Stochastic LQ Problem with Delayed Control
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- 英文论文题名：Stochastic LQ Problem with Delayed Control
- 论文作者：Yuan-Hua Ni ; Cedric Ka-Fai Yiu ; Huanshui Zhang ; Ji-Feng Zhang
- 英文论文作者：Yuan-Hua Ni ; Cedric Ka-Fai Yiu ; Huanshui Zhang ; Ji-Feng Zhang ; College of Computer and Control Engineering ; Nankai University ; Department of Applied Mathematics ; The Hong Kong Polytechnic University ; School of Control Science and Engineering ; Shandong University ; Academy of Mathematics and Systems Science ; Chinese Academy of Sciences
- 年：2017
- 作者机构：College of Computer and Control Engineering,Nankai University;Department of Applied Mathematics,The Hong Kong Polytechnic University;School of Control Science and Engineering,Shandong University;Academy of Mathematics and Systems Science,Chinese Academy of Sciences;
- 英文论文关键词：Indefinite stochastic linear-quadratic optimal control ; input delay ; forward-backward stochastic difference equation ; convexity
- 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：O232
- 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：5
- 文件大小：197k
- 原文格式：D
- 会议级别：全国
摘要
A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper,which does not require any definiteness constraint on the cost weighting matrices. Necessary and sufficient conditions are derived for the case with a fixed time-state initial pair. A set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. Furthermore, the convexity of the cost functional is fully characterized via certain properties of the solution of the Riccati-like equations.
A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper,which does not require any definiteness constraint on the cost weighting matrices. Necessary and sufficient conditions are derived for the case with a fixed time-state initial pair. A set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. Furthermore, the convexity of the cost functional is fully characterized via certain properties of the solution of the Riccati-like equations.
A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper,which does not require any definiteness constraint on the cost weighting matrices. Necessary and sufficient conditions are derived for the case with a fixed time-state initial pair. A set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. Furthermore, the convexity of the cost functional is fully characterized via certain properties of the solution of the Riccati-like equations.
引文
[1]M.Ait Rami,X.Chen,and X.Y.Zhou.Discrete-time indefinite LQ control with state and control dependent noise.Journal of Global Optimization,2002,vol.23,245-265.
[2]M.Ait Rami,X.Chen,J.B.Moore,and X.Y.Zhou.Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls.IEEE Transactions on Automatic Control,2001,vol.46,428-440.
[3]Y.Alekal,P.Brunovsky,D.H.Chyung,and E.B.Lee.The quadratic problem for systems with time delays.IEEE Transactions on Automatic Control,1971,vol.16,673-687.
[4]E.Altman,T.Basar,and R.Srikant.Congestion control as a stochastic control problem with action delays.Automatica,1999,vol.35,1937-1950.
[5]J.Baillieul and P.Antsaklis.Control and communication challenges in networked real-time systems.Proceedings of the IEEE,2007,vol.95,no.1,9-28.
[6]J.M.Bismut.Linear quadratic optimal stochastic control with random coefficients.SIAM Journal on Control and Optimization,1976,vol.14,419-444.
[7]L.Chen and Z.Wu.Maximum principle for stochastic optimal control problem with delay and application.Automatica,2010,vol.46,1074-1080.
[8]S.Chen,X.Li,and X.Y.Zhou.Stochastic linear quadratic regulators with indefinite control weight costs.SIAM Journal on Control and Optimization,1998,vol.36,1685-1702.
[9]M.C.Delfour.The linear-quadratic optimal control problem with delays in state and control variables:a state space approach.SIAM Journal on Control and Optimization,1986,vol.24,835-883.
[10]J.Hespanha,P.Naghshtabrizi,and Y.Xu.A survey of recent results in networked control systems.Proceedings of the IEEE,2007,vol.95,no.1,138-162,2007.
[11]Y.Hu and X.Y.Zhou.Indefinite stochastic Riccati equations.SIAM Journal on Control and Optimization,2003,vol.42,123-137.
[12]R.E.Kalman.Contribution to the Theory of Optimal Control.Boletin de la Sociedad Matematica Mexicana,1960,vol.5,no.2,102-119.
[13]H.N.Koivo and E.B.Lee.Controller synthesis for linear systems with retarded state and control variables and quadratic cost.Automatica,1972,vol.8,203-208.
[14]A.Kojima and S.Ishijima.Formulas on preview and delayed H∞control.IEEE Transactions on Automatic Control,2006,vol.51,1920-1937.
[15]Y.H.Ni,R.J.Elliott,and X.Li.Discrete-time mean-field stochastic linear-quadratic optimal controls,II:Infinite time horizon.Automatica,2015,vol.57,65-77.
[16]Y.H.Ni,J.F.Zhang,and X.Li.Indefinite mean-field stochastic linear-quadratic optimal control.IEEE Transactions on Automatic Control,2015,vol.60,no.7,1786-1800.
[17]W.H.Mao,F.Q.Deng,and A.H.Wan.Robust H2/H∞global linearization filter design for nonlinear stochastic timevarying delay systems.SCIENCE CHINA Information Sciences,2016,vol.59,no.3:032204,Doi:10.1007/s11432-015-5386-7.
[18]S.Peng.A general stochastic maximum principle for optimal control problems.SIAM Journal on Control and Optimization,1990,vol.28,no.4,966-979.
[19]R.Penrose.A generalized inverse of matrices.Mathematical Proceedings of the Cambridge Philosophical Society,1955,vol.52,17-19.
[20]J.R.Sun,X.Li,and J.M.Yong.Open-Loop and closedloop solvabilities for stochastic linear quadratic optimal control problems.SIAM Journal on Control and Optimization,2016,vol.54,no.5,2274-2308.
[21]J.R.Sun and J.M.Yong.Mean-field stochastic linear quadratic optimal control problems:open-loop solvabilities.To appear in ESAIM:Control Optimisation and Calculus of Variations,2016,DOI:10.1051/cocv/2016023.
[22]G.Tadmor and L.Mirkin.H∞control and estimation with preview-part I:matrix ARE solutions in continuous time.IEEE Transactions on Automatic Control,2005,vol.50,19-28.
[23]K.Watanabe and M.Ito.A process-model control for linear systems with delay.IEEE Transactions on Automatic Control,1981,vol.26,1261-1269.
[24]L.Wei,H.Zhang,and M.Fu.Quantized stabilization for stochastic discrete-time systems with multiplicative noises.International Journal of Robust and Nonlinear Control,2013,vol.23,no.6,591-601.
[25]G.C.Wang,C.H.Zhang,and W.H.Zhang.Stochastic maximum principle for mean-field type optimal control under partial information.IEEE Transactions on Automatic Control,2014,vol.59,522-528.
[26]Y.Q.Wang,D.H.Yang,J,M.Yong,and Z.Y.Yu.Exact controllability of linear stochastic differential equations and related problems.ar Xiv:1603.07789.
[27]W.M.Wonham.On a matrix Riccati equation of stochastic control.SIAM Journal on Control and Optimization,1068,vol.6,312-326.
[28]D.Yao,S.Zhang,and X.Y Zhou.Stochastic LQ control via primal—dual semidefinite programming.SIAM Review,2004,vol.46,87-111.
[29]J.M.Yong.A linear-quadratic optimal control problem for mean-field stochastic differential equations.SIAM Journal on Control Optimization,2013,vol.51,2809-2838.
[30]J.M.Yong.Linear-quadratic optimal control problems for mean-field stochastic differential equations—time-consistent solutions.Electronically published by Transactions of the American Mathematical Society,DOI:10.1090/tran/6502,2015.
[31]J.M.Yong and X.Y.Zhou.Stochastic controls:Hamiltonian systems and HJB equations.Springer,New York,1999.
[32]H.Zhang,L.Li,J.J.Xu,and M.Y.Fu.Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise.IEEE Transactions on Automatic Control,2015,vol.60,2599-2613.
[33]H.Zhang,H.Wang,and L.Li.Adapted and casual maximum principle and analytical solution to optimal control for stochastic multiplicative noise systems with multiple inputdelays.IEEE CDC,2011,2122-2127.
[2]M.Ait Rami,X.Chen,J.B.Moore,and X.Y.Zhou.Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls.IEEE Transactions on Automatic Control,2001,vol.46,428-440.
[3]Y.Alekal,P.Brunovsky,D.H.Chyung,and E.B.Lee.The quadratic problem for systems with time delays.IEEE Transactions on Automatic Control,1971,vol.16,673-687.
[4]E.Altman,T.Basar,and R.Srikant.Congestion control as a stochastic control problem with action delays.Automatica,1999,vol.35,1937-1950.
[5]J.Baillieul and P.Antsaklis.Control and communication challenges in networked real-time systems.Proceedings of the IEEE,2007,vol.95,no.1,9-28.
[6]J.M.Bismut.Linear quadratic optimal stochastic control with random coefficients.SIAM Journal on Control and Optimization,1976,vol.14,419-444.
[7]L.Chen and Z.Wu.Maximum principle for stochastic optimal control problem with delay and application.Automatica,2010,vol.46,1074-1080.
[8]S.Chen,X.Li,and X.Y.Zhou.Stochastic linear quadratic regulators with indefinite control weight costs.SIAM Journal on Control and Optimization,1998,vol.36,1685-1702.
[9]M.C.Delfour.The linear-quadratic optimal control problem with delays in state and control variables:a state space approach.SIAM Journal on Control and Optimization,1986,vol.24,835-883.
[10]J.Hespanha,P.Naghshtabrizi,and Y.Xu.A survey of recent results in networked control systems.Proceedings of the IEEE,2007,vol.95,no.1,138-162,2007.
[11]Y.Hu and X.Y.Zhou.Indefinite stochastic Riccati equations.SIAM Journal on Control and Optimization,2003,vol.42,123-137.
[12]R.E.Kalman.Contribution to the Theory of Optimal Control.Boletin de la Sociedad Matematica Mexicana,1960,vol.5,no.2,102-119.
[13]H.N.Koivo and E.B.Lee.Controller synthesis for linear systems with retarded state and control variables and quadratic cost.Automatica,1972,vol.8,203-208.
[14]A.Kojima and S.Ishijima.Formulas on preview and delayed H∞control.IEEE Transactions on Automatic Control,2006,vol.51,1920-1937.
[15]Y.H.Ni,R.J.Elliott,and X.Li.Discrete-time mean-field stochastic linear-quadratic optimal controls,II:Infinite time horizon.Automatica,2015,vol.57,65-77.
[16]Y.H.Ni,J.F.Zhang,and X.Li.Indefinite mean-field stochastic linear-quadratic optimal control.IEEE Transactions on Automatic Control,2015,vol.60,no.7,1786-1800.
[17]W.H.Mao,F.Q.Deng,and A.H.Wan.Robust H2/H∞global linearization filter design for nonlinear stochastic timevarying delay systems.SCIENCE CHINA Information Sciences,2016,vol.59,no.3:032204,Doi:10.1007/s11432-015-5386-7.
[18]S.Peng.A general stochastic maximum principle for optimal control problems.SIAM Journal on Control and Optimization,1990,vol.28,no.4,966-979.
[19]R.Penrose.A generalized inverse of matrices.Mathematical Proceedings of the Cambridge Philosophical Society,1955,vol.52,17-19.
[20]J.R.Sun,X.Li,and J.M.Yong.Open-Loop and closedloop solvabilities for stochastic linear quadratic optimal control problems.SIAM Journal on Control and Optimization,2016,vol.54,no.5,2274-2308.
[21]J.R.Sun and J.M.Yong.Mean-field stochastic linear quadratic optimal control problems:open-loop solvabilities.To appear in ESAIM:Control Optimisation and Calculus of Variations,2016,DOI:10.1051/cocv/2016023.
[22]G.Tadmor and L.Mirkin.H∞control and estimation with preview-part I:matrix ARE solutions in continuous time.IEEE Transactions on Automatic Control,2005,vol.50,19-28.
[23]K.Watanabe and M.Ito.A process-model control for linear systems with delay.IEEE Transactions on Automatic Control,1981,vol.26,1261-1269.
[24]L.Wei,H.Zhang,and M.Fu.Quantized stabilization for stochastic discrete-time systems with multiplicative noises.International Journal of Robust and Nonlinear Control,2013,vol.23,no.6,591-601.
[25]G.C.Wang,C.H.Zhang,and W.H.Zhang.Stochastic maximum principle for mean-field type optimal control under partial information.IEEE Transactions on Automatic Control,2014,vol.59,522-528.
[26]Y.Q.Wang,D.H.Yang,J,M.Yong,and Z.Y.Yu.Exact controllability of linear stochastic differential equations and related problems.ar Xiv:1603.07789.
[27]W.M.Wonham.On a matrix Riccati equation of stochastic control.SIAM Journal on Control and Optimization,1068,vol.6,312-326.
[28]D.Yao,S.Zhang,and X.Y Zhou.Stochastic LQ control via primal—dual semidefinite programming.SIAM Review,2004,vol.46,87-111.
[29]J.M.Yong.A linear-quadratic optimal control problem for mean-field stochastic differential equations.SIAM Journal on Control Optimization,2013,vol.51,2809-2838.
[30]J.M.Yong.Linear-quadratic optimal control problems for mean-field stochastic differential equations—time-consistent solutions.Electronically published by Transactions of the American Mathematical Society,DOI:10.1090/tran/6502,2015.
[31]J.M.Yong and X.Y.Zhou.Stochastic controls:Hamiltonian systems and HJB equations.Springer,New York,1999.
[32]H.Zhang,L.Li,J.J.Xu,and M.Y.Fu.Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise.IEEE Transactions on Automatic Control,2015,vol.60,2599-2613.
[33]H.Zhang,H.Wang,and L.Li.Adapted and casual maximum principle and analytical solution to optimal control for stochastic multiplicative noise systems with multiple inputdelays.IEEE CDC,2011,2122-2127.