A numerical method for hybrid optimal control based on dynamic programming
This contribution extends a numerical method for solving optimal control problems by dynamic programming to a class of hybrid dynamic systems with autonomous as well as controlled switching. The value function of the hybrid control system is calculated based on a full discretization of the state and...
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| Veröffentlicht in: | Nonlinear analysis. Hybrid systems Jg. 5; H. 2; S. 254 - 274 |
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| Sprache: | Englisch |
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01.05.2011
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| ISSN: | 1751-570X |
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| Abstract | This contribution extends a numerical method for solving optimal control problems by dynamic programming to a class of hybrid dynamic systems with autonomous as well as controlled switching. The value function of the hybrid control system is calculated based on a full discretization of the state and input spaces. A bound for the error due to discretization is obtained from modeling the error as perturbation of the continuous dynamics and the cost terms. It is shown that the bound approaches zero and that the value function of the discretized variant converges to the value function of the original problem if the discretization parameters go to zero. The performance of a numerical scheme exploiting the discretized system is illustrated for two different examples treated previously in literature. |
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| AbstractList | This contribution extends a numerical method for solving optimal control problems by dynamic programming to a class of hybrid dynamic systems with autonomous as well as controlled switching. The value function of the hybrid control system is calculated based on a full discretization of the state and input spaces. A bound for the error due to discretization is obtained from modeling the error as perturbation of the continuous dynamics and the cost terms. It is shown that the bound approaches zero and that the value function of the discretized variant converges to the value function of the original problem if the discretization parameters go to zero. The performance of a numerical scheme exploiting the discretized system is illustrated for two different examples treated previously in literature. |
| Author | Stursberg, Olaf Rungger, Matthias |
| Author_xml | – sequence: 1 givenname: Matthias surname: Rungger fullname: Rungger, Matthias email: matthias.rungger@uni-kassel.de – sequence: 2 givenname: Olaf surname: Stursberg fullname: Stursberg, Olaf email: stursberg@uni-kassel.de |
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| CitedBy_id | crossref_primary_10_1109_TFUZZ_2023_3348535 crossref_primary_10_1016_j_automatica_2016_11_040 crossref_primary_10_1155_2012_457121 crossref_primary_10_23919_JSEE_2022_000114 crossref_primary_10_1109_TNNLS_2013_2288067 crossref_primary_10_1515_auto_2018_0080 crossref_primary_10_1109_TNNLS_2015_2388672 crossref_primary_10_3390_pr8020170 crossref_primary_10_1016_j_asoc_2016_09_007 |
| Cites_doi | 10.1109/9.654885 10.1109/CDC.2009.5400068 10.1109/9.847100 10.1109/9.273337 10.1023/A:1017992615625 10.1109/CDC.1999.827981 10.1109/TAC.2007.904451 10.1007/BF01448394 10.1109/CDC.2010.5718054 10.1007/s00211-004-0555-4 10.1007/BF01442644 10.1137/040618072 10.1109/CDC.1999.832814 10.3182/20090916-3-ES-3003.00055 10.1007/s002110050241 10.1109/CDC.1999.831404 10.1109/CDC.2004.1430241 10.3182/20090916-3-ES-3003.00053 10.1109/CDC.1995.478514 10.1109/TAC.2009.2015528 10.1049/ip-cta:20050094 10.1109/MCS.2008.931718 10.1016/S1474-6670(17)36459-5 |
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| References_xml | – reference: M.S. Branicky, S.K. Mitter, Algorithms for optimal hybrid control, in: Conference on Decision and Control, 1995, pp. 2661–2666. – volume: 44 start-page: 1259 year: 2005 end-page: 1288 ident: br000035 article-title: Hybrid control systems and viscosity solutions publication-title: SIAM Journal on Control and Optimization – volume: 75 start-page: 319 year: 1997 end-page: 337 ident: br000090 article-title: An adaptive grid scheme for the discrete Hamilton–Jacobi–Bellman equation publication-title: Numerische Mathematik – reference: M. Rungger, O. Stursberg, Function approximation for the deterministic Hamilton–Jacobi–Bellman equation, in: IEEE Conference on Decision and Control, 2009, pp. 2268–2273. – reference: R.L.V. González, M.M. Tidball, On a discrete time approximation of the Hamilton–Jacobi equation of dynamic programming. Technical Report 1375, Institut National de Recherche en Informatique et en Automatique, INRIA, 1991. – reference: H.J. Sussmann, A maximum principle for hybrid optimal control problems, in: Conference on Decision and Control, 1999, pp. 425–430. – volume: 43 start-page: 31 year: 1998 end-page: 45 ident: br000030 article-title: A unified framework for hybrid control: model and optimal control theory publication-title: IEEE Transactions on Automatic Control – year: 1997 ident: br000105 article-title: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations – volume: 45 start-page: 629 year: 2000 end-page: 637 ident: br000040 article-title: Piecewise linear quadratic optimal control publication-title: IEEE Transactions on Automatic Control – reference: O. Stursberg, A graph search algorithm for optimal control of hybrid systems, in: Conference on Decision and Control, 2004, pp. 1412–1417. – volume: 52 start-page: 1587 year: 2007 end-page: 1603 ident: br000025 article-title: On the hybrid optimal control problem: theory and algorithms publication-title: IEEE Transactions on Automatic Control – reference: A. Bemporad, D. Corona, A. Giua, C. Seatzu, Optimal state-feedback quadratic regulation of linear hybrid automata, in: IFAC Conf. on Analysis and Design of Hybrid Systems, 2003, pp. 407–412. – volume: 41 start-page: 11 year: 2005 end-page: 27 ident: br000085 article-title: Optimal control of switching systems publication-title: Automatica – volume: 29 start-page: 28 year: 2009 end-page: 93 ident: br000010 article-title: Hybrid dynamical systems publication-title: IEEE Control Systems Magazine – volume: 10 start-page: 367 year: 1983 end-page: 388 ident: br000120 article-title: On a discrete approximation of the Hamilton–Jacobi equation of dynamic programming publication-title: Applied Mathematics and Optimization – reference: X. Xu, P.J. Antsaklis, Optimal control of switched autonomous systems, in: Conference on Decision and Control, 2002, pp. 4401–4406. – year: 1992 ident: br000100 article-title: Numerical Methods for Stochastic Control Problems in Continuous Time – volume: 39 start-page: 33 year: 1994 end-page: 46 ident: br000115 article-title: Numerical nonlinear regulator design publication-title: IEEE Transactions on Automatic Control – reference: M. Rungger, O. Stursberg, Continuity of the value function for exit time optimal control problems of hybrid systems, in: IEEE Conference on Decision and Control, 2010 (in press). – volume: 49 start-page: 291 year: 2001 end-page: 323 ident: br000095 article-title: Variable resolution discretization in optimal control publication-title: Machine Learning Journal – year: 1995 ident: br000005 article-title: Dynamic Programming and Optimal Control – reference: S. Hedlund, A. Rantzer, Optimal control of hybrid systems, in: Conference on Decision and Control, 1999, pp. 3972–3977. – reference: M. Rungger, O. Stursberg, Optimal control of deterministic hybrid systems using dynamic programming, in: IFAC Conf. on Analysis and Design of Hybrid Systems, 2009, pp. 316–321. – year: 1992 ident: br000145 article-title: Nonlinear Systems – start-page: 123 year: 2006 end-page: 137 ident: br000070 article-title: Optimality zone algorithms for hybrid systems: efficient algorithms for optimal location and control computation publication-title: Hybrid Systems: Computation and Control – reference: R.L.V. González, M.M. Tidball, On the rates of convergence of fully discrete solutions of Hamilton–Jacobi equations. Technical Report 1376, Institut National de Recherche en Informatique et en Automatique, INRIA, 1991. – reference: P. Riedinger, F. Kratz, C. Iung, C. Zanne, Linear quadratic optimization for hybrid systems, in: Conference on Decision and Control, 1999, pp. 3059–3064. – reference: B. Passenberg, O. Stursberg, A graph search algorithm for optimizing the discrete location sequence in hybrid optimal control problems, in: IFAC Conf. on Analysis and Design of Hybrid Systems, 2009, pp. 304–309. – reference: D.W. Moore, Simplicial mesh generation with applications, Ph.D. Thesis, Cornell University, 1992. – volume: 99 start-page: 85 year: 2004 end-page: 112 ident: br000150 article-title: Error estimation and adaptive discretization for the discrete stochastic Hamilton–Jacobi–Bellman equation publication-title: Numerische Mathematik – volume: 9 year: 1997 ident: br000160 article-title: Multidimensional triangulation and interpolation for reinforcement learning publication-title: Neural Information Processing Systems – volume: 153 start-page: 567 year: 2006 end-page: 574 ident: br000045 article-title: Relaxed dynamic programming in switching systems publication-title: IEE Proceedings — Control Theory & Applications – volume: 15 start-page: 1 year: 1987 end-page: 13 ident: br000125 article-title: A numerical approach to the infinite horizon problem of deterministic control theory publication-title: Applied Mathematics and Optimization – ident: 10.1016/j.nahs.2010.09.002_br000155 – ident: 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discretization in optimal control publication-title: Machine Learning Journal doi: 10.1023/A:1017992615625 – volume: 9 year: 1997 ident: 10.1016/j.nahs.2010.09.002_br000160 article-title: Multidimensional triangulation and interpolation for reinforcement learning publication-title: Neural Information Processing Systems – ident: 10.1016/j.nahs.2010.09.002_br000065 doi: 10.1109/CDC.1999.827981 – volume: 52 start-page: 1587 year: 2007 ident: 10.1016/j.nahs.2010.09.002_br000025_1 article-title: On the hybrid optimal control problem: theory and algorithms publication-title: IEEE Transactions on Automatic Control doi: 10.1109/TAC.2007.904451 – year: 1995 ident: 10.1016/j.nahs.2010.09.002_br000005 – volume: 10 start-page: 367 year: 1983 ident: 10.1016/j.nahs.2010.09.002_br000120 article-title: On a discrete approximation of the Hamilton–Jacobi equation of dynamic programming publication-title: Applied Mathematics and Optimization doi: 10.1007/BF01448394 – ident: 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start-page: 11 year: 2005 ident: 10.1016/j.nahs.2010.09.002_br000085 article-title: Optimal control of switching systems publication-title: Automatica – volume: 75 start-page: 319 year: 1997 ident: 10.1016/j.nahs.2010.09.002_br000090 article-title: An adaptive grid scheme for the discrete Hamilton–Jacobi–Bellman equation publication-title: Numerische Mathematik doi: 10.1007/s002110050241 – ident: 10.1016/j.nahs.2010.09.002_br000055 – ident: 10.1016/j.nahs.2010.09.002_br000020 doi: 10.1109/CDC.1999.831404 – ident: 10.1016/j.nahs.2010.09.002_br000080 doi: 10.1109/CDC.2004.1430241 – ident: 10.1016/j.nahs.2010.09.002_br000140 – ident: 10.1016/j.nahs.2010.09.002_br000075 doi: 10.3182/20090916-3-ES-3003.00053 – year: 1992 ident: 10.1016/j.nahs.2010.09.002_br000145 – year: 1992 ident: 10.1016/j.nahs.2010.09.002_br000100 – ident: 10.1016/j.nahs.2010.09.002_br000060 doi: 10.1109/CDC.1995.478514 – volume: 54 start-page: 1440 year: 2009 ident: 10.1016/j.nahs.2010.09.002_br000025_2 article-title: On the hybrid optimal control problem: theory and algorithms publication-title: IEEE Transactions on Automatic Control doi: 10.1109/TAC.2009.2015528 – volume: 153 start-page: 567 year: 2006 ident: 10.1016/j.nahs.2010.09.002_br000045 article-title: Relaxed dynamic programming in switching systems publication-title: IEE Proceedings — Control Theory & Applications doi: 10.1049/ip-cta:20050094 – volume: 29 start-page: 28 year: 2009 ident: 10.1016/j.nahs.2010.09.002_br000010 article-title: Hybrid dynamical systems publication-title: IEEE Control Systems Magazine doi: 10.1109/MCS.2008.931718 – start-page: 123 year: 2006 ident: 10.1016/j.nahs.2010.09.002_br000070 article-title: Optimality zone algorithms for hybrid systems: efficient algorithms for optimal location and control computation – ident: 10.1016/j.nahs.2010.09.002_br000050 doi: 10.1016/S1474-6670(17)36459-5 |
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| Title | A numerical method for hybrid optimal control based on dynamic programming |
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