具有SiISS逆动态的随机非线性系统的稳定性研究与控制器设计
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摘要
在利用Lyapunov稳定性理论对随机非线性系统设计全局稳定控制器时,即使对简单的线性定常系统,闭环系统也会变成高度非线性方程,而由此闭环系统所给出的输入.输出信号往往又是很复杂的随机过程,所以,“非线性”与“随机”是控制理论研究的困难所在.正是由于这一根本原因,控制理论的许多最典型而又最基础的问题长期以来悬而未决.因此,随机非线性系统控制的基础理论问题的研究具有一定挑战性,同时这一问题在实际应用中也很重要,非常值得我们深入研究.
本文主要成果包括:
1.受确定性情中积分输入状态稳定性(iISS)和随机情形中的基于Lyapunov函数的随机输入状态稳定性(SISS)的启发,基于Lyapunov函数的随机积分输入状态稳定性(SiISS)的概念首次被提出,并且得到了SilSS的两个重要性质:(i)SiISS严格弱于基于Lyapunov函数的SISS的定义;(ii)SiISS严格强于最小相位的性质,但是,如果仅仅在最小相位的假设下,不存在动态输出反馈控制律实现依概率全局镇定.
2.几乎必然有界性,一种比依概率有界性更强的概念被引入,引入这个概念的目的是为了证明闭环信号的有界性和收敛性.
3.建立了一些重要的数学工具,这些数学工具在闭环系统的有界性和收敛性分析中起着重要的作用.
4.针对具有SiISS逆动态的如下随机非线性系统:得到了一个统一的动态输出反馈控制律的设计框架,使得系统状态几乎必然调节到原点并且闭环系统的信号几乎必然有界.
When global stabilization controllers are designed by Lyapunov stabilization theory for stochastic nonlinear system, the closed-loop systems usually can be transformed into exceedingnonlinear functions even for simple time-unvarying linear systems. And the input-to-outputsignals given by the closed-loop systems usually are more complicated stochastic process, so the difficulties of the investigation for control theory are“stochastic”and“nonlinear”. It is because of this basic reason that many typical and basal problems of control theory are unfathomed for a long time. Thus, the investigation for the control theories of stochastic nonlinear systems is a challenge and worthy to investigate, which is very important in actual application.
The contributions of this work are characterized by the following novel features:
1.Motivated by the concept of integral input-to-state stability (iISS) in deterministic systems and stochastic input-to-state stability (SISS) using Lyapunov function in stochastic systems, a concept of stochastic integral input-to-state stability (SilSS) using Lyapunov function is first introduced, two important properties of SilSS are obtained: (i) SilSS is strictly weaker than SISS using Lyapunov function; (ii) SilSS is stronger than the minimumphaseproperty. However, only under the minimum-phase assumption, there is no dynamic output feedback control law for global stabilization in probability.
2.Almost sure boundedness, a reasonable and stronger concept than boundedness in probability, is introduced. The purpose of introducing the concept is to prove the boundednessand convergence of some signals in the closed-loop control system.
3.Some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed-loop system are established.
4.Considering the stochastic nonlinear systems with SilSS inverse dynamics described in the following form: a unifying framework is proposed to design a dynamic output feedback control law, which drives the states to the origin almost surely while maintaining all the closed-loop signals bounded almost surely.
本文主要成果包括:
1.受确定性情中积分输入状态稳定性(iISS)和随机情形中的基于Lyapunov函数的随机输入状态稳定性(SISS)的启发,基于Lyapunov函数的随机积分输入状态稳定性(SiISS)的概念首次被提出,并且得到了SilSS的两个重要性质:(i)SiISS严格弱于基于Lyapunov函数的SISS的定义;(ii)SiISS严格强于最小相位的性质,但是,如果仅仅在最小相位的假设下,不存在动态输出反馈控制律实现依概率全局镇定.
2.几乎必然有界性,一种比依概率有界性更强的概念被引入,引入这个概念的目的是为了证明闭环信号的有界性和收敛性.
3.建立了一些重要的数学工具,这些数学工具在闭环系统的有界性和收敛性分析中起着重要的作用.
4.针对具有SiISS逆动态的如下随机非线性系统:得到了一个统一的动态输出反馈控制律的设计框架,使得系统状态几乎必然调节到原点并且闭环系统的信号几乎必然有界.
When global stabilization controllers are designed by Lyapunov stabilization theory for stochastic nonlinear system, the closed-loop systems usually can be transformed into exceedingnonlinear functions even for simple time-unvarying linear systems. And the input-to-outputsignals given by the closed-loop systems usually are more complicated stochastic process, so the difficulties of the investigation for control theory are“stochastic”and“nonlinear”. It is because of this basic reason that many typical and basal problems of control theory are unfathomed for a long time. Thus, the investigation for the control theories of stochastic nonlinear systems is a challenge and worthy to investigate, which is very important in actual application.
The contributions of this work are characterized by the following novel features:
1.Motivated by the concept of integral input-to-state stability (iISS) in deterministic systems and stochastic input-to-state stability (SISS) using Lyapunov function in stochastic systems, a concept of stochastic integral input-to-state stability (SilSS) using Lyapunov function is first introduced, two important properties of SilSS are obtained: (i) SilSS is strictly weaker than SISS using Lyapunov function; (ii) SilSS is stronger than the minimumphaseproperty. However, only under the minimum-phase assumption, there is no dynamic output feedback control law for global stabilization in probability.
2.Almost sure boundedness, a reasonable and stronger concept than boundedness in probability, is introduced. The purpose of introducing the concept is to prove the boundednessand convergence of some signals in the closed-loop control system.
3.Some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed-loop system are established.
4.Considering the stochastic nonlinear systems with SilSS inverse dynamics described in the following form: a unifying framework is proposed to design a dynamic output feedback control law, which drives the states to the origin almost surely while maintaining all the closed-loop signals bounded almost surely.
引文
[1]D.Angeli,E.D.Sontag,and Y.Wang.A characterization of integral input-to-state stability [J].IEEE Transactions on Automatic Control,2000,45:1082-1097.
[2]P.D.Christofides and A.R.Teel.Singular perturbations and input-to state stability [J].IEEE Transactions on Automatic Control,1996,41:1645-1650.
[3]H.Deng and M.Krstic.Stochastic nonlinear stabilization,part I:a backstepping design [J].Systems and Control Letters,1997,32:143-150.
[4]H.Deng and M.Krstic.Stochastic nonlinear stabilization,part Ⅱ:inverse optimality [J].Systems and Control Letters,1997,32:151-159.
[5]H.Deng and M.Krstic.Output-feedback stochastic nonlinear stabilization [J].IEEE Transactions on Automatic Control,1999,44:328-333.
[6]H.Deng and M.Krstic.Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J].Systems and Control Letters,2000,39:173—182.
[7]H.Deng,M.Krstic,and R.J.Williams.Stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J].IEEE Transactions on Automatic Control,2001,46:1237-1253.
[8]Y.Fu,Z.Tian,and S.Shi.Output feedback stabilization for a class of stochastic time-delay nonlinear systems [J].IEEE Transactions on Automatic Control,2005,50:847-851.
[9]R.A.Gihman and A.V.Skorohod.Stochastic Differential Equations [M].New York:Springer-Verlag,1972.
[10]R.Z.Has'minskii.Stochastic Stability of Diffrential Equations [M].Norwell,Massachusetts:Kluwer Academic Publishers,1980.
[11]J.P.Hespanha and A.S.Morse.Certainty equivalence implies detectability [J].Systems and Control Letters,1999,36:1-13.
[12]J.P.Hespanha and A.S.Morse.Supervisory control of integral-input-to-state stabilizing controllers [J].Proc.5th Eur.Control Conf.,1999.
[13]X.M.Hu.On state observers for nonlinear systems [J].Systems and Control Letters,1991,17:645-473.
[14]A.Isidori.Global almost disturbance decoupling with stability for nonminimum-phase single-input single-output nonlinear systems [J].Systems and Control Letters,1996,28:115-122.
[15]H.B.Ji and H.S.Xi.Adaptive output-feedback tracking of stochastic nonlinear systems [J].IEEE Transactions on Automatic Control,2006,51:355-360.
[16]Z.P.Jiang,I.Mareels,D.J.Hill,and J.Huang.A unifying framework for global regulation via nonlinear output feedback:from ISS to iISS [J].IEEE Transactions on Automatic Control,2004,49:549-562.
[17]Z.P.Jiang,A.Teel,and L.Praly.Small-gain theorem for ISS systems and applications [J].Mathematics of Control,Signals and Systems,1994,7:95-120.
[18]H.K.Khalil.Nonlinear Systems [M].Third Edition,Upper Saddle River,NJ:Prentice-Hall,2002.
[19]F.C.Klebaner.Introduction to Stochastic Calculus with Applications [M].London,U.K.:Imperial College Press,1998.
[20]P.Kokotovic and M.Arcak.Constructive nonlinear control:a historical perspective [J].Automatica,2001,37:637-662.
[21]M.Krstic and H.Deng.Stabilization of Uncertain Nonlinear Systems [M].New York:Springer,1998.
[22]M.Krstic,I.Kanellakopoulos,and P.V.Kokotovic.Nonlinear and Adaptive Control Design [M].New York:Wiley,1995.
[23]D.Liberzon.ISS and integral-ISS disturbance attenuation with bounded controls [J].IEEE Conf.Decision and Control,1999.
[24]S.J.Liu,S.S.Ge,and J.F.Zhang.Adaptive output-feedback control for a class of uncertain stochastic nonlinear systems with time delays [J].International Journal of Control,2008,81:1210-1220.
[25]S.J.Liu,J.F.Zhang,and Z.P.Jiang.Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems [J].Automatica,2007,43:238-251.
[26]S.J.Liu,J.F.Zhang,and Z.P.Jiang.A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems [J].Acta Mathematicae Applicatae Sinica,English Series,2008,24:141-156.
[27]S.J.Liu and J.F.Zhang.Output-feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states [J].International Journal of Robust and Nonlinear Control,2008,18:665-687.
[28]S.J.Liu,Z.P.Jiang,and J.F.Zhang.Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems [J].Automatica,2008,44:1944-1957.
[29]Y.Liu,Z.G.Pan,and S.Shi.Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost [J].IEEE Transactions on Automatic Control,2003,48:509-513.
[30]Y.Liu and J.F.Zhang.Reduced-order observer-based control design for nonlinear stochastic systems.Systems and Control Letters [J],2004,52:123-135.
[31]Y.G.Liu and J.F.Zhang.Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems [J].Science in China (Series F),2004,47:527-544.
[32]Y.G.Liu and J.F.Zhang.Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics [J].SI AM Journal on Control and Optimization,2006,45:885-926.
[33]X.Mao.Stochastic Differential Equations and their Applications [M].Chichester:Horwood Publ.,1991.
[34]X.R.Mao and C.G.Yuan.Stochastic Differential Equations with Markovian Switching [M].London,U.K.:Imperial College Press,2006.
[35]R.Marino and P.Tomei.Nonlinear Control Design:Geometric,Adaptive and Robust [M].New Gersey:Prentice-Hall,1995.
[36]R.Marino and P.Tomei.Nonlinear output feedback tracking with almost disturbance decoupling [J].IEEE Transactions on Automatic Control,1999,44:18-28.
[37]Z.G.Pan and T.Basar.Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion [J].SIAM Journal of Control and Optimization,1999,37:957-995.
[38]Z.G.Pan,K.Ezal,A.Krener,and P.V.Kokotovic.Backstepping design with local optimality matching [J].IEEE Transactions on Automatic Control,2001,46:1014-1027.
[39]Z.G.Pan,Y.G.Liu,and S.J.Shi.Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics [J].Science in China(Series F),2001,44:292-308.
[40]E.D.Sontag.Smooth stabilization implies coprime factorization [J].IEEE Transactions on Automatic Control,1989,34:435-443.
[41]E.D.Sontag.Further facts about input to state stabilization [J].IEEE Transactions on Automatic Control,1990,35:473-476.
[42]E.D.Sontag.Comments on integral variants of ISS [J].Systems and Control Letters,1998,34:93-100.
[43]E.D.Sontag and A.R.Teel.Changing supply rates in input-to-state stable systems [J].IEEE Transactions on Automatic Control,1995,40:1476-1478.
[44]E.D.Sontag and Y.Wang.On characterizations of the input-to-state stability property [J].Systems and Control Letters,1995,24:351-359.
[45]E.D.Sontag and Y.Wang.New characterizations of the input to state stability property [J].IEEE Transactions on Automatic Control,1996,41:1283-1294.
[46]C.Tang and T.Basar.Stochastic stability of singularly perturbed nonlinear systems [J].Proceedings of the 40th IEEE conference on decision and control,Orlando,Florida,Dec.2001,pp.399-404.
[47]J.Tsinias.Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation [J].ESAIM Control Optim.Calc.Var.,1997,2:57-85.
[48]S.R.Wang.Basic of Probability Theory and Stochastic Process [M].Beijing,China:Science Press,1997.
[49]Z.J.Wu,X.J.Xie,and S.Y.Zhang.Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function [J].International Journal of Control,2006,79:1635-1646.
[50]Z.J.Wu,X.J.Xie,and S.Y.Zhang.Adaptive backstepping controller design using stochastic small-gain theorem [J].Automatica,2007,43:608-620.
[2]P.D.Christofides and A.R.Teel.Singular perturbations and input-to state stability [J].IEEE Transactions on Automatic Control,1996,41:1645-1650.
[3]H.Deng and M.Krstic.Stochastic nonlinear stabilization,part I:a backstepping design [J].Systems and Control Letters,1997,32:143-150.
[4]H.Deng and M.Krstic.Stochastic nonlinear stabilization,part Ⅱ:inverse optimality [J].Systems and Control Letters,1997,32:151-159.
[5]H.Deng and M.Krstic.Output-feedback stochastic nonlinear stabilization [J].IEEE Transactions on Automatic Control,1999,44:328-333.
[6]H.Deng and M.Krstic.Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J].Systems and Control Letters,2000,39:173—182.
[7]H.Deng,M.Krstic,and R.J.Williams.Stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J].IEEE Transactions on Automatic Control,2001,46:1237-1253.
[8]Y.Fu,Z.Tian,and S.Shi.Output feedback stabilization for a class of stochastic time-delay nonlinear systems [J].IEEE Transactions on Automatic Control,2005,50:847-851.
[9]R.A.Gihman and A.V.Skorohod.Stochastic Differential Equations [M].New York:Springer-Verlag,1972.
[10]R.Z.Has'minskii.Stochastic Stability of Diffrential Equations [M].Norwell,Massachusetts:Kluwer Academic Publishers,1980.
[11]J.P.Hespanha and A.S.Morse.Certainty equivalence implies detectability [J].Systems and Control Letters,1999,36:1-13.
[12]J.P.Hespanha and A.S.Morse.Supervisory control of integral-input-to-state stabilizing controllers [J].Proc.5th Eur.Control Conf.,1999.
[13]X.M.Hu.On state observers for nonlinear systems [J].Systems and Control Letters,1991,17:645-473.
[14]A.Isidori.Global almost disturbance decoupling with stability for nonminimum-phase single-input single-output nonlinear systems [J].Systems and Control Letters,1996,28:115-122.
[15]H.B.Ji and H.S.Xi.Adaptive output-feedback tracking of stochastic nonlinear systems [J].IEEE Transactions on Automatic Control,2006,51:355-360.
[16]Z.P.Jiang,I.Mareels,D.J.Hill,and J.Huang.A unifying framework for global regulation via nonlinear output feedback:from ISS to iISS [J].IEEE Transactions on Automatic Control,2004,49:549-562.
[17]Z.P.Jiang,A.Teel,and L.Praly.Small-gain theorem for ISS systems and applications [J].Mathematics of Control,Signals and Systems,1994,7:95-120.
[18]H.K.Khalil.Nonlinear Systems [M].Third Edition,Upper Saddle River,NJ:Prentice-Hall,2002.
[19]F.C.Klebaner.Introduction to Stochastic Calculus with Applications [M].London,U.K.:Imperial College Press,1998.
[20]P.Kokotovic and M.Arcak.Constructive nonlinear control:a historical perspective [J].Automatica,2001,37:637-662.
[21]M.Krstic and H.Deng.Stabilization of Uncertain Nonlinear Systems [M].New York:Springer,1998.
[22]M.Krstic,I.Kanellakopoulos,and P.V.Kokotovic.Nonlinear and Adaptive Control Design [M].New York:Wiley,1995.
[23]D.Liberzon.ISS and integral-ISS disturbance attenuation with bounded controls [J].IEEE Conf.Decision and Control,1999.
[24]S.J.Liu,S.S.Ge,and J.F.Zhang.Adaptive output-feedback control for a class of uncertain stochastic nonlinear systems with time delays [J].International Journal of Control,2008,81:1210-1220.
[25]S.J.Liu,J.F.Zhang,and Z.P.Jiang.Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems [J].Automatica,2007,43:238-251.
[26]S.J.Liu,J.F.Zhang,and Z.P.Jiang.A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems [J].Acta Mathematicae Applicatae Sinica,English Series,2008,24:141-156.
[27]S.J.Liu and J.F.Zhang.Output-feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states [J].International Journal of Robust and Nonlinear Control,2008,18:665-687.
[28]S.J.Liu,Z.P.Jiang,and J.F.Zhang.Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems [J].Automatica,2008,44:1944-1957.
[29]Y.Liu,Z.G.Pan,and S.Shi.Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost [J].IEEE Transactions on Automatic Control,2003,48:509-513.
[30]Y.Liu and J.F.Zhang.Reduced-order observer-based control design for nonlinear stochastic systems.Systems and Control Letters [J],2004,52:123-135.
[31]Y.G.Liu and J.F.Zhang.Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems [J].Science in China (Series F),2004,47:527-544.
[32]Y.G.Liu and J.F.Zhang.Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics [J].SI AM Journal on Control and Optimization,2006,45:885-926.
[33]X.Mao.Stochastic Differential Equations and their Applications [M].Chichester:Horwood Publ.,1991.
[34]X.R.Mao and C.G.Yuan.Stochastic Differential Equations with Markovian Switching [M].London,U.K.:Imperial College Press,2006.
[35]R.Marino and P.Tomei.Nonlinear Control Design:Geometric,Adaptive and Robust [M].New Gersey:Prentice-Hall,1995.
[36]R.Marino and P.Tomei.Nonlinear output feedback tracking with almost disturbance decoupling [J].IEEE Transactions on Automatic Control,1999,44:18-28.
[37]Z.G.Pan and T.Basar.Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion [J].SIAM Journal of Control and Optimization,1999,37:957-995.
[38]Z.G.Pan,K.Ezal,A.Krener,and P.V.Kokotovic.Backstepping design with local optimality matching [J].IEEE Transactions on Automatic Control,2001,46:1014-1027.
[39]Z.G.Pan,Y.G.Liu,and S.J.Shi.Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics [J].Science in China(Series F),2001,44:292-308.
[40]E.D.Sontag.Smooth stabilization implies coprime factorization [J].IEEE Transactions on Automatic Control,1989,34:435-443.
[41]E.D.Sontag.Further facts about input to state stabilization [J].IEEE Transactions on Automatic Control,1990,35:473-476.
[42]E.D.Sontag.Comments on integral variants of ISS [J].Systems and Control Letters,1998,34:93-100.
[43]E.D.Sontag and A.R.Teel.Changing supply rates in input-to-state stable systems [J].IEEE Transactions on Automatic Control,1995,40:1476-1478.
[44]E.D.Sontag and Y.Wang.On characterizations of the input-to-state stability property [J].Systems and Control Letters,1995,24:351-359.
[45]E.D.Sontag and Y.Wang.New characterizations of the input to state stability property [J].IEEE Transactions on Automatic Control,1996,41:1283-1294.
[46]C.Tang and T.Basar.Stochastic stability of singularly perturbed nonlinear systems [J].Proceedings of the 40th IEEE conference on decision and control,Orlando,Florida,Dec.2001,pp.399-404.
[47]J.Tsinias.Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation [J].ESAIM Control Optim.Calc.Var.,1997,2:57-85.
[48]S.R.Wang.Basic of Probability Theory and Stochastic Process [M].Beijing,China:Science Press,1997.
[49]Z.J.Wu,X.J.Xie,and S.Y.Zhang.Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function [J].International Journal of Control,2006,79:1635-1646.
[50]Z.J.Wu,X.J.Xie,and S.Y.Zhang.Adaptive backstepping controller design using stochastic small-gain theorem [J].Automatica,2007,43:608-620.
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