Static Output Feedback Fault Tolerant Control and Relaxed Stability Analysis of Positive Polynomial Fuzzy Systems

The increased demand for safety in control systems has motivated the research for fault-tolerant controllers (FTCs). This article presents the static output feedback FTC design and relaxed stability analysis for positive nonlinear systems on account of polynomial fuzzy models. A polynomial copositive Lyapunov function candidate is proposed for the stability analysis of positive polynomial fuzzy-model-based (PPFMB) control systems for the first time. This method develops the decision variable on the polynomial copositive Lyapunov function candidate to be a polynomial vector and also makes it independent of the form of the positive polynomial fuzzy model. Thus, it can be applied to a more extensive range of PPFMB control systems and potentially generates more relaxed stability analysis results. To achieve the stability analysis and control synthesis, the following challenges are overcome: 1) approximate the nonconvex terms in stability and positivity conditions into convex ones through the idea of matrix decomposition; 2) eliminate the nonconvex conditions resulting from the derivative of the polynomial Lyapunov function by employing an sum-of-squares (SOS) optimization method; 3) relax the results of stability analysis and control design by introducing the advanced membership-function-dependent technique to develop both the stability conditions and the positivity conditions. The relaxed conditions are obtained on the basis of SOS form so that the MATLAB third-party toolbox, SOSTOOLS, can be used to find feasible solutions. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed method.