A New Spectral Method for the Nonlinear Optimal Control Systems
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- 英文论文题名：A New Spectral Method for the Nonlinear Optimal Control Systems
- 论文作者：CHEN Xuesong ; Li Xingke ; ZHANG Lili ; CAI Shuting
- 英文论文作者：CHEN Xuesong ; Li Xingke ; ZHANG Lili ; CAI Shuting ; School of Applied Mathematics ; Guangdong University of Technology ; School of Automation ; Guangdong University of Technology
- 年：2017
- 作者机构：School of Applied Mathematics,Guangdong University of Technology;School of Automation,Guangdong University of Technology;
- 英文论文关键词：Generalized Hamilton-Jacobi-Bellman equation ; nonlinear optimal control ; Galerkin approximation ; Chebyshev polynomials
- 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：6
- 文件大小：266k
- 原文格式：D
- 会议级别：全国
摘要
A new spectral method based on Galerkin approximation solutions of the nonlinear optimal control systems is proposed in this paper. Galerkin approximation with Chebyshev polynomials(GACP) is firstly used to solve the generalized Hamilton-Jacobi-Bellman(GHJB) equation for the nonlinear optimal control systems. The proposed GACP method employs some Chebyshev global polynomials as the trial functions for discretization of GHJB equation on a well-defined region of attraction. A stable optimal solution of a nonlinear control system is finally obtained. Numerical example shows that the proposed method can efficiently solve the nonlinear optimal control problem and therefore is promising.
A new spectral method based on Galerkin approximation solutions of the nonlinear optimal control systems is proposed in this paper. Galerkin approximation with Chebyshev polynomials(GACP) is firstly used to solve the generalized Hamilton-Jacobi-Bellman(GHJB) equation for the nonlinear optimal control systems. The proposed GACP method employs some Chebyshev global polynomials as the trial functions for discretization of GHJB equation on a well-defined region of attraction. A stable optimal solution of a nonlinear control system is finally obtained. Numerical example shows that the proposed method can efficiently solve the nonlinear optimal control problem and therefore is promising.
A new spectral method based on Galerkin approximation solutions of the nonlinear optimal control systems is proposed in this paper. Galerkin approximation with Chebyshev polynomials(GACP) is firstly used to solve the generalized Hamilton-Jacobi-Bellman(GHJB) equation for the nonlinear optimal control systems. The proposed GACP method employs some Chebyshev global polynomials as the trial functions for discretization of GHJB equation on a well-defined region of attraction. A stable optimal solution of a nonlinear control system is finally obtained. Numerical example shows that the proposed method can efficiently solve the nonlinear optimal control problem and therefore is promising.
引文
[1]Q.Gong and W.Kang,“A pseudospectral method for the optimal control of constrained feedback linearizable systems,”IEEE Transactions on Automatic Control,vol.51,no.7,pp.1115-1130,2006.
[2]F.L.Lewis and D.Vrabie,Optimal control.Hoboken,New Jersey:John Wiley and Sons,2013.
[3]D.Bertsekas,Dynamic programming and optimal control.Nashua:Athena Scientific,2005.
[4]T.Bian,Y.Jiang,and Z.Jiang,“Adaptive dynamic programming and optimal control of nonlinear nonaffine systems,”Automatica,vol.50,pp.2624-2632,2014.
[5]F.Wang and H.Zhang,“Adaptive dynamic programming:an introduction,”IEEE Computational Intelligence Magazine,vol.4,pp.39-47,May 2009.
[6]W.Powell,Approximate dynamic programming:solving the curses of dimensionality.Wiley Interscience,2007.
[7]D.Bertsekas and J.Tsitsiklis,Athena scientific optimization and computation series,Neuro-dynamic programming.A-thena Scientific,1996.
[8]B.Luo,H.Wu,T.Wang,and D.Liu,“Data-based approximate policy iteration for affine nonlinear continuous-time optimal control design,”Automatica,vol.50,pp.3281-3290,2014.
[9]D.Wang and D.Liu,“Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach,”Neurocomputing,vol.78,no.1,pp.14-22,2012.
[10]H.Zhang and Q.W.etc,“A novel infinite-time optimal tracking control scheme for a class of discrete-time nonlinear systems via the greedy hdp iteration algorithm,”IEEE Transactions on Systems,Man and Cybernetics,Part B(Cybernetics),vol.38,no.4,pp.937-942,2008.
[11]R.Sutton and A.Barto,Reinforcement learning:an introduction.Cambridge,MA:MIT Press,1998.
[12]R.Bellman,Dynamic programming.Princeton University Press,1957.
[13]K.I.Markman,J.,“An iterative algorithm for solving hamilton-jacobi type equations,”SIAM Journal on Scientific Computing,vol.22,pp.312-329,2000.
[14]J.Sakamoto N.,Schaft A,“Analytical approximation methods for the stabilizing solution of the hamilton-jacobi equation,”IEEE Transactions on Automatic Control,vol.53,no.10,pp.2335-2350,2008.
[15]S.Cacace and E.etc,“A patchy dynamic programming scheme for a class of hamilton-jacobi-bellman equations,”SIAM Journal on Scientific Computing,vol.34,pp.2625-2649,2012.
[16]T.Hunting,A proof of the higher order accuracy of the patchy method for solving the Hamilton-Jacobi-Bellman equation.Phd thesis,University of California,Davis,CA,2011.
[17]R.Beard,Improving the closed-looped performance of nonlinear systems.Phd thesis,Rensselaer Polytechnic Institute,New York,1995.
[18]R.Beard and G.Sarudus,“Improving the performance of stabilizing control for nonlinear systems,”Control Systems Magazine,vol.16,pp.27-35,1996.
[19]R.Beard,G.Saridis,and J.Wen,“Galerkin approximations of the generalized hamilton-jacobi-bellman equation,”Automatica,vol.33,no.12,pp.2159-2177,1997.
[20]J.R.Beard,J.Gunther and W.Stirling,“The nonlinear projection filter,”AIAA Journal of Guidance,Control and Dynamics,vol.96,pp.582-626,1998.
[21]S.E.Smears,I.,“Discontinuous galerkin finite element approximation of nondivergence form elliptic equations with cordes coefficients,”SIAM Journal on Numerical Analysis,vol.51,no.4,pp.2088-2106,2013.
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[23]C.Aguilar and A.Krener,“Numerical solutions to the bellman equation of optimal control,”J.Optim.Theory Appl.,vol.160,pp.527-552,2014.
[24]D.E.Kirk,Optimal control theory:an introduction.Courier Corporation,2012.
[2]F.L.Lewis and D.Vrabie,Optimal control.Hoboken,New Jersey:John Wiley and Sons,2013.
[3]D.Bertsekas,Dynamic programming and optimal control.Nashua:Athena Scientific,2005.
[4]T.Bian,Y.Jiang,and Z.Jiang,“Adaptive dynamic programming and optimal control of nonlinear nonaffine systems,”Automatica,vol.50,pp.2624-2632,2014.
[5]F.Wang and H.Zhang,“Adaptive dynamic programming:an introduction,”IEEE Computational Intelligence Magazine,vol.4,pp.39-47,May 2009.
[6]W.Powell,Approximate dynamic programming:solving the curses of dimensionality.Wiley Interscience,2007.
[7]D.Bertsekas and J.Tsitsiklis,Athena scientific optimization and computation series,Neuro-dynamic programming.A-thena Scientific,1996.
[8]B.Luo,H.Wu,T.Wang,and D.Liu,“Data-based approximate policy iteration for affine nonlinear continuous-time optimal control design,”Automatica,vol.50,pp.3281-3290,2014.
[9]D.Wang and D.Liu,“Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach,”Neurocomputing,vol.78,no.1,pp.14-22,2012.
[10]H.Zhang and Q.W.etc,“A novel infinite-time optimal tracking control scheme for a class of discrete-time nonlinear systems via the greedy hdp iteration algorithm,”IEEE Transactions on Systems,Man and Cybernetics,Part B(Cybernetics),vol.38,no.4,pp.937-942,2008.
[11]R.Sutton and A.Barto,Reinforcement learning:an introduction.Cambridge,MA:MIT Press,1998.
[12]R.Bellman,Dynamic programming.Princeton University Press,1957.
[13]K.I.Markman,J.,“An iterative algorithm for solving hamilton-jacobi type equations,”SIAM Journal on Scientific Computing,vol.22,pp.312-329,2000.
[14]J.Sakamoto N.,Schaft A,“Analytical approximation methods for the stabilizing solution of the hamilton-jacobi equation,”IEEE Transactions on Automatic Control,vol.53,no.10,pp.2335-2350,2008.
[15]S.Cacace and E.etc,“A patchy dynamic programming scheme for a class of hamilton-jacobi-bellman equations,”SIAM Journal on Scientific Computing,vol.34,pp.2625-2649,2012.
[16]T.Hunting,A proof of the higher order accuracy of the patchy method for solving the Hamilton-Jacobi-Bellman equation.Phd thesis,University of California,Davis,CA,2011.
[17]R.Beard,Improving the closed-looped performance of nonlinear systems.Phd thesis,Rensselaer Polytechnic Institute,New York,1995.
[18]R.Beard and G.Sarudus,“Improving the performance of stabilizing control for nonlinear systems,”Control Systems Magazine,vol.16,pp.27-35,1996.
[19]R.Beard,G.Saridis,and J.Wen,“Galerkin approximations of the generalized hamilton-jacobi-bellman equation,”Automatica,vol.33,no.12,pp.2159-2177,1997.
[20]J.R.Beard,J.Gunther and W.Stirling,“The nonlinear projection filter,”AIAA Journal of Guidance,Control and Dynamics,vol.96,pp.582-626,1998.
[21]S.E.Smears,I.,“Discontinuous galerkin finite element approximation of nondivergence form elliptic equations with cordes coefficients,”SIAM Journal on Numerical Analysis,vol.51,no.4,pp.2088-2106,2013.
[22]I.Smears and E.Sueli,“Discontinuous galerkin finite element approximation of hamilton-jacobi-bellman equations with cordes coefficients,”SIAM Journal On Numerical Analysis,vol.52,no.2,pp.993-1016,2014.
[23]C.Aguilar and A.Krener,“Numerical solutions to the bellman equation of optimal control,”J.Optim.Theory Appl.,vol.160,pp.527-552,2014.
[24]D.E.Kirk,Optimal control theory:an introduction.Courier Corporation,2012.