非線性控制系統的穩定度判別是基於李亞普諾夫穩定準則(Lyapunov stability criterion)，保證Takagi-Sugeno模糊控制系統穩定的充分條件，就是試圖找到一個共同正定矩陣P (common P)讓所有子系統皆滿足李亞普諾夫不等式(Lyapunov inequality)，而控制器的設計方法多採用平行分配補償器(PDC)的概念，此方法稱之為共同二次李亞普諾夫函數(common quadratic Lyapunov function - CQLF)分析法，此共同正定矩陣P可透過MATLAB的線性矩陣不等式(LMI)控制工具箱的解法所求得，然而當模糊系統的規則數過多時，可能就無法找到這個共同正定矩陣P來符合所有子系統。 為了放寬此穩定度條件的保守性，近年來許多學者提出了許多有別於尋找單一共同正定矩陣P的新方法來定義李亞普諾夫函數 (Lyapunov function)，常見的有模糊二次李亞普諾夫函數(fuzzy quadratic Lyapunov function - FQLF)分析法、模糊線積分李亞普諾夫函數(fuzzy line-integral Lyapunov function - FLILF)分析法和切換式二次李亞普諾夫函數(switching quadratic Lyapunov function - SQLF)分析法來推導Takagi-Sugeno模糊控制系統的充份穩定度條件，理論上此三種穩定度條件會比傳統的條件寬鬆許多，因為傳統的穩定準則只是此三種方法的特例。 由上述三種方法，我們可以藉由修改歸屬函數微分之絕對值上界的條件來放寬模糊二次李亞普諾夫分析法，另外藉由重建切換式Takagi-Sugeno模糊模型來放寬切換式二次李亞普諾夫函數分析法，最後則是結合模糊線積分李亞普諾夫函數分析法和切換式二次李亞普諾夫函數分析法兩大概念而推導出切換式模糊線積分李亞普諾夫函數分析法(switching fuzzy line-integral Lyapunov function - SFLILF)，並以幾個例子來證明我們所提出的穩定度放寬條件的可行性。 The stability condition of nonlinear control system is based on the Lyapunov stability criterion. That tried to find a single positive-definite matrix P (common P) to satisfy all Lyapunov inequalities. Then the sufficient stability condition of Takagi-Sugeno fuzzy control system (T-S fuzzy control system) can be guaranteed. Furthermore, the controller design is using the Parallel Distributed Compensation (PDC) concept. This analysis method is so-call common quadratic Lyapunov function (CQLF) method. We use the linear matrix inequality (LMI) Control Toolbox of MATLAB to seek for a common P. However, if the number of rules of a fuzzy system is large, the common P may not be found. In order to relax the conservative of stability conditions, recent years many researchers have proposed several approaches different from a single common P. There have three common methods which redefine the new Lyapunov function are fuzzy quadratic Lyapunov function (FQLF) method, fuzzy line-integral Lyapunov function (FLILF) method and switching quadratic Lyapunov function (SQLF). Theoretically, these three methods are more relaxed than traditional method; because of the traditional analysis method is just a special case of these three methods. By the above three methods, we can revise the condition that are time-derivatives of membership functions’ absolute values to relax the FQLF method. Besides, reconstruct the switching T-S fuzzy model to relax the SQLF method. Finally is unifies the FLILF method and SQLF method concept to derive the switching fuzzy line-integral Lyapunov function (SFLILF) method. The effectiveness of the proposed approach is shown through numerical examples.
