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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 & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
References:
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References:
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[14]

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[15]

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[20]

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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]

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[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

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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]

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