T-S 模糊系統之最佳化時間控制

Abstract

本論文針對Takagi-Sugeno (T-S) 模糊模型，設計一最佳化時間控制器，T-S模糊模型可視為一多面體線性微分包圍(polytopic linear differential inclusion) 數學模型，利用李群論（Lie Algebra）的幾何特性推導最佳化時間之奇異性、存在性及切換次數。 本論文提出最佳化時間之控制器，首先針對控制器之存在性，推導T-S模糊模型之可控制性，在T-S模糊模型相關論文中，此為首次探討T-S模糊模型之控制性，並提出歸納式秩數(rank)條件式。透過最大值理論(maximum principle)，最佳化控制器為砰-砰(bang-bang)型式，在系統之奇異性之推導中，證明所提出之歸納式秩數(rank)條件式，可供設計非奇異之控制器使用;亦即，可控制之T-S模糊模型可設計最佳時間控制器，引用可解析李群論(solvable Lie algebra)於控制器之切換次數證明，此證明可提供於計算最佳演算解，讓演算法更容易找到最佳解。透過模擬本控制器設計法則均經在聯結車前進及倒車系統及多輸入系統之可控制性及最佳化時間控制器。

This dissertation investigates geometric property of time-optimal problem in Takagi-Sugeno (T-S) fuzzy model via Lie algebra. We will focus on the existence of time-optimal solution, singularity of switching function and number of switching. These inherent problems are considered because of their rich geometric properties. The necessary condition for the existence of time-optimal solution reveals the controllability of T-S fuzzy model which can be found by the generalized rank condition. The time-optimal controller can be found as the bang-bang type by applying maximum principle. In the study of singularity problem, we will focus on switching function whatever vanished on a finite time interval. The bounded number of switching can be found if the T-S model (also a nonlinear system) is solvable. This feature can be applied to solve the time-optimal problem by numerical approach. Fast response is always a considered property in this dissertation. A notion directly relate to the convergence rate of the state trajectories. A controller design of T-S fuzzy model on maximal convergence rate is introduced by the level set function. The result of maximizing the convergence rate is characterized from the maximal invariant ellipsoid. The controller is also bang-bang within both the initial states and target states belong to level set.