The control parameterization method for nonlinear optimal control: A survey

Qun Lin; Ryan Loxton; Kok Lay Teo

doi:10.3934/jimo.2014.10.275

Article Contents

2014, Volume 10, Issue 1: 275-309. Doi: 10.3934/jimo.2014.10.275

This issue Previous Article Heuristics for parallel machine scheduling with batch delivery consideration Next Article Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation

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: January 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 49M37; Secondary: 65K10, 65P99, 90C30, 93C15.

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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]	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), 679-691.doi: 10.1080/10556780500142306.
[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), 1393-1403.doi: 10.1016/j.mcm.2005.08.012.
[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), 260-291.doi: 10.1007/s10957-011-9904-5.
[35]	B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems: Series B, 16 (2011), 1101-1117.doi: 10.3934/dcdsb.2011.16.1101.
[36]	Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7 (2011), 63-81.
[37]	Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems - Series B: Applications and Algorithms, 18 (2011), 59-76.
[38]	Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica J. IFAC, 48 (2012), 2116-2129.doi: 10.1016/j.automatica.2012.06.055.
[39]	Y. Liu, A. Eberhard and K. L. Teo, A numerical method for a class of mixed switching and impulsive optimal control problems, Computers and Mathematics with Applications, 52 (2006), 625-636.doi: 10.1016/j.camwa.2006.10.001.
[40]	C. Liu, Z. Gong, E. Feng and H. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.doi: 10.3934/jimo.2009.5.835.
[41]	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 and Applications, 98 (1998), 65-82.doi: 10.1023/A:1022684730236.
[42]	R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica J. IFAC, 49 (2013), 2652-2664.doi: 10.1016/j.automatica.2013.05.027.
[43]	R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.doi: 10.3934/naco.2012.2.571.
[44]	R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44 (2008), 2923-2929.doi: 10.1016/j.automatica.2008.04.011.
[45]	R. 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), 2455-2460.doi: 10.1109/TAC.2009.2029310.
[46]	R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119.doi: 10.1109/TAC.2010.2050710.
[47]	R. Loxton, K. L. Teo and V. Rehbock, Robust suboptimal control of nonlinear systems, Applied Mathematics and Computation, 217 (2011), 6566-6576.doi: 10.1016/j.amc.2011.01.039.
[48]	R. 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), 973-980.doi: 10.1016/j.automatica.2008.10.031.
[49]	R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica J. IFAC, 45 (2009), 2250-2257.doi: 10.1016/j.automatica.2009.05.029.
[50]	D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,'' 3rd Edition, Springer, New York, 2008.
[51]	R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica J. IFAC, 28 (1992), 1113-1123.doi: 10.1016/0005-1098(92)90054-J.
[52]	J. Matula, On an extremum problem, Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 28 (1987), 376-392.doi: 10.1017/S0334270000005464.
[53]	J. Nocedal and S. J. Wright, "Numerical Optimization,'' 2nd Edition, Springer, New York, 2006.
[54]	V. Rehbock, "Tracking Control and Optimal Control,'' PhD thesis, University of Western Australia, Perth, 1994.
[55]	V. Rehbock and L. Caccetta, Two defence applications involving discrete valued optimal control, ANZIAM Journal, 44 (2002), E33-E54.doi: 10.1017/S1446181100007884.
[56]	V. Rehbock, K. L. Teo, L. S. Jennings and H. W. J. Lee, A survey of the control parametrization and control parametrization enhancing methods for constrained optimal control problems, in "Progress in Optimization: Contributions from Australasia,'' Kluwer Academic Publishers, Dordrecht, 1999.doi: 10.1007/978-1-4613-3285-5_13.
[57]	J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.doi: 10.1016/S0005-1098(03)00167-5.
[58]	T. Ruby and V. Rehbock, Numerical solutions of optimal switching control problems, in "Optimization and Control with Applications,'' Springer, New York, 2005.doi: 10.1007/0-387-24255-4_21.
[59]	T. Ruby, V. Rehbock and W. B. Lawrance, Optimal control of hybrid power systems, Dynamics of Continuous, Discrete and Impulsive Systems - Series B: Applications and Algorithms, 10 (2003), 429-439.
[60]	A. Siburian and V. Rehbock, Numerical procedure for solving a class of singular optimal control problems, Optimization Methods and Software, 19 (2004), 413-426.doi: 10.1080/10556780310001656637.
[61]	D. E. Stewart, A numerical algorithm for optimal control problems with switching costs, Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 34 (1992), 212-228.doi: 10.1017/S0334270000008730.
[62]	K. L. Teo, Control parametrization enhancing transform to optimal control problems, Nonlinear Analysis: Theory, Methods and Applications, 63 (2005), e2223-e2236.doi: 10.1016/j.na.2005.03.066.
[63]	K. L. Teo and C. J. Goh, A simple computational procedure for optimization problems with functional inequality constraints, IEEE Transactions on Automatic Control, 32 (1987), 940-941.doi: 10.1109/TAC.1987.1104471.
[64]	K. L. Teo, C. J. Goh and C. C. Lim, A computational method for a class of dynamical optimization problems in which the terminal time is conditionally free, IMA Journal of Mathematical Control and Information, 6 (1989), 81-95.doi: 10.1093/imamci/6.1.81.
[65]	K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems,'' Longman Scientific and Technical, Essex, 1991.
[66]	K. L. Teo and L. S. Jennings, Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Applications, 63 (1989), 1-22.doi: 10.1007/BF00940727.
[67]	K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control, Journal of Optimization Theory and Applications, 68 (1991), 335-357.doi: 10.1007/BF00941572.
[68]	K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 40 (1999), 314-335.doi: 10.1017/S0334270000010936.
[69]	K. L. Teo, G. Jepps, E. J. Moore and S. Hayes, A computational method for free time optimal control problems, with application to maximizing the range of an aircraft-like projectile, Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 28 (1987), 393-413.doi: 10.1017/S0334270000005476.
[70]	K. L. Teo, W. R. Lee, L. S. Jennings, S. Wang and Y. Liu, Numerical solution of an optimal control problem with variable time points in the objective function, ANZIAM Journal, 43 (2002), 463-478.
[71]	K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica J. IFAC, 29 (1993), 789-792.doi: 10.1016/0005-1098(93)90076-6.
[72]	T. L. Vincent and W. J. Grantham, "Optimality in Parametric Systems,'' John Wiley, New York, 1981.
[73]	L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718.doi: 10.3934/jimo.2009.5.705.
[74]	L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Optimal control problems arising in the zinc sulphate electrolyte purification process, Journal of Global Optimization, 54 (2012), 307-323.doi: 10.1007/s10898-012-9863-x.
[75]	K. H. Wong, L. S. Jennings and F. Benyah, The control parametrization enhancing transform for constrained time-delayed optimal control problems, ANZIAM Journal, 43 (2002), E154-E185.
[76]	S. F. Woon, V. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamics and Systems Theory, 10 (2010), 175-188.
[77]	S. F. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.doi: 10.1002/oca.1015.
[78]	C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.doi: 10.3934/jimo.2006.2.435.
[79]	C. Z. Wu, K. L. Teo and V. Rehbock, A filled function method for optimal discrete-valued control problems, Journal of Global Optimization, 44 (2009), 213-225.doi: 10.1007/s10898-008-9319-5.
[80]	R. Yu and P. Leung, Optimal partial harvesting schedule for aquaculture operations, Marine Resource Economics, 21 (2006), 301-315.
[81]	C. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518.doi: 10.1007/s10898-012-9858-7.
[82]	Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693.doi: 10.1021/ie200996f.