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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
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show all references

References:
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Automatica J. IFAC, 47 (2011), 341-347. doi: 10.1016/j.automatica.2010.10.037.  Google Scholar

[2]

Longman Scientific and Technical, Essex, 1988.  Google Scholar

[3]

World Scientific, Singapore, 2006.  Google Scholar

[4]

in "Proceedings of the 17th World Congress of the International Federation of Automatic Control,'' 2008. Google Scholar

[5]

Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 19 (1976), 316-332.  Google Scholar

[6]

Discrete Event Dynamic Systems: Theory and Applications, 15 (2005), 433-448. doi: 10.1007/s10626-005-4060-4.  Google Scholar

[7]

Journal of Computational and Applied Mathematics, 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[8]

Journal of Industrial and Management Optimization, 4 (2008), 95-105. doi: 10.3934/jimo.2008.4.95.  Google Scholar

[9]

Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144. doi: 10.1021/ie202475p.  Google Scholar

[10]

Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471.  Google Scholar

[11]

Nonlinear Analysis: Real World Applications, 14 (2013), 1536-1550. doi: 10.1016/j.nonrwa.2012.10.017.  Google Scholar

[12]

Applied Mathematics and Computation, 219 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015.  Google Scholar

[13]

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[14]

in "Proceedings of the 47th IEEE Conference on Decision and Control,'' 2008. doi: 10.1109/CDC.2008.4739025.  Google Scholar

[15]

Ocean Engineering, 35 (2008), 63-76. doi: 10.1016/j.oceaneng.2007.07.007.  Google Scholar

[16]

in "Proceedings of the 40th IEEE Conference on Decision and Control,'' 2001. Google Scholar

[17]

Mathematical and Computer Modelling, 49 (2009), 1682-1691. doi: 10.1016/j.mcm.2008.08.019.  Google Scholar

[18]

Journal of Industrial and Management Optimization, 4 (2008), 247-270. doi: 10.3934/jimo.2008.4.247.  Google Scholar

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Automatica J. IFAC, 24 (1988), 3-18. doi: 10.1016/0005-1098(88)90003-9.  Google Scholar

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Automatica J. IFAC, 45 (2009), 2692-2698. doi: 10.1016/j.automatica.2009.07.028.  Google Scholar

[21]

University of Western Australia, Perth, 2004. Google Scholar

[22]

Automatica J. IFAC, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

[23]

Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.  Google Scholar

[24]

Journal of Industrial and Management Optimization, 8 (2012), 591-609. doi: 10.3934/jimo.2012.8.591.  Google Scholar

[25]

Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855.  Google Scholar

[26]

Journal of Optimization Theory and Applications, 117 (2003), 69-92. doi: 10.1023/A:1023600422807.  Google Scholar

[27]

Computers and Mathematics with Applications, 36 (1998), 87-105. doi: 10.1016/S0898-1221(98)00131-X.  Google Scholar

[28]

Automatica J. IFAC, 37 (2001), 2017-2023. doi: 10.1016/S0005-1098(01)00159-5.  Google Scholar

[29]

Dynamic Systems and Applications, 6 (1997), 243-261.  Google Scholar

[30]

Automatica J. IFAC, 35 (1999), 1401-1407. doi: 10.1016/S0005-1098(99)00050-3.  Google Scholar

[31]

Journal of Optimization Theory and Applications, 118 (2003), 55-66. doi: 10.1023/A:1024735407694.  Google Scholar

[32]

Optimization Methods and Software, 21 (2006), 679-691. doi: 10.1080/10556780500142306.  Google Scholar

[33]

Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012.  Google Scholar

[34]

Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[35]

Discrete and Continuous Dynamical Systems: Series B, 16 (2011), 1101-1117. doi: 10.3934/dcdsb.2011.16.1101.  Google Scholar

[36]

Pacific Journal of Optimization, 7 (2011), 63-81.  Google Scholar

[37]

Dynamics of Continuous, Discrete and Impulsive Systems - Series B: Applications and Algorithms, 18 (2011), 59-76.  Google Scholar

[38]

Automatica J. IFAC, 48 (2012), 2116-2129. doi: 10.1016/j.automatica.2012.06.055.  Google Scholar

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Computers and Mathematics with Applications, 52 (2006), 625-636. doi: 10.1016/j.camwa.2006.10.001.  Google Scholar

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Journal of Industrial and Management Optimization, 5 (2009), 835-850. doi: 10.3934/jimo.2009.5.835.  Google Scholar

[41]

Journal of Optimization Theory and Applications, 98 (1998), 65-82. doi: 10.1023/A:1022684730236.  Google Scholar

[42]

Automatica J. IFAC, 49 (2013), 2652-2664. doi: 10.1016/j.automatica.2013.05.027.  Google Scholar

[43]

Numerical Algebra, Control and Optimization, 2 (2012), 571-599. doi: 10.3934/naco.2012.2.571.  Google Scholar

[44]

Automatica J. IFAC, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[45]

IEEE Transactions on Automatic Control, 54 (2009), 2455-2460. doi: 10.1109/TAC.2009.2029310.  Google Scholar

[46]

IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710.  Google Scholar

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Applied Mathematics and Computation, 217 (2011), 6566-6576. doi: 10.1016/j.amc.2011.01.039.  Google Scholar

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Automatica J. IFAC, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

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Springer, New York, 2008.  Google Scholar

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Automatica J. IFAC, 28 (1992), 1113-1123. doi: 10.1016/0005-1098(92)90054-J.  Google Scholar

[52]

Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 28 (1987), 376-392. doi: 10.1017/S0334270000005464.  Google Scholar

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Springer, New York, 2006.  Google Scholar

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PhD thesis, University of Western Australia, Perth, 1994. Google Scholar

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ANZIAM Journal, 44 (2002), E33-E54. doi: 10.1017/S1446181100007884.  Google Scholar

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in "Progress in Optimization: Contributions from Australasia,'' Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4613-3285-5_13.  Google Scholar

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Automatica J. IFAC, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar

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in "Optimization and Control with Applications,'' Springer, New York, 2005. doi: 10.1007/0-387-24255-4_21.  Google Scholar

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Dynamics of Continuous, Discrete and Impulsive Systems - Series B: Applications and Algorithms, 10 (2003), 429-439.  Google Scholar

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Optimization Methods and Software, 19 (2004), 413-426. doi: 10.1080/10556780310001656637.  Google Scholar

[61]

Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 34 (1992), 212-228. doi: 10.1017/S0334270000008730.  Google Scholar

[62]

Nonlinear Analysis: Theory, Methods and Applications, 63 (2005), e2223-e2236. doi: 10.1016/j.na.2005.03.066.  Google Scholar

[63]

IEEE Transactions on Automatic Control, 32 (1987), 940-941. doi: 10.1109/TAC.1987.1104471.  Google Scholar

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IMA Journal of Mathematical Control and Information, 6 (1989), 81-95. doi: 10.1093/imamci/6.1.81.  Google Scholar

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Longman Scientific and Technical, Essex, 1991.  Google Scholar

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Journal of Optimization Theory and Applications, 63 (1989), 1-22. doi: 10.1007/BF00940727.  Google Scholar

[67]

Journal of Optimization Theory and Applications, 68 (1991), 335-357. doi: 10.1007/BF00941572.  Google Scholar

[68]

Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.  Google Scholar

[69]

Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 28 (1987), 393-413. doi: 10.1017/S0334270000005476.  Google Scholar

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ANZIAM Journal, 43 (2002), 463-478.  Google Scholar

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Automatica J. IFAC, 29 (1993), 789-792. doi: 10.1016/0005-1098(93)90076-6.  Google Scholar

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John Wiley, New York, 1981.  Google Scholar

[73]

Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

[74]

Journal of Global Optimization, 54 (2012), 307-323. doi: 10.1007/s10898-012-9863-x.  Google Scholar

[75]

ANZIAM Journal, 43 (2002), E154-E185.  Google Scholar

[76]

Nonlinear Dynamics and Systems Theory, 10 (2010), 175-188.  Google Scholar

[77]

Optimal Control Applications and Methods, 33 (2012), 576-594. doi: 10.1002/oca.1015.  Google Scholar

[78]

Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435.  Google Scholar

[79]

Journal of Global Optimization, 44 (2009), 213-225. doi: 10.1007/s10898-008-9319-5.  Google Scholar

[80]

Marine Resource Economics, 21 (2006), 301-315. Google Scholar

[81]

Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7.  Google Scholar

[82]

Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693. doi: 10.1021/ie200996f.  Google Scholar

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