基于T-S模型的奇异摄动系统的采样鲁棒H_∞控制
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摘要
基于T-S模型,本论文对基于采样数据的非线性奇异摄动系统的鲁棒H_∞控制问题进行研究.对于T-S模型中规则后件中的各个局部线性模型,利用“输入滞后”(inputdelay)方法,将基于采样数据的离散形式的控制律转化为带滞后的连续形式的控制律.在此控制律之下,局部闭环系统成为一个变时滞的连续奇异摄动系统.在对此闭环系统的稳定性和L_2增益特征进行分析的基础上,以LMI的形式给出了满足要求的控制律满足的条件.然后将各个局部线性模型的控制律利用模糊推理方法“合成”为系统总的控制律。
Based on T-S fuzzy model, state-feedback robust H_∞control problem for nonlinear singularly perturbed systems is studied. For each individual local linear model in T-S fuzzy model, the recent "input delay" approach to sampled-data control is applied, where the closed-loop system is represented as a continuous one with time-varying input delay. Linear matrix inequalities (LMIs) for solution of robust H_∞control problem are derived via input-output approach to stability and L_2-gain analysis of time-delay systems. The resulting overall controller, which is nonlinear in general, is a fuzzy "blending" of each individual linear controller.
Based on T-S fuzzy model, state-feedback robust H_∞control problem for nonlinear singularly perturbed systems is studied. For each individual local linear model in T-S fuzzy model, the recent "input delay" approach to sampled-data control is applied, where the closed-loop system is represented as a continuous one with time-varying input delay. Linear matrix inequalities (LMIs) for solution of robust H_∞control problem are derived via input-output approach to stability and L_2-gain analysis of time-delay systems. The resulting overall controller, which is nonlinear in general, is a fuzzy "blending" of each individual linear controller.
引文
[1] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design [M]. New York: Academic, 1986.
[2] KLIMUSHEV A I, KRASOVSKIIN K. Uncertain asymptotic stability of systems of differential equations with a small parameter in the derivative terms [J]. J. of Applied Mathematics and Mechanics, 1962, 25(9): 1011- 1025.
[3] SANDELL N R. Robust stability of systems with application to singular perturbations [J].Automatica, 1979, 15(4): 467-470.
[4] CHEN B, LIN C. On the stability bounds of singularly perturbed systems [J]. IEEE Trans.on Automatic Control, 1990, 35(11): 1265-1270.
[5] FENG W. Characterization and computation for the bound e* in linear time-invariant singularly perturbed systems [J]. Systems & Control Letters, 1988, 11(2): 195-202.
[6] SEN S, DATTA K B. Stability bounds of singularity perturbed systems [J]. IEEE Trans on Automatic Control , 1993, 38(2): 302-304.
[7] CHEN S J, LIN J L. Maximal stability bounds of singularly perturbed systems [J]. J. of Franklin Institute, 1999, 336(8): 1209-1218.
[8] LIN C L, CHEN B S. On the design of stabilizing controllers for singularly perturbed systems [J]. IEEE Trans. on Automatic Control, 1992, 37(11): 1828-1834.
[9] CHIOU J C, KUNG F C, LI T H S. An infinite e 2bound stabilization design for a class of singularly perturbed systems [J]. IEEE Trans on Circuits and Systems-I: Fundamental Theory and Applications, 1999, 46(12): 1507-1510.
[10] SHAHRUZ S M. Design of H_∞ optimal compensators for singularly perturbed systems [A]. Proc. of IEEE Conference on Decision and Control [C]. New York: IEEE Press, 1989,2397-2398.
[11] FRIDMAN E. Near-optimal H_∞ control of linear singularly perturbed systems [J]. IEEE Trans, on Automatic Control, 1996, 41(2): 236-240.
[12] MUKAIDANI H, XU H, MIZUKAMI K. Recursive approach of H_∞ control problems for singularly perturbed systems under perfect and imperfect state measurements [J]. Int. J.System Sciences, 1999, 30(5): 467-477.
[13] TAN W, LEUNG T, TU Q. H_∞ control for singularly perturbed systems [J]. Automatica,1998, 34(2): 255-260.
[14] TONG V V, MAHMOUD E S. H_∞ control for singularly perturbed systems via game theory [A]. Proc of Asilomar Conf on Signals , Systems and Computers [C]. 1992, 415-419.
[15] DRAGAN V. Asymptotic expansions for game- theoretic Riccati equations and stabilization with disturbance attenuation for singularly perturbed systems [J]. Systems & Control Letters,1993, 126(6): 455-463.
[16] PAN Z, BASAR T. H_∞ optimal control for singularly perturbed systems - Part I : perfect state measurements [J]. Automatica, 1993, 29(2): 401-423.
[17] PAN Z , BASAR T. H_∞ optimal control for singularly perturbed systems - Part II: Imperfect state measurements [J]. IEEE Trans. on Automatic Control, 1994, 39(2): 280-299.
[18] XU H, MIZUKAMI K. Nonstandard extension of H_∞ optimal con2 trol for singularly perturbed systems [A]. Proc. of the 7 th Int. Symp on Dynamic Games and Applications [C].Cambridge, Boston: Birkhauser, 1996, 931-948.
[19] DRAGAN V. H_∞ norms and distubance attenuation for systems with fast transient [J]. IEEE Trans. on Automatic Control, 1996, 41(5): 747-750.
[20] SHI P, DRAGAN V. Asymptotic H_∞ control of singularly perturbed systems with parametric uncertainties [J]. IEEE Trans. on Automatic Control, 1999, 44(9): 1738-1742.
[21] DJEMAI M, BARBOT J.-P. and KHALIL H. Digital multirate control for a class of nonlinear singualrly perturbed systems [J]. Int. J. Control, 1999, 72(10): 851-865.
[22] BAMIEH B, PEARSON J, FRANCIS B, and TANNENBAUM A. A lifting tech- nique for linear periodic systems [J]. System & Control Letters, 1991, 17: 79-88.
[23]YAMAMOTO Y,New approach to sampled-data control systems--A function space method [A].Proc.29th Conf.Decision and Control[C],Honolulu,HI,1990,1882-1887.
[24]PAN Z,BASAR T.H_∞ optimal control for singularly perturbed systems with sampled state measurements[A],in Advances in Dynamic Games and Applications[M],T.Basar and A.Haurie,Eds.Boston,MA:Birkh(a|¨)user,1994,1:23-55.
[25]FRIDMAN E,SEURET A,and RICHARD J P.Robust sampled-data stabilization of linear systems:An input delay approach[J].Automatica,2004,40:1441-1446.
[26]FRIDMAN E.Robust sampled-data H_∞ control of linear singularly perturbed systems[J].IEEE Tran.on Automatic Control,2006,51(3):470-475.
[27]GRUJIC L T.Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems[J].Int.J.Control,1981,33(3):481-504.
[28]SABERI A,KHALIL H.Quadratic-type Lyapunov functions for singularly perturbed systems [J].IEEE Trans.on Automatic Control,1984,29(6):542-550.
[29]KHALIL H K.Stability analysis of nonlinear multiparameter singularly perturbed systems [J].IEEE Trans.on Automatic Control,1987,32(3):260-263.
[30]TUAN H D,HOSOE S.Multivariable circle criteria for multiparameter singularly perturbed systems[J].IEEE Trans.on Automatic Control,2000,45(4):720-725.
[31]PAN Z,BASAR T.Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements[J].Int.J.Robust and Nonlinear Control,1996,6(7):585-608.
[32]SOBOLEV V A.Integral manifolds and decomposition of singularly perturbed systems[J].Systems & Control Letters,1984,5(3):169-179.
[33]KHORASANI K.On linearization of nonlinear singularly perturbed systems[J].IEEE Trans.on Automatic Control,1987,32(3):256-259.
[34]CHEN C C,HSIEH J G,SU J P.Global stabilization of nonlinear singularly perturbed systems with fast actuators:exact design manifold approach[J].Int.J.Control,1994,25(4):753-762.
[35]SHARKEY P M,O'REILL Y J.Exact design manifold control of a class of nonlinear singularly perturbed systems[J].IEEE Trans.on Automcatic Control,1987,32(10):933-935.
[36]MOALLEM M,KHORASANI K,PATEL R V.An integral manifold approach for tip-position tracking of flexible multi-link manipulators[J].IEEE Trans.on Robotics and Automation,1997,13(6):823-837.
[37]SUN Y J,CHENG J S,HSIEH J G.Feedback control of a class of nonlinear singularly perturbed systems with time delay[J].Int.J.of Systems Science,1996,27(6):589-596.
[38]Al-ASHOOR R A,KHORASANI K.A decentralized indirect adaptive control for a class of two-time-scale nonlinear systems with application to flexible-joint manipulators[J].IEEE Trans on Industrial Electronics,1999,46(5):1019-1029.
[39]TAKAGI T,SUGENO M.Fuzzy identification of systems and its applications to modelling and control[J].IEEE Trans.Syst.,Man,Cybern.,1985,15(1):116-132.
[40]TANAKA K,SUGENO M.Stability analysis and design of fuzzy control systems[J].Fuzzy Sets Syst.,1992,45(2):135-156.
[41]MA X-J,SUN Z-Q.Output tracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model[J].IEEE Trans.Syst.,Man,Cybern.,2000,30(1):47-59.
[42]BEGOVICH O,SANCHEZ E N,MALDONADO M.Takagi-Sugeno fuzzy scheme for realtime trajectory tracking of an underactuated robot[J].IEEE Trans.Contr.Syst.,2002,10(1):14-20.
[43]ASSAWINCHAICHOTE W,NGUANG S K,SHI P.H_∞ output feedback control design for uncertain fuzzy singularly perturbed systems:an LMI approach[J].Automatica,2004,40:2147-2152.
[44]LI T-H S,LIN K-J.Stabilization of singularly perturbed fuzzy systems]J].IEEE Trans.Fuzzy Syst.,2004,12(5):579-595.
[45]LIU H P,SUN F C,SUN Z Q.Stability analysis and synthesis of fuzzy singularly perturbed systems[J].IEEE Trans.Fuzzy Syst.,2005,13(2):273-284.
[46]LIU H P,SUN F C,HU Y N.H_∞ control for fuzzy singularly perturbed systems[J].Fuzzy Sets and Systems,2005,155:272-291
[47]LANGARI R,LI W.Analysis and efficient implementation of fuzzy logic control algorithms [A].Proc.of 1996 NAFIPS Biennial Conf.of the North American[C].1996,1-4.
[48]CHATURVDI P,RASTGOUFARD P,NALLAN H C.Piece-wise stability analysis of nonlinear robot arm via singular perturbations[A].Proc.of the 26th IEEE Southeastern Symponium on System Theory[C].IEEE Press,1994,59-63.
[49]SICILIANO B,BOOK W J.A singnular perturbation approach to control of lightweight flexible manipulators[J].Int.J.of Robotics Research,1988,17(4):79-90.
[50]邱晓红,高金源,张林昌.飞行轨迹指令综合跟踪控制器设计[J].航空学报,1997,18(4):407-411.
[51]HWANG J H.Drum boiler reduced model:a singular perturbation method[A].Proc.of Int.Conf.on Industrial Electronics,Control and Instrumentation[C].1994,3:1960-1964.
[52]SONER H M.Singular perturbations in manufacturing[J].SIAM J.Control and Optimization,1993,31(1):132-146.
[53]YANG J M,WU J,WU L.Nonlinear characteristics of the phase locked loop AC motor speed regulation system[J].J.of South China University of Technology,2000,8(9):32-39.
[54]CHANG H C,ALUKO M.Multi-scale analysis of exotic dynamics in surface catalyzed reactions-Ⅰ[J].Chemical Engineering Science,1984,39(1):37-50.
[55]CHOW J H.Time Scale Modelling of Dynamic Networks[M].New York:Springer-Verlag,1982.
[56]CHRISTOFIDES P D,DAOUTIDIS P.Feedback control of two-time-scale nonlinear systems [J].Int.J.Control,1996,63(5):965-994.
[57]DA D,CORLESS M.Exponential stability of a class of nonlinear singularly perturbed systems with marginally stable boundary layer systems[A].in Proc.the American Control Conference[C],San Francisco,California,USA,1993,pp.3101-3106.
[58]刘华平,孙富春,何克忠,孙增圻.奇异摄动控制系统:理论与应用[J].控制理论与应用,2003,20(1):1-7.
[59]HUANG Y-P,ZHOU K.Robust stability of uncertain time-delay systems[J].IEEE Trans.Autom.Control,2000,45(11):2169-2173.
[60]GU K,KHARITONOV V,CHEN J.Stability of Time-Delay Systems[M].Boston,MA:Birkh(a|¨)user,2003.
[61]WONHAM W M.Linear Multivariable Control:A Geometric Approach[M].2nd ed.,New York:Springer-verlag,1979.
[62]KHALIL H K.Nonlinear Systems[M].New York:Macmillan,1992.
[63]俞立.鲁棒控制-线性矩阵不等式处理方法[M].北京:清华大学出版社,2002.
[2] KLIMUSHEV A I, KRASOVSKIIN K. Uncertain asymptotic stability of systems of differential equations with a small parameter in the derivative terms [J]. J. of Applied Mathematics and Mechanics, 1962, 25(9): 1011- 1025.
[3] SANDELL N R. Robust stability of systems with application to singular perturbations [J].Automatica, 1979, 15(4): 467-470.
[4] CHEN B, LIN C. On the stability bounds of singularly perturbed systems [J]. IEEE Trans.on Automatic Control, 1990, 35(11): 1265-1270.
[5] FENG W. Characterization and computation for the bound e* in linear time-invariant singularly perturbed systems [J]. Systems & Control Letters, 1988, 11(2): 195-202.
[6] SEN S, DATTA K B. Stability bounds of singularity perturbed systems [J]. IEEE Trans on Automatic Control , 1993, 38(2): 302-304.
[7] CHEN S J, LIN J L. Maximal stability bounds of singularly perturbed systems [J]. J. of Franklin Institute, 1999, 336(8): 1209-1218.
[8] LIN C L, CHEN B S. On the design of stabilizing controllers for singularly perturbed systems [J]. IEEE Trans. on Automatic Control, 1992, 37(11): 1828-1834.
[9] CHIOU J C, KUNG F C, LI T H S. An infinite e 2bound stabilization design for a class of singularly perturbed systems [J]. IEEE Trans on Circuits and Systems-I: Fundamental Theory and Applications, 1999, 46(12): 1507-1510.
[10] SHAHRUZ S M. Design of H_∞ optimal compensators for singularly perturbed systems [A]. Proc. of IEEE Conference on Decision and Control [C]. New York: IEEE Press, 1989,2397-2398.
[11] FRIDMAN E. Near-optimal H_∞ control of linear singularly perturbed systems [J]. IEEE Trans, on Automatic Control, 1996, 41(2): 236-240.
[12] MUKAIDANI H, XU H, MIZUKAMI K. Recursive approach of H_∞ control problems for singularly perturbed systems under perfect and imperfect state measurements [J]. Int. J.System Sciences, 1999, 30(5): 467-477.
[13] TAN W, LEUNG T, TU Q. H_∞ control for singularly perturbed systems [J]. Automatica,1998, 34(2): 255-260.
[14] TONG V V, MAHMOUD E S. H_∞ control for singularly perturbed systems via game theory [A]. Proc of Asilomar Conf on Signals , Systems and Computers [C]. 1992, 415-419.
[15] DRAGAN V. Asymptotic expansions for game- theoretic Riccati equations and stabilization with disturbance attenuation for singularly perturbed systems [J]. Systems & Control Letters,1993, 126(6): 455-463.
[16] PAN Z, BASAR T. H_∞ optimal control for singularly perturbed systems - Part I : perfect state measurements [J]. Automatica, 1993, 29(2): 401-423.
[17] PAN Z , BASAR T. H_∞ optimal control for singularly perturbed systems - Part II: Imperfect state measurements [J]. IEEE Trans. on Automatic Control, 1994, 39(2): 280-299.
[18] XU H, MIZUKAMI K. Nonstandard extension of H_∞ optimal con2 trol for singularly perturbed systems [A]. Proc. of the 7 th Int. Symp on Dynamic Games and Applications [C].Cambridge, Boston: Birkhauser, 1996, 931-948.
[19] DRAGAN V. H_∞ norms and distubance attenuation for systems with fast transient [J]. IEEE Trans. on Automatic Control, 1996, 41(5): 747-750.
[20] SHI P, DRAGAN V. Asymptotic H_∞ control of singularly perturbed systems with parametric uncertainties [J]. IEEE Trans. on Automatic Control, 1999, 44(9): 1738-1742.
[21] DJEMAI M, BARBOT J.-P. and KHALIL H. Digital multirate control for a class of nonlinear singualrly perturbed systems [J]. Int. J. Control, 1999, 72(10): 851-865.
[22] BAMIEH B, PEARSON J, FRANCIS B, and TANNENBAUM A. A lifting tech- nique for linear periodic systems [J]. System & Control Letters, 1991, 17: 79-88.
[23]YAMAMOTO Y,New approach to sampled-data control systems--A function space method [A].Proc.29th Conf.Decision and Control[C],Honolulu,HI,1990,1882-1887.
[24]PAN Z,BASAR T.H_∞ optimal control for singularly perturbed systems with sampled state measurements[A],in Advances in Dynamic Games and Applications[M],T.Basar and A.Haurie,Eds.Boston,MA:Birkh(a|¨)user,1994,1:23-55.
[25]FRIDMAN E,SEURET A,and RICHARD J P.Robust sampled-data stabilization of linear systems:An input delay approach[J].Automatica,2004,40:1441-1446.
[26]FRIDMAN E.Robust sampled-data H_∞ control of linear singularly perturbed systems[J].IEEE Tran.on Automatic Control,2006,51(3):470-475.
[27]GRUJIC L T.Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems[J].Int.J.Control,1981,33(3):481-504.
[28]SABERI A,KHALIL H.Quadratic-type Lyapunov functions for singularly perturbed systems [J].IEEE Trans.on Automatic Control,1984,29(6):542-550.
[29]KHALIL H K.Stability analysis of nonlinear multiparameter singularly perturbed systems [J].IEEE Trans.on Automatic Control,1987,32(3):260-263.
[30]TUAN H D,HOSOE S.Multivariable circle criteria for multiparameter singularly perturbed systems[J].IEEE Trans.on Automatic Control,2000,45(4):720-725.
[31]PAN Z,BASAR T.Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements[J].Int.J.Robust and Nonlinear Control,1996,6(7):585-608.
[32]SOBOLEV V A.Integral manifolds and decomposition of singularly perturbed systems[J].Systems & Control Letters,1984,5(3):169-179.
[33]KHORASANI K.On linearization of nonlinear singularly perturbed systems[J].IEEE Trans.on Automatic Control,1987,32(3):256-259.
[34]CHEN C C,HSIEH J G,SU J P.Global stabilization of nonlinear singularly perturbed systems with fast actuators:exact design manifold approach[J].Int.J.Control,1994,25(4):753-762.
[35]SHARKEY P M,O'REILL Y J.Exact design manifold control of a class of nonlinear singularly perturbed systems[J].IEEE Trans.on Automcatic Control,1987,32(10):933-935.
[36]MOALLEM M,KHORASANI K,PATEL R V.An integral manifold approach for tip-position tracking of flexible multi-link manipulators[J].IEEE Trans.on Robotics and Automation,1997,13(6):823-837.
[37]SUN Y J,CHENG J S,HSIEH J G.Feedback control of a class of nonlinear singularly perturbed systems with time delay[J].Int.J.of Systems Science,1996,27(6):589-596.
[38]Al-ASHOOR R A,KHORASANI K.A decentralized indirect adaptive control for a class of two-time-scale nonlinear systems with application to flexible-joint manipulators[J].IEEE Trans on Industrial Electronics,1999,46(5):1019-1029.
[39]TAKAGI T,SUGENO M.Fuzzy identification of systems and its applications to modelling and control[J].IEEE Trans.Syst.,Man,Cybern.,1985,15(1):116-132.
[40]TANAKA K,SUGENO M.Stability analysis and design of fuzzy control systems[J].Fuzzy Sets Syst.,1992,45(2):135-156.
[41]MA X-J,SUN Z-Q.Output tracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model[J].IEEE Trans.Syst.,Man,Cybern.,2000,30(1):47-59.
[42]BEGOVICH O,SANCHEZ E N,MALDONADO M.Takagi-Sugeno fuzzy scheme for realtime trajectory tracking of an underactuated robot[J].IEEE Trans.Contr.Syst.,2002,10(1):14-20.
[43]ASSAWINCHAICHOTE W,NGUANG S K,SHI P.H_∞ output feedback control design for uncertain fuzzy singularly perturbed systems:an LMI approach[J].Automatica,2004,40:2147-2152.
[44]LI T-H S,LIN K-J.Stabilization of singularly perturbed fuzzy systems]J].IEEE Trans.Fuzzy Syst.,2004,12(5):579-595.
[45]LIU H P,SUN F C,SUN Z Q.Stability analysis and synthesis of fuzzy singularly perturbed systems[J].IEEE Trans.Fuzzy Syst.,2005,13(2):273-284.
[46]LIU H P,SUN F C,HU Y N.H_∞ control for fuzzy singularly perturbed systems[J].Fuzzy Sets and Systems,2005,155:272-291
[47]LANGARI R,LI W.Analysis and efficient implementation of fuzzy logic control algorithms [A].Proc.of 1996 NAFIPS Biennial Conf.of the North American[C].1996,1-4.
[48]CHATURVDI P,RASTGOUFARD P,NALLAN H C.Piece-wise stability analysis of nonlinear robot arm via singular perturbations[A].Proc.of the 26th IEEE Southeastern Symponium on System Theory[C].IEEE Press,1994,59-63.
[49]SICILIANO B,BOOK W J.A singnular perturbation approach to control of lightweight flexible manipulators[J].Int.J.of Robotics Research,1988,17(4):79-90.
[50]邱晓红,高金源,张林昌.飞行轨迹指令综合跟踪控制器设计[J].航空学报,1997,18(4):407-411.
[51]HWANG J H.Drum boiler reduced model:a singular perturbation method[A].Proc.of Int.Conf.on Industrial Electronics,Control and Instrumentation[C].1994,3:1960-1964.
[52]SONER H M.Singular perturbations in manufacturing[J].SIAM J.Control and Optimization,1993,31(1):132-146.
[53]YANG J M,WU J,WU L.Nonlinear characteristics of the phase locked loop AC motor speed regulation system[J].J.of South China University of Technology,2000,8(9):32-39.
[54]CHANG H C,ALUKO M.Multi-scale analysis of exotic dynamics in surface catalyzed reactions-Ⅰ[J].Chemical Engineering Science,1984,39(1):37-50.
[55]CHOW J H.Time Scale Modelling of Dynamic Networks[M].New York:Springer-Verlag,1982.
[56]CHRISTOFIDES P D,DAOUTIDIS P.Feedback control of two-time-scale nonlinear systems [J].Int.J.Control,1996,63(5):965-994.
[57]DA D,CORLESS M.Exponential stability of a class of nonlinear singularly perturbed systems with marginally stable boundary layer systems[A].in Proc.the American Control Conference[C],San Francisco,California,USA,1993,pp.3101-3106.
[58]刘华平,孙富春,何克忠,孙增圻.奇异摄动控制系统:理论与应用[J].控制理论与应用,2003,20(1):1-7.
[59]HUANG Y-P,ZHOU K.Robust stability of uncertain time-delay systems[J].IEEE Trans.Autom.Control,2000,45(11):2169-2173.
[60]GU K,KHARITONOV V,CHEN J.Stability of Time-Delay Systems[M].Boston,MA:Birkh(a|¨)user,2003.
[61]WONHAM W M.Linear Multivariable Control:A Geometric Approach[M].2nd ed.,New York:Springer-verlag,1979.
[62]KHALIL H K.Nonlinear Systems[M].New York:Macmillan,1992.
[63]俞立.鲁棒控制-线性矩阵不等式处理方法[M].北京:清华大学出版社,2002.