This paper presents a descriptor form design methodology for polynomial fuzzy-model-based (FMB) control systems. To begin with, the closed-loop polynomial fuzzy system is cast into the descriptor form representation. Applying the commonly used Lyapunov function of the polynomial fuzzy model, the first sum-of-squares (SOS) based stabilization control design approach is proposed. Through the redundancy of descriptor form representation, polynomial fuzzy slack matrices are brought into the stabilization analysis for relaxation. Applying fuzzy slack matrices for the stabilization analysis causes the double fuzzy summation problem which can be seen as the copositivity problem. Therefore, the copositivity relaxation is applied in the stabilization control design. In addition, for the special cases that all membership functions are functions of the states being not related to the inputs, the second SOS-based stabilization control design approach is proposed by applying a novel fuzzy Lyapunov function. As the novel fuzzy Lyapunov function contains the commonly used Lyapunov function as a special case, the second design approach is guaranteed to be less conservative than the first one. Since the novel fuzzy Lyapunov function is applied, the time derivatives of membership functions are necessary to be considered in the stabilization analysis. Hence, the sector nonlinearity technique is applied to deal with the rest part of the membership function time derivative after extracting the polynomial common factors. Finally, three examples are given to shown the effectiveness and relaxation of the proposed descriptor form control design methodology.

本文提出了一种基于多项式模糊模型 (FMB) 控制系统的描述符形式设计方法。首先，将闭环多项式模糊系统转换为描述符形式表示。应用多项式模糊模型的常用李雅普诺夫函数，提出了基于第一平方和（SOS）的稳定控制设计方法。通过描述符形式表示的冗余，将多项式模糊松弛矩阵引入到松弛的镇定分析中。将模糊松弛矩阵应用于稳定性分析会导致双重模糊求和问题，可以将其视为协同问题。因此，在稳定控制设计中应用了共性松弛。此外，对于所有隶属函数都是与输入无关的状态函数的特殊情况，提出了第二种基于 SOS 的稳定控制设计方法，即应用新的模糊 Lyapunov 函数。由于新颖的模糊李雅普诺夫函数包含常用的李雅普诺夫函数作为特例，因此第二种设计方法保证比第一种方法保守。由于应用了新的模糊李雅普诺夫函数，因此在稳定性分析中必须考虑隶属函数的时间导数。因此，在提取多项式公因子后，应用扇区非线性技术处理隶属函数时间导数的其余部分。最后，