Control Design Of Stochastic Fuzzy Systems Based On Parameterized Linear Matrix Inequality Techniques

During the last decades, there have been many achievements in theresearch of fuzzy control. The well-known Takagi-Sugeno (T-S) fuzzymodel is extensively used to handle complex nonlinear systems. Tanakaand his co-workers have proposed a framework of control design of fuzzycontroller based on parallel distributed compensation (PDC) and linearmatrix inequality (LMI). Because this framework is flexible to deal withglobal stability and design multi-variable system controller by means ofLyapunov function, it draws many interests from wide fields. Recently,Tanaka's approach is extended to stochastic fuzzy systems. However, thestability conditions of T-S fuzzy systems based on quadratic Lyapunovfunction are quite conservative since they usually depend on the existenceof a common positive matrix. In order to release the conservation, thispaper applies parameterized linear matrix inequality (PLMI) techniquesto Lyapunov stability analysis of stochastic fuzzy systems. We obtain aclass of PLMI & LMI conditions for the stochastically asymptotically stable of stochastic T-S fuzzy systems. In addition, we investigate theproblems of state feedback H_∞control and robust H_∞control for stochasticfuzzy systems.Applying PLMI techniques and Lyapunov stability theorems ofnonlinear stochastic systems, we introduce a class of PLMI conditions forasymptotically stable of stochastic T-S fuzzy control systems. Moreover,these PLMI characterizations are reduced to some pure LMI programs,which could be solved by Matlab LMI Toolbox. Both theoretical analysisand numerical examples show that the proposed system control designmethod in this paper gives more feasible solutions than the existingresults in the literature do.Furthermore, we investigate the problem of state feedback H_∞control of T-S fuzzy control systems which suffer both externaldisturbances and state-dependent noise. By using of PLMI techniques andLyapunov stability theorems of nonlinear stochastic systems, we derive aclass of conditions for the existence of state feedback H_∞controller andoptimal H_∞controller in the terms of PLMI as well as those of LMI. Asimulation example illustrates the application of the proposed methods.In addition, we attack the problem of robust H_∞control of uncertainstochastic T-S fuzzy control systems, whose parameters involve bothdeterministic uncertainties and stochastic uncertainties. By using of PLMItechniques and Lyapunov stability theorems of nonlinear stochastic systems, we obtain a class of PLMI sufficient conditions for robuststochastic stability and existence of robust H_∞controller and optimal H_∞controller. Then these PLMI characterizations are reduced to pure LMIprograms. Finally, a numerical example is given to demonstrate theeffectiveness of the methods.