不确定时滞非线性系统的鲁棒稳定性研究
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摘要
众所周知非线性系统广泛存在于客观世界，时滞为一般系统所固有，而不确定性在所难免，因此对不确定时滞非线性系统的鲁棒稳定性进行研究显得特别重要。故不确定时滞非线性系统的鲁棒稳定性研究是当前控制界研究的热点和难点之一。本文归纳了现有一些文献所提出的鲁棒稳定性定理，并在此基础上对时滞非线性系统的鲁棒稳定性分析及控制方面做了一些工作。
首先，在Lyapunov稳定性理论基础上，针对T-S动态模糊模型，归纳了利用线性矩阵不等式和并行分布补偿原理(PDC)及模糊模型相除补偿方法(CDF)分析模糊系统所得到的稳定性定理和保证系统稳定的模糊控制器设计，对上述方法做了一个比较分析和概括总结；基于T-S动态模糊模型，归纳了不确定连续模糊和离散模糊非线性系统的鲁捧稳定性定理。
其次，对具有时滞的非线性系统，利用模糊T-S模型对其进行模糊建模，采用分段Lyapunov函数方法和不等式处理技巧，得到系统鲁棒稳定的充分条件，并在此基础上提出了模糊控制器的设计方法，使得整个系统是鲁棒稳定的。
最后，介绍了一类同时存在两种不确定性的非线性时滞系统鲁棒稳定性条件及保证系统鲁棒稳定的模糊控制器设计方法，对该鲁棒稳定性条件的证明做了很详细的推导运算。
As we all know that nonlinear systems exist in the world extensively, time delays are inherent in systems and uncertainties are not avoidable. So it is important to investigate the robust stability analysis of uncertain nonlinear systems with time delays. Now it is one of the hotspots and also the difficulties in control domain. This paper summarizes some of theories of robust stability and based on the existing this, some work has been done in this paper on robust stability analysis and control of time-delay nonlinear systems.
Firstly, On the basis of Lyapunov stability theories, The theorems of stability using the approach of linear matrix inequality and the principle of PDC and the concept of CDF (compensation and division for fuzzy model) are summarized in this paper and a kind of fuzzy controller which makes whole systems gradual stabilizing is got based on the T-S fuzzy dynamic model. I compare the above-mentioned methods and draw a conclusion after carefully analyzing. And the theorems of robust stability for uncertain continuous and discrete fuzzy nonlinear systems are summarized based on the T-S fuzzy dynamic model.
Secondly, The fuzzy T-S model is used to approximate to time-delay nonlinear systems. And the sufficient condition of robust stability for time-delay nonlinear systems is put forward based on piecewise Lyapunov function and inequality method. In addition, a kind of fuzzy controller which makes whole systems stabilizing is designed in the paper.
At last, the robust stability condition of time-delay nonlinear systems with two kinds of uncertain is summarized and a kind of fuzzy controller that makes the whole systems robust stability is introduced. At the same time, detailed illation of proof for this robust stability condition is put forward in the paper.
首先，在Lyapunov稳定性理论基础上，针对T-S动态模糊模型，归纳了利用线性矩阵不等式和并行分布补偿原理(PDC)及模糊模型相除补偿方法(CDF)分析模糊系统所得到的稳定性定理和保证系统稳定的模糊控制器设计，对上述方法做了一个比较分析和概括总结；基于T-S动态模糊模型，归纳了不确定连续模糊和离散模糊非线性系统的鲁捧稳定性定理。
其次，对具有时滞的非线性系统，利用模糊T-S模型对其进行模糊建模，采用分段Lyapunov函数方法和不等式处理技巧，得到系统鲁棒稳定的充分条件，并在此基础上提出了模糊控制器的设计方法，使得整个系统是鲁棒稳定的。
最后，介绍了一类同时存在两种不确定性的非线性时滞系统鲁棒稳定性条件及保证系统鲁棒稳定的模糊控制器设计方法，对该鲁棒稳定性条件的证明做了很详细的推导运算。
As we all know that nonlinear systems exist in the world extensively, time delays are inherent in systems and uncertainties are not avoidable. So it is important to investigate the robust stability analysis of uncertain nonlinear systems with time delays. Now it is one of the hotspots and also the difficulties in control domain. This paper summarizes some of theories of robust stability and based on the existing this, some work has been done in this paper on robust stability analysis and control of time-delay nonlinear systems.
Firstly, On the basis of Lyapunov stability theories, The theorems of stability using the approach of linear matrix inequality and the principle of PDC and the concept of CDF (compensation and division for fuzzy model) are summarized in this paper and a kind of fuzzy controller which makes whole systems gradual stabilizing is got based on the T-S fuzzy dynamic model. I compare the above-mentioned methods and draw a conclusion after carefully analyzing. And the theorems of robust stability for uncertain continuous and discrete fuzzy nonlinear systems are summarized based on the T-S fuzzy dynamic model.
Secondly, The fuzzy T-S model is used to approximate to time-delay nonlinear systems. And the sufficient condition of robust stability for time-delay nonlinear systems is put forward based on piecewise Lyapunov function and inequality method. In addition, a kind of fuzzy controller which makes whole systems stabilizing is designed in the paper.
At last, the robust stability condition of time-delay nonlinear systems with two kinds of uncertain is summarized and a kind of fuzzy controller that makes the whole systems robust stability is introduced. At the same time, detailed illation of proof for this robust stability condition is put forward in the paper.
引文
[1] Cruz.J.B. and Perkins, W.R. A new approach to the sensitivity problem in multivariable feedback systems. IEEE Trans. Automat. Contr., 1964,vol.Ac-9:216-223
[2] Zames.G. Functional analysis applied to nonlinear feedback systems. IEEE Trans. Circuits and Systems, 1963,vol.cs-10:392～404
[3] Kalman.R.Z.. When is a linear control system optimal? Trans.ASME,ser.D.Basic Engr.,1964, Vol.86:51～60
[4] Rosenbrock,H.H.. The stability of multivariable systems. IEEE Trans. Automat. Contr.,1976, Vol.AC-17:105～107
[5] Youla,D.C.,Jabr,H.A. and Bongiorno.J.J. Modern wiener Hopf design of optimal controllers. part II :the multivariable case. IEEE Trans. Automat. Contr., 1976,Vol.AC-21:319～338
[6] .Kucera, V. Discrete linear control: the polynomial equation approach ..New York Wiley, 1979.
[7] Safonobv M.G.. Stability and robustness of multivariable feedback systems. MIT ,press,1980.
[8] Zames,G.. Feedback and optimal sensitivity :model reference transformations, multiplicative seminorms and approximate inverses. IEEE Trans. Automat. Contr., 1981,Vol.AC-26: 301-320
[9] Leitmann G. Guaranteed asymptotic stability for some linear systems with bounded uncertainties, ASME J. Dynamical Systems, Measurement and Control, 1979,101(1): 212-216
[10] Gutman S, Leitmann G. Stabilizing feedback control for dynamical systems with bounded uncertainty. Proc. IEEE Conf. Decision and Control, Clearwater, 1976: 94-99
[11] Barmish, B.R.and Leitmann,G.(1982). On ultimate bounded control of uncertain system in the absence of matching conditions. IEEE Trans. Auto. Contr.,27(1): 153-157.
[12] Barmish, B.R.(1983).Stabilization of uncertain system via linear control. IEEE Trans. Auto.Contr.,28: 848-850.
[13] Chen,Y.H.and Leitmann,G.(1987).Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control, 45(5): 1527-1542.
[14] Leitmann G, Pandey S. A controller for a class of mismatched uncertain systems. Proc. IEEE Conf. Decision and Control, Tampa, Florida .1989:59-64
[15] Peterson I.R, Hollot C.V. A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 1986,22(4):397～411
[16] Khargonekar,P.P., Petersen,I.R and Zhou,K.(1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and Hco control theory. IEEE Trans. Auto.Contr., 35(3): 356-361.
[17] Peres,P. L.D., Geromel, J.C.and Bemussou,J.(1991). Qudratic stabilizability of linear uncertain systems in convex-bounded domains. Automatica, 29(2):491～495.
[18] Germel J.C, Peres P. L.D and Bernussou J. On a convex parameter space method for linear control design of uncertain systems. SIAM J.Con.And Optim. 1991,29(2):381～402
[19] S.Phoojaruechanachai and K.Furata. Memotyless stabilization of uncertain linear systems including time-varying state delays. IEEE Transactions on Automatic Control, 1992, 37(7): 1022～1026
[20] Li Xi, C.E. Souza. Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica, 1997,33(9): 1657～1662
[21] Yu Li. Stability robustness bounds for linear systems with delayed perturbations. Journal of the Franklin institute, 1999, 336(5):755～765
[22] Xi Li, de Souza C. E. Delay-dependent robust stability and stabilization of Uncertain linear delay systems: a linear matrix inequality approach. IEEE Trans. Automat. Control, 1997,42(8):1144～1148
[23] J.S.Luo, P.P.J.VanDenbosch Independent delay stability criteria for uncertaion linear state space models. Automatica, 1997,33(2): 171～179
[24] S.Huang, W. Ren. Longitudinal control with time-delay in Platooning. IEEE Proc. Control Theory and Applications., 1998,148(2): 211～217
[25] H.Trinh, M.Aldeen. On the stability of linear systems with delayed perturbations. IEEE Transactions on Automatic Control, 1994,39(29): 1948～1951
[26] Li Xi, C.E.Souza.. Robust stabilization and H∞ control of uncertain linear time delay systems. Proceeding of IFAC 13th Triennial world Congress. San Francisco USA. 1996:113～118
[27] Vladimir L. Kharitonov, Daniel Melchor-Aguilar. On delay dependent stability conditions. Systems Contrrol& Letters, 2000,40(1):71～76
[28] S. Wong and D. Seborg. Control strategy for single-input single-output nonlinear systems with time delays. Int J. Control. 1988, 48:2303～2327
[29] M. Henson and D. Seborg. Time delay compensation for nonlinear process. Ind. Eng. Chen. Res. 1994, 33:1493～1500
[30] O Toshiki and Watanabe A. Input-output linearization of nonlinear systems with time delays in state variables. Int. J. Sys. Sci., 1998, 29:573～578
[31] 苏宏业，褚健 一类非线性时滞系统的稳定化控制器设计研究．自动化学报，1998，24(1)：40～35
[32] Mrdjan Jankovic. Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Contr., 2001, 46(7): 1048～1060
[33] 段广仁 线性系统理论(第二版)，哈尔滨工业大学出版社，2004年
[34] 段平 基于LMI的时滞系统的稳定性研究，硕士学位论文，西南交通大学，2004，4
[35] 王德进．H_2和H_∞。优化控制理论，哈尔滨工业大学出版社，2001年．
[36] 孙增圻 张再兴等 智能控制理论与技术，清华大学出版社，2004年，2
[37] R. A. Horn, C. A. Johson. Matrix Analysis. New York: Cambridge Univ. Press.1985: 132～138
[38] S. J. Xu and Mo Daroucch. On the Robustness of Linear Systems with Nonlinear Uncertain Parameters. Automatic, 1998, 34(8): 1005~1008
[39] I. R.. Petersen, C. V. Hollot. A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems. Automatic, 1986, 22(2): 397～411
[40] 陈东彦，徐世杰，邵成勋．线性时滞不确定系统的稳定性分析，控制理论与应用，1999，16(5)：754～756
[41] B. Xu. Stability Robustness Bounds for Linear Systems with Multiple time-Varying Delayed Perturbations. International Journal of Systems Science, 1997, 28: 1311～1317
[42] 蒋培刚，苏宏业，褚健．线性不确定时滞系统的时滞依赖鲁棒镇定方法研究，控制与决策，1999，14(2)：115～119
[43] 佟绍成，王涛等 模糊控制系统的设计及稳定性分析，科学出版社，2004
[44] 吴忠强，高静美等 一种新型模糊控制器设计与稳定性分析，系统仿真学报，2003，15(7)：1054～1056
[45] 吴忠强，高静美等 离散模糊系统的新型模糊控制器与观测器设计，系统仿真学报，2002，14(7)：955～957
[46] Cao S G, Rees N M, Feng G. Quadratic stability analysis and design of continuous-time fuzzy control systems [J]. International Journal of Systems Science, 1996,27(2): 193～203.
[47] Cao S G, Rees N M, Feng G. H∞ control of nonlinear continuous-time systems based on dynamical fuzzy models [J]. International Journal of Systems Science, 1996,27 (9): 823～830.
[48] 年晓红 Lurie型控制系统的鲁棒决定稳定性，控制理论与应用，1995,12(5)：641～645
[49] 魏新江，井元伟 一类不确定时滞系统的模糊鲁棒控制及其稳定性分析，东北大学学报，2004，25(12)：1123～1126
[2] Zames.G. Functional analysis applied to nonlinear feedback systems. IEEE Trans. Circuits and Systems, 1963,vol.cs-10:392～404
[3] Kalman.R.Z.. When is a linear control system optimal? Trans.ASME,ser.D.Basic Engr.,1964, Vol.86:51～60
[4] Rosenbrock,H.H.. The stability of multivariable systems. IEEE Trans. Automat. Contr.,1976, Vol.AC-17:105～107
[5] Youla,D.C.,Jabr,H.A. and Bongiorno.J.J. Modern wiener Hopf design of optimal controllers. part II :the multivariable case. IEEE Trans. Automat. Contr., 1976,Vol.AC-21:319～338
[6] .Kucera, V. Discrete linear control: the polynomial equation approach ..New York Wiley, 1979.
[7] Safonobv M.G.. Stability and robustness of multivariable feedback systems. MIT ,press,1980.
[8] Zames,G.. Feedback and optimal sensitivity :model reference transformations, multiplicative seminorms and approximate inverses. IEEE Trans. Automat. Contr., 1981,Vol.AC-26: 301-320
[9] Leitmann G. Guaranteed asymptotic stability for some linear systems with bounded uncertainties, ASME J. Dynamical Systems, Measurement and Control, 1979,101(1): 212-216
[10] Gutman S, Leitmann G. Stabilizing feedback control for dynamical systems with bounded uncertainty. Proc. IEEE Conf. Decision and Control, Clearwater, 1976: 94-99
[11] Barmish, B.R.and Leitmann,G.(1982). On ultimate bounded control of uncertain system in the absence of matching conditions. IEEE Trans. Auto. Contr.,27(1): 153-157.
[12] Barmish, B.R.(1983).Stabilization of uncertain system via linear control. IEEE Trans. Auto.Contr.,28: 848-850.
[13] Chen,Y.H.and Leitmann,G.(1987).Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control, 45(5): 1527-1542.
[14] Leitmann G, Pandey S. A controller for a class of mismatched uncertain systems. Proc. IEEE Conf. Decision and Control, Tampa, Florida .1989:59-64
[15] Peterson I.R, Hollot C.V. A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 1986,22(4):397～411
[16] Khargonekar,P.P., Petersen,I.R and Zhou,K.(1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and Hco control theory. IEEE Trans. Auto.Contr., 35(3): 356-361.
[17] Peres,P. L.D., Geromel, J.C.and Bemussou,J.(1991). Qudratic stabilizability of linear uncertain systems in convex-bounded domains. Automatica, 29(2):491～495.
[18] Germel J.C, Peres P. L.D and Bernussou J. On a convex parameter space method for linear control design of uncertain systems. SIAM J.Con.And Optim. 1991,29(2):381～402
[19] S.Phoojaruechanachai and K.Furata. Memotyless stabilization of uncertain linear systems including time-varying state delays. IEEE Transactions on Automatic Control, 1992, 37(7): 1022～1026
[20] Li Xi, C.E. Souza. Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica, 1997,33(9): 1657～1662
[21] Yu Li. Stability robustness bounds for linear systems with delayed perturbations. Journal of the Franklin institute, 1999, 336(5):755～765
[22] Xi Li, de Souza C. E. Delay-dependent robust stability and stabilization of Uncertain linear delay systems: a linear matrix inequality approach. IEEE Trans. Automat. Control, 1997,42(8):1144～1148
[23] J.S.Luo, P.P.J.VanDenbosch Independent delay stability criteria for uncertaion linear state space models. Automatica, 1997,33(2): 171～179
[24] S.Huang, W. Ren. Longitudinal control with time-delay in Platooning. IEEE Proc. Control Theory and Applications., 1998,148(2): 211～217
[25] H.Trinh, M.Aldeen. On the stability of linear systems with delayed perturbations. IEEE Transactions on Automatic Control, 1994,39(29): 1948～1951
[26] Li Xi, C.E.Souza.. Robust stabilization and H∞ control of uncertain linear time delay systems. Proceeding of IFAC 13th Triennial world Congress. San Francisco USA. 1996:113～118
[27] Vladimir L. Kharitonov, Daniel Melchor-Aguilar. On delay dependent stability conditions. Systems Contrrol& Letters, 2000,40(1):71～76
[28] S. Wong and D. Seborg. Control strategy for single-input single-output nonlinear systems with time delays. Int J. Control. 1988, 48:2303～2327
[29] M. Henson and D. Seborg. Time delay compensation for nonlinear process. Ind. Eng. Chen. Res. 1994, 33:1493～1500
[30] O Toshiki and Watanabe A. Input-output linearization of nonlinear systems with time delays in state variables. Int. J. Sys. Sci., 1998, 29:573～578
[31] 苏宏业，褚健 一类非线性时滞系统的稳定化控制器设计研究．自动化学报，1998，24(1)：40～35
[32] Mrdjan Jankovic. Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. Automat. Contr., 2001, 46(7): 1048～1060
[33] 段广仁 线性系统理论(第二版)，哈尔滨工业大学出版社，2004年
[34] 段平 基于LMI的时滞系统的稳定性研究，硕士学位论文，西南交通大学，2004，4
[35] 王德进．H_2和H_∞。优化控制理论，哈尔滨工业大学出版社，2001年．
[36] 孙增圻 张再兴等 智能控制理论与技术，清华大学出版社，2004年，2
[37] R. A. Horn, C. A. Johson. Matrix Analysis. New York: Cambridge Univ. Press.1985: 132～138
[38] S. J. Xu and Mo Daroucch. On the Robustness of Linear Systems with Nonlinear Uncertain Parameters. Automatic, 1998, 34(8): 1005~1008
[39] I. R.. Petersen, C. V. Hollot. A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems. Automatic, 1986, 22(2): 397～411
[40] 陈东彦，徐世杰，邵成勋．线性时滞不确定系统的稳定性分析，控制理论与应用，1999，16(5)：754～756
[41] B. Xu. Stability Robustness Bounds for Linear Systems with Multiple time-Varying Delayed Perturbations. International Journal of Systems Science, 1997, 28: 1311～1317
[42] 蒋培刚，苏宏业，褚健．线性不确定时滞系统的时滞依赖鲁棒镇定方法研究，控制与决策，1999，14(2)：115～119
[43] 佟绍成，王涛等 模糊控制系统的设计及稳定性分析，科学出版社，2004
[44] 吴忠强，高静美等 一种新型模糊控制器设计与稳定性分析，系统仿真学报，2003，15(7)：1054～1056
[45] 吴忠强，高静美等 离散模糊系统的新型模糊控制器与观测器设计，系统仿真学报，2002，14(7)：955～957
[46] Cao S G, Rees N M, Feng G. Quadratic stability analysis and design of continuous-time fuzzy control systems [J]. International Journal of Systems Science, 1996,27(2): 193～203.
[47] Cao S G, Rees N M, Feng G. H∞ control of nonlinear continuous-time systems based on dynamical fuzzy models [J]. International Journal of Systems Science, 1996,27 (9): 823～830.
[48] 年晓红 Lurie型控制系统的鲁棒决定稳定性，控制理论与应用，1995,12(5)：641～645
[49] 魏新江，井元伟 一类不确定时滞系统的模糊鲁棒控制及其稳定性分析，东北大学学报，2004，25(12)：1123～1126