本論文係針對殊異擾動模糊系統，提出一系列之模糊控制穩定法則，並且保證整個控制系統之 控制性能。首先運用高木-菅野模糊模型(Takagi-Sugeno Fuzzy Model)，將非線性殊異擾動系統建構成殊異擾動模糊系統及其相對應之模糊快速及慢速子系統。接著，提出狀態回授模糊控制法、組合式模糊控制法、直接式模糊控制法、以觀測器為基礎之模糊控制法以及靜態與動態輸出回授模糊控制等方法。藉由李亞普諾夫穩定法則，整個回授系統之穩定性判斷條件可簡化成線性矩陣不等式。基於考慮殊異擾動參數(Epsilon)之上界問題，本文對所有模糊控制器均提出其相對之代數不等式以求得Epsilon之上界值Epsilon*，故所設計之模糊控制器均保證在Epsilon belong to (0, Epsilon*) 之範圍內整個殊異擾動模糊系統之穩定性。同時，所導出之H infinity 控制性能可使外部雜訊之影響降至最小。更進一步，當殊異擾動模糊系統存在不確定參數時，本文提出可穩定之模糊控制器設計法。最後，基於考慮殊異擾動模糊系統之暫態響應行為，本文亦提出具極點安置特性之動態輸出回授模糊控制法。本論文之貢獻為所設計之模糊控制器的增益矩陣均可藉由線性矩陣不等式求得，同時亦可計算出殊異擾動參數之上界值。本論文舉出非線性電路與直流伺服馬達驅動之倒單擺系統來驗證所提模糊控制法則之適用性與有效性。

　This dissertation presents several fuzzy control designs to stabilize the singularly perturbed fuzzy systems with guaranteed H infinity control performance. Firstly, by using Takagi-Sugeno (T-S) fuzzy model, we transform a nonlinear singular perturbation system into a singularly perturbed fuzzy system, where the corresponding fuzzy slow and fast subsystems of the original singularly perturbed fuzzy system are also obtained. Then we propose a series of fuzzy control designs including state feedback fuzzy control, composite fuzzy control, direct fuzzy control, observer-based fuzzy control, static and dynamic output feedback fuzzy controls. Based on the Lyapunov stability criterion, the stability conditions are reduced to a linear matrix inequality (LMI) problem. By the guarantee Epsilon-bound issue, the allowable perturbation bound Epsilon* can be determined by some simple algebraic inequalities such that the proposed fuzzy control stabilizes the singularly perturbed fuzzy systems for all Epsilon belong to (0, Epsilon*). Moreover, the developed criterion guarantees a minimum disturbance. Furthermore, the singularly perturbed fuzzy system with parametric uncertainties is also examined. For better transient behavior of singularly perturbed fuzzy system, the pole placement problem is also developed. Some practical systems, for example, the nonlinear circuit and the DC motor driver inverted pendulum systems, are given to illustrate the validity of the proposed schemes. All the simulation results demonstrate the proposed fuzzy control designs are feasible and satisfactory. The main contribution of this dissertation is that all the proposed controller gain matrices can be determined by the LMIs and the corresponding Epsilon upper bound can be calculated.

[1] Alexandridis, A. T., “Design of output feedback controllers and output observers,” IEE Proc. Contr. Theory Appl. vol. 146, pp. 108-112, 1999.
[2] Al-Holou, N., Lahdhiri, T., Joo, D. S., Weaver, J. and Al-Abbas, F., “Sliding mode neural network inference fuzzy logic control for active suspension systems,” IEEE Trans. Fuzzy Syst., vol. 10, pp. 234-246, 2002.
[3] Arzelier, D., Henrion D. and Peaucelle D., “Robust stabilization of a polytope of matrices,” Int. J. Control, vol. 75, pp. 744-752, 2002.
[4] Askarpour, S. and Qwens, T. J., “Eigenstructure assignment by output feedback: The case of common open- and Closed-loop characteristic vectors,” IEE Proc. Contr. Theory Appl. vol. 146, pp. 37-40, Jan., 1999.
[5] Assawinchaichote, W. and Nguang, S. S., “Fuzzy observer-based controller design for singularly perturbed nonlinear systems: An LMI approach,” Proc. IEEE Conf. Decision and Control, pp. 2165-2170, 2002.
[6] Assawinchaichote, W. and Nguang, S. S., “Fuzzy output feedback control design for singularly perturbed systems: An LMI approach,” Proc. IEEE Conf. Decision and Control, pp. 863-868, 2003.
[7] Assawinchaichote, W. and Nguang, S. S., “ fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: An LMI approach,” IEEE Trans. Syst., Man, Cybern., vol. 34, pp. 579-588, 2004.
[8] Assawinchaichote, W. and Nguang, S. S., “ filtering for fuzzy singularly perturbed systems with pole placement constraints: An LMI approach,” IEEE Trans. Signal Process., vol. 52, pp. 1659-1667, 2004.
[9] Assawinchaichote, W. and Nguang, S. S., “ output feedback control design for uncertain fuzzy singularly perturbed systems: an LMI approach,” Automatica, vol. 40, pp. 2147-2152, 2004.
[10] Bialas, S., “A necessary and sufficient condition for the stability of interval matrices,” Int. J. Contr. vol. 37, No. 4, pp. 717-722, 1983.
[11] Boyed, S., Ghaoui, L. El, Feron, E., and Balakrishnan, V., Linear matrix inequalities in systems and control theory. Philadelphia, PA: SIAM, 1994.
[12] Cao, Y. Y., and Frank P. M., “Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models,” Fuzzy Sets and Systems, vol. 124, pp. 213-229, 2001.
[13] Cao, Y. Y., Lam, J. and Sun Y. X., “Static output feedback stabilization: an ILMI approach,” Automatica, vol. 34, pp. 1641-1645, 1998.
[14] Chen, B. S., Lee, C. H. and Chang, Y. C., “ tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 32-43, 1996.
[15]Chen, B. S. and Lin, C. L., “On the stability bounds of singularly perturbed systems,” IEEE Trans. Automat. Contr., vol. 35, pp. 1265-1270, 1990.
[16] Chen, B. S., Tseng, C. S. and Uang, H. J., “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 571-585, 1999.
[17] Chen, B. S., Tseng, C. S. and Uang, H. J., “Mixed fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 249-265, 2000.
[18] Chen, B. S., Uang, H. J. and Tseng, C. S., “Robust tracking enhancement of robot systems including motor dynamics: a fuzzy-based dynamic game approach,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 538-552, 1998.
[19] Chen, S. J. and Lin, J. L., “Maximal stability bounds of singularly perturbed systems,” Journal of Franklin Institute, pp. 1209-1218, 1999.
[20] Chilali, M. and Gahinet, P., “ design with pole placement constraints: an LMI approach,” IEEE Trans. Automat. Contr., vol. 41, pp. 358-367, 1996.
[21] Chilali, M., Gahinet, P. and Apkarian P., “Robust pole placement in LMI regions,” IEEE Trans. Automat. Contr., vol. 44, pp. 2257-2270, 1999.
[22] Chiou, J. S., Kung, F. C. and Li, T. H. S., “An infinite -bound stabilization design for a class of singularly perturbed systems,” IEEE Trans. Circuits Syst. I, vol. 46, pp. 1507-1510, 1999.
[23] Chiou, J. S., Kung, F. C. and Li, T. H. S., “Robust stabilization of a class of singularly perturbed discrete bilinear Systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 1187-1191, 2000.
[24] Choi, H. H. and Chung, M. J., “Robust observer-based controller design for linear uncertain time-delay systems,” Automatica, vol. 33, pp. 1749-1752, 1997.
[25] Concha, J., Cipriano A. and Vidal, R., “Design of fuzzy controllers based on stability analysis,” Fuzzy Sets and Systems, vol. 121, pp. 25-38, 2001.
[26] Crusius, C. A. R. and Trofino, A., “Sufficient LMI conditions for output feedback control problems,” IEEE Trans. Automat. Contr., vol. 44, pp. 1053-1057, 1999.
[27] Farinwata, S. S., Filev, D. and Langari, R., Fuzzy control: Synthesis and analysis. New York: John Wiley & Sons, 2000.
[28] Gahinet, P., Nemirovski, A., Laub, A. and Chilali, M., The LMI control toolbox. Natick, MA: The Mathworks, 1995.
[29] Gajic, Z., Petkovski, DJ. and Harkara, N., “The recursive algorithm for the optimal static output feedback control problem of linear singularly perturbed systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 465-468, 1989.
[30] Guerra, T. M. and Vermeiren, L., “Control laws for Takagi-Sugeno fuzzy models,” Fuzzy Sets and Systems, vol. 120, pp. 95-108, 2001.
[31] Gupta, M. M. and Yamakawa, T., Fuzzy computing theory, hardware and application. Elsevier Science Publishers, 1988.
[32] Hamzaoui, A., Essounbouli, N., Benmahammed, K. and Zaytoon, J., “State observer based robust adaptive fuzzy controller for nonlinear uncertain and perturbed systems,” IEEE Trans. Syst., Man, Cybern., vol. 34, pp. 942-950, 2004.
[33] Han, Z. X., Feng, G., Walcott, B. L. and Ma, J., “Dynamic output feedback controller design for fuzzy systems,” IEEE Trans.Syst.,Man, Cybern., vol. 30, pp. 204-210, 2000.
[34] Horn, R. A. and Johnson, C. R., Matrix Analysis. New York: Cambridge University Press, 1988.
[35] Huang, J. C., Wang, H. S. and Chang, F. R., “Robust control for uncertain linear time-invariant descriptor systems,” IEE Proc. Contr. Theory Appl. vol. 147, pp. 648-654, Nov., 2000.
[36] Ishihara, J. Y. and Terra, M. H., “On the Lyapunov theorem for singular systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 1926-1930, 2002.
[37] Isidori, A., Nonlinear control systems. New York: Springer-Verlag, 1995.
[38] Isidori, A. and Asolfi, A., “Disturbance attenuation and control via measurement feedback in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1283-1293, 1992.
[39] Jiang, C. S., “New method of robust pole assignment by output feedback,” IEE Proc. Contr. Theory Appl. vol. 138, pp. 172-176, 1991.
[40] Joh, J. J., Chen, Y. H. and Langari, R., “On the stability issues of linear Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 402-410, 1998.
[41] Juang, Y. T. and Shao, C. S., “Stability analysis of dynamic interval systems,” Int. J. Contr. vol. 49, pp. 1410-1408, 1989.
[42] Kafri, W. S. and Abed, E. H., “Stability analysis of discrete-time singularly perturbed systems,” IEEE Circuits Syst. I, vol. 43, pp. 848-850, 1996.
[43] Khalil H. K., “On the robustness of output feedback control methods to modeling errors,” IEEE Trans. Automat. Contr., vol. 26, pp. 524-526, 1981.
[44] Khalil, H. K., “Output feedback control of linear two-time-scale systems,” IEEE Trans. Automat. Contr., vol. 32, pp. 784-792, Sep. 1987.
[45] Khalil, H. K., Nonlinear systems. Upper Saddle River, NJ: Prentice Hall, 1996.
[46] Kim, E., “A new computational approach to stability analysis and synthesis of linguistic fuzzy control system,” IEEE Trans. Fuzzy Syst., vol. 12, pp. 379-388, 2004.
[47] Kim, E., “Output feedback tracking control of robot manipulators with model uncertainty via adaptive fuzzy logic,” IEEE Trans. Fuzzy Syst.,
vol. 12, pp. 368-378, 2004.
[48] Kim, E. and Kim, S., “Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: continuous case,” IEEE Trans. Fuzzy Syst., vol. 10, pp. 391-400, 2002.
[49] Kiriakidis, K., “Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 269-277, 2001.
[50] Kiriakidis, K., Grivas, A., and Tzes, A., “Quadratic stability analysis of the Takagi-Sudeno fuzzy model,” Fuzzy Sets and Systems, vol. 98, pp. 1-14, 1998.
[51] Kokotovic, P. V., Khalil, H. K. and O’Reilly, J., Singularly perturbation methods in control: analysis and design. New York: Academic, 1986.
[52] Kokotovic, P. V., O’Malley, R. E., Jr., and Sannuti, P., “Singular perturbations and order reduction in control theory- An overview,” Automatica, vol. 12, pp. 123-132, 1976.
[53] Kuschewski, J. G., Hui, S. and Zak, S. H., “Application of feedforward neural networks to dynamical system identification and control,” IEEE Trans. Contr. Syst. Technol., vol. 1, pp. 37-49, 1993.
[54] Lee, H. J., Park, J. B. and Chen, G., “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 369-379, 2001.
[55] Lee, K. R., Jeung, E. T. and Park, H. B., “Robust fuzzy control for uncertain nonlinear systems via state feedback: an LMI approach,” Fuzzy Sets and Systems, vol. 120, pp. 123-134, 2001.
[56] Li, N., Li, S. Y., Xi, Y. G. and Ge, S. S., “Stability analysis of T-S fuzzy system based on observers,” International Journal of Fuzzy Systems, vol. 5, pp. 22-30, 2003.
[57] Li, T. H. S., Chiou, J. S. and Kung, F. C., “Stability bounds of singularly perturbed discrete systems,” IEEE Trans. Automat. Contr., vol. 44, pp. 1934-1938, 1999.
[58] Li, T. H. S. and Chiou, J. H., “A new Dstability criterion of multiparameter singularly perturbed discrete systems,” IEEE Trans. Circuits & Systems – Part I, vol. 49, pp. 1226-1330, 2002.
[59] Li, T. H. S. and Lin, K. J., “Stabilization of singularly perturbed fuzzy systems”, IEEE Trans. Fuzzy Syst., vol. 12, pp. 579-595, 2004.
[60] Li, T. H. S., Wang, M. S. and Sun, Y. Y., “Robust dynamic output feedback sliding mode control of singular perturbation systems,” JSME International Journal, vol. 38, pp. 719-726, 1995.
[61] Lin, C., Wang, Q. G. and Lee, T. H., “An improvement on multivariable PID controller design via iterative LMI approach,” Automatica, vol. 40, pp. 519-525, 2004.
[62] Lin, J. L. and Chen, S. J., “Robust analysis of uncertain linear singular systems with output feedback control,” IEEE Trans. Automat. Contr., vol. 44, pp. 1924-1929, 1999.
[63] Liu, H., Sun, F. and Sun, Z., “Stability analysis and Synthesis of fuzzy singularly perturbed systems,” IEEE Trans. Fuzzy Syst., vol. 13, pp. 273-284, 2005.
[64] Lo, J. C. and Lin, M. L., “Observer based robust control for fuzzy systems using two-step procedure,” IEEE Trans. Fuzzy Syst., vol. 12, pp. 350-359, 2004.
[65] Ma, X.-J., Sun, Z.-Q. and He, Y.-Y., “Analysis and design of fuzzy controller and fuzzy observer,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 41-51, 1998.
[66] Ma, X. J. and Sun, Z. Q., “Output tracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model,” IEEE Trans. Syst., Man, Cybern., vol. 30, pp. 47-59, 2000.
[67] Ma, X. J. and Sun, Z. Q., “Analysis and design of fuzzy reduced-dimensional observer and fuzzy functional observer,” Fuzzy Sets and Systems, vol. 120, pp. 35-63, 2001.
[68] Marino, R. and Tomei P., “Global adaptive output feedback control of nonlinear systems, Part I: Linear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 17-32, 1993.
[69] Marino, R. and Tomei P., “Global adaptive output feedback control of nonlinear systems, Part II: Nonlinear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 33-48, 1993.
[70] Mattei, M., “Robust multivariable PID control for linear parameter varying systems,” Automatica, vol. 37, pp. 1997-2003, 2001.
[71] Moheimani, S. O. R. and Petersen, I. R., “Optimal guaranteed cost control of uncertain systems via static and dynamic output feedback,” Automatica, vol. 32, pp. 575-579, 1996.
[72] Naidu, D. S., “Singular perturbations and time scales in control theory and applications: an overview,” Dynamics of Continuous, Discrete and Impulsive Systems,” vol. 9, pp. 233-278, 2002.
[73] Nguang, S. K., “Robust nonlinear output feedback control,” IEEE Trans. Automat. Contr., vol. 41, pp. 1003-1007, 1996.
[74] Oloomi, H. and Sawan, M. E., “The observer-based controller design of discrete-time singularly perturbed systems,” IEEE Trans. Automat. Contr., vol. 32, pp. 246-248, 1987.
[75] O’Reilly, J., “Full-order observers for a class of singularly perturbed linear time-varying systems,” Int. J. Contr., vol. 30, pp. 745-756, 1979.
[76] Pan, Z. and Basar, T., “ -optimal control for singularly perturbed systems- Part I: Perfect state measurements,” Automatica, vol. 29, pp. 401-423, 1993.
[77] Pan, Z. and Basar, T., “ -optimal control for singularly perturbed systems- Part II: Imperfect state measurements,” IEEE Trans. Automat. Contr., vol. 39, pp. 280-299, 1994.
[78] Park, J., Kim, J. and Park, D., “LMI-based design of stabilizing fuzzy controllers for nonlinear systems described by Takagi-Sugeno fuzzy model,” Fuzzy Sets and Systems, vol. 122, pp. 73-82, 2001.
[79] Qin, S. J. and Borders, G., “A multi-region fuzzy logic controller for nonlinear process control,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 74-81, 1994.
[80] Ramos, D. C. W. and Peres, P. L. D., “An LMI Condition for the Robust Stability of Uncertain Continuous-Time Linear Systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 675-678, 2002.
[81] Saksena, V. R., O’Reilly, J. and Kokotovic, P. V., “Singular perturbations and time scale methods in control theory: survey 1976-1983,” Automatica, vol. 20, pp. 273-293, 1984.
[82] Sastry, S., Nonlinear systems analysis, stability and control. New York: Springer-Verlag, 1999.
[83] Scherer, C., Gahinet, P. and Chilali, “Multiobjective output feedback control via LMI optimization,” IEEE Trans. Automat. Contr., vol. 42, pp. 896-911, 1997.
[84] Sen, S. and Datta, K. B., “Stability Bounds of Singularity perturbed systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 302-304, 1993.
[85] Shao, Z. H., “Robust stability of two-time-scale systems with nonlinear uncertainties,” IEEE Trans. Automat. Contr., vol. 49, pp. 258-261, 2004.
[86] Soh, C. B., “Correcting Argoun’s approach for the stability of interval matrices,” Int. J. Contr. vol. 51, pp. 1151-1154, 1990.
[87] Soh, Y. C. and Evans, R. J., “Stability analysis of interval matrices-continuous and discrete systems,” Int. J. Contr. vol. 47, pp. 25-32,
1988.
[88] Takagi, T. and Sugeno, M., “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, pp. 116-132, 1985.
[89] Tanaka, K., Hori, T. and Wang, H. O., “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy
Syst., vol. 11, pp. 582-589, 2003.
[90] Tanaka, K., Ikeda, T. and Wang, H. O., “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, control theory and linear matrix inequalities,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 1-13, 1996.
[91] Tanaka, K., Ikeda, T. and Wang, H. O., “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250-265, 1998.
[92] Tanaka, K. and Wang, H. O., Fuzzy control systems design and analysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001.
[93] Teixeira, M. C. M., Assuncao, E. and Avellar, R. G., “On relaxed LMI-based design for fuzzy regulators and fuzzy observers,” IEEE Trans.
Fuzzy Syst., vol. 11, pp. 613-623, 2003.
[94] Tseng, C. S., Chen, B. S. and Uang, H. J., “Fuzzy tracking control
design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 381-392, 2001.
[95] Tuan, H. D. and Hosoe, S., “On linear robust controllers for a class of nolinear singular perturbed systems,” Automatica, vol. 29, pp. 735-
739, 1999.
[96] Vidyasagar, M., “Robust stabilization of singularly perturbed systems,” Syst. Contr. Lett. vol. 5, pp.413-418, 1985
[97] Vidyasagar, M., Nonlinear systems analysis. Englewood Cliffs, NJ: Prentice Hall, 1993.
[98] Wang, Dianhui and Soh, C. B., “On regularizing singular systems by decentralized output feedback,” IEEE Trans. Automat. Contr., vol. 44, pp. 148-152, 1999.
[99] Wang, H. O., Tanaka, K. and Griffin, M. F., “An approach to fuzzy control of nonlinear system: stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14-23, 1996.
[100]Wang, W. J. and Luoh, L., “Stability and stabilization of fuzzy large scale systems,” IEEE Trans. Fuzzy Syst., vol. 12, pp. 309-315, 2004.
[101]Weinmann A., Uncertain models and robust control. New York: Springer Verlag, 1991.
[102]Wen, C. and Fong, I. K., “Parametric uncertainty bounds for performance robustness of linear systems with output feedback,” IEE Proc. Contr. Theory Appl. vol. 143, pp. 509-513, 1996.
[103]Yoneyama, J., Nishikawa, M., Katayama, H. and Ichikawa, A., “Output stabilization of Takagi-Sugeno fuzzy systems,” Fuzzy Sets and Systems, vol. 111, pp. 253-266, 2000.
[104]Yoneyama, J., Nishikawa, M., Katayama, H. and Ichikawa, A., “Design of utput feedback controllers for Takagi-Sugeno fuzzy systems,” Fuzzy Sets and Systems, vol. 121, pp. 127-148, 2001.
[105]Zak, S. H., “Stabilizing fuzzy system models using linear controllers,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 236-240, 1999.
[106]Zhang, F., Wang, Q.-G. and Lee, T. H. “On the design of multivariable PID controllers via LMI approach,” Automatica, vol. 38, pp. 517-526,
2002.
[107]Zhang, J. M., Li, R. H. and Zhang P. A., “Stability analysis and symmetric design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 120, pp. 65-72, 2001.
[108]Zhou, K. and Doyle, J. C. Essentials of robust control. Upper Saddle River, NJ: Prentice-Hall, 1998.