臺灣博碩士論文加值系統

摘 要
本文提出一個改良型多變數適應性模糊小腦模式(CMAC)控制系統，以解決一類非線性系統的追蹤控制問題。 首先，整合模糊邏輯和小腦模式演算，建構一個可降低輸入維度，簡化系統結構的多變數模糊小腦模式單元(FCMAC)，用於逼近具不確定性的非線性多變數系統模式，以產生理想的多變數控制輸入。 其次，針對一類單變數(SISO)的非線性系統，應用上述的模糊小腦模式單元，設計適當的適應律及控制律並結合具滑動面特性的輸出回授，發展成一單變數(MISO)適應性模糊小腦模式控制系統，以調適非線性系統的不確定性，進行線上參數自動調整，免於耗時的先備性離線學習。 此外，為了降低模糊小腦模式單元的逼近誤差，以增加系統的控制精度及確保閉回路系統的穩定，遂引入一傳統的切換式強健補償器，初期完成一閉回路漸進穩定之適應性模糊小腦模式控制系統。 後來為了改善因不連續的切換補償作用所衍生控制信號之顫動現象，進而提出一平滑式強健補償器來替代原有切換補償器，完成一改良型單變數(MISO)適應性模糊小腦模式控制系統。

最後，拓展上述所有單變數(MISO)的理論和應用，完成一改良型多變數(MIMO)適應性模糊小腦模式控制系統，作為本文之主要結果之一。 藉由完備的李雅普諾夫穩定度分析，證明所有閉回路信號是有界的，且追蹤誤差至少可指數收斂至一殘局，其大小可藉由調整參數任意控制。雖然追蹤精度略為降低，但控制信號的品質卻可大大提升。 經由多個應用問題的模擬結果，驗證了本文所提方法的正確性及可行性。

ABSTRACT

In this thesis, a modified multivariable adaptive fuzzy cerebellar model articulation controller (CMAC) control scheme is proposed to solve the tracking problem for a class of nonlinear systems. Firstly, a fuzzy CMAC (FCMAC) that merges fuzzy logic and CMAC algorithm such that the input space dimension and the complicated structure in CMAC can be simplified. The FCMAC module is used to approximate a nonlinear multivariable (multi-input multi-output (MIMO)) system involving uncertainty to create the desired ideal control inputs. Next, suitable control and adaptive laws with output feedback based on sliding surface concept are incorporated with FCMAC into a multi-input single-output (MISO) adaptive FCMAC (AFCMAC) control system, to tune all of the control gains on-line, thereby accommodating the uncertainty of nonlinear systems without prior off-line learning phase. Particularly, to reduce the approximation error, improve the tracking accuracy, and guarantee the closed-loop stability, the conventional switching robust compensation is adopted. Furthermore, to overcome the chattering problem associated with discontinuity derived from switching action, a smooth compensation is then proposed, completing the modified MISO AFCMAC control scheme.

Eventually, the theories and applications concerning the modified MISO AFCMAC control scheme is further to extend successfully to the modified MIMO AFCMAC control scheme as the main results of this work. By integrated Lyapunov stability analysis, it is guaranteed that all of the closed-loop signals are bounded, and the tracking errors converge exponentially to a residual set, whose size can be adjusted by changing the design parameters. On the whole, although the tracking precision is reduced slightly, the control signal’s quality can be improved greatly. Finally, simulation results for its applications to several examples are presented to demonstrate the validity and applicability of the methodologies proposed in this thesis.

TABLE OF CONTENTS

ACKNOWLEDGMENT (Chinese)..................................................................................................................................... I
ABSTRACT (Chinese).............................................................................................................................................................. II
ABSTRACT (English)............................................................................................................................................................... III
TABLE OF CONTENTS............................................................................................................................................................. IV
LIST OF FIGURES........................................................................................................................................................................ VII
LIST OF TABLES......................................................................................................................................................................... X
LIST OF NOTATIONS.............................................................................................................................................................. XIV

CHAPTER 1
INTRODUCTION…….....…………………................................................................................................................................... 1
1.1 Motivation and Related Researches………………………………….…………………….…..……….….. 1
1.2 Thesis Contributions ……………………………………….………………………………………….……..…… 4
1.3 Thesis Organization ……………………………………….…………………………………………….………… 5

CHAPTER 2
FUZZY CMAC DESIGN………………………………………………………………………..………………………….. 9
2.1 Basic CMAC Design…………………………………………………………………………………………..…… 10
2.2 Fuzzy CMAC Design……………………………………………………………………………………….…....… 14
2.3 Concluding Remarks…………………………………………………………………………………....………..… 16

CHAPTER 3
ADAPTIVE FUZZY CMAC CONTROLLER DESIGN………………………………………………………… 18
3.1 Problem Description……………………………………………………………………………………………… 19
3.2 AFCMAC Design…………………………………………………………………………………………..……… 20
3.3 AFCMAC with Switching Compensation (AFCMAC)……………………….………………… 24
3.4 AFCMAC with Smooth Compensation (Modified AFCMAC)……………………………... 27
3.5 Applications of AFCMAC………………………………………………………………………………..…… 33
3.5.1 Example 1: Inverted Pendulum with Friction………………………………………………… 33
3.5.2 Example 2: One-Link Rigid Robotic Manipulator………………………………………… 44
3.5.3 Example 3: Duffing forced-oscillation system………………………………………….…… 47
3.5.4 Example 4: Third-order Nonlinear System………………………………………………….… 49
3.6 Concluding Remarks………………………………………………………………………………………..…… 53

CHAPTER 4
MULTIVARIABLE ADAPTIVE FUZZY CMAC CONTROLLER DESIGN……………………..…..… 54
4.1 Problem Description…………………………………………………………………………………………....… 55
4.2 Multivariable AFCMAC Design……………………………………………………………………….…… 57
4.3 Multivariable AFCMAC with Switching Compensation (MIMO AFCMAC)…….… 62
4.4 Multivariable AFCMAC with Smooth Compensation (Modified MIMO AFCMAC)……………………………………………………………………………………………………………… 65
4.5 Applications of the Modified MIMO AFCMAC…………………………………………………… 72
4.5.1 Example 1: Multivariable Affined Squared Nonlinear Systems…………….……… 72
4.5.2 Example 2: Tracking Control for a Tri-wheeled Mobile Robot……………..…….… 84
4.6 Concluding Remarks…………………………………………………………………………………….…..…… 94
CHAPTER 5
CONCLUSIONS AND FUTURE WORKS……………………………………………………..……………………… 95
5.1 Concluding Remarks………………………………………………………………………………………......… 95
5.2 Suggestions for Future Research…………………………………………………………………………… 97
REFERENCES……………………………………………………………………………………..………………..…….…… 99

LIST OF FIGURES

Fig. 1.1 Skeleton diagram of the modified MIMO AFCMAC control scheme.………. 3

Fig. 2.1 Schematic diagram of fuzzy sets integrated with CMAC.…….....………………... 11
Fig. 2.2 Sketch of CMAC integrated with fuzzy sets. ………………………………………….... 14

Fig. 3.1 Architecture of the AFCMAC control schemes.…….....……..………………………. 25
Fig. 3.2 Diagram of the inverted pendulum.…….....…………………………………………….…… 33
Fig. 3.3(a) Tracking response of Case 1 in Example 1.…….....………………….………………… 36
Fig. 3.3(b) Tracking error of Case 1 in Example 1.…….....……………….…………….…………… 37
Fig. 3.3(c) Control input of Case 1 in Example 1.…….....……………………………..……..……… 37
Fig. 3.3(d) Tracking error of Case 2 in Example 1.……………………….……………......………… 38
Fig. 3.3(e) Control input of Case 2 in Example 1.…………………………………………..………… 38
Fig. 3.3(f) Updated CMAC weights of Case 2 in Example 1.…….....……………………..…… 39
Fig. 3.3(g) Updated approximation error bound of Case 2 in Example 1.……......………. 39
Fig. 3.3(h) Tracking error of Case 3 in Example 1.…….....………………………………………..… 40
Fig. 3.4(a) Tracking error of Case 4 in Example 1.…….......................................................………… 41
Fig. 3.4(b) Control input (u) of Case 4 in Example 1.…….....………………………………….…… 41
Fig. 3.4(c) Control input (uAFCMAC) of Case 4 in Example 1.……................................…….…… 42
Fig. 3.4(d) Control input (uAR) of Case 4 in Example 1.……..........................................….….…… 42
Fig. 3.4(e) Tracking error of Case 4 in Example 1.…….....………………….……………….….…… 43
Fig. 3.4(f) Control input (u) of Case 4 in Example 1.…….....………………………….……….…… 43
Fig. 3.5(a) Tracking response of Example 2.……......................................................................….……… 45
Fig. 3.5(b) Tracking error of Example 2.…….....……………………………………………………..….… 46
Fig. 3.5(c) Control input of Example 2.…….....………………………………………………………..…… 46
Fig. 3.6(a) Tracking response of Example 3.……..................................................................…….……… 48
Fig. 3.6(b) Tracking error of Example 3.…….....…………………………………………….…………….. 48
Fig. 3.6(c) Control input of Example 3.……...................................................................................………… 49
Fig. 3.7(a) Tracking response of Example 4.…….....………………………………………………..…… 51
Fig. 3.7(b) Tracking error of Example 4.…….....………………………………………………………...… 51
Fig. 3.7(c) Control input of Example 4.…….......................................................................................……… 52
Fig. 3.7(d) Tracking error of Example.............................................................................................................. 52

Fig. 4.1 Architecture of the MIMO AFCMAC control schemes............................................. 63
Fig. 4.2 Illustration of residual set region (for n=2).......................................................................... 72
Fig. 4.3(a) Tracking responses of Case 1 in Example 1....................................................................... 76
Fig. 4.3(b) Tracking errors of Case 1 in Example 1................................................................................ 76
Fig. 4.3(c) Control inputs of Case 1 in Example 1................................................................................... 77
Fig. 4.3(d) Updated approximation error bounds of Case 1 in Example 1............................ 77
Fig. 4.3(e) Update CMAC weights ( ) of Case 1 in Example 1.............................................. 78
Fig. 4.3(f) Update CMAC weights ( ) of Case 1 in Example 1.............................................. 78
Fig. 4.3(g) Update CMAC weights ( ) of Case 1 in Example 1............................................. 79
Fig. 4.4(a) Tracking responses of Case 2 in Example 1...................................................................... 79
Fig. 4.4(b) Tracking errors of Case 2 in Example 1............................................................................... 80
Fig. 4.4(c) Control inputs of Case 2 in Example 1.................................................................................. 80
Fig. 4.4(d) Updated approximation error bound of Case 2 in Example 1.............................. 81
Fig. 4.5(a) Tracking responses of Case 3 in Example 1...................................................................... 81
Fig. 4.5(b) Tracking errors of Case 3 in Example 1............................................................................... 82
Fig. 4.5(c) Control inputs of Case 3 in Example 1................................................................................... 82
Fig. 4.5(d) Updated approximation error bounds of Case 3 in Example 1............................ 83
Fig. 4.5(e) Convergence on Case 3 in Example 1..................................................................................... 83
Fig. 4.6 A three-wheeled mobile robot........................................................................................................ 84
Fig. 4.7 The configuration of the mobile robot..................................................................................... 84
Fig. 4.8 Block diagram of trajectory tracking control design.................................................... 86
Fig. 4.9(a) Tracking responses of Case 1 in Example 2...................................................................... 88
Fig. 4.9(b) Tracking trajectory of Case 1 in Example 2...................................................................... 89
Fig. 4.9(c) Tracking errors of Case 1 in Example 2................................................................................ 89
Fig. 4.9(d) Control inputs (ui) of Case 1 in Example 2........................................................................ 90
Fig. 4.9(e) Control inputs ( ) of Case 1 in Example 2........................................................................ 90
Fig. 4.9(f) Updated approximation error bounds of Case 1 in Example 2............................. 91
Fig. 4.9(g) Control inputs (ui) of Case 2 in Example 2........................................................................ 91
Fig. 4.9(h) Tracking errors of Case 2 in Example 2............................................................................... 92
Fig. 4.9(i) Control inputs of Case 2 in Example 2................................................................................... 92
Fig. 4.9(j) Updated approximation error bound of Case 3 in Example 2............................... 93


LIST OF TABLES

Table 2.1 Relationship between fuzzy rules and CMAC.................................................................. 66

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