A neighboring extremal solution for an optimal switched impulsive control problem

Previous Article
Canonical duality solution for alternating support vector machine
JIMO Home
This Issue
Next Article
Single-period inventory model with discrete stochastic demand based on prospect theory

July 2012, 8(3): 591-609. doi: 10.3934/jimo.2012.8.591

A neighboring extremal solution for an optimal switched impulsive control problem

Canghua Jiang ^1, , Kok Lay Teo ^2, , Ryan Loxton ^3, and Guang-Ren Duan ^1,

1.	Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China, China
2.	Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845, Australia
3.	Department of Mathematics and Statistics, Curtin University, Perth 6845

Received May 2011 Revised November 2011 Published June 2012

Abstract
Related Papers

This paper presents a neighboring extremal solution for a class of optimal switched impulsive control problems with perturbations in the initial state, terminal condition and system's parameters. The sequence of mode's switching is pre-specified, and the decision variables, i.e. the switching times and parameters of the system involved, have inequality constraints. It is assumed that the active status of these constraints is unchanged with the perturbations. We derive this solution by expanding the necessary conditions for optimality to first-order and then solving the resulting multiple-point boundary-value problem by the backward sweep technique. Numerical simulations are presented to illustrate this solution method.

Keywords: Optimal control, impulsive control, switched systems, neighboring extremals, nonlinear dynamic systems..

Mathematics Subject Classification: Primary: 90C31, 90C59; Secondary: 49N25, 49N3.

Citation: Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial and Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

References:

[1]	J. T. Betts, "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming," 2^nd edition, Advances in Design and Control, 19, SIAM, Philadelphia, PA, 2010.
[2]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control: Optimization, Estimation, and Control," Revised printing, Hemisphere Publishing Corp. Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975.
[3]	R. Ghaemi, J. Sun and I. V. Kolmanovsky, An integrated perturbation analysis and sequential quadratic programming approach for model predictive control, Automatica, 45 (2009), 2412-2418. doi: 10.1016/j.automatica.2009.06.028.
[4]	_____, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints, IEEE Transactions on Automatic Control, 54 (2009), 2674-2679. doi: 10.1109/TAC.2009.2031576.
[5]	S. Gros, B. Chachuat and D. Bonvin, Neighbouring-extremal control for singular dynamic optimisation problems. Part II. Multiple-input systems, International Journal of Control, 82 (2009), 1193-1211. doi: 10.1080/00207170802460032.
[6]	S. Gros, B. Srinivasan, B. Chachuat and D. Bonvin, Neighbouring-extremal control for singular dynamic optimisation problems. Part I. Single-input systems, International Journal of Control, 82 (2009), 1099-1112. doi: 10.1080/00207170802460024.
[7]	C. Y.-F. Ho, B. W.-K. Ling, Y.-Q. Liu, P. K.-S. Tam and K.-L. Teo, Optimal PWM control of switched-capacitor DC-DC power converters via model transformation and enhancing control techniques, IEEE Transactions on Circuits and Systems. I. Regular Papers, 55 (2008), 1382-1391. doi: 10.1109/TCSI.2008.916442.
[8]	B. Kugelmann and H. J. Pesch, New general guidance method in constrained optimal control. I. Numerical method, Journal of Optimization Theory & Applications, 67 (1990), 421-435. doi: 10.1007/BF00939642.
[9]	Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 18 (2011), 59-76.
[10]	Y. Liu, K. L. Teo, L. S. Jennings and S. Wang, On a class of optimal control problems with state jumps, Journal of Optimization Theory & Applications, 98 (1998), 65-82. doi: 10.1023/A:1022684730236.
[11]	R. C. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE Transactions on Automatic Control, 54 (2009), 2455-2460. doi: 10.1109/TAC.2009.2029310.
[12]	R. C. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica J. IFAC, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031.
[13]	K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283.
[14]	_____, Sensitivity analysis for optimal control problems subject to higher order state constraints, Annals of Operations Research, 101 (2001), 43-73. doi: 10.1023/A:1010956104457.
[15]	H. J. Pesch, Real-time computation of feedback controls for constrained optimal control problems. I. Neighbouring extremals, Optimal Control Applications & Methods, 10 (1989), 129-145. doi: 10.1002/oca.4660100205.
[16]	_____, Real-time computation of feedback controls for constrained optimal control problems. II. A correction method based on multiple shooting, Optimal Control Applications & Methods, 10 (1989), 147-171. doi: 10.1002/oca.4660100206.
[17]	K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[18]	C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.
[19]	R. Yu and P. Leung, Optimal partial harvesting schedule for aquaculture operations, Marine Resource Economics, 21 (2006), 301-315.

show all references

References:

[1]	J. T. Betts, "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming," 2^nd edition, Advances in Design and Control, 19, SIAM, Philadelphia, PA, 2010.
[2]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control: Optimization, Estimation, and Control," Revised printing, Hemisphere Publishing Corp. Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975.
[3]	R. Ghaemi, J. Sun and I. V. Kolmanovsky, An integrated perturbation analysis and sequential quadratic programming approach for model predictive control, Automatica, 45 (2009), 2412-2418. doi: 10.1016/j.automatica.2009.06.028.
[4]	_____, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints, IEEE Transactions on Automatic Control, 54 (2009), 2674-2679. doi: 10.1109/TAC.2009.2031576.
[5]	S. Gros, B. Chachuat and D. Bonvin, Neighbouring-extremal control for singular dynamic optimisation problems. Part II. Multiple-input systems, International Journal of Control, 82 (2009), 1193-1211. doi: 10.1080/00207170802460032.
[6]	S. Gros, B. Srinivasan, B. Chachuat and D. Bonvin, Neighbouring-extremal control for singular dynamic optimisation problems. Part I. Single-input systems, International Journal of Control, 82 (2009), 1099-1112. doi: 10.1080/00207170802460024.
[7]	C. Y.-F. Ho, B. W.-K. Ling, Y.-Q. Liu, P. K.-S. Tam and K.-L. Teo, Optimal PWM control of switched-capacitor DC-DC power converters via model transformation and enhancing control techniques, IEEE Transactions on Circuits and Systems. I. Regular Papers, 55 (2008), 1382-1391. doi: 10.1109/TCSI.2008.916442.
[8]	B. Kugelmann and H. J. Pesch, New general guidance method in constrained optimal control. I. Numerical method, Journal of Optimization Theory & Applications, 67 (1990), 421-435. doi: 10.1007/BF00939642.
[9]	Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 18 (2011), 59-76.
[10]	Y. Liu, K. L. Teo, L. S. Jennings and S. Wang, On a class of optimal control problems with state jumps, Journal of Optimization Theory & Applications, 98 (1998), 65-82. doi: 10.1023/A:1022684730236.
[11]	R. C. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE Transactions on Automatic Control, 54 (2009), 2455-2460. doi: 10.1109/TAC.2009.2029310.
[12]	R. C. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica J. IFAC, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031.
[13]	K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Computational Optimization and Applications, 5 (1996), 253-283.
[14]	_____, Sensitivity analysis for optimal control problems subject to higher order state constraints, Annals of Operations Research, 101 (2001), 43-73. doi: 10.1023/A:1010956104457.
[15]	H. J. Pesch, Real-time computation of feedback controls for constrained optimal control problems. I. Neighbouring extremals, Optimal Control Applications & Methods, 10 (1989), 129-145. doi: 10.1002/oca.4660100205.
[16]	_____, Real-time computation of feedback controls for constrained optimal control problems. II. A correction method based on multiple shooting, Optimal Control Applications & Methods, 10 (1989), 147-171. doi: 10.1002/oca.4660100206.
[17]	K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[18]	C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.
[19]	R. Yu and P. Leung, Optimal partial harvesting schedule for aquaculture operations, Marine Resource Economics, 21 (2006), 301-315.

[1]	Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007
[2]	Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[3]	María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1
[4]	Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[5]	Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial and Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042
[6]	Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1
[7]	Haijun Sun, Xinquan Zhang. Guaranteed cost control of discrete-time switched saturated systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4515-4522. doi: 10.3934/dcdsb.2020300
[8]	Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415
[9]	C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial and Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435
[10]	Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185
[11]	Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164
[12]	Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial and Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1
[13]	Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
[14]	K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050
[15]	Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial and Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
[16]	Philippe Jouan, Said Naciri. Asymptotic stability of uniformly bounded nonlinear switched systems. Mathematical Control and Related Fields, 2013, 3 (3) : 323-345. doi: 10.3934/mcrf.2013.3.323
[17]	Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020
[18]	Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control and Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[19]	Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
[20]	Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial and Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63

2021 Impact Factor: 1.411

Tools

Metrics

Other articles
by authors

on AIMS
Canghua Jiang Kok Lay Teo Ryan Loxton Guang-Ren Duan
on Google Scholar
Canghua Jiang Kok Lay Teo Ryan Loxton Guang-Ren Duan

[Back to Top]