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.

提出了几种改善高木-菅野模糊描述符系统 (TSFDS) 稳定条件的方法。首先，我们证明了通过改进的非二次模糊 Lyapunov 函数和 PDC 控制器，TSFDS 的稳定性问题被重新表述为检查“三重”模糊和的负性，然后 T-S 模糊系统的松弛方法可以直接应用于描述符系统。在后续的文章中，分别基于松弛决策变量和波利亚定理将两种宽松的方法扩展到 TSFDS，但这些条件仅是充分的。其次，我们设计了一个同时由导数矩阵和状态矩阵的隶属函数组成的非二次模糊 Lyapunov 函数，并推广了之前相关的 Lyapunov 函数。然后使用非 PDC 控制器，TSFDS的充分条件和渐近必要条件是基于波利亚定理提出的。所有条件都转换为 LMI，仿真示例说明了这些方法的改进和有效性。