Research On Control Approach Of Fuzzy Discrete Systems With Time Delays

Discrete time-delay systems are universal in many physical processes. Time delays usually invite a mass of complicated phenomena of closed-loop systems. Therefore, the academic research on delayed systems and their industrial applications have attracted considerable attention. On the other hand, as an omnipotent approximator of kinds of nonlinear systems, Takagi-Sugeno (T-S) fuzzy model is a hotspot in the fuzzy control field. Based on the common quadratic Lyapunov function (CQLF) theory, it becomes a popular and reliable method to deal with stability analysis and control law design for the T-S fuzzy discrete systems with time delays employing linear matrix inequalities (LMI) technique. However, this method has somewhat conservativeness since it may not be able to solve the LMIs for some special fuzzy systems and the difficulty of the calculation will be increased due to the large number of premise variables and rules of fuzzy systems. In order to find more relaxed method, several multiple Lyapunov function approaches have been proposed, such as the piecewise quadratic Lyapunov function (PQLF) approach and parameter-dependent Lyapunov function (PDLF) approach. Nevertheless, these approaches also have some shortcomings.Motivated by the aforementioned concerns, this paper investigates the fuzzy control for a class of T-S discrete systems with time delays based on the Lyapunov function approach. First, a novel nonquadratic Lyapunov function is proposed in delay-independent cases to overcome the difficulty of solving series of quadratic matrix inequalities (QMI) when the traditional PDLF is adopted. The stability analysis and the design way of H∞control law are derived in the form of LMI via a nonparallel distributed compensation (non-PDC) scheme. The new conclusion under a special situation is also suitable for a PDC law. Furthermore, this kind of new Lyapunov function approach is also utilized in delay-dependent cases and compared with the traditional PDLF method both in theoretical analysis and numerical examples. Finally, by introducing Hale transformation, a new stability criteria for open-loop discrete systems with time delays is put forward by the traditional PDLF approach combined with a matrix transformation which could be be viewed as an extension to the Schur complement. And a novel H∞control law is obtained via cone complementary linearization (CCL) iterative algorithm. The simulation results illustrate that such novel sufficient conditions are less conservative than previous results obtained within the quadratic framework but also have good control performances.