are used to provide the additional degree of freedom which improves the conservativeness of the proposed results. Finally, a numerical example is illustrated to show the use of the main results.

Introduction

Switched systems are often encountered in many practical systems including automated highway systems, automotive engine control systems, chemical process, constrained robotics, power systems and power electronics, robot manufacture, and stepper motors. A switched system consists of several subsystems and a discrete switching signal. Hence a switched system is a hybrid system. Switching among subsystems may produce many complicate nonlinear system behaviors, such as chaos and multiple limit cycles [1], [2]. It is also well known that the existence of time delay in a system may cause instability or bad performance in closed loop control systems. Time-delay phenomena are usually confronted in many practical systems, such as AIDS epidemic, chemical engineering systems, hydraulic systems, inferred grinding model, neural network, nuclear reactor, population dynamic model, and rolling mill [3], [4], [5]. There are two interesting facts in switched system: the first one is that stability of a switched system can be achieved by choosing the switching signal even when each subsystem is unstable [6], [7], [8], [9]. The another one is that the stable property for each subsystem cannot imply that the overall switched system is also stable under arbitrary switching signal [10], [11]. Hence stability and stabilization problems for switched systems with time delay have been investigated in recent years [1], [6], [7], [8], [9], [10], [11]. Filtering problem is playing an important role in control applications and signal processing over past years, because it has been applied in practical engineering such as biomedical systems, manufacturing, process control, mechatronics [12]. The celebrated Kalman filter assumes that the system model and statistical information of noise are precisely known. However, these assumptions are actually questionable and usually not practical. Hence H_∞ filtering technology was introduced in [13] and has been provided in diverse systems, such as fuzzy systems [14], [15], networked systems [12], [16], [17], nonlinear stochastic systems [18], singular systems [19], continuous delay systems [20], and discrete-time systems [21], [22], [23], [24], [25], [26]. The H_∞ filtering scheme is developed to design a signal estimator for a given system, such that L₂ gain of filtering error is less than a prescribed level [23]. In [16] and [20], a Markovian approach was developed to guarantee the H_∞ filtering for time-delay systems. In [23], a delay partition approach was used to guarantee the reliable H_∞ filtering for discrete time-delay systems with randomly occurred nonlinearities. In practical point of view, signal propagation cannot always be guaranteed in fixed speed. Hence interval time-varying delay will be more suitable than constant time delay. Furthermore linear fractional perturbations are the generalized perturbed forms for general parameter perturbations in [6], [7], [14], [27], [28]. To the best of authors’ knowledge, there are no research articles considering the $H_{\infty }$ filtering problem about discrete-time switched systems with interval time-varying delay and linear fractional perturbations. The stable switched $H_{\infty }$ filter and the minimization of disturbation attenuation for the filtering error system will be constructed and achieved, respectively. In this paper, some nonegative inequality and the free weighting matrix technique are provided to design the switched $H_{\infty }$ filter and improve the conservativeness for the discrete-time switched systems. A numerical example with simulation is illustrated to demonstrate the use of obtained results in this paper.The notation used throughout this paper is as follows. For a matrix $A$, we denote the transpose by $A^T$, symmetric positive (negative) definite by $A>0$ ($A<0$), $n\times m$ dimension by $A_{(n\times m)}$. $A⩽B$ means that matrix $B-A$ is symmetric positive semi-definite. $I$ denotes the identity matrix. For a vector $x$, we denote the Euclidean norm by $\Vert x\Vert$. Define $L_2(0\text{,}\infty )=\left\{w(k):{\sum }_{k=0}^{\infty }{w}^T(k)w(k)<\infty \right\}$.