LMI在一类模糊控制系统稳定性分析中的应用
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摘要
线性矩阵不等式(LMI)方法是一种凸优化方法。由于内点法等算法日益成熟，并成为解决LMI问题的强有力的工具，线性矩阵不等式方法在控制理论中得到了越来越广泛的应用和发展。T-S模型是模糊控制方法中最流行的和最有发展前途的研究平台之一，而模糊控制系统的稳定性分析和设计方法一直是最重要的研究课题。本论文在归纳了线性矩阵不等式的特点和应用，总结了模糊系统稳定性分析的方法后，利用线性矩阵不等式(LMI)和并行分布补偿原理，针对非线性系统的T-S模型，提出了新的模糊控制系统的稳定性条件和分析方法。仿真结果验证了该方法的有效性。
In this paper, it is studied to use Linear Matrix Inequality (LMI) in fuzzy control field. As a convex method, LMI becomes more and more important in control field. First, this paper summarizes characteristics and uses of LMI. Then this paper concludes the methods in analyzing stability of fuzzy system. At last, this paper is concerned with the quadratic stability conditions of T-S fuzzy control systems that relax the existing conditions. New approaches to relaxed quadratic stability and stability conditions of fuzzy control systems are suggested. The new approaches relaxed the conservatism of the previous works by considering the interactions among the fuzzy subsystems hi a matrix and solving it via linear matrix inequalities. The examples and simulations can demonstrate the efficiency of the new approaches.
In this paper, it is studied to use Linear Matrix Inequality (LMI) in fuzzy control field. As a convex method, LMI becomes more and more important in control field. First, this paper summarizes characteristics and uses of LMI. Then this paper concludes the methods in analyzing stability of fuzzy system. At last, this paper is concerned with the quadratic stability conditions of T-S fuzzy control systems that relax the existing conditions. New approaches to relaxed quadratic stability and stability conditions of fuzzy control systems are suggested. The new approaches relaxed the conservatism of the previous works by considering the interactions among the fuzzy subsystems hi a matrix and solving it via linear matrix inequalities. The examples and simulations can demonstrate the efficiency of the new approaches.
引文
1．孙增圻．智能控制理论与技术．清华大学出版社．1997．4
2．胡寿松．自动控制原理(第三版)．国防工业出版社．1994
3．胡寿松，王执铨，胡维理．最有控制理论与系统(第一版)．东南大学出版社．1995
4．楼顺天，施阳．基于Matlab的系统分析与设计．西安交通大学出版社．1999．6
5．徐昕．Matlab工具箱应用指南．电子工业出版社．2000
6．蔡自兴．智能控制．国防工业出版社．1998
7．冯泰校．线性规划及其理论基础．中央广播电视大学出版社．2000．9
8．胡运权．运筹学教程．清华大学出版社．1998
9．何平，王鸿绪．模糊控制器的设计及应用．科学出版社．1997
10．谢振华，范训礼，曲建岭，王磊，耿昌茂．基于LMI的模糊控制器设计方法．控制与决策．Vol．15：549-552．2000
11.L.A.Zadeh. Fuzzy sets. Inform. Contr. Vol.8: 338-353. 1965
12.E.H.Mamdani. Applications of fuzzy algorithms for control of simple dynamic plant. Proc. Inst. Elect. Eng. Vol. 121: 1585-1588. 1974
13.L.P.Holmblad, J.J.Φsterguard. Control of a cement kiln by fuzzy logic. Fuzzy Information and Decision Processes. Noth-Holland. 1982
14.Stephen Boyd. Notes on Interior-Point Methods for Positive-Definite Programming. Stanford University. 1993
15.Tetsuya Iwasaki. Control System Design via LMIs. J.SICE. Vol.34: 164-169. 1995
16.P.Gahinet, P.Apkarian. A linear matrix inequality approach to H∞ control. Int. J. Robust Nonlinear Control. 421-448. 1994
17.T.Terano, K.Asai, M.Sugeno. Applied Fuzzy Systems. New York: Academic. 1994
18.R.Palm, D.Driankov, H.Hellendoom. Model Based Fuzzy Control. Berlin, Germany: Springer-Verlag. 1996
19.T.Takagi, M.Sugeno. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man. Vol. 15: 116-132. 1985
20． K.Tanaka , M.Sugeno . Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. Vol.45: 136-156． 1992
21． K.Tanaka . A Theory of Advanced Fuzzy Control . Tokyo , Japan : Kyuoristsu. 1994
22． K.Tanaka, M.Sano. A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer . IEEE Trans. Fuzzy Syst. 119-134． 1994． 5
23． H.O.Wang, K.Tanaka, M.F.Griffin. An approach to fuzzy control of nonlinear systems: Stability and design issues. IEEE Trans. Fuzzy. Syst. Vol.4: 14-23． 1996
24． K.Tanaka , T.Ikeda , H.O.Wang . A unified approach to controlling chaos via LMI-based fuzzy control system design. IEEE Trans. Circuits. Syst. Vol.45: 1021-1040． 1998
25． K.Tanaka , T.Kosaki . Design of a stable fuzzy controller for an articulated vehicle. IEEE Trans. Syst. Man. Cybern. Vol.27． 1997
26． M.A.L.Thathachar, P.Viswanath. On the Stability of fuzzy systems. IEEE Trans. Fuzzy Syst. Vol.5: 145-151． 1997
27． S.H.Zak. Stabilizing fuzzy system models using linear controllers. IEEE Trans. Fuzzy Syst. Vol.7: 236-240． 1999
28． X.Ma , Z.Sun , Y.He . Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans. Fuzzy Syst. Vol.6: 41-51． 1998
29． S.Cao, N.W.Rees, G.Feng. Analysis and design of fuzzy control systems using dynamic fuzzy-state space models. IEEE Trans. Fuzzy Syst. Vol.7: 192-200． 1999
30． K.S.Narendra , J.Balakrishnan . A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans. Automat. Contr. Vol.39: 2469-2471． 1994
31． K.Tanaka, T.Ikeda, H.O.Wang. Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. Vol.6: 250-265． 1998
32． KTanaka, T.Ikeda, H.O.Wang. An LMI approach to fuzzy controller designs based on the relaxed stability conditions . Proc. IEEE Int. Conf. Fuzzy Syst. Spain. 171-176． 1997
33． S.Boyd , L.Ghaoui, E.Feron, V.Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM. 1994
34． A.Graham. Kronecker Product and Matrix Calculus with Applications. New York: Wiley. 1981
35． D.Filev. Algebraic design of fuzzy logic controllers. IEEE int. Symp. Intell. Contr. 253-256． 1996
36． H.O.W.ang , K.Tanaka , M.F.Griffin . Parallel distributed compensation of nonlinear systems by Takagi and Sugeno's fuzzy model . Proc.FUZZ-IEEE'95． Yokohama, Japan. 531-538． 1995
37． K.Tanaka, T.Ikeda, H.O.Wang. Robust stabilization of a class of uncertain nonlinear system via fuzzy control. IEEE Trans. Fuzzy Syst. Vol.4: 1-13． 1996
38． S.Tsa, Y.Fung. Synthesis and stability analysis of linguistic fuzzy controlled systems. FUZZ-IEEE'98． Vol.1: 366-371． Anchorage, AK. 1998
39． D.G.Luenberger. Dynamic equations in descriptor form. IEEE Trans. Automat. Contr. Vol.AC-22: 312-321． 1977
40． S.Hong , R.Langari. Synthesis of an LMI-based fuzzy control system with guaranteed optimal H∞ performance. FUZZ-IEEE'98． Anchorage, AK. 422-427． 1998
41． H.O.Wang. T-S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems. 18th Int. Conf. North Amer. Fuzzy Inform. Processing Soc. San Antonio. 1999
42． T.Taniguchi , K.Tanaka, K.Yamafuji, H.O.Wang. Fuzzy descriptor systems: Stability analysis and design via LMIs . 1999 Amer. Contr. Conf. San Diego. 1827-1831． 1999
43． Euntai.Kim , Heejin.Lee . New Approaches to Relaxed Quadratic Stability Condition of Fuzzy Control Systems. IEEE Trans. Fuzzy Syst. Vol. 8: 523-534． 2000
2．胡寿松．自动控制原理(第三版)．国防工业出版社．1994
3．胡寿松，王执铨，胡维理．最有控制理论与系统(第一版)．东南大学出版社．1995
4．楼顺天，施阳．基于Matlab的系统分析与设计．西安交通大学出版社．1999．6
5．徐昕．Matlab工具箱应用指南．电子工业出版社．2000
6．蔡自兴．智能控制．国防工业出版社．1998
7．冯泰校．线性规划及其理论基础．中央广播电视大学出版社．2000．9
8．胡运权．运筹学教程．清华大学出版社．1998
9．何平，王鸿绪．模糊控制器的设计及应用．科学出版社．1997
10．谢振华，范训礼，曲建岭，王磊，耿昌茂．基于LMI的模糊控制器设计方法．控制与决策．Vol．15：549-552．2000
11.L.A.Zadeh. Fuzzy sets. Inform. Contr. Vol.8: 338-353. 1965
12.E.H.Mamdani. Applications of fuzzy algorithms for control of simple dynamic plant. Proc. Inst. Elect. Eng. Vol. 121: 1585-1588. 1974
13.L.P.Holmblad, J.J.Φsterguard. Control of a cement kiln by fuzzy logic. Fuzzy Information and Decision Processes. Noth-Holland. 1982
14.Stephen Boyd. Notes on Interior-Point Methods for Positive-Definite Programming. Stanford University. 1993
15.Tetsuya Iwasaki. Control System Design via LMIs. J.SICE. Vol.34: 164-169. 1995
16.P.Gahinet, P.Apkarian. A linear matrix inequality approach to H∞ control. Int. J. Robust Nonlinear Control. 421-448. 1994
17.T.Terano, K.Asai, M.Sugeno. Applied Fuzzy Systems. New York: Academic. 1994
18.R.Palm, D.Driankov, H.Hellendoom. Model Based Fuzzy Control. Berlin, Germany: Springer-Verlag. 1996
19.T.Takagi, M.Sugeno. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man. Vol. 15: 116-132. 1985
20． K.Tanaka , M.Sugeno . Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. Vol.45: 136-156． 1992
21． K.Tanaka . A Theory of Advanced Fuzzy Control . Tokyo , Japan : Kyuoristsu. 1994
22． K.Tanaka, M.Sano. A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer . IEEE Trans. Fuzzy Syst. 119-134． 1994． 5
23． H.O.Wang, K.Tanaka, M.F.Griffin. An approach to fuzzy control of nonlinear systems: Stability and design issues. IEEE Trans. Fuzzy. Syst. Vol.4: 14-23． 1996
24． K.Tanaka , T.Ikeda , H.O.Wang . A unified approach to controlling chaos via LMI-based fuzzy control system design. IEEE Trans. Circuits. Syst. Vol.45: 1021-1040． 1998
25． K.Tanaka , T.Kosaki . Design of a stable fuzzy controller for an articulated vehicle. IEEE Trans. Syst. Man. Cybern. Vol.27． 1997
26． M.A.L.Thathachar, P.Viswanath. On the Stability of fuzzy systems. IEEE Trans. Fuzzy Syst. Vol.5: 145-151． 1997
27． S.H.Zak. Stabilizing fuzzy system models using linear controllers. IEEE Trans. Fuzzy Syst. Vol.7: 236-240． 1999
28． X.Ma , Z.Sun , Y.He . Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans. Fuzzy Syst. Vol.6: 41-51． 1998
29． S.Cao, N.W.Rees, G.Feng. Analysis and design of fuzzy control systems using dynamic fuzzy-state space models. IEEE Trans. Fuzzy Syst. Vol.7: 192-200． 1999
30． K.S.Narendra , J.Balakrishnan . A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans. Automat. Contr. Vol.39: 2469-2471． 1994
31． K.Tanaka, T.Ikeda, H.O.Wang. Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. Vol.6: 250-265． 1998
32． KTanaka, T.Ikeda, H.O.Wang. An LMI approach to fuzzy controller designs based on the relaxed stability conditions . Proc. IEEE Int. Conf. Fuzzy Syst. Spain. 171-176． 1997
33． S.Boyd , L.Ghaoui, E.Feron, V.Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM. 1994
34． A.Graham. Kronecker Product and Matrix Calculus with Applications. New York: Wiley. 1981
35． D.Filev. Algebraic design of fuzzy logic controllers. IEEE int. Symp. Intell. Contr. 253-256． 1996
36． H.O.W.ang , K.Tanaka , M.F.Griffin . Parallel distributed compensation of nonlinear systems by Takagi and Sugeno's fuzzy model . Proc.FUZZ-IEEE'95． Yokohama, Japan. 531-538． 1995
37． K.Tanaka, T.Ikeda, H.O.Wang. Robust stabilization of a class of uncertain nonlinear system via fuzzy control. IEEE Trans. Fuzzy Syst. Vol.4: 1-13． 1996
38． S.Tsa, Y.Fung. Synthesis and stability analysis of linguistic fuzzy controlled systems. FUZZ-IEEE'98． Vol.1: 366-371． Anchorage, AK. 1998
39． D.G.Luenberger. Dynamic equations in descriptor form. IEEE Trans. Automat. Contr. Vol.AC-22: 312-321． 1977
40． S.Hong , R.Langari. Synthesis of an LMI-based fuzzy control system with guaranteed optimal H∞ performance. FUZZ-IEEE'98． Anchorage, AK. 422-427． 1998
41． H.O.Wang. T-S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems. 18th Int. Conf. North Amer. Fuzzy Inform. Processing Soc. San Antonio. 1999
42． T.Taniguchi , K.Tanaka, K.Yamafuji, H.O.Wang. Fuzzy descriptor systems: Stability analysis and design via LMIs . 1999 Amer. Contr. Conf. San Diego. 1827-1831． 1999
43． Euntai.Kim , Heejin.Lee . New Approaches to Relaxed Quadratic Stability Condition of Fuzzy Control Systems. IEEE Trans. Fuzzy Syst. Vol. 8: 523-534． 2000