最後，再將所得之李亞普諾夫函數經平方和檢測方法改寫為純量形式，以平方和檢測法去檢驗其系統之穩定性，藉此確保我們的閉迴路系統的穩定性與狀態回授估測器追蹤狀態的性能，在論文的第五章我們分別提供控制系統與狀態回授估測系統各2個數值分析的例子，來證明其有效性。;In this thesis, a polynomial nonlinear system, modelled by T-S fuzzy model with added disturbances, is studied. Based on non-quadratic, homogeneous Lyapunov function, both controller and observer are considered in the analysis where Euler′s homogeneous polynomial theorem is used to avoid the derivative term dot Q(x) that is seen in the existing papers.
After some background reviewed, we started with fuzzy system models established by Taylor series. To tackle the derivative Lyapunov dot Q(x) terms and the zero row structure in the input matrix B(x) in the existing papers, Euler homogeneous polynomial theory is applied to derive the stabilization condition in LMI formulation and then converted into SOS form so that SOSTOOLS is used to for synthesis analysis.
Finally, Sum of Square is applied to solve for the Lyapunov Q(x) and controller/observer gains, thereby ensuring the stability of the closed-loop feedback system as well as the observed-state feedback control system. Several examples are provided in Chapter 5 to demonstrate the analysis is effective.