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January  2014, 10(1): 275-309. doi: 10.3934/jimo.2014.10.275

The control parameterization method for nonlinear optimal control: A survey

Qun Lin 1, , Ryan Loxton 1, and  Kok Lay Teo 2,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845
2. 

Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, W.A. 6845

Received  January 2013 Revised  July 2013 Published  October 2013

The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.
Keywords: switching times, time-scaling transformation, control parameterization, Optimal control, state constraints..
Mathematics Subject Classification: Primary: 49M37; Secondary: 65K10, 65P99, 90C30, 93C1.
Citation: Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial and Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
References:
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show all references

References:
[1]

B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints, Automatica J. IFAC, 47 (2011), 341-347. doi: 10.1016/j.automatica.2010.10.037.

[2]

N. U. Ahmed, "Elements of Finite-Dimensional Systems and Control Theory,'' Longman Scientific and Technical, Essex, 1988.

[3]

N. U. Ahmed, "Dynamic Systems and Control with Applications,'' World Scientific, Singapore, 2006.

[4]

Z. Benayache, G. Besançon and D. Georges, A new nonlinear control methodology for irrigation canals based on a delayed input model, in "Proceedings of the 17th World Congress of the International Federation of Automatic Control,'' 2008.

[5]

J. M. Blatt, Optimal control with a cost of switching control, Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 19 (1976), 316-332.

[6]

M. Boccadoro, Y. Wardi, M. Egerstedt and E. Verriest, Optimal control of switching surfaces in hybrid dynamical systems, Discrete Event Dynamic Systems: Theory and Applications, 15 (2005), 433-448. doi: 10.1007/s10626-005-4060-4.

[7]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control, Journal of Computational and Applied Mathematics, 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.

[8]

L. Caccetta, I. Loosen and V. Rehbock, Computational aspects of the optimal transit path problem, Journal of Industrial and Management Optimization, 4 (2008), 95-105. doi: 10.3934/jimo.2008.4.95.

[9]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A max-min control problem arising in gradient elution chromatography, Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144. doi: 10.1021/ie202475p.

[10]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471.

[11]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A class of optimal state-delay control problems, Nonlinear Analysis: Real World Applications, 14 (2013), 1536-1550. doi: 10.1016/j.nonrwa.2012.10.017.

[12]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 219 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015.

[13]

Q. Q. Chai, C. H. Yang, K. L. Teo and W. H. Gui, Optimal control of an industrial-scale evaporation process: Sodium aluminate solution, Control Engineering Practice, 20 (2012), 618-628. doi: 10.1016/j.conengprac.2012.03.001.

[14]

B. Christiansen, H. Maurer and O. Zirn, Optimal control of a voice-coil-motor with Coulombic friction, in "Proceedings of the 47th IEEE Conference on Decision and Control,'' 2008. doi: 10.1109/CDC.2008.4739025.

[15]

M. Chyba, T. Haberkorn, R. N. Smith and S. K. Choi, Design and implementation of time efficient trajectories for autonomous underwater vehicles, Ocean Engineering, 35 (2008), 63-76. doi: 10.1016/j.oceaneng.2007.07.007.

[16]

J. Y. Dieulot and J. P. Richard, Tracking control of a nonlinear system with input-dependent delay, in "Proceedings of the 40th IEEE Conference on Decision and Control,'' 2001.

[17]

B. Farhadinia, K. L. Teo and R. Loxton, A computational method for a class of non-standard time optimal control problems involving multiple time horizons, Mathematical and Computer Modelling, 49 (2009), 1682-1691. doi: 10.1016/j.mcm.2008.08.019.

[18]

M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270. doi: 10.3934/jimo.2008.4.247.

[19]

C. J. Goh and K. L. Teo, Control parametrization: A unified approach to optimal control problems with general constraints, Automatica J. IFAC, 24 (1988), 3-18. doi: 10.1016/0005-1098(88)90003-9.

[20]

P. G. Howlett, P. J. Pudney and X. Vu, Local energy minimization in optimal train control, Automatica J. IFAC, 45 (2009), 2692-2698. doi: 10.1016/j.automatica.2009.07.028.

[21]

L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, "MISER3 Optimal Control Software: Theory and User Manual,'' University of Western Australia, Perth, 2004.

[22]

L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z.

[23]

C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.

[24]

C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609. doi: 10.3934/jimo.2012.8.591.

[25]

K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855.

[26]

C. Y. Kaya and J. L. Noakes, Computational method for time-optimal switching control, Journal of Optimization Theory and Applications, 117 (2003), 69-92. doi: 10.1023/A:1023600422807.

[27]

H. W. J. Lee, K. L. Teo and X. Q. Cai, An optimal control approach to nonlinear mixed integer programming problems, Computers and Mathematics with Applications, 36 (1998), 87-105. doi: 10.1016/S0898-1221(98)00131-X.

[28]

H. W. J. Lee, K. L. Teo and A. E. B. Lim, Sensor scheduling in continuous time, Automatica J. IFAC, 37 (2001), 2017-2023. doi: 10.1016/S0005-1098(01)00159-5.

[29]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.

[30]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica J. IFAC, 35 (1999), 1401-1407. doi: 10.1016/S0005-1098(99)00050-3.

[31]

H. W. J. Lee and K. L. Teo, Control parametrization enhancing technique for solving a special ODE class with state dependent switch, Journal of Optimization Theory and Applications, 118 (2003), 55-66. doi: 10.1023/A:1024735407694.

[32]

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