The singular optimal control of switched systems

This chapter studies a singular case of Optimal Control Problems(OCPs) governed by a class of switched control systems. We proposea new mathematical formalism for this type of switched dynamic systemsand study OCPs with a quadratic cost functionals. The original sophisticatedoptimization problem is...

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Fecha de publicación:
2017
Institución:
Universidad de Medellín
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Repositorio UDEM
Idioma:
eng
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oai:repository.udem.edu.co:11407/4381
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http://hdl.handle.net/11407/4381
Palabra clave:
Optimal control
Singularities
Switched dynamic systems
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oai_identifier_str oai:repository.udem.edu.co:11407/4381
network_acronym_str REPOUDEM2
network_name_str Repositorio UDEM
repository_id_str
dc.title.spa.fl_str_mv The singular optimal control of switched systems
title The singular optimal control of switched systems
spellingShingle The singular optimal control of switched systems
Optimal control
Singularities
Switched dynamic systems
title_short The singular optimal control of switched systems
title_full The singular optimal control of switched systems
title_fullStr The singular optimal control of switched systems
title_full_unstemmed The singular optimal control of switched systems
title_sort The singular optimal control of switched systems
dc.contributor.affiliation.spa.fl_str_mv Azhmyakov, V., Department of Basic Sciences, Universidad de Medellin, Medellin, Colombia
Velez, C.M., Department of Mathematical Science, Univeridad EAFIT, Medellin, Colombia
dc.subject.keyword.eng.fl_str_mv Optimal control
Singularities
Switched dynamic systems
topic Optimal control
Singularities
Switched dynamic systems
description This chapter studies a singular case of Optimal Control Problems(OCPs) governed by a class of switched control systems. We proposea new mathematical formalism for this type of switched dynamic systemsand study OCPs with a quadratic cost functionals. The original sophisticatedoptimization problem is next replaced by an auxiliary "weaklyrelaxed" OCP. Our main result includes a formal proof of the local convexityproperty of the obtained auxiliary OCP. The convex structure ofthe OCP implies a possibility to apply a variety of powerful and relativelysimple optimization schemes to the sophisticated singular OCP involvingswitched dynamics. The conceptual numerical approach we finallydevelop includes an optimal switching times selection ("timing") and asimultaneous optimal switched modes sequence scheduling ("sequencing"). © 2017 Nova Science Publishers, Inc. All rights reserved.
publishDate 2017
dc.date.accessioned.none.fl_str_mv 2017-12-19T19:36:52Z
dc.date.available.none.fl_str_mv 2017-12-19T19:36:52Z
dc.date.created.none.fl_str_mv 2017
dc.type.eng.fl_str_mv Book Chapter
dc.type.coarversion.fl_str_mv http://purl.org/coar/version/c_970fb48d4fbd8a85
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dc.type.driver.none.fl_str_mv info:eu-repo/semantics/bookPart
dc.identifier.isbn.none.fl_str_mv 9781536109924; 9781536109795
dc.identifier.uri.none.fl_str_mv http://hdl.handle.net/11407/4381
dc.identifier.reponame.spa.fl_str_mv reponame:Repositorio Institucional Universidad de Medellín
dc.identifier.instname.spa.fl_str_mv instname:Universidad de Medellín
identifier_str_mv 9781536109924; 9781536109795
reponame:Repositorio Institucional Universidad de Medellín
instname:Universidad de Medellín
url http://hdl.handle.net/11407/4381
dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.isversionof.spa.fl_str_mv https://www.scopus.com/inward/record.uri?eid=2-s2.0-85020138152&partnerID=40&md5=70ac329a297f65c5da5a77ebe1b215b4
dc.relation.ispartofes.spa.fl_str_mv Advances in Communications and Media Research
Advances in Communications and Media Research Volume 12, 1 January 2017, Pages 127-143
dc.relation.references.spa.fl_str_mv Armijo, L. (1966). Minimization of functions having lipschitz continuous first partial derivatives. Pacific Journal of Mathematics, 16(1), 1-3.
Atkinson, K., & Han, W. (2001). Theoretical Numerical Analysis.
Axelsson, H., Wardi, Y., Egerstedt, M., & Verriest, E. I. (2008). Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. Journal of Optimization Theory and Applications, 136(2), 167-186. doi:10.1007/s10957-007-9305-y
Azhmyakov, V., Basin, M., & Reincke-Collon, C. (2014). Optimal LQ-type switched control design for a class of linear systems with piecewise constant inputs. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), , 19 6976-6981.
Azhmyakov, V., Basin, M. V., & Raisch, J. (2012). A proximal point based approach to optimal control of affine switched systems. Discrete Event Dynamic Systems: Theory and Applications, 22(1), 61-81. doi:10.1007/s10626-011-0109-8
Azhmyakov, V., Boltyanski, V. G., & Poznyak, A. (2008). Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems, 2(4), 1089-1097. doi:10.1016/j.nahs.2008.09.003
Azhmyakov, V., Cabrera, J., & Poznyak, A. (0000). Optimal fixed - levels control for non - linear systems with quadratic cost functionals. Optimal Control Applications and Methods.
Azhmyakov, V., Egerstedt, M., Fridman, L., & Poznyak, A. (2009). Continuity properties of nonlinear affine control systems: Applications to hybrid and sliding mode dynamics. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), 3(PART 1) 204-209.
Azhmyakov, V., & Juarez, R. (2015). On the projected gradient method for switched - mode systems optimization. In: Proceedings of the 5th IFAC Conference on Analysis and Design of Hybrid Systems.
Azhmyakov, V., & Schmidt, W. (2006). Approximations of relaxed optimal control problems. Journal of Optimization Theory and Applications, 130(1), 61-77. doi:10.1007/s10957-006-9085-9
Bello Cruz, J. Y., & De Oliveira, C. W. (2014). On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Hilbert Spaces, 1-18.
Benoist, J., & Hiriart-Urruty, J. -. (1996). What is the subdifferential of the closed convex hull of a function? SIAM Journal on Mathematical Analysis, 27(6), 1661-1679. doi:10.1137/S0036141094265936
Bertsekas, D. P. (1995). Nonlinear Programming.
Betts, J. T. (2001). Practical Methods for Optimal Control using Nonlinear Programming.
Boltyanski, V., & Poznyak, A. (2012). The robust maximum principle. The Robust Maximum Principle.
Bonnard, B., & Chyba, M. (2003). Singular Trajectories and their Role in Control Theory.
Burachik, R., Drummond, L. M., Iusem, A. N., & Svaiter, B. F. (1995). Full convergence of the steepest descent method with inexact line searches. Optimization, 32(2), 137-146. doi:10.1080/02331939508844042
Cesari, L. (1983). Optimization - Theory and Applications.
Clarke, F. H. (1983). Optimization and nonsmooth analysis. Optimization and Nonsmooth Analysis.
Clarke, F. H., Ledyaev, Y. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth Analysis and Control Theory.
Dmitruk, A. V. (1990). Maximum principle for a general optimal control problem with state and regular mixed constraints. Optimality of Control Dynamical Systems, 14, 26-42.
Dmitruk, A. V. (2009). On the development of pontryagin's maximum principle in the works of A.ya. dubovitskii and A.A. milyutin. Control and Cybernetics, 38(4), 923-957.
Egerstedt, M., Wardi, Y., & Axelsson, H. (2006). Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51(1), 110-115. doi:10.1109/TAC.2005.861711
Fattorini, H. O. (1999). Infinite Dimensional Optimization and Control Theory
Gill, P. E., Murray, W., & Wright, M. H. (1981). Practical Optimization
Goldstein, A. A. (1964). Convex programming in hilbert space. Bulletin of the American Mathematical Society, 70(5), 709-710. doi:10.1090/S0002-9904-1964-11178-2
Hale, J. K. (1969). Ordinary Differential Equations.
Hale, M., & Wardi, Y. (2014). Mode scheduling under dwell time constraints in switched-mode systems. Paper presented at the Proceedings of the American Control Conference, 3954-3959. doi:10.1109/ACC.2014.6858763
Hiriart-Urruty, J. -., & Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms.
Ioffe, A. D., & Tihomirov, V. M. (1979). Theory of Extremal Problems.
Li, D. (1995). Zero duality gap for a class of nonconvex optimization problems. Journal of Optimization Theory and Applications, 85(2), 309-324. doi:10.1007/BF02192229
Lincoln, B., & Rantzer, A. (2001). Optimizing linear system switching. Paper presented at the Proceedings of the IEEE Conference on Decision and Control, 3 2063-2068.
Mitsos, A., Chachuat, B., & Barton, P. I. (2009). Mccormick-based relaxations of algorithms. SIAM Journal on Optimization, 20(2), 573-601. doi:10.1137/080717341
Polak, E. (1997). Optimization, springer-verlag.
Rockafellar, R. T., & Wets, R. J. -. (1998). Variational analysis. Grundlehren Math.Wiss., 317.
Roubíček, T. (1997). Relaxation in Optimization Theory and Variational Calculus
Scott, J. K., & Barton, P. I. (2013). Convex and concave relaxations for the parametric solutions of semi-explicit index-one differential-algebraic equations. Journal of Optimization Theory and Applications, 156(3), 617-649. doi:10.1007/s10957-012-0149-8
Teo, K. L., Goh, C. J., & Wong, K. H. (1991). A Unified Computational Approach to Optimal Control Problems.
Wardi, Y. (2012). Optimal control of switched-mode dynamical systems. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), 4-8.
Wardi, Y., Egerstedt, M., & Hale, M. (0000). Switched-mode systems: Gradient-descent algorithms with armijo step sizes. Discrete Event Dynamic Systems, to Appear.
Wardi, Y., Egerstedt, M., & Twu, P. (2012). A controlled-precision algorithm for mode-switching optimization. Paper presented at the Proceedings of the IEEE Conference on Decision and Control, 713-718. doi:10.1109/CDC.2012.6426621
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_16ec
rights_invalid_str_mv http://purl.org/coar/access_right/c_16ec
dc.publisher.spa.fl_str_mv Nova Science Publishers, Inc.
dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias Básicas
dc.source.spa.fl_str_mv Scopus
institution Universidad de Medellín
repository.name.fl_str_mv Repositorio Institucional Universidad de Medellin
repository.mail.fl_str_mv repositorio@udem.edu.co
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spelling 2017-12-19T19:36:52Z2017-12-19T19:36:52Z20179781536109924; 9781536109795http://hdl.handle.net/11407/4381reponame:Repositorio Institucional Universidad de Medellíninstname:Universidad de MedellínThis chapter studies a singular case of Optimal Control Problems(OCPs) governed by a class of switched control systems. We proposea new mathematical formalism for this type of switched dynamic systemsand study OCPs with a quadratic cost functionals. The original sophisticatedoptimization problem is next replaced by an auxiliary "weaklyrelaxed" OCP. Our main result includes a formal proof of the local convexityproperty of the obtained auxiliary OCP. The convex structure ofthe OCP implies a possibility to apply a variety of powerful and relativelysimple optimization schemes to the sophisticated singular OCP involvingswitched dynamics. The conceptual numerical approach we finallydevelop includes an optimal switching times selection ("timing") and asimultaneous optimal switched modes sequence scheduling ("sequencing"). © 2017 Nova Science Publishers, Inc. All rights reserved.engNova Science Publishers, Inc.Facultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85020138152&partnerID=40&md5=70ac329a297f65c5da5a77ebe1b215b4Advances in Communications and Media ResearchAdvances in Communications and Media Research Volume 12, 1 January 2017, Pages 127-143Armijo, L. (1966). Minimization of functions having lipschitz continuous first partial derivatives. Pacific Journal of Mathematics, 16(1), 1-3.Atkinson, K., & Han, W. (2001). Theoretical Numerical Analysis.Axelsson, H., Wardi, Y., Egerstedt, M., & Verriest, E. I. (2008). Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. Journal of Optimization Theory and Applications, 136(2), 167-186. doi:10.1007/s10957-007-9305-yAzhmyakov, V., Basin, M., & Reincke-Collon, C. (2014). Optimal LQ-type switched control design for a class of linear systems with piecewise constant inputs. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), , 19 6976-6981.Azhmyakov, V., Basin, M. V., & Raisch, J. (2012). A proximal point based approach to optimal control of affine switched systems. Discrete Event Dynamic Systems: Theory and Applications, 22(1), 61-81. doi:10.1007/s10626-011-0109-8Azhmyakov, V., Boltyanski, V. G., & Poznyak, A. (2008). Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems, 2(4), 1089-1097. doi:10.1016/j.nahs.2008.09.003Azhmyakov, V., Cabrera, J., & Poznyak, A. (0000). Optimal fixed - levels control for non - linear systems with quadratic cost functionals. Optimal Control Applications and Methods.Azhmyakov, V., Egerstedt, M., Fridman, L., & Poznyak, A. (2009). Continuity properties of nonlinear affine control systems: Applications to hybrid and sliding mode dynamics. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), 3(PART 1) 204-209.Azhmyakov, V., & Juarez, R. (2015). On the projected gradient method for switched - mode systems optimization. In: Proceedings of the 5th IFAC Conference on Analysis and Design of Hybrid Systems.Azhmyakov, V., & Schmidt, W. (2006). Approximations of relaxed optimal control problems. Journal of Optimization Theory and Applications, 130(1), 61-77. doi:10.1007/s10957-006-9085-9Bello Cruz, J. Y., & De Oliveira, C. W. (2014). On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Hilbert Spaces, 1-18.Benoist, J., & Hiriart-Urruty, J. -. (1996). What is the subdifferential of the closed convex hull of a function? SIAM Journal on Mathematical Analysis, 27(6), 1661-1679. doi:10.1137/S0036141094265936Bertsekas, D. P. (1995). Nonlinear Programming.Betts, J. T. (2001). Practical Methods for Optimal Control using Nonlinear Programming.Boltyanski, V., & Poznyak, A. (2012). The robust maximum principle. The Robust Maximum Principle.Bonnard, B., & Chyba, M. (2003). Singular Trajectories and their Role in Control Theory.Burachik, R., Drummond, L. M., Iusem, A. N., & Svaiter, B. F. (1995). Full convergence of the steepest descent method with inexact line searches. Optimization, 32(2), 137-146. doi:10.1080/02331939508844042Cesari, L. (1983). Optimization - Theory and Applications.Clarke, F. H. (1983). Optimization and nonsmooth analysis. Optimization and Nonsmooth Analysis.Clarke, F. H., Ledyaev, Y. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth Analysis and Control Theory.Dmitruk, A. V. (1990). Maximum principle for a general optimal control problem with state and regular mixed constraints. Optimality of Control Dynamical Systems, 14, 26-42.Dmitruk, A. V. (2009). On the development of pontryagin's maximum principle in the works of A.ya. dubovitskii and A.A. milyutin. Control and Cybernetics, 38(4), 923-957.Egerstedt, M., Wardi, Y., & Axelsson, H. (2006). Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51(1), 110-115. doi:10.1109/TAC.2005.861711Fattorini, H. O. (1999). Infinite Dimensional Optimization and Control TheoryGill, P. E., Murray, W., & Wright, M. H. (1981). Practical OptimizationGoldstein, A. A. (1964). Convex programming in hilbert space. Bulletin of the American Mathematical Society, 70(5), 709-710. doi:10.1090/S0002-9904-1964-11178-2Hale, J. K. (1969). Ordinary Differential Equations.Hale, M., & Wardi, Y. (2014). Mode scheduling under dwell time constraints in switched-mode systems. Paper presented at the Proceedings of the American Control Conference, 3954-3959. doi:10.1109/ACC.2014.6858763Hiriart-Urruty, J. -., & Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms.Ioffe, A. D., & Tihomirov, V. M. (1979). Theory of Extremal Problems.Li, D. (1995). Zero duality gap for a class of nonconvex optimization problems. Journal of Optimization Theory and Applications, 85(2), 309-324. doi:10.1007/BF02192229Lincoln, B., & Rantzer, A. (2001). Optimizing linear system switching. Paper presented at the Proceedings of the IEEE Conference on Decision and Control, 3 2063-2068.Mitsos, A., Chachuat, B., & Barton, P. I. (2009). Mccormick-based relaxations of algorithms. SIAM Journal on Optimization, 20(2), 573-601. doi:10.1137/080717341Polak, E. (1997). Optimization, springer-verlag.Rockafellar, R. T., & Wets, R. J. -. (1998). Variational analysis. Grundlehren Math.Wiss., 317.Roubíček, T. (1997). Relaxation in Optimization Theory and Variational CalculusScott, J. K., & Barton, P. I. (2013). Convex and concave relaxations for the parametric solutions of semi-explicit index-one differential-algebraic equations. Journal of Optimization Theory and Applications, 156(3), 617-649. doi:10.1007/s10957-012-0149-8Teo, K. L., Goh, C. J., & Wong, K. H. (1991). A Unified Computational Approach to Optimal Control Problems.Wardi, Y. (2012). Optimal control of switched-mode dynamical systems. Paper presented at the IFAC Proceedings Volumes (IFAC-PapersOnline), 4-8.Wardi, Y., Egerstedt, M., & Hale, M. (0000). Switched-mode systems: Gradient-descent algorithms with armijo step sizes. Discrete Event Dynamic Systems, to Appear.Wardi, Y., Egerstedt, M., & Twu, P. (2012). A controlled-precision algorithm for mode-switching optimization. Paper presented at the Proceedings of the IEEE Conference on Decision and Control, 713-718. doi:10.1109/CDC.2012.6426621ScopusThe singular optimal control of switched systemsBook Chapterinfo:eu-repo/semantics/bookParthttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_3248Azhmyakov, V., Department of Basic Sciences, Universidad de Medellin, Medellin, ColombiaVelez, C.M., Department of Mathematical Science, Univeridad EAFIT, Medellin, ColombiaAzhmyakov V.Velez C.M.Department of Basic Sciences, Universidad de Medellin, Medellin, ColombiaDepartment of Mathematical Science, Univeridad EAFIT, Medellin, ColombiaOptimal controlSingularitiesSwitched dynamic systemsThis chapter studies a singular case of Optimal Control Problems(OCPs) governed by a class of switched control systems. We proposea new mathematical formalism for this type of switched dynamic systemsand study OCPs with a quadratic cost functionals. The original sophisticatedoptimization problem is next replaced by an auxiliary "weaklyrelaxed" OCP. Our main result includes a formal proof of the local convexityproperty of the obtained auxiliary OCP. The convex structure ofthe OCP implies a possibility to apply a variety of powerful and relativelysimple optimization schemes to the sophisticated singular OCP involvingswitched dynamics. The conceptual numerical approach we finallydevelop includes an optimal switching times selection ("timing") and asimultaneous optimal switched modes sequence scheduling ("sequencing"). © 2017 Nova Science Publishers, Inc. All rights reserved.http://purl.org/coar/access_right/c_16ec11407/4381oai:repository.udem.edu.co:11407/43812020-05-27 15:42:02.974Repositorio Institucional Universidad de Medellinrepositorio@udem.edu.co

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