×
Lin, Qun; Loxton, Ryan; Teo, Kok Lay; Wu, Yong Hong

Optimal control computation for nonlinear systems with state-dependent stopping criteria. (English) Zbl 1258.49051

Automatica 48, No. 9, 2116-2129 (2012).
Summary: In this paper, we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion. This stopping criterion is defined by a smooth surface in the state space; when the state trajectory hits this surface, the governing dynamic system stops. By restricting the controls to piecewise constant functions, we derive a finite-dimensional approximation of the optimal control problem. We then develop an efficient computational method, based on nonlinear programming, for solving the approximate problem. We conclude the paper with four numerical examples.

Cited in 31 Documents

MSC:

49M37 Numerical methods based on nonlinear programming
49K15 Optimality conditions for problems involving ordinary differential equations
90C30 Nonlinear programming

Keywords:

nonlinear optimal control; control parameterization; nonlinear programming; time-scaling transformation

Software:

NLPQLP; MISER3
PDF BibTeX XML Cite
\textit{Q. Lin} et al., Automatica 48, No. 9, 2116--2129 (2012; Zbl 1258.49051)
Full Text: DOI Link

References:

[1] Ahmed, N. U., Elements of finite-dimensional systems and control theory (1988), Longman Scientific and Technical: Longman Scientific and Technical Essex · Zbl 0658.93002
[2] Bulirsch, R.; Nerz, E.; Pesch, H. J.; von Stryk, O., Combining direct and indirect methods in optimal control: range maximization of a hang glider, (Bulirsch, R.; Miele, A.; Stoer, J.; Well, K. H., Optimal control: calculus of variations, optimal control theory and numerical methods, Vol. 111 (1993), Birkhäuser), 273-288 · Zbl 0808.65067
[3] Chyba, M.; Haberkorn, T.; Singh, S. B.; Smith, R. N.; Choi, S. K., Increasing underwater vehicle autonomy by reducing energy consumption, Ocean Engineering, 36, 62-73 (2009)
[4] Gerdts, M.; Kunkel, M., A nonsmooth Newton’s method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4, 247-270 (2008) · Zbl 1157.49036
[5] Hager, W. W., Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik, 87, 247-282 (2000) · Zbl 0991.49020
[6] Hindmarsh, A. C., Large ordinary differential equation systems and software, IEEE Control Systems Magazine, 2, 24-30 (1982)
[8] Kaya, C. Y.; Martínez, J. M., Euler discretization and inexact restoration for optimal control, Journal of Optimization Theory and Applications, 134, 191-206 (2007) · Zbl 1135.49019
[9] Kaya, C. Y.; Noakes, J. L., Computational method for time-optimal switching control, Journal of Optimization Theory and Applications, 117, 69-92 (2003) · Zbl 1029.49029
[10] Lee, H. W.J.; Teo, K. L.; Rehbock, V.; Jennings, L. S., Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6, 243-262 (1997) · Zbl 0894.49018
[11] Lin, Q.; Loxton, R.; Teo, K. L.; Wu, Y. H., A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7, 63-81 (2011) · Zbl 1211.49041
[12] Loxton, R.; Teo, K. L.; Rehbock, V., Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44, 2923-2929 (2008) · Zbl 1160.49033
[13] Luenberger, D. G.; Ye, Y., Linear and nonlinear programming (2008), Springer: Springer New York · Zbl 1207.90003
[14] Luus, R., Iterative dynamic programming (2000), Chapman and Hall: Chapman and Hall Boca Raton · Zbl 1070.49001
[15] Nocedal, J.; Wright, S. J., Numerical optimization (2006), Springer: Springer New York · Zbl 1104.65059
[17] Teo, K. L.; Goh, C. J.; Lim, C. C., 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, 81-95 (1989) · Zbl 0673.49013
[18] Teo, K. L.; Goh, C. J.; Wong, K. H., A unified computational approach to optimal control problems (1991), Longman Scientific and Technical: Longman Scientific and Technical Essex · Zbl 0747.49005
[19] Teo, K. L.; Jepps, G.; Moore, E. J.; Hayes, S., 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, 393-413 (1987) · Zbl 0654.70031
[20] Vanderbei, R. J., Case studies in trajectory optimization: trains, planes, and other pastimes, Optimization and Engineering, 2, 215-243 (2001) · Zbl 1011.49026
[21] Vincent, T. L.; Grantham, W. J., Optimality in parametric systems (1981), John Wiley: John Wiley New York · Zbl 0485.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.