多項式模糊系統穩定性分析

Stability Analysis of Polynomial Fuzzy Systems

何冠賢(Guan-Sian He)

指導教授 ： 余國瑞

國立中正大學/工學院/光機電整合工程研究所/碩士(2012年)

摘要

這篇論文利用多項式模糊系統的概念，來建立四輪全向型行動機器人(ODMR)的多項式模糊模型，再根據平行分部補償(PDC)來設計行動機器人的多項式模糊控制器，此控制器使整個閉迴路行動機器人能追蹤參考軌跡，以達到控制追蹤的目的，為了確保使用的控制器穩定性，利用多項式尼亞布諾夫函數(polynomial Laypunov function)所推導的平方和(sum of square)不等式條件來驗證系統的全域穩定．
利用平方和(SOS)方法，推導出非線性系統輸出和輸入限制設計條件，藉由平方和條件得到之控制器增益能夠滿足閉迴路系統最大輸入或輸出限制，相較於傳統線性矩陣不等式(LMI)的輸入和輸出限制條件中，本論文所提出的平方和限制條件包含了傳統LMI的不等式條件，以至於平方和方法將更一般性(general)以及能求解的空間(feasible)更大．
　　尤其這篇論文的焦點是在於提出了利用多項式尼亞布諾夫函數推導出當非線性系統具時間延遲的穩定和穩定化條件，而且傳統LMI所利用二次尼亞布諾夫函數為多項式尼亞布諾夫函數的一個特例，再將多項式模糊系統所使用的SOS的不等式條件，可使用最近發展在MATLAB的SOSTOOLS來做數值解，為了陳述本論文設計方法的有效性，將會提出例子來與傳統方法做模擬比較驗證．

關鍵字

多項式糢糊系統 ； 輸出輸入限制 ； 平方和 ； 多項式時間延遲系統 ； 多項式糢糊控制器 ； 多項式尼亞布諾夫函數 ； 穩定性 ； 穩定化 ； 時間延遲

並列摘要

This study presents a polynomial fuzzy model and a path controller design for a nonlinear four-wheeled omnidirectional mobile robot (ODMR) using polynomial fuzzy systems. A polynomial controller was designed according to the parallel distributed compensation (PDC) from the given polynomial fuzzy model of the ODMR. This proposed controller is capable of driving the closed-loop system states of the ODMR to follow reference trajectory commands. We used stability conditions that were represented by the sum of squares (SOS) to guarantee global stability.
　In addition, we derived the limitation conditions represented in term of SOS for control input and output using a polynomial Lyapunov function. The stable polynomial controller satisfied the constraint on the control input and output. These proposed SOS-based constraint conditions are more general and relaxed than are current linear matrix inequality (LMI)-based constraint conditions.
　This study focuses on developing methods for stability analysis and stabilization based on the SOS approach and that depend on the size of the time-delay. A polynomial Lyapunov function was applied to derive the stability and stabilization time-delay conditions of the nonlinear time-delay systems, and contained quadratic Lyapunov functions as a special case. Finally, computer simulations showed that the SOS-based approaches were more effective than were the LMI-based approaches.

並列關鍵字

Polynomial fuzzy system ； Constraints of input and output ； Sum of square ； Polynomial fuzzy control system of time delay ； Polynomial fuzzy controller ； Polynomial Lyapunov function ； Stability ； Stabilization ； Time delay

參考文獻

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