On the linear quadratic dynamic optimization problems with fixed-levels control functions

This paper deals with a constrained LQ-type optimal control problem (OCP) in the presence of fixed levels input restrictions. We consider control processes governed by linear differential equations with a priori known control switching structure. The set of admissible inputs reflects some important...

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Fecha de publicación:
2017
Institución:
Universidad de Medellín
Repositorio:
Repositorio UDEM
Idioma:
eng
OAI Identifier:
oai:repository.udem.edu.co:11407/4256
Acceso en línea:
http://hdl.handle.net/11407/4256
Palabra clave:
Convex optimization
Numerical methods
Optimal control
Systems theory
Convex optimization
Differential equations
Optimal control systems
Optimization
System theory
Control functions
Control process
Control switching
Engineering applications
Linear differential equation
Linear quadratic
Optimal control problem
Optimal controls
Numerical methods
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http://purl.org/coar/access_right/c_16ec
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oai_identifier_str oai:repository.udem.edu.co:11407/4256
network_acronym_str REPOUDEM2
network_name_str Repositorio UDEM
repository_id_str
dc.title.spa.fl_str_mv On the linear quadratic dynamic optimization problems with fixed-levels control functions
title On the linear quadratic dynamic optimization problems with fixed-levels control functions
spellingShingle On the linear quadratic dynamic optimization problems with fixed-levels control functions
Convex optimization
Numerical methods
Optimal control
Systems theory
Convex optimization
Differential equations
Optimal control systems
Optimization
System theory
Control functions
Control process
Control switching
Engineering applications
Linear differential equation
Linear quadratic
Optimal control problem
Optimal controls
Numerical methods
title_short On the linear quadratic dynamic optimization problems with fixed-levels control functions
title_full On the linear quadratic dynamic optimization problems with fixed-levels control functions
title_fullStr On the linear quadratic dynamic optimization problems with fixed-levels control functions
title_full_unstemmed On the linear quadratic dynamic optimization problems with fixed-levels control functions
title_sort On the linear quadratic dynamic optimization problems with fixed-levels control functions
dc.contributor.affiliation.spa.fl_str_mv Azhmyakov, V., Department of Basic Sciences, Universidad de Medellin, Medellin, Colombia
Trujillo, L.A.G., School of Ingeniering, Universidad de Medellin, Medellin, Colombia
dc.subject.keyword.eng.fl_str_mv Convex optimization
Numerical methods
Optimal control
Systems theory
Convex optimization
Differential equations
Optimal control systems
Optimization
System theory
Control functions
Control process
Control switching
Engineering applications
Linear differential equation
Linear quadratic
Optimal control problem
Optimal controls
Numerical methods
topic Convex optimization
Numerical methods
Optimal control
Systems theory
Convex optimization
Differential equations
Optimal control systems
Optimization
System theory
Control functions
Control process
Control switching
Engineering applications
Linear differential equation
Linear quadratic
Optimal control problem
Optimal controls
Numerical methods
description This paper deals with a constrained LQ-type optimal control problem (OCP) in the presence of fixed levels input restrictions. We consider control processes governed by linear differential equations with a priori known control switching structure. The set of admissible inputs reflects some important natural engineering applications and moreover, can also be interpreted as a result of a quantization procedure applied to the original dynamic system. We propose a novel implementable algorithm that makes it possible to calculate a (numerically consistent) approximative solution to the constrained LQ-type OCPs under consideration. Our contribution mainly discusses theoretic aspects of the proposed solution scheme and contains an illustrative numerical example. © 2017, Forum-Editrice Universitaria Udinese SRL. All rights reserved.
publishDate 2017
dc.date.accessioned.none.fl_str_mv 2017-12-19T19:36:42Z
dc.date.available.none.fl_str_mv 2017-12-19T19:36:42Z
dc.date.created.none.fl_str_mv 2017
dc.type.eng.fl_str_mv Article
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dc.type.coar.fl_str_mv http://purl.org/coar/resource_type/c_6501
http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.driver.none.fl_str_mv info:eu-repo/semantics/article
dc.identifier.issn.none.fl_str_mv 11268042
dc.identifier.uri.none.fl_str_mv http://hdl.handle.net/11407/4256
dc.identifier.reponame.spa.fl_str_mv reponame:Repositorio Institucional Universidad de Medellín
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identifier_str_mv 11268042
reponame:Repositorio Institucional Universidad de Medellín
instname:Universidad de Medellín
url http://hdl.handle.net/11407/4256
dc.language.iso.none.fl_str_mv eng
language eng
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dc.relation.ispartofes.spa.fl_str_mv Italian Journal of Pure and Applied Mathematics
Italian Journal of Pure and Applied Mathematics Volume 2017, Issue 37, January 2017, Pages 219-237
dc.relation.references.spa.fl_str_mv Ali, U., Cai, H., Mostofi, Y., & Wardi, Y. (2016). Motion and communication co-optimization with path planning and online channel prediction. Paper presented at the Proceedings of the American Control Conference, , 2016-July 7079-7084. doi:10.1109/ACC.2016.7526789
Armijo, L. (1966). Minimization of functions having lipschitz continuous first partial derivatives. Pacific Journal of Mathematics, 16(1), 1-3.
Azhmyakov, V., Basin, M. V., & Gil García, A. E. (2014). Optimal control processes associated with a class of discontinuous control systems: Applications to sliding mode dynamics. Kybernetika, 50(1), 5-18. doi:10.14736/kyb-2014-1-0005
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., Galvan-Guerra, R., & Egerstedt, M. (2009). Hybrid LQ-optimization using dynamic programming. Paper presented at the Proceedings of the American Control Conference, 3617-3623. doi:10.1109/ACC.2009.5160100
Azhmyakov, V., Galván-Guerra, R., & Egerstedt, M. (2010). On the LQ-based optimization techniques for impulsive hybrid control systems. Paper presented at the Proceedings of the 2010 American Control Conference, ACC 2010, 129-135.
Azhmyakov, V., & Juarez, R. (2015). On the projected gradient methods for switched - mode systems optimization. IFAC-PapersOnLine, 48(27), 181-186. doi:10.1016/j.ifacol.2015.11.172
Azhmyakov, V., Juarez, R., & Pickl, S. (2015). On the local convexity of singular optimal control problems associated with the switched-mode dynamic systems. IFAC-PapersOnLine, 28(25), 271-276. doi:10.1016/j.ifacol.2015.11.099
Azhmyakov, V., Martinez, J. C., & Poznyak, A. (2016). Optimal fixed-levels control for nonlinear systems with quadratic cost-functionals. Optimal Control Applications and Methods, 37(5), 1035-1055. doi:10.1002/oca.2223
Azhmyakov, V., & Raisch, J. (2008). Convex control systems and convex optimal control problems with constraints. IEEE Transactions on Automatic Control, 53(4), 993-998. doi:10.1109/TAC.2008.919848
Azhmyakov, V., Serrezuela, R. R., & Trujillo, L. A. G. (2014). Approximations based optimal control design for a class of switched dynamic systems. Paper presented at the IECON Proceedings (Industrial Electronics Conference), 90-95. doi:10.1109/IECON.2014.7048482
Banks, S. P., & Khathur, S. A. (1989). Structure and control of piecewise-linear systems. International Journal of Control, 50(2), 667-686. doi:10.1080/00207178908953388
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.
Bonilla, M., Malabre, M., & Azhmyakov, V. (2015). An implicit systems characterization of a class of impulsive linear switched control processes. part 1: Modeling. Nonlinear Analysis: Hybrid Systems, 15, 157-170. doi:10.1016/j.nahs.2014.04.002
Branicky, M. S., Borkar, V. S., & Mitter, S. K. (1998). A unified framework for hybrid control: Model and optimal control theory. IEEE Transactions on Automatic Control, 43(1), 31-45. doi:10.1109/9.654885
Brockett, R. W., & Liberzon, D. (2000). Quantized feedback stabilization of linear systems. IEEE Transactions on Automatic Control, 45(7), 1279-1289. doi:10.1109/9.867021
Cassandras, C. G., Pepyne, D. L., & Wardi, Y. (2001). Optimal control of a class of hybrid systems. IEEE Transactions on Automatic Control, 46(3), 398-415. doi:10.1109/9.911417
Ding, X. -., Wardi, Y., & Egerstedt, M. (2009). On-line optimization of switched-mode dynamical systems. IEEE Transactions on Automatic Control, 54(9), 2266-2271. doi:10.1109/TAC.2009.2026864
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.
Galván-Guerra, R., Azhmyakov, V., & Egerstedt, M. (2011). Optimization of multiagent systems with increasing state dimensions: Hybrid LQ approach. Paper presented at the Proceedings of the American Control Conference, 881-887.
Garavello, M., & Piccoli, B. (2005). Hybrid necessary principle. SIAM Journal on Control and Optimization, 43(5), 1867-1887. doi:10.1137/S0363012903416219
Gill, P. E., Murray, W., & Wright, M. H. (1981). Practical Optimization.
Goebel, R., & Subbotin, M. (2007). Continuous time linear quadratic regulator with control constraints via convex duality. IEEE Transactions on Automatic Control, 52(5), 886-892. doi:10.1109/TAC.2007.895915
Goodwin, G. C., Seron, M. M., & De Doná, J. A. (2005). Constrained Control & Estimation - an Optimization Perspective.
Kirk, D. E. (2004). Optimal Control Theory: An Introduction
Kojima, A., & Morari, M. (2004). LQ control for constrained continuous-time systems. Automatica, 40(7), 1143-1155. doi:10.1016/j.automatica.2004.02.007
Lozovanu, D., & Pickl, S. (2015). Determining the optimal strategies for discrete control problems on stochastic networks with discounted costs. Discrete Applied Mathematics, 182, 169-180. doi:10.1016/j.dam.2014.09.009
Lozovanu, D., & Pickl, S. (2014). Optimization of Stochastic Discrete Systems and Control on Complex Networks: Computational Networks.
Lygeros, J. (2004). Lecture Notes on Hybrid Systems.
Martinez, J. C., & Azhmyakov, V. (2013). Optimal switched-type control design for a class of nonlinear systems. Paper presented at the IEEE International Conference on Automation Science and Engineering, 1069-1074. doi:10.1109/CoASE.2013.6653904
Polak, E. (1997). Optimization.
Poznyak, A., Polyakov, A., & Azhmyakov, V. (2014). Attractive ellipsoids in robust control. Attractive Ellipsoids in Robust Control.
Pytlak, R. (1999). Numerical methods for optimal control problems with state constraints. Numerical Methods for Optimal Control Problems with State Constraints.
Rantzer, A., & Johansson, M. (2000). Piecewise linear quadratic optimal control. IEEE Transactions on Automatic Control, 45(4), 629-637. doi:10.1109/9.847100
Rockafellar, R. T. (1970). Convex Analysis.
Shaikh, M. S., & Caines, P. E. (2007). On the hybrid optimal control problem: Theory and algorithms. IEEE Transactions on Automatic Control, 52(9), 1587-1603. doi:10.1109/TAC.2007.904451
Stanton, S. A., & Marchand, B. G. (2010). Finite set control transcription for optimal control applications. Journal of Spacecraft and Rockets, 47(3), 457-471. doi:10.2514/1.44056
Stoer, J., & Bulirsch, R. (1980). Introduction to Numerical Analysis.
Taringoo, F., & Caines, P. E. (2009). The sensitivity of hybrid systems optimal cost functions with respect to switching manifold parameters doi:10.1007/978-3-642-00602-9_38
Teo, K. L., Goh, C. J., & Wong, K. H. (1991). A Unified Computational Approach to Optimal Control Problems.
Verriest, E. I. (2009). Multi-mode multi-dimensional systems with poissonian sequencing. The Brockett Legacy Issue of Communications in Information and Systems, 9(1), 77-102.
Wonham, W. M. (1985). Linear Multivariable Control: A Geometric Approach.
Xu, X., & Antsaklis, P. J. (2003). Optimal control of hybrid autonomous systems with state jumps. Paper presented at the Proceedings of the American Control Conference, 6 5191-5196.
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 Forum-Editrice Universitaria Udinese SRL
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
_version_ 1808481185414774784
spelling 2017-12-19T19:36:42Z2017-12-19T19:36:42Z201711268042http://hdl.handle.net/11407/4256reponame:Repositorio Institucional Universidad de Medellíninstname:Universidad de MedellínThis paper deals with a constrained LQ-type optimal control problem (OCP) in the presence of fixed levels input restrictions. We consider control processes governed by linear differential equations with a priori known control switching structure. The set of admissible inputs reflects some important natural engineering applications and moreover, can also be interpreted as a result of a quantization procedure applied to the original dynamic system. We propose a novel implementable algorithm that makes it possible to calculate a (numerically consistent) approximative solution to the constrained LQ-type OCPs under consideration. Our contribution mainly discusses theoretic aspects of the proposed solution scheme and contains an illustrative numerical example. © 2017, Forum-Editrice Universitaria Udinese SRL. All rights reserved.engForum-Editrice Universitaria Udinese SRLFacultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85016603353&partnerID=40&md5=e7a77b88d1c0caa2f5a939eb4d300620Italian Journal of Pure and Applied MathematicsItalian Journal of Pure and Applied Mathematics Volume 2017, Issue 37, January 2017, Pages 219-237Ali, U., Cai, H., Mostofi, Y., & Wardi, Y. (2016). Motion and communication co-optimization with path planning and online channel prediction. Paper presented at the Proceedings of the American Control Conference, , 2016-July 7079-7084. doi:10.1109/ACC.2016.7526789Armijo, L. (1966). Minimization of functions having lipschitz continuous first partial derivatives. Pacific Journal of Mathematics, 16(1), 1-3.Azhmyakov, V., Basin, M. V., & Gil García, A. E. (2014). Optimal control processes associated with a class of discontinuous control systems: Applications to sliding mode dynamics. Kybernetika, 50(1), 5-18. doi:10.14736/kyb-2014-1-0005Azhmyakov, 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., Galvan-Guerra, R., & Egerstedt, M. (2009). Hybrid LQ-optimization using dynamic programming. Paper presented at the Proceedings of the American Control Conference, 3617-3623. doi:10.1109/ACC.2009.5160100Azhmyakov, V., Galván-Guerra, R., & Egerstedt, M. (2010). On the LQ-based optimization techniques for impulsive hybrid control systems. Paper presented at the Proceedings of the 2010 American Control Conference, ACC 2010, 129-135.Azhmyakov, V., & Juarez, R. (2015). On the projected gradient methods for switched - mode systems optimization. IFAC-PapersOnLine, 48(27), 181-186. doi:10.1016/j.ifacol.2015.11.172Azhmyakov, V., Juarez, R., & Pickl, S. (2015). On the local convexity of singular optimal control problems associated with the switched-mode dynamic systems. IFAC-PapersOnLine, 28(25), 271-276. doi:10.1016/j.ifacol.2015.11.099Azhmyakov, V., Martinez, J. C., & Poznyak, A. (2016). Optimal fixed-levels control for nonlinear systems with quadratic cost-functionals. Optimal Control Applications and Methods, 37(5), 1035-1055. doi:10.1002/oca.2223Azhmyakov, V., & Raisch, J. (2008). Convex control systems and convex optimal control problems with constraints. IEEE Transactions on Automatic Control, 53(4), 993-998. doi:10.1109/TAC.2008.919848Azhmyakov, V., Serrezuela, R. R., & Trujillo, L. A. G. (2014). Approximations based optimal control design for a class of switched dynamic systems. Paper presented at the IECON Proceedings (Industrial Electronics Conference), 90-95. doi:10.1109/IECON.2014.7048482Banks, S. P., & Khathur, S. A. (1989). Structure and control of piecewise-linear systems. International Journal of Control, 50(2), 667-686. doi:10.1080/00207178908953388Betts, J. T. (2001). Practical Methods for Optimal Control using Nonlinear Programming.Boltyanski, V., & Poznyak, A. (2012). The robust maximum principle. The Robust Maximum Principle.Bonilla, M., Malabre, M., & Azhmyakov, V. (2015). An implicit systems characterization of a class of impulsive linear switched control processes. part 1: Modeling. Nonlinear Analysis: Hybrid Systems, 15, 157-170. doi:10.1016/j.nahs.2014.04.002Branicky, M. S., Borkar, V. S., & Mitter, S. K. (1998). A unified framework for hybrid control: Model and optimal control theory. IEEE Transactions on Automatic Control, 43(1), 31-45. doi:10.1109/9.654885Brockett, R. W., & Liberzon, D. (2000). Quantized feedback stabilization of linear systems. IEEE Transactions on Automatic Control, 45(7), 1279-1289. doi:10.1109/9.867021Cassandras, C. G., Pepyne, D. L., & Wardi, Y. (2001). Optimal control of a class of hybrid systems. IEEE Transactions on Automatic Control, 46(3), 398-415. doi:10.1109/9.911417Ding, X. -., Wardi, Y., & Egerstedt, M. (2009). On-line optimization of switched-mode dynamical systems. IEEE Transactions on Automatic Control, 54(9), 2266-2271. doi:10.1109/TAC.2009.2026864Egerstedt, 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 Theory.Galván-Guerra, R., Azhmyakov, V., & Egerstedt, M. (2011). Optimization of multiagent systems with increasing state dimensions: Hybrid LQ approach. Paper presented at the Proceedings of the American Control Conference, 881-887.Garavello, M., & Piccoli, B. (2005). Hybrid necessary principle. SIAM Journal on Control and Optimization, 43(5), 1867-1887. doi:10.1137/S0363012903416219Gill, P. E., Murray, W., & Wright, M. H. (1981). Practical Optimization.Goebel, R., & Subbotin, M. (2007). Continuous time linear quadratic regulator with control constraints via convex duality. IEEE Transactions on Automatic Control, 52(5), 886-892. doi:10.1109/TAC.2007.895915Goodwin, G. C., Seron, M. M., & De Doná, J. A. (2005). Constrained Control & Estimation - an Optimization Perspective.Kirk, D. E. (2004). Optimal Control Theory: An IntroductionKojima, A., & Morari, M. (2004). LQ control for constrained continuous-time systems. Automatica, 40(7), 1143-1155. doi:10.1016/j.automatica.2004.02.007Lozovanu, D., & Pickl, S. (2015). Determining the optimal strategies for discrete control problems on stochastic networks with discounted costs. Discrete Applied Mathematics, 182, 169-180. doi:10.1016/j.dam.2014.09.009Lozovanu, D., & Pickl, S. (2014). Optimization of Stochastic Discrete Systems and Control on Complex Networks: Computational Networks.Lygeros, J. (2004). Lecture Notes on Hybrid Systems.Martinez, J. C., & Azhmyakov, V. (2013). Optimal switched-type control design for a class of nonlinear systems. Paper presented at the IEEE International Conference on Automation Science and Engineering, 1069-1074. doi:10.1109/CoASE.2013.6653904Polak, E. (1997). Optimization.Poznyak, A., Polyakov, A., & Azhmyakov, V. (2014). Attractive ellipsoids in robust control. Attractive Ellipsoids in Robust Control.Pytlak, R. (1999). Numerical methods for optimal control problems with state constraints. Numerical Methods for Optimal Control Problems with State Constraints.Rantzer, A., & Johansson, M. (2000). Piecewise linear quadratic optimal control. IEEE Transactions on Automatic Control, 45(4), 629-637. doi:10.1109/9.847100Rockafellar, R. T. (1970). Convex Analysis.Shaikh, M. S., & Caines, P. E. (2007). On the hybrid optimal control problem: Theory and algorithms. IEEE Transactions on Automatic Control, 52(9), 1587-1603. doi:10.1109/TAC.2007.904451Stanton, S. A., & Marchand, B. G. (2010). Finite set control transcription for optimal control applications. Journal of Spacecraft and Rockets, 47(3), 457-471. doi:10.2514/1.44056Stoer, J., & Bulirsch, R. (1980). Introduction to Numerical Analysis.Taringoo, F., & Caines, P. E. (2009). The sensitivity of hybrid systems optimal cost functions with respect to switching manifold parameters doi:10.1007/978-3-642-00602-9_38Teo, K. L., Goh, C. J., & Wong, K. H. (1991). A Unified Computational Approach to Optimal Control Problems.Verriest, E. I. (2009). Multi-mode multi-dimensional systems with poissonian sequencing. The Brockett Legacy Issue of Communications in Information and Systems, 9(1), 77-102.Wonham, W. M. (1985). Linear Multivariable Control: A Geometric Approach.Xu, X., & Antsaklis, P. J. (2003). Optimal control of hybrid autonomous systems with state jumps. Paper presented at the Proceedings of the American Control Conference, 6 5191-5196.ScopusOn the linear quadratic dynamic optimization problems with fixed-levels control functionsArticleinfo:eu-repo/semantics/articlehttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Azhmyakov, V., Department of Basic Sciences, Universidad de Medellin, Medellin, ColombiaTrujillo, L.A.G., School of Ingeniering, Universidad de Medellin, Medellin, ColombiaAzhmyakov V.Trujillo L.A.G.Department of Basic Sciences, Universidad de Medellin, Medellin, ColombiaSchool of Ingeniering, Universidad de Medellin, Medellin, ColombiaConvex optimizationNumerical methodsOptimal controlSystems theoryConvex optimizationDifferential equationsOptimal control systemsOptimizationSystem theoryControl functionsControl processControl switchingEngineering applicationsLinear differential equationLinear quadraticOptimal control problemOptimal controlsNumerical methodsThis paper deals with a constrained LQ-type optimal control problem (OCP) in the presence of fixed levels input restrictions. We consider control processes governed by linear differential equations with a priori known control switching structure. The set of admissible inputs reflects some important natural engineering applications and moreover, can also be interpreted as a result of a quantization procedure applied to the original dynamic system. We propose a novel implementable algorithm that makes it possible to calculate a (numerically consistent) approximative solution to the constrained LQ-type OCPs under consideration. Our contribution mainly discusses theoretic aspects of the proposed solution scheme and contains an illustrative numerical example. © 2017, Forum-Editrice Universitaria Udinese SRL. All rights reserved.http://purl.org/coar/access_right/c_16ec11407/4256oai:repository.udem.edu.co:11407/42562020-05-27 19:16:07.663Repositorio Institucional Universidad de Medellinrepositorio@udem.edu.co

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