Improvements on stability conditions and control design of Takagi-Sugeno fuzzy descriptor systems

Several methods to improve stabilization conditions of Takagi-Sugeno fuzzy descriptor systems (TSFDS) are presented. First, we prove that with a modified non-quadratic fuzzy Lyapunov function and a PDC controller, stabilization problems of TSFDS are reformulated as checking negativity of "triple" fuzzy summations, and then relaxed methods of T-S fuzzy systems can be directly applied to descriptor systems. In the sequel, two relaxed methods are extended to TSFDS based on slack decision variables and Polya's Theorem, respectively, but these conditions are only sufficient. Second, we design a non-quadratic fuzzy Lyapunov function which simultaneously consists of membership functions of derivative matrices and state matrices, and it generalizes previous related Lyapunov functions. Then with a non-PDC controller, not only sufficient but asymptotically necessary conditions for TSFDS are presented based on Polya's theorem. All conditions are cast into LMIs, and simulation examples illustrate improvements and effectiveness of these methods.