模糊邏輯控制系統之穩定度分析與應用

國

立

交

通

大

學

電控工程研究所

博

士

論

文

模 糊 邏 輯 控 制 系 統 之 穩 定 度 分 析 與 應 用

The Stability Analysis and its Application in Fuzzy Control Systems

研 究 生：馬立山

指導教授：吳炳飛 教授

模 糊 邏 輯 控 制 系 統 之 穩 定 度 分 析 與 應 用

The Stability Analysis and its Application in Fuzzy Control Systems

研 究 生：馬立山 Student：Li-Shan Ma

指導教授：吳炳飛 Advisor：Prof. Bing-Fei Wu

國 立 交 通 大 學

電控工程研究所

博 士 論 文

A Dissertation

Submitted to Institute of Electrical and Control Engineering College of Electrical Engineering

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Electrical and Control Engineering

Dec. 2009

Hsinchu, Taiwan, Republic of China

模 糊 邏 輯 控 制 系 統 之 穩 定 度 分 析 與 應 用

研究生：馬立山 指導教授：吳炳飛教授

國立交通大學電控工程研究所博士班

中文摘要

在本篇論文中，我們分析了 P 與 PD 型之模糊邏輯控制系統之絕對穩定度，另外也提出了一 種基於模糊邏輯控制系統之應用，即只利用傳輸一狀態之數值信號，並利用適應性模糊類神經 觀測器 (AFNO)去同步一類的未知混沌系統。關於穩定度分析，包括兩種狀況：確定與非確定性 受控體。而穩定度分析包括以下參數：參考輸入、致動增益、區間(Interval )受控體參數。對確 定性受控體而言，我們利用 Popov 或線性化的方法，針對 P 與 PD 型之模糊邏輯控制系統，在不 同參考輸入信號與致動增益下，作絕對穩定度分析，另外，關於模糊邏輯控制系統在參數空間 之穩態誤差也可被分析。針對非確定性受控體，我們利用基於 Lur’e 系統之參數化強健 Popov 準 則，來作 P 型模糊邏輯控制系統之絕對穩定度分析，而關於非確定性受控體之 PD 型分析，在我 們方法中，PD 型之模糊邏輯控制器，為一種單一輸入之 PD 型模糊邏輯控制器，而且此控制器 可被轉成一種特殊 P 型模糊邏輯控制器，而再作進一步分析。與之前研究不同的是，我們利用 參數化強健 Popov 準則，可針對非零之參考輸入，且非確定性之受控體，作絕對穩定度分析。 我們亦利用 PSPICE 元件，設計了一個模糊電流控制 RC 電路，透過數值與 PSPICE 模擬驗證我 們所作分析之結果。另外，在模擬例子中，我們也利用不同平衡點的觀念，解釋模糊邏輯控制 系統之震盪機制。最後，我們也比較幾種非確定性系統之絕對穩定度準則，驗證我們的分析的 有效性。另一方面，模糊邏輯控制系統也可以被設計用來智慧化同步混沌信號，其應用主要觀

念為只藉傳輸一狀態之數值信號，並利用 AFNO 去同步一類的未知混沌系統，如果此一非線性 混沌系統可以藉由微分幾何的方法，被轉換成標準的 Lur’e 系統，則此方法便可以被應用來作同 步。值得一提的是，在這一個方法中，AFNO 之適應性模糊類神經(FNN)可以被線上即時調整權 重，去對傳送端之非線性項作建模。另外，藉由傳送端傳送一個狀態並利用接收端之觀測器可 以對傳送端未知之所有狀態作重建，當所有狀態被觀測到，傳送端與接收端便達到同步。AFNO 可以線上適應性估測傳送端之狀態，即使傳送端已經切換到另一個混沌系統，接收端之 AFNO 還可以與新的混沌系統達到同步。另外一方面，即使存在建模誤差或外加有界干擾，AFNO 亦 可強健的達到同步。模擬結果驗證 ANFO 對混沌系統之同步應用是有效的。

The Stability Analysis and its Application in Fuzzy Control Systems

Student：Li-Shan Ma Advisor：Prof. Bing-Fei Wu

Institute of Electrical and Control Engineering

National Chiao Tung University

ABSTRACT

This thesis analyzes the absolute stability in P and PD type fuzzy logic control systems with both certain and uncertain linear plants. In addition, the adaptive fuzzy-neural observer (AFNO) is applies to synchronize a class of unknown chaotic systems via scalar transmitting signal only. Stability analysis includes the reference input, actuator gain and interval plant parameters. For certain linear plants, the stability (i.e. the stable equilibriums of error) in P and PD types is analyzed with the Popov or linearization methods under various reference inputs and actuator gains. The steady state errors of fuzzy control systems are also addressed in the parameter plane. The parametric robust Popov criterion for parametric absolute stability based on Lur’e systems is also applied to the stability analysis of P type fuzzy control systems with uncertain plants. The PD type fuzzy logic controller in our approach is a single-input fuzzy logic controller and is transformed into the P type for analysis. In our work, the absolute stability analysis of fuzzy control systems is given with respect to a non-zero reference input and an uncertain linear plant with the parametric robust Popov criterion unlike previous works. Moreover, a fuzzy current controlled RC circuit is designed with PSPICE models. Both numerical and PSPICE simulations are provided to verify the analytical results. Furthermore, the oscillation mechanism in fuzzy control systems is specified with various equilibrium points of view in the simulation example. Eventually, the comparisons are also given to show the effectiveness of the analysis method. On the other hand, the fuzzy control system can be applied to synchronize the chaotic signals in the master end intelligently. With a scalar transmitting signal only, the AFNO is utilized to synchronize a class of unknown chaotic systems. The proposed method can be used for synchronization if nonlinear chaotic systems can be transformed into the canonical form of Lur’e system type by the differential geometric method. In this approach, the adaptive fuzzy-neural network (FNN) in AFNO is adopted on line to model the nonlinear term in the master end. Additionally, the

master’s unknown states can be reconstructed from one transmitted state using observer design in the slave end. Synchronization is achieved when all states are observed. The utilized scheme can adaptively estimate the transmitter states on line, even if the transmitter is changed into another chaotic system. On the other hand, the robustness of AFNO can be guaranteed with respect to the modeling error, and external bounded disturbance. Simulation results confirm that the AFNO design is valid for the application of chaos synchronization.

致 謝

修習博士學位這一路走來，有太多我需要感謝的人!尤其是我的論文指導教授 吳炳飛老師。 感謝老師在學術研究上給予學生的啟發、鼓勵與陪伴，在此學生要由衷的向老師表示謝意和敬 意。 感謝口試委員 鄧清政教授、張志永教授、涂世雄教授、鄭泗東教授，彭昭暐教授在百忙之中， 願意撥冗參與口試，並給予論文寶貴的建議。 在研究過程中，也要特別感謝學長暐哥一路的陪伴。特別是在研究上的討論與建議，常常可 以給我新的啟發與觀念的成長。 感謝CSSP 實驗室的同學與學弟妹多年來提供的協助，尤其您們的勤奮和努力，常是我學習的 指標，尤其是重甫、世孟、欣翰、全財等。 學生也要感謝碩士班的指導教授 涂世雄老師，奠定學生作研究的方法與態度。另外老師為人 處世的言教與身教，也深深地影響著學生。 同時，也要感謝建國科技大學同仁們的支持與鼓勵，儘管是一句關心與問候，也常能激勵我， 在繁重的學校工作外，還能努力堅持於相關研究工作。 在求學過程中，我也要感謝洪宣天神父，透過與神父定期的靈修談話，讓自己不斷調整與天 主、與人的關係。在談話的過程中，神父也不斷從福音的角度鼓勵我前進，在生活中不斷發現 天主的召叫。 感謝父親與母親，從小到大對我與弟妹的栽培，使我們能得到良好的教育，希望未來能對兩 位老人家回報一二。我也要感謝岳父母，對小孩的悉心照顧，分擔我們夫妻倆對小孩的生活照

顧壓力。 另外，家中兩位寶貝兒子慕恩及瀚恩，雖然經常在我趕研究進度時候，意猶未盡地霸佔電腦， 令人幾近抓狂，但您們卻也是鼓勵我奮鬥的泉源。您們的調皮，其實透露了無限的創造力，想 到此時，真的會會心一笑。 多年來，太太 憶如在我就讀研究所博士班期間，協助照料子女，尤其在留職停薪那一年，協 助分擔家中許多事務，免除我後顧之憂，使我能一心向學，更是讓我無以回報。 因篇幅有限，還有很多曾經教導我的師長、幫助我的同仁、鼓勵我朋友，無法一一致意，謹 在此表達由衷的感謝，謝謝您們。 博士求學過程中真的是一個漫長且艱辛的路程，有時看不到希望，甚至必須在絕望中持續奮 鬥，不知何時光明會出現，尤其投稿論文被拒絕，畢業遙遙無期的時候。雖然未來前途也充滿 了挑戰，希望透過如此一個真實的經驗，讓我體察到，即使未來處在困境中，仍要持續奮鬥， 因為奮鬥才有希望。 最後將論文獻給所有關心、支持及協助我的人 立山 於交大CSSP 實驗室 11/6/2009

Contents

摘要...iii

ABSTRACT ...v

誌謝 ...vii

Contents...ix

List of Figures...xiii

List of Tables...xvii

Chapter 1 Introduction...1

1.1 Motivation...1

1.2 Organizations of the Dissertation...9

Chapter 2 The Fuzzy Logic Control Systems...10

2.1 Fuzzy Logic Controller...10

2.2 P Type Fuzzy Logic Control System...11

2.3 PD Type Fuzzy Logic Control System...12

2.3.1 Calculation of Signed Distance...13

2.3.2 The Presentation of the SFLC System...13

2.3.3 The Analytic Representation of the SFLC System ...14

Chapter 3 Equilibrium Points and Stability Analysis in P and PD Type Fuzzy

Control Systems...20

3.1 Equilibrium Point Analysis for P Type Fuzzy Control Systems with Linear

Plants...20

3.2 Stability Analysis for P Type Fuzzy Control Systems with a Certain Linear

Plant...21

3.2.1 Frequency Domain Approach...21

3.3 Stability Analysis for P Type Fuzzy Control Systems with an Uncertain Linear

Plant...22

3.4 Transformation SFLC from PD to P Type...26

3.5 Equilibrium Point Analysis for PD Type Fuzzy Control Systems with Linear

Plants...27

3.6 Stability Analysis for PD Type Fuzzy Control Systems with Linear

Plants...28

3.6.1 Frequency Domain Approach...28

3.6.2 Time Domain Approach...28

3.7 Stability Analysis for PD Type Fuzzy Control Systems with Uncertain Linear

Plants...29

Chapter 4 Fuzzy Current Control RC Circuit System Design...31

4.1 The Block Diagram of the Fuzzy Current Control RC Circuit System...31

4.2 Circuit Plant...32

4.3 Fuzzy Logic Controller Circuit...32

4.4 The Overall Design Circuit...32

4.4.1 Voltage Controlled Current Circuit...33

4.4.2 Current Amplifier...33

4.4.3 PD type Signal Generation...33

Chapter 5 Simulation Results...38

5.1 P Type Example Demonstrations...39

5.1.1 Certain Linear Circuit Plant...39

5.1.2

Mechanism of Oscillations in the Fuzzy Control system...40

5.1.3

Alternative Control Function...41

5.2 PD Type Example Demonstrations …...42

5.2.1 Certain Linear Circuit Plant...42

5.2.2

Alternative Control Function. ...42

5.2.3

Uncertain Linear Circuit Pl

ant

...43

Chapter 6 Comparisons with Other Approaches...56

6.1 Robust Lur’e Test...56

6.2 Robust Circle Criterion...57

6.3 Robust Popov Criterion ...58

6.4 Parametric Robust Popov Criterion...58

6.5 A Brief Summary on Comparisons ...59

Chapter 7 Application: Observer-Based Synchronization for a Class of Unknown

Chaotic Systems with Adaptive Fuzzy-Neural Network...69

7.1 Overview ...69

7.2 Overall Structure of Adaptive Synchronization with Fuzzy-Neural Observer

Design...70

7.2.1 Introduction of Overall Structure...70

7.2.2 Dynamics of the Master and Slave Ends...70

7.3 Adaptive Fuzzy-Neural Network Observer Design...71

7.3.1 Fuzzy-Neural Network...72

7.3.2 Adaptive Fuzzy-Neural Network Observer...73

7.4 Simulation Results ...75

7.4.1

Example 1...75

7.4.2 Example 2...77

7.5 Conclusion Remarks...79

Chapter 8 Conclusions...91

Reference ...93

VITA...103

Publication List...104

List of Figures

Fig. 2.1 The P type fuzzy control system...15

Fig. 2.2 The membership functions of the fuzzy logic controller...15

Fig. 2.3 The control function of the fuzzy logic controller...16

Fig 2.4 The single-input fuzzy logic control system...17

Fig. 2.5 The skew-symmetric property in

(e,e)

and the calculation of signed distance..18

Fig. 2.6 The control function of the fuzzy logic controller in SFLC...18

Fig. 2.7 The transition formation in the transformation...19

Fig.3.1 The transformed SFLC with the special P type fuzzy control system

formation...30

Fig. 4.1 The block diagram of a fuzzy current control RC circuit system...35

Fig. 4.2 The RC circuit plant...35

Fig. 4.3 The designed fuzzy current control RC circuit system

.

...36

Fig. 4.4 The control function of a fuzzy controller with circuit design parameters...37

Fig. 5.1 The membership functions of the fuzzy control system...45

Fig. 5.2 The fuzzy control function with PSPICE simulation by Table 5.2 parameters..45

Fig. 5.3 The equilibrium stability of the P type fuzzy control system by Table 5.1 for

, where o indicates a stable equilibrium, and

( ,r K) ×

denotes an unstable

equilibrium...47

Fig. 5.4 (a) The phase plane of

( , )e e

when

( ,r K)=(0.2, 5)

; (b) The time waveform

when

( ,r K)=(0.2, 5)

; (c) PSPICE waveform when

( ,r K)=(0.2, 5)

...48

Fig. 5.5 (a) The phase plane of

( , )e e

when

( ,r K)=(0.2, 4)

; (b) The time waveform

when

( ,r K)=(0.2, 4)

; (c) PSPICE waveform when

( ,r K)=(0.2, 4)

...50

Fig. 5.6 The equilibrium with the stability of the alternative fuzzy controller by Table

5.3 for

( ,

, where o denotes a stable equilibrium, and

×

indicates an

unstable equilibrium...51

)

r K

Fig. 5.7 The Popov plots for the P type fuzzy control system with uncertain circuit

plant...51

Fig. 5.8 The equilibriums with the stability of the PD type fuzzy control system by

Table 5.1 for

, where o indicates a stable equilibrium, and denotes an

unstable equilibrium...52

( ,r K) ×

Fig. 5.9 (a) The time waveform when

( ,r K)=(0.2,10)

(b) PSPICE waveform when

...53

( ,r K)=(0.2,10)

Fig. 5.10 (a)The time waveform when

( ,r K)=(0.2, 9)

; (b) PSPICE waveform when

...54

( ,r K)=(0.2, 9)

Fig. 5.11 Equilibrium with the stability of the PD type fuzzy control systems in Table

5.3 for

( ,

, where o denotes a stable equilibrium, and

×

indicates an

unstable equilibrium...54

)

r K

Fig. 5.12 The Popov plots for the PD type fuzzy control systems with the uncertain

circuit plant...55

Fig. 6.1 The robust Lur’e test...60

Fig. 6.2 The sector bound from the robust Lur’e test and the control surface of the fuzzy

logic controller...60

Fig. 6.3 The time waveform of the stable test case respect to the robust Lur’e test...61

Fig. 6.4 The time waveform of the unstable test case respect to the robust Lur’e test....62

Fig. 6.5 Robust circle criterion...62

Fig. 6.6 The sector bound from the robust circle criterion and control surface of fuzzy

logic controller...63

Fig. 6.7 The time waveform of the stable test case respect to the robust circle

Fig. 6.8 The time waveform of the unstable test case respect to the robust circle

criterion...64

Fig. 6.9 Robust Popov criterion...65

Fig. 6.10 The sector bound from the robust Popov criterion and control surface of fuzzy

logic controller...65

Fig. 6.11 Parametric robust Popov criterion for the reference inputs...66

Fig. 6.12 The time waveform of the stable test case respect to the parametric robust

Popov criterion with a bounded pulse reference...66

Fig. 6.13 The time waveform of the stable test case respect to the parametric robust

Popov criterion with the reference input

r=990

...67

Fig. 6.14 The time waveform of the stable test case respect to the parametric robust

Popov criterion with the reference input

r= −990

...67

Fig. 7.1 The overall structure of synchronization with AFNO...81

Fig. 7.2 The fuzzy-neural approximator...81

Fig. 7.3 The first states

xM1

and

xˆS1

in Chua’s circuit and AFNO under different initial

conditions...83

Fig. 7.4 The second states

xM2

and

xˆS2

in Chua’s circuit and AFNO under different

initial conditions...83

Fig. 7.5 The third states

xM3

and

xˆS3

in Chua’s circuit and AFNO under different

initial conditions...84

Fig. 7.6 The first states

xM1

and

xˆS1

in Chua’s circuit and AFNO under different

disturbances...84

Fig. 7.7 The second states

xM2

and

xˆS2

in Chua’s circuit and AFNO under different

disturbances...85

Fig. 7.8 The third states

xM3

and

xˆS3

in Chua’s circuit and AFNO under different

Fig. 7.9 The structure of synchronization with the switched masters...86

Fig. 7.10 The first states in Chua’s circuit, Rössler system and AFNO under different

initial conditions and switched masters: (a) actual figure size (b) enlarged

figure size of local region...87

Fig. 7.11 The second states in Chua’s circuit, Rössler system and AFNO under different

initial conditions and switched masters...88

Fig. 7.12 The third states in Chua’s circuit, Rössler system and AFNO under different

initial conditions and switched masters...88

Fig. 7.13 The first states in Chua’s circuit, Rössler system and AFNO under different

disturbances and switched masters...89

Fig. 7.14 The second states in Chua’s circuit, Rössler system and AFNO under different

disturbances and switched masters...89

Fig. 7.15 The third states in Chua’s circuit, Rössler system and AFNO under different

List of Tables

Table 2.1 Rules of the fuzzy logic controller...16

Table 2.2 Parameters of the fuzzy logic controller...16

Table 2.3 Rules of conventional FLC with control error defined as

ed

...17

Table 2.4 Rules of SFLC...17

Table 5.1 Parameters of the fuzzy logic controller in simulations...44

Table 5.2 Parameters of the fuzzy current control RC circuit system...46

Table 5.3 Alternative parameters of the fuzzy logic controller...50

Table 6.1 Parameters of fuzzy logic controller for the robust Lur’e test...61

Table 6.2 Parameters of fuzzy Logic controller for the robust circle criterion ...63

Table 6.3 The validity of the different robust stability tests...68

Table 7.1 Three cases of the initial conditions...82

Chapter 1

Introduction

1.1 Motivation

Fuzzy logic controller (FLC) has become a conventionally adopted control algorithm, and has been employed in various industrial applications [1], since Mamdani [2] proposed the first linguistic FLC based on expert experience to control a laboratory steam engine. The FLC design does not require an accurate mathematical model. Unlike traditional nonlinear controllers, FLC can work with imprecise inputs, and can deal with nonlinearity and uncertainty. Therefore, many studies are devoted to this field. Conversely, since the accurate mathematical model is not required to design FLC, the design procedure is still based on trial and error. Hence, the stability and performance of FLC cannot be guaranteed. Systematic analysis and synthesis schemes [3]-[26] have recently been developed to improve this issue.

Some methods [3]-[10] adopt the Takagi-Sugeno (T-S) fuzzy models to determine the stability of fuzzy control systems by the Lyapunov function or linear matrix inequality (LMI). The overall plant is first represented as a T-S fuzzy model by a fuzzy blending of each linear system model. The controller is then designed based on this T-S fuzzy model by Lyapunov function or LMI. However, an appropriate fuzzy model may be difficult to formulate for an arbitrary nonlinear dynamic system. Additionally, a common Lyapunov function for general cases, and an existing positive-definite matrix, are both difficult to obtain. Besides the T-S fuzzy model, Lyapunov functions are also adopted to design and analyze the robust PD fuzzy

Popov-Lyapunov approach [11]. In addition, the stability on the T-S fuzzy model is analyzed by the Kharitonov theorem incorporated with the Schur and Hurwitz criterions [12]. Recently, the developments of fuzzy logic control designs almost focus on the T-S fuzzy models control. The stability analyses all apply the time-domain LMI approach. The main research directions include model uncertainties [13]-[20] and time-delay [21]-[23] or both [24], [25]. The stability issues due to the reference input influence are not to be discussed in the T-S fuzzy models control.

Kickert and Mamdani [26] first applied the describing function approach (DF) to analyze the stability of fuzzy control systems by granting fuzzy control systems as a multi-level relay model. The describing function of FLC can, under reasonable assumptions, be obtained to predict the existence of a limit cycle in fuzzy logic control systems [27], [28]. DF provides an approximate approach to obtain the stability of unforced fuzzy control systems. DF may yield inaccurate or incorrect analysis results, because it is an aggressive and approximate approach. In other words, under some assumptions, DF can only be applied to analyze fuzzy system stability successfully. Additionally, the steady state error and transient response of fuzzy control systems with the sinusoidal and exponential input describing functions techniques are analyzed in [29] and [30], respectively.

The choice of parameters in fuzzy control systems with phase plane approach was proposed in [31]-[33]. Then, the phase plane analysis can be utilized to design fuzzy rules, or measure the performance and stability of a specific set of fuzzy rules. Phase plane analysis is a simple graphical approach, in which the system trajectories are inspected to provide information on system stability and performance. However, it is restricted to second order dynamic systems.

The extension of classical circle criteria is also applied to analyze the stability of linear systems with fuzzy logic controllers [34], [35]. The extended circle criteria can be employed to test the SISO and MIMO systems [34]. The extended circle criteria for MISO and MIMO

are presented in [35] for testing the robust stability in PI, such as fuzzy control systems with uncertain plant gains. This algorithm limits the nonlinearity of fuzzy controller to the sector bound.

The Popov is a frequency domain stability criterion for closed loop nonlinear systems of Lur’etype.Fuzzy controlsystemscan beregarded asLur’etypesystems.Kandeletal.[36] adopted the Popov criterion to analyze the stability of fuzzy control systems with controller as multi-level relay. Furutani et al. [37] utilized the shifted Popov criterion to manage the fuzzy controller with both time-variant and time-invariant parts. However, the Popov criteria applied to the stability analyzes on the fuzzy logic control do not consider the effect of reference input.

On theotherhand,thelatestresearch developmentson theLur’esystemsstability analyzes concentrate on the systems with model uncertainties [38]-[41] and time-delay [42]-[43] or both [44]-[46]. The main approaches include the time-domain LMI [38]-[44] and the classical frequency-domain [45], [46] methods. The stability issues due to the reference input influence are not even discussed except in [51]. By [51], we can predict that the stability of fuzzy control systems will crash due to reference input shift, so it is important to take the reference inputs as one of the parameters for stability analyzes of fuzzy control systems.

In short,therecentstability analysisdevelopmentson theLur’etype systemsalmostalways use the time-domain LMI approach. The concerned issues are on uncertainties and time-delay orboth.However,thedevelopmentdirectionsdon’tconcern thereference inputinfluenceon stability.

Other investigations on fuzzy logic control systems can be described as follows. Butkiewicz [47] investigated the steady error of a fuzzy control system with respect to different fuzzy reasoning processes [47]. Tao and Taur [48] designed a robust complexity-reduced PID-like fuzzy controller for a plant with fuzzy linear model in [48].

Malki et al. [49] derived a fuzzy PD controller from the conventional continuous-time linear PD controller [49], in which the proportional and derivative gains are a nonlinear function of the input signal. The stability of this new type fuzzy PD controller is ensured by the small gain theorem. Taur and Tao [50] analyzed and designed region-wise linear fuzzy controllers (RLFC) [50], and found that the RLFCs generally performed better than the PD controllers.

Our work analyzes the absolute stability in P and PD type fuzzy control systems with both certain and uncertain linear plants. The control functions in P and PD type fuzzy controllers are known to be piecewise linear, and can be described with mathematical equations. The equilibrium points of each piecewise linear surface in a P type fuzzy control system with a certain linear plant can be calculated by this description. The unique error equilibrium point of the overall system can be obtained by determining whether the error equilibrium point located in its own error region. Therefore, the error equilibrium points in the reference and actuator gain parameter space can be analyzed. Additionally, the absolute stability can be analyzed using the frequency and time domain approaches. Since a P type fuzzy control system is a Lur’esystem,itsstability can betested by thePopov criteriain thefrequency domain.In the time domain, the stability can be tested by linearizing the system with regard to the equilibrium point. Conversely, the stability of a P type fuzzy control system can be tested by the parametric robust Popov criterion [51] incorporated with the Kharitonov theorem for uncertain linear plant and interval parameters, including actuator gain, reference input and plant parameters. Notably, the actuator gain can be included in one of the plant parameters. For a PD type fuzzy control system, single-input fuzzy logic controller (SFLC) [52] is introduced into our analysis. In a certain linear plant situation, the equilibrium point of fuzzy control systems can be analyzed using the same P type fuzzy analysis concepts. A PD type fuzzy control system with an SFLC controller can be transformed into a P type system, so that its stability can be analyzed with the Popov and linearization methods. The parametric

absolute stability ofLur’esystemscan also beapplied to atransformed PD typefuzzy control system when the plant is uncertain. For comparison with theoretical analysis, a fuzzy current controlled RC circuit is designed with a PSPICE model. Simulation results including both numerical and PSPICE confirm the theoretical analysis. Additionally, the mechanism of oscillations in fuzzy control systems is interpreted with a viewpoint of equilibrium points in a simulation example. Finally, the comparisons also are made to exhibit the effectiveness of the analysis method. The applied method parametric robust Popov criterion will be compared with therobustLur’etest[54],therobustcirclecriterion [54],and therobustPopov criterion [54]. In compared methods, the stability of uncertain fuzzy control systems which are considered as stable by compared methods will crash under the effect of the reference inputs. On the other hand, by the applied analysis method, the stability can be guaranteed for the certain interval reference inputs. In summary, this study can provide a valuable reference in designing fuzzy control systems.

In conclusion, the stability analysis is extended to a non-zero reference input and an uncertain linear plant. This is in contrast to the approach employed by Kim et al. [27], in which DF is derived and applied to analyze the stability of fuzzy control systems for zero reference inputs and certain linear plants. The DF method may yield inaccurate or incorrect analysis results without restricted assumptions. By contrast, the Popov criterion based on the Kharitonov theory can guarantee an exact stability investigation. Moreover, SFLC [52] is applied in the analysis of a PD type fuzzy control system. SFLC is an efficient FLC, owing to its 1-D fuzzy rules only. By this feature, the SLFC can be implemented as an analog circuit and applied for high frequency control. This work first investigates the steady state error and robust stability analysis for linear plants using the proposed structure transformation. Additionally, an analog fuzzy control system is designed with a PSPICE model to verify the analysis results. Finally, the explanations for unstable oscillations in fuzzy control systems are

presented with the equilibrium concept.

On the other hands, a kind of the applications based on fuzzy control systems is addressed in this thesis. In this application, the adaptive fuzzy-neural observer (AFNO) is applies to synchronize a class of unknown chaotic systems with a scalar transmitted signal only. The synchronization of chaotic systems has been extensively studied and given its potential application to security communications. Synchronization means that the master and slave have identical states as time goes to infinity. Pecora and Carroll first considered the synchronization of chaotic systems [55], in which the drive-response concept is introduced to achieve synchronization by a scalar transmitted signal. Perfectly identical parameters cannot be achieved in real applications. Therefore, the nonlinear robust control [56,57] concept is employed to chaos synchronization with previous known states within the margin of synchronization error. An adaptive recurrent neural controller can be utilized to synchronize with respect to unknown systems [58,59]. However, all states should be measurable with this algorithm. In contrast, the nonlinear observer is designed to synchronize chaotic systems [60,61,62]. Morgül and Solak [62] presented global synchronization is possible for a system with Brunowsky canonical form. Grassi and Mascolo [61] provided a systematic method for synchronizing using a scale transmitted signal. Message-free synchronization has been developed to permit communication with masking message in chaotic signals [63]. Messages can be extracted with message-free synchronization. Moreover, Boutayeb [60] proposed a scheme which is provided to synchronize and extract message simultaneously. Nevertheless, these systems do not consider the robustness of the state observer with respect to parameters mismatch [60,61,62]. Adaptive sliding observer design [64,65] can handle parameters mismatch. Furthermore, a robust observer [66] is designed for synchronization using the Takagi-Sugeno fuzzy model and the LMI approach. Millerioux and Daafouz recently introduced the input-independent global chaos synchronization [67]. In this method, the added

message does not affect the synchronization if the observer gain is appropriately designed. Other studies consider nonlinear observer designs for chaos synchronization [68,69]. However, by the methods of previous descriptions, the chaotic systems should be known previously before synchronization design. Recently, the system identification approaches [70,71,72] have been introduced for a scale signal identification and chaos synchronization respectively. In [71], the system identification concepts are applied to approximate the chaotic signal.Theproposed identification schemeassumesaLur’etypesystem asareferencemodel. This allows us to separate the identification process into two parts, adjusting alternatively the parameters of the linear and the nonlinear part. For modeling the linear system, the autoregressive moving average (ARMA) approach is utilized. On the other hand, the genetic algorithm is applied to optimize the break points parameters of nonlinear static functions to approximate nonlinear mapping. However, this approach is based on off-line identification, and it is not an on-line tuning scheme. Furthermore, the order in linear part identification should be by trial and error. The identification results just imitate the transmission signal and the other states in the master end cannot be achieved to synchronize simultaneously. In addition, the simulation results of this approach seem not very well. According to [70], the recursive identification is applied for chaos synchronization when the slave has exactly identical structure to the master system, but its parameters are unknown. It is shown that the unknown slave system parameters can be found by the concepts of adaptive synchronization. In other words, when the unknown slave system parameters are found, the synchronization is achieved. However, the structures in the master and slave ends should be known previously and exactly the same, although the parameters in the slave end can be estimated by recursive identification. The discussion of robustness is not included too. More recently, an alternative indirect Takagi–Sugeno fuzzy model based adaptive fuzzy observer design has been applied to chaos synchronization under assumptions that states are unmeasurable and parameters are

unknown [72]. The adaptive law is designed to estimate the unknown parameters in the T-S fuzzy model of the slave end. When the unknown parameters are estimated correctly, the synchronization is achieved. However, the form of the T-S fuzzy model should be known first, and then the adaptive fuzzy observer is designed by the T-S fuzzy model. In addition, the discussion of robustness is not included.

This investigation achieves synchronization with respect to a class of unknown master chaotic systems by introducing the concepts of AFNO [73], Brunowsky canonical form [62] and Lur’esystems[74]. The proposed system includes a chaos master with canonical form and the slave with AFNO. The AFNO combines a FNN and a linear observer. In this design, the slave should synchronize with the master by a scale transmitted signal .This approach employs adaptive FNN to model the nonlinear term of the master end. The output of the adaptive FNN, robust input and a transmitted state are sent to the linear observer to estimate the states of the slave. The master and slave achieve synchronization when all states are estimated at the slave. Additionally, the adaptive laws are needed to update the weights of the FNN, when the reconstructed and transmitted states differ from each other.

The benefits of provided AFNO for synchronization can be stated as follows. AFNO is first applied to chaos synchronization with only one transmitted signal. Since AFNO is on line learning at the slave, the synchronization can be achieved respect to a switched unknown chaotic system with the Lur’e type. Additionally, the adaptability for parameters change or even system switched in the mater and the robustness for modeling error and external bounded disturbance are also given. AFNO also has FNN’s inherent properties of fault-tolerance, parallelism learning, linguistic information and logic control. By comparing with [70,71,72], our presentation provides the on-line, robust and adaptive synchronization for a class of chaotic systems. The form of nonlinear functions in the master end cannot be known in previous due to soft computing with FNN for fitting it in the slave end.

1.2 Organizations of the Dissertation

This thesis is organized as follows. Chapter 1 is an introduction. Chapter 2 describes the P and PD type fuzzy control systems. Chapter 3 analyzes the equilibrium points and stability in P type fuzzy control system. Chapter 4 then performs the same analyses in a PD type fuzzy control system. Chapter 5 provides simulation results with Matlab and PSPICE simulators. In Chapter 6, the comparisons are made to show the superiority of the applied analysis method. Furthermore, in Chapter 7, the observer-based synchronization for a class of unknown chaotic systems with adaptive fuzzy-neural network is presented. Finally, some conclusions are given in Chapter 8.

Chapter 2

The Fuzzy Logic Control Systems

2.1 Fuzzy Logic Controller

Both P and PD type fuzzy logic control systems include a linear plant with time-invariant uncertainty, adjustable actuator gain and reference input. Moreover, the fuzzy logic controllers are the cores of systems. An FLC can be taken as multiple bends of piecewise linear functions, since it has singleton and specific membership functions. Hence, a fuzzy logiccontrolsystem can betreated asaLur’etype system.

Consider the fuzzy logic control system in Fig. 2.1. The IF-THEN rules in single input fuzzy logic controller can be described as:

: i

Rule If e is M , then_i uf is u ,i (1)

where e is the control error and M and_i u denote fuzzy sets. If a singleton is applied in a_i

fuzzifier, then the product inference and center average are formulated in the inference engine and defuzzifier, respectively. The output of the fuzzy logic controller can be represented as

( ) f i i i u 



 e u , (2) where ( ) ( ) ( ) i i j j M e e M e  



.

For simplification, this study uses the fuzzy rules and membership functions listed in Table 2.1 [27] and Fig. 2.2 are adopted in this thesis, respectively. Table 2.2 presents the fuzzy controller parameters. Figure 2.3 shows the control function of the fuzzy controller, which can

be described as:

 

 













2 2 2 3 1 1 1 2 0 1 1 1 1 2 1 2 2 3 2 1: , , , 2: , , , ( ) 3: , , , 4: , , , 5: , , , f segment k e c e a a segment k e c e a a u e segment k e e a a segment k e c e a a segment k e c e a a      _{ }    _    _{   }        (3) where 1 2 3 0 a   ,a a 0 b   ,₁ b₂ b₃ c₁ b₂ k a_{1 2}, c₂  b₃ k a_{2 3}, 1 0 1 b k a  , 2 1 1 2 1 b b k a a    , and 3 2 2 3 2 b b k a a    .

Remark 2.1: The assumptions 0   a₁ a_{2 } a_n and 0   b₁ b_{2 } b_n are satisfied

for n multiple bends of a control function. The control output of the static fuzzy system is

given by:













1 0 1 1 1 , , , ( ) , , , , , , n n n n f n n n n k e c e a a u e k e e a a k e c e a a         _         (4) where c_n b_n1k a_{n n1}, 1 1 n n n n n b b k a a      ,and n1, 2,3, , n.

The control function  satisfies





2

ˆ ˆ ˆ

0e (e e ) ( )e k e e( ) ,  e ,  eˆ , (5)

where (0) 0 , k0and  indicates some neighborhood of e0.

2.2 P Type Fuzzy Logic Control System

Figure 2.1 illustrates a P type fuzzy control system with a fuzzy logic controller, a parametric linear time-invariant system and adjustable parameters, which include actuator

linear function, and is depicted in Fig. 2.3.

The linear plant H s p( , ) shown in Fig. 2.1 can be presented as





1

( , ) ( ) ( ) ( )

H s p C p sI A p B p , (6)

where _{A p}( )_{ and}n n _{A p( ) is a stable matrix; B p  ;( )} n1 _{C p( ) , the parameter}1n vector p exists in a compact and simple connected region _l_.

The transfer functionG s p K( , , ) with amplifier gain K can be stated as





1

( , , ) ( ) ( ) ( , )

G s p K C p sI A p B p K (7)

where _{B p K( , )KB p( ) , and}n1 _{K. The overall static fuzzy logic control system in}

Fig. 2.1 can be described as:

( ) ( , ) _f

x_A p x B p K u ,

( )

y C p x , (8)

where the control input u_f ( )e ; the control error e r y  , _xn_{, e and y;} the reference input r is a constant value, and r is a constant value, and r.

The closed loop system is given by





( ) ( , ) ( )

x A p x B p K  r C p x . (9)

The error equilibrium points and relative stability under the influence of parameters including actuator gain K, reference input r and time invariant uncertainty in linear plants are addressed. The parameter vector is defined as ( , , )r p K .

2.3 PD Type Fuzzy Logic Control System

This subchapter discusses the PD type SFLC depicted in Fig. 2.4. The SFLC’s output u_f

as shown in Table 2.3 [52], is significantly reduced into 1-D space, as in Table 2.4, owing to the single input and skew-symmetric property. Due to the skew-symmetric property of the rule table, (e ,e) can be split into five regions. Figure 2.5 illustrates an example of this division of(e ,e). The reduced 1-D rules improve the efficiency of the controller by saving time cost for a look up rule table, although it also adds the calculation time of signed distance. Therefore, the SFLC is suitable for implementation in circuit control. The SFLC is introduced in this subchapter for further equilibrium points and stability analysis in the following subchapters.

2.3.1 Calculation of signed distance

The control error in SFLC is defined as ( ) d

e t   .y r (10)

The switching line s as shown in Fig. 2.5 is given by_l

: 0

l d d

s ee  . (11)

The signed perpendicular distance D of general point_S Q e e( , )_d  to a switching line is_d calculated as follows: 2 sgn( ) 1 d d s l e e D s D        _{, (12)} where 2 1 d d e e D      

is shown in Fig. 2.5 and sgn( ) 1 for 0 1 for 0 l l l s s s   _{ }  .

The control output u_f (D_S)is defined according to the control rule in SFLC as given in Table 2.4 and Fig. 2.4.

2.3.2 The presentation of the SFLC system

The SFLC system can be described as:

( ) ( , ) _f

( )

y C p x , (13)

where the control input u_f (D_s).

2.3.3 The analytic representation of the SFLC system

If Tables 2.2, 2.4 and Fig. 2.2 are applied into the controller in SFLC, then the control function ( ) of the fuzzy controller is as displayed in Fig. 2.6. The surface of the fuzzy controller in SFLC is typically oddly symmetrical; therefore, the control force is given by

( ) ( ) ( ) f s S u D   D  , (14) where 2 1 s e e D         _.

In the following analysis, this representation as illustrated in Fig. 2.7 is applied to PD type analysis. In Chapter 3, the SFLC system is reformatted as a special P type fuzzy control system, and is employed to analyze the equilibrium

Fig. 2.1 The P type fuzzy control system.

e

(a) f

u

(b)

Table 2.1

Rules of the fuzzy logic controller

e NBE NME NSE ZRE PSE PME PBE

f

u _{NBU NMU NSU ZRU PSU PMU PBU}

Table 2.2

Parameters of the fuzzy logic controller

NBE NME NSE ZRE PSE PME PBE

e

3

a

 a₂ a_{1 0} a₁ a₂ a₃

NBU NMU NSU ZRU PSU PMU PBU

f u 3 b  b₂ b_{1 0} b₁ b₂ b₃

e

( )

e



1

a

a

2 a3 1

a



2

a



3

a



1 b 2 b 3 b 1 b  2

b



3

b



0 f u k e 1 1 f u  k e c 2 2 f u  k e c

y

r

_d

y

dt 2 1 1



1 s I ( ) B p

x

C p( ) ( ) A p f

u

s

D

( , , )

G s p K

( )  ( , ) B pK d

e

Fig. 2.4 The single-input fuzzy logic control system.

Table 2.3

Rules of conventional FLC with control error defined as ed

d e d e Table 2.4 Rules of SFLC S

D _{NBE NME NSE ZRE PSE PME PBE}

f

Fig. 2.5 The skew-symmetric property in (e ,e) and the calculation of signed distance. s D 1

a

a

2 a3 1 a  2

a



3 a  1 b 2 b 3 b 1 b  2

b



3

b



( )

D

_s



y

r

_d

y

dt

e

₁

s I

 ( ) B p

x

C p( ) ( ) A p f

u

S

D









( , , )

G s p K

( )



( , ) B pK 2

1











2

1

1









Chapter 3

Equilibrium Points and Stability Analysis in

P and PD Type Fuzzy Control Systems

3.1 Equilibrium Point Analysis for P Type Fuzzy Control Systems

with Linear Plants

This subchapter presents the analysis of error equilibrium points and stability in P type fuzzy control systems. The equilibrium point in fuzzy control systems can be derived when equilibrium points can be solved. Moreover, the stability of the equilibrium point can be judged with the linearizing system around the equilibrium or the Popov criterion in the following subchapter. If the error equilibrium points of the overall system are stable, then the steady state error can be derived from this result.

By (9), let x_0, then





( ) 0

Ax B K r Cx  . (15)

If _A1 _{exists, then (16) is obtained.}

1 _{( ) ( ) 0}

x A B K_  e _{ , (16)}

where e r Cx  .

Multiply the result by C in (16), and let Cx r e , then

1 _{( ) ( ) 0}

e r CA B K_{ }  e _{ . (17)}

e

e , can be determined from (16) and (17), respectively.

Assumption 3.1: The unique solution exists in (17). In other words, an error equilibrium

point uniquely exists.

Under Assumption 3.1, the error equilibrium points can be solved from (18) by replacing (4) in each segment.













1 1 1 0 1 1 1 1 ( )( ) 0 , , ( )( ) 0 , , ( )( ) 0 , . e e n n n n e e e e n n n n e r CA B K k e c if e a a e r CA B K k e if e a a e r CA B K k e c if e a a                     1,2,3, . n _ (18)

One of these error equilibrium points is the unique point of the overall system. The unique point is identified by checking whether _ee _{is located in its own error region.}

3.2. Stability Analysis for P Type Fuzzy Control Systems with a

Certain Linear Plant

In the certain linear plant case, the stability can be determined by the time or frequency domain approaches proposed in [51]. In the time domain approach, the eigenvalues of the linearizied system (8) can be applied to determine the stability. In the frequency domain, the Popov criterion is utilized to test stability.

3.2.1 Frequency domain approach

Consider the error dynamic system for a given parameter vector ( , , )r p K .

ˆ ( )ˆ ( ) (ˆ ( ) )ˆ xA p x B p C p x , (19) where ˆ_{x x x r p K }e( , , )_, and ˆ( _{C p x}( ) )ˆ _{C p x e r p K}( )ˆ e( , , ) _{e r p K}e( , , )   _   _{ } _.

( , , ) ( ) ( , , )

e e

e r p K  r C p x r p K . (20)

The error dynamic system is also of Lur’e type. The function  satisfies the followingˆ

sector condition if _{e r p K}e( , , )_{ .}

2

ˆˆˆ ˆ

0_{e e}( )_{k e r p K e}[ ( , , )]e _{,  eˆ , (21)}

where ˆ_{e e e r p K }e( , , )_{and k 0.}

By the Popov criterion, (19) is absolutely stable for a given ( , , )r p K , if there exists a real number v v r p K( , , )satisfying 1 Re[(1 ) ( , , )] 0 [ ( , , )]e j v G j p K k e r p K     ,   , (22) where _{G s p K( , , )C p sI A p( )[  ( )]}1_{B p K( , ).}

3.2.2 Time domain approach

Under an arbitrary parameter vector ( , , )r p K , if an equilibrium state _{x r p K}e( , , ) _{of the} system exists, then the stability can be determined from the linearization of (9) near the state equilibrium point.

Remark 3.1: If the unique state equilibrium is stable, then the steady state error in fuzzy

control systems can be obtained from the state equilibrium by _ee _{ r Cx}e_.

3.3 Stability Analysis for P Type Fuzzy Control Systems with an

Uncertain Linear Plant

In this subchapter, the parametric absolute stability can be tested using the parametric robust Popov criterion incorporated with Kharitonov theorem, when the parameter vector

( , , )r p K R_ref   , where R_ref [ , ]r r .

subchapter, because fuzzy control function ( ) is sometimes impossible to obtain mathematically, and parameters ( , , )r p K vary in a range in real application. Therefore, the stability analysis by the parametric robust Popov criterion in [51] is adopted to handle this situation.

Applying Theorem 1 in [51], let’sconsider the uncertain P type fuzzy control system (9) satisfying the following conditions. Then, the P type fuzzy control system is parametric absolute stable.

(1) If the fuzzy controller is continuous, and for some neighborhood  of e0

satisfies

2

ˆ ˆ ˆ

0e[ (e e ) ( )]e k e e( ) ,  e ,  eˆ , and (0) 0 , (23)

where k e( ) is a positive number depending on e. (2) If _{( )} 1_{( ) ( , )} 1 ₀ (0) C p A p B p K k     ,  p  (24)

holds, for any ( , , )r p K Rref    and any  satisfying the sector condition (23), there exists a solution _{e e r p K}e( , , ) _{of (17) in }e_{( , , )r p K ,} where









1 0 1 0 , ( ( ) ( ) ( , ) 0) ( ) ( , , ) , ( ( ) ( ) ( , ) 0) ( ) e r _{r when r C p A p B p K} p r p K r r when r C p A p B p K p        _         _      (25) and 1 0( ) 1p C p A( ) ( ) ( , ) (0)p B p K k

 _{ }  _{. A more detail proof on (15) and (16) can be}

referred in the Lemma 1 of [51].

(3) If for a given region R_ref of r and for any p , the condition e( )

R p

 is satisfied, and a real number v_o v r p K_o( , , ) exists such that the following inequality holds

1 Re[(1 ) ( , , )] 0 ( , , ) o R j v G j p K k r p K      ,   , (26) where





( ) max ( ) : e( , , ) R R k p  k e e r p K , (27) and e( , , ) R r p K

 represents the region containing _{e r p K}e( , , ) _{for all}

ref

r R .

Remark 3.2: k r p K is hard to find, so we suppose that_R( , , ) Rref . Moreover, assume that for any p , G(0, , ) 0p K  , *_{( ) max}



_{( ) :}



R ref

k p  k e e R , and there exists a real

number v_o v r p K_o( , , )letting the inequality hold.

* 1 Re[(1 _o) ( , , )] 0 R j v G j p K k      ,   . (28) The P type fuzzy control system is then parametric absolute stable. [51]

Remark 3.3:

(1) This test can be extended to the general P type fuzzy control functions design. (2) The assumption in Remark 3.2 does not lose generality, since most systems have

(0, , ) 0

G p K  .

(3) The effect of Kcan be combined into plant parameters p.

The existence of v_o v p_o( ) for every p should be guaranteed in (28). This is

generally a difficult problem. Therefore, the parametric robust Popov criterion incorporated with Kharitonov [51], [53], [54] for interval Lur’e systems is introduced into a parametric absolute stable analysis.

Consider the following as a family of interval plants ( ) ( , , ) ( ) Q s G s p K P s  , (29)

where Q s( ) and P s( ) belong to the families of real interval polynomials Q( )s and P( )s , respectively.



( ): ( ) 0 1 , i i, i , 0, , ,



s Q s Q s    q q s q s_ and q _q q _ for all i 

Q( )=  

and



( ): ( ) _{0 1} n, , , 0, , .



n j j j

s P s P s _{   }p p s p s and p _p p _ for all j_ n

  P( )=   (30) ( ), i Q K s i1, 2,3, 4 and j( ), P

K s j1, 2,3, 4 represent the Kharitonov polynomials associated with Q( )s and P( )s , respectively. The Kharitonov systems associated with G s p K( , , ) are defined as the 16 plants of the following set,





( ) ( ) : : , 1, 2,3, 4 , ( ) i Q K j P K s G s i j K s     _  _     (31) where 1 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; Q K s  q q s q s   q s q s q s q s _ 2 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; Q K s  q q s q s  q s q s q s q s _ 3 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; Q K s _{ }q q s q s _ _q s _q s _q s _q s _ 4 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; Q K s  q q s q s  q s q s q s q s _ 1 2 3 4 5 6 0 1 2 3 5 6 ( ) ; P K s  p p s p s   p s p s p s p s _ 2 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; P K s  p p s p s   p s p s p s p s _ 3 2 3 4 5 6 0 1 2 3 4 5 6 ( ) ; P K s _{ }p p s p s _  _p s _p s _p s _p s _ 4 2 3 4 5 6 0 1 2 3 4 5 6 ( ) . P K s _{ }p p s p s _  _p s _p s _p s _p s _

A P type fuzzy control system is absolutely stable in sector

 

0, k for all

( ) ( , , )

G s G s p K , if a real v can be obtained by verifying the robust Popov condition for_o

( ) _K( )

G s G s to satisfy inequality (28).

Remark 3.4:

(1) The previous descriptions imply that only 16 Popov plots need to be drawn from family ( )

K

Popov condition (28) holds for the whole familyG s( ).

(2) The P type fuzzy control systems of Lur’e type can be tested by the parametric robust Popov criterion. By [51], [53], [54], the criterion incorporated with Kharitonov for interval Lur’e systems can be considered here for parametric absolute stability analysis of P type fuzzy control systems.

3.4 Transformation SFLC from PD to P Type

In the following, the SFLC is transformed from PD to P type, so that the equilibrium point and stability can be analyzed by the transformed special P type fuzzy logic control system. From Fig. 2.4, the factor

2

1

1 of SFLC is integrated into both the proportional and

derivative factors. The  and  in Fig. 2.7 are then defined as

2 , 1      and 2 1 1     . (32) Assumption 3.2: CB0.

According to Assumption 3.2 and Fig 2.7, the following derivation can be obtained.

.

e r y r Cx    (33)

By differentiating both sides, then

( _f) .

e Cx C Ax Bu CAx (34) From (33) and (34), then

1

( ) ( ) ,

e e r Cx CAx r C x

           (35)

where C₁(CCA), and rr.

After transformation, the transformed plant in Fig. 3.1 can be obtained

1 1

( , , ) ( )[ ( )] ( , ).

PD

From Fig. 3.1, the special P type transformation from the SFLC system can be described as:

( ) ( , ) _f

xA p x B p K u ,

1( )

yC p x, (37)

where the control input u_f  ( ), and control error  r y .

The transfer function H_PD( , )s p of the transformed plant in Fig. 3.1 can be described as





1 1

( , ) ( ) ( ) ( )

PD

H s p C p sI A p B p , (38)

3.5 Equilibrium Point Analysis for PD Type Fuzzy Control

Systems with Linear Plants

From Fig. 3.1, the equilibrium point can be analyzed

( ) ( , ) ( ). xA p x B p K   (39) Let x0, 0A p x B p K( )  ( , ) ( ).  (40) If _A1_{( )p exists, then} 1_{( ) ( , ) ( ) 0.} x A  p B p K   (41)

By multiplying the result of (40) by Cand using (35), then

1

( ) ( ) ( ) ( , ) ( ) 0

C p x C p A p B p K  e e

   _ (42)

When t , x0 and e0 are implied. By e0,

1

( ) ( ) ( , ) ( ) 0

e e

e _{ }r C p A p B p K e _{ . (43)}

Remark 3.5: The error equilibrium point of the PD type fuzzy control system is

e

e  e

d

e

3.6 Stability Analysis for PD Type Fuzzy Control Systems with

Linear Plants

The transformed P type of SFLC in Fig. 3.1 can be employed to analyze the stability of SFLC for a given ( , , )r p K .

3.6.1 Frequency domain approach

Consider the error dynamic system in Fig. 3.1 for the given parameter vector( , , )r p K .

1 ( ) ( , ) ( ( ) ) x A p x B p K  C p x, (45) where _{x x x r p K}_{ }e( , , )_, 1 1 ( _{C p x}( ) ) _{C p x e r p K}( ) e( , , ) _{e r p K}e( , , )   _   _{ }  _, and ( , , ) 1( ) ( , , ) e e e r p K  rC p x r p K . (46)

The error dynamic system is also of Lur’e type. The function satisfies the following sector condition, if _{e r p K}e( , , )_.

2

0_{e e}_( )_{ k e r p K e}[ ( , , )]e , _{ e} _{, (47)}

where _{e e e r p K }e( , , )_{, and k 0.}

From the Popov criterion, (39) is absolutely stable for a given( , , )r p K , if a real number

0 0( , , ) v v r p K satisfying 0 1 Re[(1 ) ( , , )] 0 [ ( , , )] PD e j v G j p K k e r p K      ,   . (48)

3.6.2 Time domain approach

Consider an arbitrary parameter vector ( , , )r p K in SFLC. Suppose that an equilibrium state _{x r p K}e( , , ) _{of the system exists. The stability can be determined by the linearization of}

(37) near the error equilibrium point.

3.7 Stability Analysis for PD Type Fuzzy Control Systems with

Uncertain Linear Plants

Since the transformed SFLC is a special P type fuzzy control system as shown in Fig. 3.1, the parametric Popov criterion [51] incorporated with Kharitonov theorem is adopted to analyze the stability of PD type fuzzy control systems with uncertainties.

y

r r







y 

1

s I

 ( ) B p

x

C p

₁

( )

( )

Ap

f

u

( , , )

PD

G s p K

( )





( , )

B p K

( , )

PD

H s p

Chapter 4

Fuzzy Current Control RC Circuit System

Design

The temperature control is an important issue in many industrial processes or medical applications. The temperature controls systems are analogous to RC electrical circuits and are governed by the following third-order equation (49) [75]. In our design, FLC is applied to control the RC electrical circuits to reach the specified output voltage. In other words, it is similar to regulate the temperature to desired set point. This chapter specifies fuzzy current control RC circuit systems of P and PD types for verifying the theoretical analysis using PSPICE simulation.

In this chapter, the circuit structure is specified first. The fuzzy logic controller is then designed to construct the fuzzy control function, which is mapping I/O relation of the fuzzy controller. Finally, some components of the overall structure of the fuzzy logic control system are introduced.

4.1 The Block Diagram of the Fuzzy Current Control RC Circuit

System

Figure 4.1 depicts the block diagram of a fuzzy current control RC circuit. The control objective of this system is to track a dc constant reference voltage r. To avoid the loading