Robust H∞ filter design for discrete-time switched systems with interval time-varying delay and linear fractional perturbations: LMI optimization approach
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 filtering problem about discrete-time switched systems with interval time-varying delay and linear fractional perturbations. The stable switched 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 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 , we denote the transpose by , symmetric positive (negative) definite by (), dimension by . means that matrix is symmetric positive semi-definite. denotes the identity matrix. For a vector , we denote the Euclidean norm by . Define .
Section snippets
Problem statement and preliminaries
Consider the following uncertain discrete-time switched time-delay system:where , is the state defined by , , is the measured ouput, is the signal to be estimated, and is the disturbance input which belongs to , is an initial state function, time-varying delay
Illustrative example
Consider system (1) with the following parameters:,
Conclusion
The robust switched filtering for the discrete-time switched systems with interval time-varying delay and linear fractional perturbations has been considered in this paper. Some nonnegative inequalities and the free weighting matrix technique have provided the extra degree of freedom to improve the conservativeness of the proposed results. In addition, a switched filter can be selected by a convex optimization problem with a set of LMI constraints for filtering error system. A numerical
Acknowledgment
The research reported here was supported by the National Science Council of Taiwan, ROC under Grant no. NSC 101-2221-E-507-002.
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