具LMI數量減少效益之Takagi-Sugeno模糊大型系統控制器合成

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張偉(Wei Chang) 查詢紙本館藏

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論文名稱

具LMI數量減少效益之Takagi-Sugeno模糊大型系統控制器合成
(Controller Synthesis of Large-Scale Takagi-Sugeno Fuzzy Systems with Reduced Numbers of LMI)

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摘要(中)

我們可以發現許多物理系統是由許多較小的子系統經由彼此間的網路連結所組成，在此，我們統稱它為大型系統(large-scale systems)。針對此系統，可發現非常多模糊控制文獻探討其內部網路連結與其控制法則間的關係；由這些文獻研究結果來看，針對大型系統之模糊控制器設計時，所面臨之最重要且最困難的問題，大多來自如何適切處理其內部錯綜複雜的網路連結，因為在此系統之中，每個子系統都會與系統內其他子系統有所連結，所以整個大型系統將包含為數眾多的子系統連結線路所產生的動態需要處理。就本論文所採用的Takagi-Sugeno (T-S) 模糊控制方法論來說，常使用來處理此網路連結的方法有二。首先，第一種方法是找一組線性方程式來替換代表原網路動態的非線性方程式，此組線性方程式所模擬的動態，可被看成其所代表非線性方程式動態的上界，因此藉由其所設計的控制器，將具備穩定原非線性方程式中，所有低於此上界的系統動態。此方法不需要對網路連結做線性化，因此有效避免線性化網路連結時所產生的問題(如以下第二種方法中，將提到的規則爆炸問題)，但其所需付出的代價是必須對原網路動態具備一定程度的了解，才能給出一組合適的線性方程式來表示其上界，因此，由於資訊上的不足，一般都需要在控制器求出並合成進受控系統後，在模擬程序中確認所給出的線性方程式，是否真能代表其上界，若確定結果為否，則須重新制定一組線性方程式並重新求解控制器。此外，有文獻提出了第二種處理內部網路動態的方法，此方法運用T-S 模糊控制方法論中最常被使用的線性化工具(兩種方法，分別為sector nonlinearity方法與local approximation in fuzzy partition spaces方法)，來線性化這些可能由一組非線性方程式所描述的網路連結。在此方法架構之下，使用者可將原本複雜的非線性方程式，轉換成一組由模糊函數決定其加權數(我們稱此數量為模糊數fuzzy rule number)的數組線性方程式，且一旦成功轉換後，即可運用一系列成熟發展的工具來協助使用者設計控制器，也不須在模擬程序中確認任何假設前提。但此方法運用在大型系統控制器設計上，將面臨一個大問題，當組成此大型系統的子系統數量增加時，其內部的網路連結數量將大量的上升，此時使用者將需要非常大量的模糊數才能模擬原系統的動態，這就是我們在智慧型系統中常看到的規則爆炸(rule-explosion)問題，並造成控制器求解上的困難。

因此，本論文提出了數種新式控制器設計程序，來解決以上所提的兩種問題。首先，就以上以線性方程式來替換代表原網路動態的非線性方程式方法之缺點，我們提出一組具備強健性(robustness)與H-infinity 概念的控制器方法，在此我們依舊需要由使用者給定一組邊界條件，並於模擬程序中做確認動作，但我們所採用的控制程序，將減少使用第一種方法時所產生控制器求解上的保守性。接著，我們發展了一個新的控制器設計程序，在此程序中，使用者運用上一段落中所提的第二種方法，即使用sector nonlinearity方法與local approximation in fuzzy partition spaces方法將網路連結直接線性化，但在我們的控制器設計程序中，可有效降低規則爆炸問題的發生，因為我們對網路連結線性化所產生的模糊數，做了特別的處理(藉由一特殊推導與S-procedure的運用)，因此不會因為受控系統子系統數量上升，而大量增加後續控制器設計所需求解的穩定條件之數量;此方法的優點是，我們保留了此領域最常被使用的模糊化方法，但卻沒有過去因受控系統組成子系統數量上升，而須面臨的規則爆炸問題。

總結來說，本論文最大的貢獻是我們所提出的控制器方法，特別適用於受控系統由大量子系統所組成的情況，因為以設計者的觀點來看，若是基於T-S模糊系統與線性矩陣不等式(Linear Matrix Inequality: LMI)方法來對一大型系統做控制，如何使其方法能在計算機上處理一真正“大型”的系統，將是此方法日後能否確實投入實際運用的關鍵，因此，我們希望本論文之研究結果，能協助模糊控制社群在此研究方向上更進一步，並殷切期盼看到其成果早日投入實際運用之中。

摘要(英)

There are many physical systems which consist of multiple subsystems and are linked via a network of interconnections. These systems are called large-scale systems and they, and especially their inherent control design problems, have been considered by many fuzzy control papers. From the results of these researches, we know that the most important and difficult part of designing a fuzzy controller for large-scale systems is handling the nonlinear interconnections. Since each subsystem has several interconnections with the other subsystems, an entire nonlinear large-scale system contains a lot of interconnections. In general, if a nonlinear large-scale system is transformed into a Takagi-Sugeno (T-S) fuzzy system, one of two methods have been used to cope with the nonlinear interconnections before designing the fuzzy controller to stabilize the system. The first method is to set some specific bounded conditions in which the interconnections must satisfy. As a result, the nonlinear interconnections can be transformed into one set of linear functions. Then, the closed loop system can be solved by the Linear Matrix Inequality (LMI) method. Unfortunately, if the dynamics of the interconnections are not known in advance, then the interconnections’ bounded conditions must be checked in the simulation process. The second method is to linearize those nonlinear interconnections using the most popular ‘sector nonlinearity’ method or ‘local approximation in fuzzy partition spaces’ method. However, if the system consists of a large number of subsystems and each interconnection is transformed into a set of fuzzy rules, then the ‘rule-explosion’ problem may happen.

This dissertation proposed several novel methods to solve the above two problems. Firstly, regarding to the problem of bounding constraint of interconnections, we introduce the robustness and H-infinity concept into the control design in order to decrease the conservatism. With the aids of robustness and H-infinity controller design methods, the conservatism caused by the norm inequality may be overcome and the corresponding controllers are obtained easily. Secondly, we develop one novel control design process. In this process, the ‘sector nonlinearity’ or ‘local approximation in fuzzy partition spaces’ methods are used to transform the original system into a T-S fuzzy system, and a Parallel Distributed Compensation (PDC) type fuzzy controller is designed. However, it should be emphasized that, in this process, the ‘rule-explosion’ problem is avoided because of a special derivation which eliminates the fuzzy rules generated by the interconnections, after which the S-procedure is used to obtain the stabilization conditions.

In conclusion, the proposed fuzzy control design processes in this dissertation are especially useful when the number of subsystems in the large-scale system is large. This is the most important contribution of this dissertation. Finally, we provide several numerical simulations in this dissertation in order to show the applications of the present fuzzy controller design approach.

關鍵字(中)

★ 大型系統
★ Takagi-Sugeno 模糊系統
★ H-infinity 控制
★ S程序
★ 線性矩陣不等式

關鍵字(英)

★ Large-Scale Systems
★ Takagi-Sugeno Fuzzy Systems
★ H-infinity Control
★ S-procedure
★ Linear Matrix Inequality

論文目次

I.System Structure and Problems Formulation

1. Introduction 1
1.1 Motivation of This Dissertation 1
1.2 Outline of This Dissertation 6
1.3 Contributions 9

2. Modeling of Takagi-Sugeno Fuzzy Systems
and Problems Formulation 11
2.1 Modeling of Takagi-Sugeno Fuzzy Systems 12
2.2 Parallel Distributed Compensation 19
2.3 Problems Formulation 20

II. Fuzzy Controller Design for Large-Scale System

3. Mathematic Tools Review for the Control
Designing in Large-Scale Systems 25
3.1 Useful Mathematic Tools 26
3.1.1 Linear Matrix Inequality 26
3.1.2 S-Procedure 30
3.2 Summary 45

4. Fuzzy Control Design: by dividing interconnections
into communicating and non-communicating division 47

4.1 System Description and Problem Formulation 48
4.2 Preliminaries and Notations 50
4.3 Decentralized Fuzzy Control Design 54
4.4 A Numerical Example 62
4.5 Summary 68

5. Fuzzy control Design: by transforming nonlinear interconnections into one weighted interconnection with bounded conditions 71
5.1 System Description and Problem Formulation 72
5.2 Results and Controller Synthesis 73
5.3 Examples 90
5.3.1 Balancing Double-Inverted Pendulums 90
5.3.2 Stabilization regions based on relaxed
stabilization conditions 98
5.3.3 Comparison of computation loads
with one related paper 99
5.4 Summary 101

6. Fuzzy Control Design: by a special derivation
and the S-procedure method 103
6.1 System Description and Problem Formulation 104
6.2 Results and Controller Synthesis 104
6.3 Reduction of the number of LMIs 112
6.4 Robust control with modeling error 117
6.5 Examples and simulations 121
6.6 Summary 129

III. Simulations and Conclusions

7. Simulations: Comparison of Different type of Fuzzy Controller Design Methods for the Number-Reduction
Effect of LMIs 131
7.1 Comparison of computation loads
with one related paper 132
7.2 Summary 136

8. Conclusions and Future Works 137
8.1 Conclusions 137
8.2 Future Works 138

Appendix 141
Bibliography 145
Publications 153

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指導教授

王文俊(Wen-June Wang)

審核日期

2014-7-21

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