臺灣博碩士論文加值系統

在高速度系統需求之製造工業快速發展下，衍生出之溫升問題正受到前所未有的矚目。因而，針對處理溫升現象之數學分析及研究在許多相關應用領域上是一有趣的話題。因此，在此討論偏微分方程之熱傳方程式。之後，我們探究方式是以數值方法求解出微分方程，再而處理溫升(blow-up)問題。在近幾年，模糊控制已經廣泛應用於處理非線性系統模型。其中，T-S模糊方法已應用於非線性系統上。此基本觀念是將一非線性系統分解成數個線性子系統和配合權重函數。至於穩定性條件則使用Lyapunov求得，可確保閉迴路系統穩定無虞，此充分條件可轉換成線性矩陣不等式之型式，以強而有利的數值工具獲得控制增益。本論文中，探討邊界控制和effective control來處理非線性熱傳方程式之溫升控制問題皆是建構於T-S模糊模型上。雖然以模糊模式可精確表示成藉以有限差分法所形成原系統模型，但其模糊規則數是以非線性項的個數成指數增加。因此，一個嶄新的想法因而產生：針對一函數以傅立葉級數展開來取得操作點，再以Zak線性化方法來獲得模糊模型，歸屬函數的取得是定義從目前的狀態到各個操作點之間的距離。最後，以倒傳遞類神經網路來調整模糊控制器的等級函數來獲取較快速的暫態響應，此處等級函數，乃源於控制規則庫之歸屬函數，是利用倒傳遞網路來取得最佳的數值。

Under the fastest-growing high-speed system manufacture industry, the blowup phenomena is gaining more attention than before. Therefore, the mathematical analysis of the blow-up phenomena gives an interesting application to those fields. We are mainly concerned with the PDE of heat equation. Then, we will apply an
efficient numerical method for estimating the blow-up time of the solutions of ODEs, and the blow-up problem of PDEs. In recent years, fuzzy control has been widely applied to deal with nonlinear systems. The T-S fuzzy approach has been extensively used to model nonlinear systems. The basic idea for the approach is to decompose the model of a nonlinear system into a set of linear subsystems with associated
nonlinear weighting functions. Lyapunov method is used to obtain the stability conditions which ensure the stability of the closed loop system, these adequate conditions can be transformed as LMIs and solved using powerful numerical methods
to obtain the control gains. In this thesis, we will discuss boundary control and smart control to deal with blow-up problem are developed based on T-S fuzzy. Although the fuzzy modeling can exactly represent the original model obtained by
FDM, its rule number will increase exponentially with respect to the number of nonlinear terms. A brand new idea is brought up: we thought of a function of x and can be expanded in a Fourier-series and obtained operation points, then Zak
linearization is used to obtain the fuzzy model, where the membership functions are defined by the distance from the current state to each operation point. Finally, the proposed intelligent methods using Back-Propagation algorithm of neural-networks is used to adjust membership function of the fuzzy controller to improve the system performance.

Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Organization of This Thesis . . . . . . . . . . . . . . . . . . . . . . . 2
2 Spectrum Analysis for Linear Heat Equations 4
3 Stability of Nonlinear Heat Equations Using FDM 9
3.1 Blow-up Phenomenon of Nonlinear Heat Equations . . . . . . . . . . 10
3.2 Finite Difference Method (FDM) . . . . . . . . . . . . . . . . . . . . 12
3.3 FDM of Nonlinear Heat Equations . . . . . . . . . . . . . . . . . . . 13
3.3.1 Fuzzy Model and Stabilization . . . . . . . . . . . . . . . . . . 15
3.4 An Effective Control for the Nonlinear Heat Equation . . . . . . . . . 16
3.4.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 19
4 Approximate Fuzzy Model of Heat Equations 25
4.1 Membership functions of Fuzzy Models . . . . . . . . . . . . . . . . . 32
4.2 Membership Functions Tuned by BPN . . . . . . . . . . . . . . . . . 32
4.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Conclusions and Future Works 41
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A The Membership Functions of T-S Fuzzy Model 46
B Operation Points of Nine Basis Functions 48
C The Membership Functions and System Matrices of Nonlinear System
49
References 43

List of Figures
3.1 The blow up point of nonlinear heat equations . . . . . . . . . . . . . 10
3.2 The response of blows-up at x = 0.5m . . . . . . . . . . . . . . . . . . 12
3.3 Discretization of a function u(t; x) . . . . . . . . . . . . . . . . . . . . 13
3.4 The boundary control of a rod . . . . . . . . . . . . . . . . . . . . . . 14
3.5 The effective control at x = 0.5m . . . . . . . . . . . . . . . . . . . . 19
3.6 The respective states of heat equation using smart control and boundary
control at d = 0.125m, d = 0.25m, and d = 0.375m, respectively. . 20
3.7 The respective states of heat equation using smart control and boundary
control at d = 0.5m, d = 0.625m, and d = 0.75m, respectively. . . 21
3.8 The respective state of heat equation using smart control and boundary
control at d = 0.875m. . . . . . . . . . . . . . . . . . . . . . . . . 21
3.9 The control force v for smart control . . . . . . . . . . . . . . . . . . 22
3.10 The control force v for boundary control . . . . . . . . . . . . . . . . 22
3.11 The temperature distribution u(t, x) using boundary control. (from
t=0~0.5sec) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.12 The temperature distribution u(t; x) using smart control. (from t=0~0.5sec) 24
4.1 Nine basis functions are used as operation points. . . . . . . . . . . . 27
4.2 The concept of nine-order and four-order error. . . . . . . . . . . . . 30
4.3 Nine basis functions are used as operation points. . . . . . . . . . . . 31
4.4 Flow-chart of the Back-Propagation Algorithm . . . . . . . . . . . . . 33
4.5 The closed-loop of Back-Propagation Neural Network. . . . . . . . . . 34
4.6 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 0sec. . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 2sec. . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 4sec. . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.9 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 6sec. . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.10 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 8sec. . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.11 The temperature distribution of approximate FD Model, fuzzy and
exact T-S fuzzy at 10sec. . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.12 The transient responses of neural-network and fuzzy model for u1, u2
and u3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.13 The transient responses of using neural-network and fuzzy model controller
for u4, u5 and u6. . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.14 The transient response of using neural-network and fuzzy model for
u7, u8 and u9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.15 The control input v of using fuzzy . . . . . . . . . . . . . . . . . . . . 39
4.16 The control input v of neural-network . . . . . . . . . . . . . . . . . . 40
4.17 The performance index of error function . . . . . . . . . . . . . . . . 40

List of Tables
3.1 The response at x = 0.5m. . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 For each matrix elements. . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 For each matrix
elements. . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Error values of exact lower-order model and fuzzy models. . . . . . . 31
C.1 For each subsystem matrices. . . . . . . . . . . . . . . . . . . . . . . 51
C.2 For each subsystem matrices. . . . . . . . . . . . . . . . . . . . . . . 52

References

[1] P. Grindrod, Patterns and Waves. Clarendon Press, Oxford, 1991.
[2] A. Friedman, Variational Principles and Free-Boundary-Value Problem. Wiley
NY, 1982.
[3] S. H. Zak, System and Control. New York: Oxford, 2003.
[4] M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models ,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 133 - 142, Apr. 1999.
[5] W. Z. Su and Z. B. lin, “Design and analysis of finite difference domain decomposition algorithms for the two-dimensional heat equation,” IEEE Conference on Algorithms and Architectures for Parallel Processing, pp. 174 - 178, Oct. 2002.
[6] H. O. Wang, K. Tanaka, and M. Griffin, “Parallel distributed compensation of
nonlinear systems by Takagi-Sugeno fuzzy model,” IEEE Conference on Fuzzy
Systems, vol. 2, pp. 531-538, Mar. 1995.
[7] Yu. Xinghuo, M. O. Efe, and O. Kaynak, “A general backpropagation algorithm for feedforward neural networks learning,” IEEE Transactions on Neural Networks, vol. 3, no. 1, pp. 251-254, Jan. 2002.
[8] K. Tanaka, “Stability and stabilizability of fuzzy-neural-linear control systems,”
IEEE Transactions on Fuzzy Systems, vol. 3, pp. 438-447, Nov. 1995.
[9] C. Hirota and K. Ozawa, “Numerical method of estimating the blow-up time
and rate of the solution of oridnary differential equations-An applicaiton to the
blows-up problems of partial differential equations,”Journal of Computational
and Applied Mathematics, vol. 193, pp. 614-637, Sep. 2006.
[10] K. Ishige and H. Yagisita, “Blow-up problems for a semilinear heat equation
with large diffusion,”Journal of Difference Equations and Applications, vol. 212,
pp. 114-128, May. 2005.
[11] K. Tanaka, T. Ikeda, and H. O. Wang, “Design of fuzzy control systems based
on relaxed LMI stability conditions,”Proceedings of the 35th IEEE Conference
on, vol. 1, pp. 598-603, 1996.
[12] K. Y. Lian, C. H. Su , and C. S. Huang, “Performance Enhancement for T-S
Fuzzy Control Using Neural Networks,”IEEE Transactions on Fuzzy systems,
vol. 14, Aug. 2006.
[13] K. Tanaka and C. S. Wang, Fuzzy control systems analysis and design: A linear
matrix inequality approach. New York: Wiley, 2001.
[14] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications
to modeling and control,” IEEE Transactions on Systems, vol. 15, no. 1, pp.
116-132, Jan.-Feb. 1985.
[15] K. Tanaka, T. Taniguchi, and H. O. Wang, “Model-Based fuzzy control of
TORA system: Fuzzy regulator and fuzzy observer design via LMIs that represent
decay rate, disturbance rejection, robustness, optimality,” Seventh IEEE
International Conference on Fuzzy Systems, vol. 1, pp. 313-318, May. 1998.
[16] P. V. O’Neil, Advanced Engineering Mathematics. California: Wadsworth, 1991.
[17] R. H. Adams, Sobolev Spaces. Academic Press, NY, 1975.