American Institute of Mathematical Sciences

Advanced
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

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 & 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, 19 (2010).   Google Scholar

[2]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control: Optimization, Estimation, and Control,", Revised printing, (1975).   Google Scholar

[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.  doi: 10.1016/j.automatica.2009.06.028.  Google Scholar

[4]

_____, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints,, IEEE Transactions on Automatic Control, 54 (2009), 2674.  doi: 10.1109/TAC.2009.2031576.  Google Scholar

[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.  doi: 10.1080/00207170802460032.  Google Scholar

[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.  doi: 10.1080/00207170802460024.  Google Scholar

[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.  doi: 10.1109/TCSI.2008.916442.  Google Scholar

[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.  doi: 10.1007/BF00939642.  Google Scholar

[9]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems,, Dynamics of Continuous, 18 (2011), 59.   Google Scholar

[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.  doi: 10.1023/A:1022684730236.  Google Scholar

[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.  doi: 10.1109/TAC.2009.2029310.  Google Scholar

[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.  doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

[13]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.   Google Scholar

[14]

_____, Sensitivity analysis for optimal control problems subject to higher order state constraints,, Annals of Operations Research, 101 (2001), 43.  doi: 10.1023/A:1010956104457.  Google Scholar

[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.  doi: 10.1002/oca.4660100205.  Google Scholar

[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.  doi: 10.1002/oca.4660100206.  Google Scholar

[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 (1991).   Google Scholar

[18]

C. Z. Wu and K. L. Teo, Global impulsive optimal control computation,, Journal of Industrial and Management Optimization, 2 (2006), 435.   Google Scholar

[19]

R. Yu and P. Leung, Optimal partial harvesting schedule for aquaculture operations,, Marine Resource Economics, 21 (2006), 301.   Google Scholar

show all references

References:
[1]

J. T. Betts, "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming,", 2nd edition, 19 (2010).   Google Scholar

[2]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control: Optimization, Estimation, and Control,", Revised printing, (1975).   Google Scholar

[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.  doi: 10.1016/j.automatica.2009.06.028.  Google Scholar

[4]

_____, Neighboring extremal solution for nonlinear discrete-time optimal control problems with state inequality constraints,, IEEE Transactions on Automatic Control, 54 (2009), 2674.  doi: 10.1109/TAC.2009.2031576.  Google Scholar

[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.  doi: 10.1080/00207170802460032.  Google Scholar

[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.  doi: 10.1080/00207170802460024.  Google Scholar

[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.  doi: 10.1109/TCSI.2008.916442.  Google Scholar

[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.  doi: 10.1007/BF00939642.  Google Scholar

[9]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems,, Dynamics of Continuous, 18 (2011), 59.   Google Scholar

[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.  doi: 10.1023/A:1022684730236.  Google Scholar

[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.  doi: 10.1109/TAC.2009.2029310.  Google Scholar

[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.  doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

[13]

K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.   Google Scholar

[14]

_____, Sensitivity analysis for optimal control problems subject to higher order state constraints,, Annals of Operations Research, 101 (2001), 43.  doi: 10.1023/A:1010956104457.  Google Scholar

[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.  doi: 10.1002/oca.4660100205.  Google Scholar

[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.  doi: 10.1002/oca.4660100206.  Google Scholar

[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 (1991).   Google Scholar

[18]

C. Z. Wu and K. L. Teo, Global impulsive optimal control computation,, Journal of Industrial and Management Optimization, 2 (2006), 435.   Google Scholar

[19]

R. Yu and P. Leung, Optimal partial harvesting schedule for aquaculture operations,, Marine Resource Economics, 21 (2006), 301.   Google Scholar

[1]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[2]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[3]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[4]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[5]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[6]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[7]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[8]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[9]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[10]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[11]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[12]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[13]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[14]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[15]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[16]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[17]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027
[18]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[19]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029
[20]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences