The Robustness Analysis Of The Discrete Time Linear Systems With Polytopic Uncertainty

Polytopic uncertain systems are an important area of study in robust control domain. It is advanced research subject of robust control theory that parameter dependent Lyapunov stability idea is used to analyze and synthesize the polytopic uncertain systems. The problem of robustness analysis of discrete time linear systems with polytopic uncertainty is discussed by using linear matrix inequality (LMI) and parameter dependent Lyapunov function stability method in this paper. Based on previous research, the following three aspects are investigated.First, the problem of robust stability of linear discrete time systems with polytopic uncertainty is investigated. By constructing parameter-dependent Lyapunov function (PDLF) and employing linear matrix inequalities (LMIs) technique, the parameter dependent linear matrix inequality is expanded, and a sufficient condition of robust stability for systems is proposed. The presented criterion has lower conservativeness.Second, the problem of robust H_∞performance analysis of discrete time linear systems with polytopic uncertainty is studied. By improving the degree of parameter dependent linear matrix inequality with dependence on the uncertain parameters and constructing PDLF, the LMI representations of robust H_∞performance criteria are given. For the number of matrix vertices N = 2 of the uncertainty domain, we compare and analyze the source of conservativity of the given methods.Finally, the problem of Gl₂ performance analysis of discrete time linear systems with polytopic uncertainty is investigated. Based on the simplified Gl₂ analysis theorem, the more general LMI conditions of Gl₂ performance analysis are given by improving the degree of parameter dependent linear matrix inequality with dependence on the uncertain parameters and constructing PDLF. In addition, the LMI representations for the different degrees are presented. Comparison analysis shows that the conservativeness of the presented conditions decreases with the increase of the degrees. Furthermore, the Gl₂ performance analysis conditions with less conservativeness are given by introducing extra matrix variables to parameter dependent linear matrix inequality.The results in this paper are expressed as LMIs, therefore Matlab can be used to obtain the solutions conveniently. Numerical examples are given to illustrate the feasibility and effectiveness of the presented methods.