Finite-time stability of stochastic nonlinear systems with Markovian switching
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- 英文论文题名：Finite-time stability of stochastic nonlinear systems with Markovian switching
- 论文作者：Juliang Yin ; Xin Yu ; Suiyang Khoo
- 英文论文作者：Juliang Yin ; Xin Yu ; Suiyang Khoo ; Department of Statistics ; Jinan University ; School of Electrical and Information Engineering ; Jiangsu University ; School of Engineering ; Deakin University
- 年：2017
- 作者机构：Department of Statistics,Jinan University;School of Electrical and Information Engineering,Jiangsu University;School of Engineering,Deakin University;
- 英文论文关键词：Finite-time stability ; Lyapunov stability ; Stochastic nonlinear systems ; Markovian switching
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O211.62
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：219k
- 原文格式：D
- 会议级别：全国
摘要
This paper presents a new Lyapunov theorem on almost surely finite-time stability for stochastic nonlinear systems with Markovian switching. Unlike the work in [3] that consider finite-time stability and stabilisation of conventional stochastic differential equation(SDE) systems, this paper aims to propose a weaker finite-time stability theory for a more general class of SDE systems with Markovian switching. A lemma is presented to discuss conditions that ensure the existence of a unique strong solution for such SDE systems with Markovian switching. Extended Comparison Principle and Bihari's inequality are derived,which relaxes some previous conditions and play an important role in the proof of the new Lyapunov theorem. Weaker conditions are proposed to ensure finite-time stability in probability one with supportive examples. Two simulation examples are given to illustrate the theoretical analysis.
This paper presents a new Lyapunov theorem on almost surely finite-time stability for stochastic nonlinear systems with Markovian switching. Unlike the work in [3] that consider finite-time stability and stabilisation of conventional stochastic differential equation(SDE) systems, this paper aims to propose a weaker finite-time stability theory for a more general class of SDE systems with Markovian switching. A lemma is presented to discuss conditions that ensure the existence of a unique strong solution for such SDE systems with Markovian switching. Extended Comparison Principle and Bihari's inequality are derived,which relaxes some previous conditions and play an important role in the proof of the new Lyapunov theorem. Weaker conditions are proposed to ensure finite-time stability in probability one with supportive examples. Two simulation examples are given to illustrate the theoretical analysis.
This paper presents a new Lyapunov theorem on almost surely finite-time stability for stochastic nonlinear systems with Markovian switching. Unlike the work in [3] that consider finite-time stability and stabilisation of conventional stochastic differential equation(SDE) systems, this paper aims to propose a weaker finite-time stability theory for a more general class of SDE systems with Markovian switching. A lemma is presented to discuss conditions that ensure the existence of a unique strong solution for such SDE systems with Markovian switching. Extended Comparison Principle and Bihari's inequality are derived,which relaxes some previous conditions and play an important role in the proof of the new Lyapunov theorem. Weaker conditions are proposed to ensure finite-time stability in probability one with supportive examples. Two simulation examples are given to illustrate the theoretical analysis.
引文
[1]X.Mao and C.Yuan,Stochastic differential equations with Markovian switching,Imperial Collage Press,2006.
[2]G.George Yin and Chao Zhu,Hybrid switching diffusions:Properties and applications,Springer,2010.
[3]J.Yin,S.Khoo,Z.Man&X.Yu,Finite-time stability and instability of stochastic nonlinear systems,Automatica,47,2671–2677,2011.
[4]S.P.Bhat&D.S.Bernstein,Finite-time stability of continuous autonomous systems,SIAM Journal on Control and Optimization,38(3),751–766,2000.
[5]E.Moulay&W.Perruquetti,Finite-time stability and stabilization of a class of continuous systems,Journal of Mathematical Analysis and Applications,323,1430–1443,2006.
[6]Y.Hong,Z.P.Jiang&G.Feng,Finite-time input-to-state stability and applications to finite-time control design,SIAM Journal on Control and Optimization,48,4395–4418,2010.
[7]A.Levant&L.M.Fridman,Accuracy of Homogeneous Sliding Modes in the Presence of Fast Actuators,IEEE Transactions on Automatic Control,55(3),810–814,2010.
[8]F.Amato,R.Ambrosino&C.Cosentino,Finite-time stability of linear time-varying systems with jumps,Automatica,45(5),1354–1358,2009.
[9]L.Weiss&E.F.Infante,Finite time stability under perturbing forces and on product spaces,IEEE Transactions on Automatic Control,12(1),54–59,1967.
[10]H.J.Kushner,Finite time stochastic stability and the analysis of tracking systems,IEEE Transactions on Automatic Control,11(2),219–227,1966.
[11]R.Situ,Theory of stochastic differential equations with jumps and applications:Mathematical and analysis techniques with applications to engineering,USA,New York:Springer,2005.
[12]M.H.Wassim&S.C.Vijay,Nonlinear Dynamical Systems and Control:A Lyapunov-Based Approach,Princeton University Press,2008.
[13]A.F.Filippov,Differential Equations with Discontinuous Righthand Sides,Kluwer Academic Publishers,1988.
[14]A.V.Skorokhod,Asymptotic methods in the Theory of Stochastic Differential Equations,American Mathematical Society,1989.
[15]K.H.Jack,Ordinary differential equations,Wiley Interscience,1969.
[16]W.A.Coppel,Stability and Asymptotic Behaviour of Differential Equations,D.C.Heath and Company,1965.
[17]I.I.Gikhman&A.V.Skorokhod,The theory of stochastic processes I,Springer-Verlag Berlin Heidelberg,2007.
[18]S.Khoo,J.Yin,Z.Man&X.Yu,Finite-time stabilization of stochastic nonlinear systems in strict-feedback form,Automatica,49,1403–1410,2013.
[19]H.Deng,M.Krstic&R.J.Williams,Stabilization of Stochastic Nonlinear Systems Driven by Noise of Unkown Covariance,IEEE Transactions on Automatic Control,46,1237-1253,2001.
[20]H.Deng&M.Krstic,Stochastic nonlinear stabilization–I:A backstepping design,Systems&Control Letters,32,143-150,1997.
[21]H.Deng&M.Krstic,Stochastic nonlinear stabilization–II:Inverse optimality,Systems&Control Letters,32,151-159,1997.
[22]H.Deng&M.Krstic,Output-Feedback Stochastic Nonlinear Stabilization,IEEE Transactions on Automatic Control,44(2),328-333,1999.
[23]H.Deng&M.Krstic,Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance,Systems&Control Letters,39,173-182,2000.
[24]R.Z.Has’minskii,Stochastic stability of differential equations,Kluwer Academic Publishers,1980.
[25]S.J.Liu,J.F.Zhang&Z.P.Jiang,Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems,Automatica,44,1944-1957,2008.
[26]S.J.Liu,S.S.Ge&J.F.Zhang,Adaptive output-feedback control for a class of uncertain stochastic nonlinear systems with time delays,International Journal of Control,81(8),1210-1220,2008.
[27]Z.Pan&T.Basar,Backstepping Controller Design for Nonlinear Stochastic Systems Under a Risk-sensitive Cost Criterion,Siam Journal of Control and Optimization,37,957-995,1999.
[28]Z.J.Wu,X.J.Xie,P.Shi&Y.Xia,Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching,Automatica,45(4),997-1004,2009.
[29]X.Yu&X.J.Xie,Output Feedback Regulation of Stochastic Nonlinear Systems With Stochastic i ISS Inverse Dynamics,IEEE Transactions on Automatic Control,55(2),304-320,2010.
[30]X.Yu,X.J.Xie&N.Duan,Small-gain control method for stochastic nonlinear systems with stochastic i ISS inverse dynamic,Automatica,46(11),1790-1798,2010.
[31]X.Mao,Stochastic Differential Equations and Applications,Horwood,2008.
[32]L.C.G.Rogers&D.Williams,Diffusions,Marko Processes and Martingales,Cambridge,2000.
[33]Y.Toshio,On the successive approximation of solutions of stochastic differential equations Journal of Mathematics Kyoto University,21(3),501-515,1981.
[2]G.George Yin and Chao Zhu,Hybrid switching diffusions:Properties and applications,Springer,2010.
[3]J.Yin,S.Khoo,Z.Man&X.Yu,Finite-time stability and instability of stochastic nonlinear systems,Automatica,47,2671–2677,2011.
[4]S.P.Bhat&D.S.Bernstein,Finite-time stability of continuous autonomous systems,SIAM Journal on Control and Optimization,38(3),751–766,2000.
[5]E.Moulay&W.Perruquetti,Finite-time stability and stabilization of a class of continuous systems,Journal of Mathematical Analysis and Applications,323,1430–1443,2006.
[6]Y.Hong,Z.P.Jiang&G.Feng,Finite-time input-to-state stability and applications to finite-time control design,SIAM Journal on Control and Optimization,48,4395–4418,2010.
[7]A.Levant&L.M.Fridman,Accuracy of Homogeneous Sliding Modes in the Presence of Fast Actuators,IEEE Transactions on Automatic Control,55(3),810–814,2010.
[8]F.Amato,R.Ambrosino&C.Cosentino,Finite-time stability of linear time-varying systems with jumps,Automatica,45(5),1354–1358,2009.
[9]L.Weiss&E.F.Infante,Finite time stability under perturbing forces and on product spaces,IEEE Transactions on Automatic Control,12(1),54–59,1967.
[10]H.J.Kushner,Finite time stochastic stability and the analysis of tracking systems,IEEE Transactions on Automatic Control,11(2),219–227,1966.
[11]R.Situ,Theory of stochastic differential equations with jumps and applications:Mathematical and analysis techniques with applications to engineering,USA,New York:Springer,2005.
[12]M.H.Wassim&S.C.Vijay,Nonlinear Dynamical Systems and Control:A Lyapunov-Based Approach,Princeton University Press,2008.
[13]A.F.Filippov,Differential Equations with Discontinuous Righthand Sides,Kluwer Academic Publishers,1988.
[14]A.V.Skorokhod,Asymptotic methods in the Theory of Stochastic Differential Equations,American Mathematical Society,1989.
[15]K.H.Jack,Ordinary differential equations,Wiley Interscience,1969.
[16]W.A.Coppel,Stability and Asymptotic Behaviour of Differential Equations,D.C.Heath and Company,1965.
[17]I.I.Gikhman&A.V.Skorokhod,The theory of stochastic processes I,Springer-Verlag Berlin Heidelberg,2007.
[18]S.Khoo,J.Yin,Z.Man&X.Yu,Finite-time stabilization of stochastic nonlinear systems in strict-feedback form,Automatica,49,1403–1410,2013.
[19]H.Deng,M.Krstic&R.J.Williams,Stabilization of Stochastic Nonlinear Systems Driven by Noise of Unkown Covariance,IEEE Transactions on Automatic Control,46,1237-1253,2001.
[20]H.Deng&M.Krstic,Stochastic nonlinear stabilization–I:A backstepping design,Systems&Control Letters,32,143-150,1997.
[21]H.Deng&M.Krstic,Stochastic nonlinear stabilization–II:Inverse optimality,Systems&Control Letters,32,151-159,1997.
[22]H.Deng&M.Krstic,Output-Feedback Stochastic Nonlinear Stabilization,IEEE Transactions on Automatic Control,44(2),328-333,1999.
[23]H.Deng&M.Krstic,Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance,Systems&Control Letters,39,173-182,2000.
[24]R.Z.Has’minskii,Stochastic stability of differential equations,Kluwer Academic Publishers,1980.
[25]S.J.Liu,J.F.Zhang&Z.P.Jiang,Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems,Automatica,44,1944-1957,2008.
[26]S.J.Liu,S.S.Ge&J.F.Zhang,Adaptive output-feedback control for a class of uncertain stochastic nonlinear systems with time delays,International Journal of Control,81(8),1210-1220,2008.
[27]Z.Pan&T.Basar,Backstepping Controller Design for Nonlinear Stochastic Systems Under a Risk-sensitive Cost Criterion,Siam Journal of Control and Optimization,37,957-995,1999.
[28]Z.J.Wu,X.J.Xie,P.Shi&Y.Xia,Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching,Automatica,45(4),997-1004,2009.
[29]X.Yu&X.J.Xie,Output Feedback Regulation of Stochastic Nonlinear Systems With Stochastic i ISS Inverse Dynamics,IEEE Transactions on Automatic Control,55(2),304-320,2010.
[30]X.Yu,X.J.Xie&N.Duan,Small-gain control method for stochastic nonlinear systems with stochastic i ISS inverse dynamic,Automatica,46(11),1790-1798,2010.
[31]X.Mao,Stochastic Differential Equations and Applications,Horwood,2008.
[32]L.C.G.Rogers&D.Williams,Diffusions,Marko Processes and Martingales,Cambridge,2000.
[33]Y.Toshio,On the successive approximation of solutions of stochastic differential equations Journal of Mathematics Kyoto University,21(3),501-515,1981.