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

A neighboring extremal solution for an optimal switched impulsive control problem

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

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.
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," 2nd 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," 2nd 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.

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