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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

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.
Citation: Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & 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. doi: 10.1016/j.automatica.2010.10.037. Google Scholar

[2]

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[3]

N. U. Ahmed, "Dynamic Systems and Control with Applications,'', World Scientific, (2006). Google Scholar

[4]

Z. Benayache, G. Besançon and D. Georges, A new nonlinear control methodology for irrigation canals based on a delayed input model,, in, (2008). Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s10626-005-4060-4. Google Scholar

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[8]

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[11]

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[12]

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[13]

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[14]

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[15]

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[16]

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[17]

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[18]

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[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. doi: 10.1016/0005-1098(88)90003-9. Google Scholar

[20]

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[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, (2004). Google Scholar

[22]

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

[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. doi: 10.1007/s10957-012-0006-9. Google Scholar

[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. doi: 10.3934/jimo.2012.8.591. Google Scholar

[25]

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

[26]

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

[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. doi: 10.1016/S0898-1221(98)00131-X. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1016/S0005-1098(99)00050-3. Google Scholar

[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. doi: 10.1023/A:1024735407694. Google Scholar

[32]

H. W. J. Lee and K. H. Wong, Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints,, Optimization Methods and Software, 21 (2006), 679. doi: 10.1080/10556780500142306. Google Scholar

[33]

R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems,, Mathematical and Computer Modelling, 43 (2006), 1393. doi: 10.1016/j.mcm.2005.08.012. Google Scholar

[34]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem,, Journal of Optimization Theory and Applications, 151 (2011), 260. doi: 10.1007/s10957-011-9904-5. Google Scholar

[35]

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