Stability Analysis of Grey Discrete Time Time-Delay Systems: A Sufficient Condition

Abstract

Uncertainties in a control system may be the results modeling errors, measurement errors, parameter variations and a linearization approximation. Most physical dynamical systems and industrial process can be described as discrete time uncertain subsystems. Similarly, the unavoidable computation delay may cause a delay time, which can be considered as timedelay in the input part of the original systems. The stability of systems with parameter perturbations must be investigated. The problem of robust stability analysis of a nominally stable system subject to perturbations has attracted wide attention (Mori and Kokame, 1989). Stability analysis attempts to decide whether a system that is pushed slightly from a steadystate will return to that steady state. The robust stability of linear continuous time-delay system has been examined (Su and Hwang, 1992; Liu, 2001). The stability analysis of an interval system is very valuable for the robustness analysis of nominally stable system subject to model perturbations. Therefore, there has been considerable interest in the stability analysis of interval systems (Jiang, 1987; Chou and Chen, 1990; Chen, 1992). Time-delay is often encountered in various engineering systems, such as the turboject engine, microwave oscillator, nuclear reactor, rolling mill, chemical process, manual control, and long transmission lines in pneumatic and hydraulic systems. It is frequently a source of the generation of oscillation and a source of instability in many control systems. Hence, stability testing for time-delay has received considerable attention (Mori, et al., 1982; Su, et al., 1988; Hmamed, 1991). The time-delay system has been investigated (Mahmoud, et al., 2007; Hassan and Boukas, 2007). Grey system theory was initiated in the beginning of 1980s (Deng, 1982). Since then the research on theory development and applications is progressing. The state-of-the-art development of grey system theory and its application is addressed (Wevers, 2007). It aims to highlight and analysis the perspective both of grey system theory and of the grey system methods. Grey control problems for the discrete time are also discussed (Zhou and Deng, 1986; Liu and Shyr, 2005). A sufficient condition for the stability of grey discrete time systems with time-delay is proposed in this article. The proposed stability criteria are simple