不确定非线性时滞系统的输出反馈控制研究
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摘要
非线性时滞系统广泛存在于现实生产和生活中，时滞的出现给系统良好的性能带来严重破坏.因此，如何处理存在时滞情况下的非线性系统的稳定性以及控制律的设计问题，成为控制理论和控制工程中研究的主要领域.与此同时，伴随着含有不确定性的非线性系统研究的深入，有关非线性不确定系统的鲁棒控制、自适应控制等研究也取得了很好的成果.另外，有关时滞问题的研究也在研究人员的努力下逐步得到完善，其中主要包括Lyapunov-Krasovskii和Lyapunov-Razumikhin方法.因此，结合以上方法和理论，含有不确定性的非线性时滞系统的研究有了更为广阔的空间.
本文主要研究了两类非线性时滞系统输出反馈控制问题，在这两类非线性时滞系统中，带有广泛的不确定性，主要包含未知不确定性参数、不确定控制系数、未建模逆动态、不确定非线性时滞项、未知时变时滞项、不确定输出动态等.针对这些特征，结合反步设计法、改变供能函数思想、 Nussbaum函数法、在线估计方法，构造合适的控制器，使其形成的闭环系统在一定意义下是稳定的.具体地讲，本文主要研究了以下两类问题：
1.一类非线性时滞系统的鲁棒自适应控制问题
在第三章中，研究对象是一类含有未建模动态、不确定参数、不确定非线性时滞项的非线性时滞系统的鲁棒自适应控制器设计问题.该问题研究的关键在于如何构造一个关于不确定参数和输出变量的观测器，需要在此基础上结合反步法来设计控制律和自适应律，其中自适应律的设计利用了调节函数的思想和方法.利用改变供能函数的思想有效地处理了未建模动态的影响，并通过构造级联系统的Lyapunov函数来证明由控制器所构造的闭环系统在半全局有界的意义下是稳定的.
2.带有未知控制系数的非线性时滞系统的鲁棒自适应控制问题
在第四章中，研究对象是带有未知控制系数的非线性时滞系统，同时它还包含未建模逆动态、不确定输出动态.通过构造一合适的观测器，结合原有系统并利用反步法构造系统的控制律.对于未知控制系数，将利用Nussbaum技术，该方法对于系统的控制律的设计起到关键的作用.对于未建模动态，同样利用改变供能函数思想的方法来有效地处理它的影响.在这一系统中，显著特征是在输出变量中含有未建模输出动态，这对于构造整个级联系统的Lyapunov函数来说带来了一定的复杂性，需要增加关于未建模输出的Lyapunov函数项.通过利用这一Lyapunov函数，有效地证明了所设计的控制律对于整个闭环系统而言，使得所有信号是有界的，并且输出变量y(t)是渐近趋于零的.
Nonlinear time delay systems have existed in actual production and life widely，whichdestroy good performance for nonlinear system seriously. So, how to deal with stability anddesign control law for nonlinear time delay systems, has been important research area for controltheory and control engineering. At the same time, with the development of deeply researchingfor nonlinear systems including uncertainty, it gets good research achievements on robust controland adaptive control for nonlinear system. In addition， the technology on time delay problemalso become more perfect with gradually researching by researchers， including bothLyapunov-Krasovskii and Lyapunov-Razumikhin method. Therefore, together with the waysand theory, it has widely space to research nonlinear uncertainty time-delay system.
The text mainly researches output feedback control problem of two kinds of nonlinear timedelay systems. The two kinds of systems include wide uncertainties, for example, uncertainparameter, uncertain control coefficient, unmodeled inverse dynamics, uncertain nonlinear timedelay, unknown time-variant time delay, uncertain output dynamics and so on. For the systems,together with backstepping method,changing supply function method, Nussbaum functionmethod and online estimating method, it can get proper controller, which makes the closedsystem stability in a certain sense.Specifically speaking，the text mainly researches two kinds ofproblems：
1. Robust adaptive control problem for a class of nonlinear time delay system
In chapter3，the research problem is on robust adaptive control problem of nonlinear timedelay system including unmodeled dynamics, uncertain parameter，uncertain nonlinear timedelay.The research key of the problem is how to structure a observer on output variable anduncertain parameter. The designing of control law and adaptive law is used backsteppingmethod. And designing adaptive law is made use of tuning function idea. With changing supplyfunction idea, it can deal with the effect of unmodeled dynamics efficiently, and using theLyapunov function of system, it can prove the designed control law is stable in the semi-globalmeaning for the closed loop system.
2. Robust control problem for nonlinear time delay system with unknown control coefficient
In chapter4， research object contains not only unknown control coefficient but alsounmodeled inverse dynamics and uncertain output dynamics. With a proper observer andbackstepping designing method, it can formulate the control law of system. For unknown controlcoefficient, the technology of Nussbaum function is vital for the control law of designing. Forthe unmodeled dynamics, it is the same as the first part with changing supply function idea to deal with. In the system, obvious feature is unmodeled output dynamics in the output variable. Itbrings some difficulty for Lyapunov function structure, which needs to increase a term forLyapunov function. Using it, it can prove the designed control law makes all the signals boundedand output variable y(t) intend to zero gradually for the closed loop system.
本文主要研究了两类非线性时滞系统输出反馈控制问题，在这两类非线性时滞系统中，带有广泛的不确定性，主要包含未知不确定性参数、不确定控制系数、未建模逆动态、不确定非线性时滞项、未知时变时滞项、不确定输出动态等.针对这些特征，结合反步设计法、改变供能函数思想、 Nussbaum函数法、在线估计方法，构造合适的控制器，使其形成的闭环系统在一定意义下是稳定的.具体地讲，本文主要研究了以下两类问题：
1.一类非线性时滞系统的鲁棒自适应控制问题
在第三章中，研究对象是一类含有未建模动态、不确定参数、不确定非线性时滞项的非线性时滞系统的鲁棒自适应控制器设计问题.该问题研究的关键在于如何构造一个关于不确定参数和输出变量的观测器，需要在此基础上结合反步法来设计控制律和自适应律，其中自适应律的设计利用了调节函数的思想和方法.利用改变供能函数的思想有效地处理了未建模动态的影响，并通过构造级联系统的Lyapunov函数来证明由控制器所构造的闭环系统在半全局有界的意义下是稳定的.
2.带有未知控制系数的非线性时滞系统的鲁棒自适应控制问题
在第四章中，研究对象是带有未知控制系数的非线性时滞系统，同时它还包含未建模逆动态、不确定输出动态.通过构造一合适的观测器，结合原有系统并利用反步法构造系统的控制律.对于未知控制系数，将利用Nussbaum技术，该方法对于系统的控制律的设计起到关键的作用.对于未建模动态，同样利用改变供能函数思想的方法来有效地处理它的影响.在这一系统中，显著特征是在输出变量中含有未建模输出动态，这对于构造整个级联系统的Lyapunov函数来说带来了一定的复杂性，需要增加关于未建模输出的Lyapunov函数项.通过利用这一Lyapunov函数，有效地证明了所设计的控制律对于整个闭环系统而言，使得所有信号是有界的，并且输出变量y(t)是渐近趋于零的.
Nonlinear time delay systems have existed in actual production and life widely，whichdestroy good performance for nonlinear system seriously. So, how to deal with stability anddesign control law for nonlinear time delay systems, has been important research area for controltheory and control engineering. At the same time, with the development of deeply researchingfor nonlinear systems including uncertainty, it gets good research achievements on robust controland adaptive control for nonlinear system. In addition， the technology on time delay problemalso become more perfect with gradually researching by researchers， including bothLyapunov-Krasovskii and Lyapunov-Razumikhin method. Therefore, together with the waysand theory, it has widely space to research nonlinear uncertainty time-delay system.
The text mainly researches output feedback control problem of two kinds of nonlinear timedelay systems. The two kinds of systems include wide uncertainties, for example, uncertainparameter, uncertain control coefficient, unmodeled inverse dynamics, uncertain nonlinear timedelay, unknown time-variant time delay, uncertain output dynamics and so on. For the systems,together with backstepping method,changing supply function method, Nussbaum functionmethod and online estimating method, it can get proper controller, which makes the closedsystem stability in a certain sense.Specifically speaking，the text mainly researches two kinds ofproblems：
1. Robust adaptive control problem for a class of nonlinear time delay system
In chapter3，the research problem is on robust adaptive control problem of nonlinear timedelay system including unmodeled dynamics, uncertain parameter，uncertain nonlinear timedelay.The research key of the problem is how to structure a observer on output variable anduncertain parameter. The designing of control law and adaptive law is used backsteppingmethod. And designing adaptive law is made use of tuning function idea. With changing supplyfunction idea, it can deal with the effect of unmodeled dynamics efficiently, and using theLyapunov function of system, it can prove the designed control law is stable in the semi-globalmeaning for the closed loop system.
2. Robust control problem for nonlinear time delay system with unknown control coefficient
In chapter4， research object contains not only unknown control coefficient but alsounmodeled inverse dynamics and uncertain output dynamics. With a proper observer andbackstepping designing method, it can formulate the control law of system. For unknown controlcoefficient, the technology of Nussbaum function is vital for the control law of designing. Forthe unmodeled dynamics, it is the same as the first part with changing supply function idea to deal with. In the system, obvious feature is unmodeled output dynamics in the output variable. Itbrings some difficulty for Lyapunov function structure, which needs to increase a term forLyapunov function. Using it, it can prove the designed control law makes all the signals boundedand output variable y(t) intend to zero gradually for the closed loop system.
引文
[1]刘祖润.线性时滞系统一致渐进稳定的时滞界[J].电路与系统学报,1999,4(3):92-96.
[2]张先明,吴敏，何勇.线性时滞系统的时滞相关稳定性[J].电路与系统学报,2003,3(8):118-120.
[3] C. Hua, C. Long, and X. Guan. Robust stabilization of uncertain dynamic time delay systemswith unknown bounds of uncertainties [C]. Proc. Amer. Control Conf.,2002,3365-3370.
[4] C. Hua, X. Guan, and S. Peng. Robust stabilization of a class of nonlinear time-delaysystems[J]. Appl. Math. Comput.,2004,3(155):737-752.
[5] C. Hua, X. Guan, and S.Peng. Robust backstepping control for a class of time delayedsystems[J]. IEEE Trans. Autom. Control,2005,(50):894-899.
[6] M.Jankovic. Control Lyapunov-Razumikhin functions and robust stabilization of time delaysystems [J].IEEE Trans. Autom. Control,2001,7(46):1048-1060.
[7]M. Jankovic. Stabilization of nonlinear time delay systems with delay-independentfeedback[C]. America: American control conference,2005.
[8]W.Chen,Z.Guan, and X.Lu. Delay-dependent output feedback guaranteed cost control foruncertain time-delay systems [J].Autom.,2004,40(10):1263-1268.
[9]Y.Fu. Output feedback stabilization for time-delay nonlinear systems [J].Acta AutomaticSinica,2002,28(5):802-805.
[10]Krstic M, Kanellakopulos I, and Kokotovic P V. Nonlinear and adaptive control design
[M].NewYork:Wiley,1995.
[11] C. Hua, G. Feng., and X. Guan.Robust controller design of a class of nonlinear time delaysystems via backstepping method[J]. Automatica,2008，(44):567-573.
[12]汤红吉,张小美,陈树中.一类不确定离散时滞系统的鲁棒状态反馈控制[J].华东师范大学学报(自然科学版),2005,5:97-102.
[13]丛屾,费树岷,费吉庆.时滞切换系统动态反馈H_∞控制[J].控制理论与应用,2007,1,(24):132-136.
[14]刘坤,刘贺平,赵立英.时滞随机系统的时滞相关稳定性[J].北京科技大学学报,2009,4(31):516-519.
[15]邵诚.具有未建模动态系统的鲁棒自适应控制[J].辽宁大学学报(自然科学版),1995,1(22):47-55.
[16] Z. Jiang, Praly.Design of robust adaptive controllers for nonlinear systems with dynamicuncertainties[J]. Automatic,1998,7(34):825-840.
[17] Z. Jiang. A combined backstepping and small-gain approach to adaptive output feedbackcontrol[J]. Automatica,1999,(35):1131-1139.
[18] M．Jankovic, R.Sepulchre,and P.Kokotovic, Clf based designs with robustnedd to dynamicinput uncertainties[J].Systems and Control Letters,1999,(37):45-54.
[19] Z. Jiang, Iven Mareels.Robust nonlinear integral control[J].IEEE Transactions onautomatic control,2001,8(46):1336-1342.
[20] E. Sontag, A. Teel.Changing supply functions in input/state stable systems[J]. IEEETransactions on automatic control,1995,(40):1476-1478.
[21] L.Praly, Y.Wang.Stabilization in spite of matched unmodeled dynamics and an equivalentdefinition of input-to-state stability[J].Mathematics of Control, Signals and Systems,1996,9:1-33.
[22] Z. Jiang, I. Mareels, and David J. Hill. A unifying framework for global regulation vianonlinear output feedback:from iss to iIss[J]. IEEE Transactions on automatic control,2004,49(4):549-561.
[23] L.Praly, Z.Jiang. Stabilization by output feedback for systems with ISS inversedynamics[J].System control letter,1993,21:19-33.
[24] S.Tong,Y.Li. Observer-based fuzzy adaptive robust control of nonlinear systems with timedelays and unmodeled dynamics[J].Neurocomputing,2010,(74):369-378.
[25]黄在堂.时滞随机微分系统的动力学性质研究[D].华南理工大学,2011.
[26]袁琦,徐迎晓.多智能体协同进化算法及其在物流配送中的应用[J].科技管理研究,2010,3:144-146.
[27]D.Angeli, E.D.Sontag, and Y.Wang.A characterization of integral input to statestability[J].IEEE Trans. Automat. Control,2000,45:1082-1097.
[28] C.Hua,X.Guan, and S.Peng.Robust output feedback tracking control for time-delaynonlinear systems using neural network[J]. IEEE Transactions on neural networks,2007,18(2):495-505.
[29] Z.Yuan, Z.Wang, and Q.Chen. Adaptive output feedback control for uncertainnonholonomic chained systems [J]. Cent.South Univ.Technol,2010,17:572-579.
[30] Z.Xi, G.Feng, and Z.Jiang. Output feedback exponential stabilization of uncertain chainedsystems [J]. Journal of the Franklin Institute,2007,34:36-57.
[31] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic. Nonlinear and adptive controldesign[M].Canada,1995.
[2]张先明,吴敏，何勇.线性时滞系统的时滞相关稳定性[J].电路与系统学报,2003,3(8):118-120.
[3] C. Hua, C. Long, and X. Guan. Robust stabilization of uncertain dynamic time delay systemswith unknown bounds of uncertainties [C]. Proc. Amer. Control Conf.,2002,3365-3370.
[4] C. Hua, X. Guan, and S. Peng. Robust stabilization of a class of nonlinear time-delaysystems[J]. Appl. Math. Comput.,2004,3(155):737-752.
[5] C. Hua, X. Guan, and S.Peng. Robust backstepping control for a class of time delayedsystems[J]. IEEE Trans. Autom. Control,2005,(50):894-899.
[6] M.Jankovic. Control Lyapunov-Razumikhin functions and robust stabilization of time delaysystems [J].IEEE Trans. Autom. Control,2001,7(46):1048-1060.
[7]M. Jankovic. Stabilization of nonlinear time delay systems with delay-independentfeedback[C]. America: American control conference,2005.
[8]W.Chen,Z.Guan, and X.Lu. Delay-dependent output feedback guaranteed cost control foruncertain time-delay systems [J].Autom.,2004,40(10):1263-1268.
[9]Y.Fu. Output feedback stabilization for time-delay nonlinear systems [J].Acta AutomaticSinica,2002,28(5):802-805.
[10]Krstic M, Kanellakopulos I, and Kokotovic P V. Nonlinear and adaptive control design
[M].NewYork:Wiley,1995.
[11] C. Hua, G. Feng., and X. Guan.Robust controller design of a class of nonlinear time delaysystems via backstepping method[J]. Automatica,2008，(44):567-573.
[12]汤红吉,张小美,陈树中.一类不确定离散时滞系统的鲁棒状态反馈控制[J].华东师范大学学报(自然科学版),2005,5:97-102.
[13]丛屾,费树岷,费吉庆.时滞切换系统动态反馈H_∞控制[J].控制理论与应用,2007,1,(24):132-136.
[14]刘坤,刘贺平,赵立英.时滞随机系统的时滞相关稳定性[J].北京科技大学学报,2009,4(31):516-519.
[15]邵诚.具有未建模动态系统的鲁棒自适应控制[J].辽宁大学学报(自然科学版),1995,1(22):47-55.
[16] Z. Jiang, Praly.Design of robust adaptive controllers for nonlinear systems with dynamicuncertainties[J]. Automatic,1998,7(34):825-840.
[17] Z. Jiang. A combined backstepping and small-gain approach to adaptive output feedbackcontrol[J]. Automatica,1999,(35):1131-1139.
[18] M．Jankovic, R.Sepulchre,and P.Kokotovic, Clf based designs with robustnedd to dynamicinput uncertainties[J].Systems and Control Letters,1999,(37):45-54.
[19] Z. Jiang, Iven Mareels.Robust nonlinear integral control[J].IEEE Transactions onautomatic control,2001,8(46):1336-1342.
[20] E. Sontag, A. Teel.Changing supply functions in input/state stable systems[J]. IEEETransactions on automatic control,1995,(40):1476-1478.
[21] L.Praly, Y.Wang.Stabilization in spite of matched unmodeled dynamics and an equivalentdefinition of input-to-state stability[J].Mathematics of Control, Signals and Systems,1996,9:1-33.
[22] Z. Jiang, I. Mareels, and David J. Hill. A unifying framework for global regulation vianonlinear output feedback:from iss to iIss[J]. IEEE Transactions on automatic control,2004,49(4):549-561.
[23] L.Praly, Z.Jiang. Stabilization by output feedback for systems with ISS inversedynamics[J].System control letter,1993,21:19-33.
[24] S.Tong,Y.Li. Observer-based fuzzy adaptive robust control of nonlinear systems with timedelays and unmodeled dynamics[J].Neurocomputing,2010,(74):369-378.
[25]黄在堂.时滞随机微分系统的动力学性质研究[D].华南理工大学,2011.
[26]袁琦,徐迎晓.多智能体协同进化算法及其在物流配送中的应用[J].科技管理研究,2010,3:144-146.
[27]D.Angeli, E.D.Sontag, and Y.Wang.A characterization of integral input to statestability[J].IEEE Trans. Automat. Control,2000,45:1082-1097.
[28] C.Hua,X.Guan, and S.Peng.Robust output feedback tracking control for time-delaynonlinear systems using neural network[J]. IEEE Transactions on neural networks,2007,18(2):495-505.
[29] Z.Yuan, Z.Wang, and Q.Chen. Adaptive output feedback control for uncertainnonholonomic chained systems [J]. Cent.South Univ.Technol,2010,17:572-579.
[30] Z.Xi, G.Feng, and Z.Jiang. Output feedback exponential stabilization of uncertain chainedsystems [J]. Journal of the Franklin Institute,2007,34:36-57.
[31] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic. Nonlinear and adptive controldesign[M].Canada,1995.
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