利用SDRE策略探討非線性系統之全域穩定度

Abstract

本論文主要探討二階非線性系統以及雙二階非線性系統在SDRE控制方法下可達全域漸進穩定度的議題。首先，在使用多重李亞普諾夫函數（multiple Lyapunov functions ，MLFs）技術下，本論文所考慮的二階系統可藉由選取合適的狀態相關係數（state-dependent coefficient，SDC）矩陣使系統達到全域漸進穩定。接著，經由適當的選取SDC矩陣，本論文探討的雙二階系統所對應SDRE的解析解可以很明確的表示出來。且在使用特定的李雅普諾夫函數以及拉塞爾不變集理論（LaSalle's invariant set theorem）下，可以求得該雙二階系統全域穩定的條件。最後，透過例子說明本論文的分析結果。

This thesis addresses the globally asymptotic stability issues for a class of second-order nonlinear systems and a class of two second-order nonlinear systems using the state-dependent Riccati equation (SDRE) approach. First, with the help of multiple Lyapunov functions (MLFs) technique, it will be shown that the considered class of second-order system is always globally asymptotically stable (GAS) by suitable choosing state-dependent coefficient (SDC) matrices. Next, by appropriate selecting the SDC matrices, we explicitly write down the analytic solution of the associated SDRE for the considered two second-order nonlinear systems and then explore the global stability condition using a specific Lyapunov function and LaSalle's invariant set theorem. Finally, the analytic results are illustrated by several examples.