Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T10:21:07.303Z Has data issue: false hasContentIssue false
  • English
  • Français

A COMBINED ADAPTIVE CONTROL PARAMETRIZATION AND HOMOTOPY CONTINUATION TECHNIQUE FOR THE NUMERICAL SOLUTION OF BANG–BANG OPTIMAL CONTROL PROBLEMS

Part of: Nonlinear algebraic or transcendental equations Existence theories Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  09 October 2014

M. A. MEHRPOUYA ,
M. SHAMSI  and
M. RAZZAGHI
M. A. MEHRPOUYA
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran, Iran email [email protected], [email protected]
M. SHAMSI*
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran, Iran email [email protected], [email protected]
M. RAZZAGHI
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an efficient computational procedure for the solution of bang–bang optimal control problems. The method is based on a well-known adaptive control parametrization method, which is one of the direct methods for numerical solution of optimal control problems. First, the adaptive control parametrization method is reviewed and then its advantages and disadvantages are illustrated. In order to resolve the need for a priori knowledge about the structure of optimal control and for resolving the sensitivity to an initial guess, a homotopy continuation technique is combined with the adaptive control parametrization method. The present combined method does not require any assumptions on the control structure and the number of switching points. In addition, the switching points are captured accurately; also, efficiency of the method is reported through illustrative examples.

Keywords

bang–bang optimal controlswitching pointnumerical methodscontrol parametrization methodhomotopy continuation technique

MSC classification

Secondary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 65H20: Global methods, including homotopy approaches 49M37: Methods of nonlinear programming type
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 1 , July 2014 , pp. 48 - 65
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Allgower, E. L. and Georg, K., “Introduction to numerical continuation methods”, in: Classics in applied mathematics, Volume 45 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003); doi:10.1137/1.9780898719154.Google Scholar
Bertrand, R. and Epenoy, R., “New smoothing techniques for solving bang–bang optimal control problems – numerical results and statistical interpretation”, Optim. Control Appl. Methods 23 (2002) 171–197; doi:10.1002/oca.709.Google Scholar
Betts, J. T., “Survey of numerical methods for trajectory optimization”, J. Guid. Control Dyn. 21 (1998) 193–207; doi:10.2514/2.4231.Google Scholar
Bryson, A. E. Jr and Ho, Yu Chi, Applied optimal control: optimization, estimation and control (CRC Press, Washington, DC, 1975).Google Scholar
Bulirsch, R., Montrone, F. and Pesch, H. J., “Abort landing in the presence of windshear as a minimax optimal control problem, part 2: multiple shooting and homotopy”, J. Opt. Theory Appl. 70 (1991) 223–254; doi:10.1007/BF00940625.Google Scholar
Burghes, D. N. and Graham, A., Control and optimal control theories with applications (Horwood Publishing Limited, Chichester, 2004); doi:10.1533/9780857099495.Google Scholar
Cerf, M., Haberkorn, T. and Trlat, E., “Continuation from a flat to a round earth model in the coplanar orbit transfer problem”, Optim. Control Appl. Methods 33 (2012) 654–675; doi:10.1002/oca.1016.CrossRefGoogle Scholar
Conway, B. A., “A survey of methods available for the numerical optimization of continuous dynamic systems”, J. Optim. Theory Appl. 152 (2012) 271–306; doi:10.1007/s10957-011-9918-z.Google Scholar
Dormand, J. R. and Prince, P. J., “A family of embedded Runge–Kutta formulae”, J. Comput. Appl. Math. 6 (1980) 19–26; doi:10.1016/0771-050X(80)90013-3.Google Scholar
Ehtamo, H., Raivio, T. and Hamalainen, R. P., “A continuation method for minimum time problems”, Technical Report, System Analysis Laboratory, Helsinki University of Technology, 2000.Google Scholar
Ficken, F. A., “The continuation method for functional equations”, Comm. Pure Appl. Math. 4 (1951) 435–456; doi:10.1002/cpa.3160040405.Google Scholar
Gergaud, J. and Haberkorn, T., “Homotopy method for minimum consumption orbit transfer problem”, ESAIM Control Optim. Calc. Var. 12 (2006) 294–310; doi:10.1051/cocv:2006003.Google Scholar
Gui, W. H., Shen, X. Y., Chen, N., Yang, C. H. and Wang, L. Y., “Optimal control of multiple-time delayed systems based on the control parameterization method”, ANZIAM J. 53 (2012) 68–86; doi:10.1017/S1446181112000053.Google Scholar
Guo, T., Jiang, F. and Li, J., “Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization”, Acta Astronaut. 71 (2012) 38–50; doi:10.1016/j.actaastro.2011.08.008.Google Scholar
Haberkorn, T., Martinon, P. and Gergaud, J., “Low-thrust minimum-fuel orbital transfer: a homotopic approach”, J. Guid. Control Dyn. 27 (2004) 1046–1060; doi:10.2514/1.4022.Google Scholar
Hermant, A., “Optimal control of the atmospheric reentry of a space shuttle by an homotopy method”, Optim. Control Appl. Methods 32 (2011) 627–646; doi:10.1002/oca.961.Google Scholar
Hu, G. S., Ong, C. J. and Teo, C. L., “Minimum-time control of a crane with simultaneous traverse and hoisting motions”, J. Optim. Theory Appl. 120 (2004) 395–416; doi:10.1023/B:JOTA.0000015690.02820.ea.Google Scholar
Huang, H.-P. and Harris McClamroch, N., “Time-optimal control for a robotic contour following problem”, IEEE J. Robot. Autom. 4 (1988) 140–149; doi:10.1109/56.2077.CrossRefGoogle Scholar
Jiang, F., Baoyin, H. and Li, J., “Practical techniques for low-thrust trajectory optimization with homotopic approach”, J. Guid. Control Dyn. 35 (2012) 245–258; doi:10.2514/1.52476.Google Scholar
Kaya, C. Y. and Noakes, J. L., “Computations and time-optimal controls”, Optim. Control Appl. Methods 17 (1996) 171–185; doi:10.1002/(SICI)1099-1514(199607/09)17:3¡171::AID-OCA571¿3.0.CO;2-9.Google Scholar
Kaya, C. Y. and Noakes, J. L., “Computational method for time-optimal switching control”, J. Optim. Theory Appl. 117 (2003) 69–92; doi:10.1023/A:1023600422807.Google Scholar
Kaya, C. Y., Lucas, S. K. and Simakov, S. T., “Computations for bang–bang constrained optimal control using a mathematical programming formulation”, Optim. Control Appl. Methods 25 (2004) 295–308; doi:10.1002/oca.749.Google Scholar
Kim, J.-H. R., Maurer, H., Astrov, Yu. A., Bode, M. and Purwins, H.-G., “High-speed switch-on of a semiconductor gas discharge image converter using optimal control methods”, J. Comput. Phys. 170 (2001) 395–414; doi:10.1006/jcph.2001.6741.Google Scholar
Ledzewicz, U. and Schättler, H., “Analysis of a cell-cycle specific model for cancer chemotherapy”, J. Biol. Systems 10 (2002) 183–206; doi:10.1142/S0218339002000597.Google Scholar
Ledzewicz, U. and Schättler, H., “Optimal bang–bang controls for a two-compartment model in cancer chemotherapy”, J. Optim. Theory Appl. 114 (2002) 609–637; doi:10.1023/A:1016027113579.Google Scholar
Lee, H. W. J., Teo, K. L., Rehbock, V. and Jennings, L. S., “Control parametrization enhancing technique for optimal discrete-valued control problems”, Automatica 35 (1999) 1401–1407; doi:10.1016/S0005-1098(99)00050-3.Google Scholar
Li, B., Teo, K. L., Zhao, G. H. and Duan, G. R., “An efficient computational approach to a class of minmax optimal control problems with applications”, ANZIAM J. 51 (2009) 162–177; doi:10.1017/S1446181110000040.Google Scholar
Li, R., Feng, Z. G., Teo, K. L. and Duan, G. R., “Tracking control of linear switched systems”, ANZIAM J. 49 (2008) 187–203; doi:10.1017/S1446181100012773.Google Scholar
Lin, Q., Loxton, R. and Teo, K. L., “The control parameterization method for nonlinear optimal control: a survey”, J. Ind. Manag. Optim. 10 (2014) 275–309; doi:10.3934/jimo.2014.10.275.CrossRefGoogle Scholar
Lin, Q., Loxton, R., Teo, K. L. and Wu, Y. H., “A new computational method for a class of free terminal time optimal control problems”, Pac. J. Optim. 7 (2011) 63–81.Google Scholar
Lin, Q., Loxton, R., Teo, K. L. and Wu, Y. H., “Optimal control computation for nonlinear systems with state-dependent stopping criteria”, Automatica 48 (2012) 2116–2129; doi:10.1016/j.automatica.2012.06.055.Google Scholar
Loxton, R., Lin, Q. and Teo, K. L., “Minimizing control variation in nonlinear optimal control”, Automatica 49 (2013) 2652–2664; doi:10.1016/j.automatica.2013.05.027.Google Scholar
Loxton, R. C., Teo, K. L., Rehbock, V. and Yiu, K. F. C., “Optimal control problems with a continuous inequality constraint on the state and the control”, Automatica 45 (2009) 2250–2257; doi:10.1016/j.automatica.2009.05.029.Google Scholar
Martinon, P. and Gergaud, J., “Using switching detection and variational equations for the shooting method”, Optim. Control Appl. Methods 28 (2007) 95–116; doi:10.1002/oca.794.CrossRefGoogle Scholar
Maurer, H. and Osmolovskii, N. P., “Second order sufficient conditions for time-optimal bang–bang control”, SIAM J. Control Optim. 42 (2004) 2239–2263; doi:10.1137/S0363012902402578.Google Scholar
Navabi, M. R., Shamsi, M. and Dehghan, M., “Numerical solution of the controlled Rayleigh nonlinear oscillator by the direct spectral method”, J. Vib. Control 14 (2008) 795–806; doi:10.1177/1077546307084239.Google Scholar
Oberle, H. J. and Grimm, W., “BNDSCO – a program for the numerical solution of optimal control problems”, Technical Report 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.Google Scholar
Rheinboldt, W. C., “Numerical continuation methods: a perspective”, J. Comput. Appl. Math. 124(1–2) (2000) 229–244; doi:10.1016/S0377-0427(00)00428-3.Google Scholar
Richter, S. L. and DeCarlo, R. A., “Continuation methods: theory and applications”, IEEE Trans. Syst. Man Cybern. SMC-13 (1983) 459–464; doi:10.1109/TSMC.1983.6313131.Google Scholar
Shampine, L. F., Gladwell, I. and Thompson, S., Solving ODEs with MATLAB (Cambridge University Press, Cambridge, 2003); doi:10.1017/CBO9780511615542.Google Scholar
Shamsi, M., “A modified pseudospectral scheme for accurate solution of bang–bang optimal control problems”, Optim. Control Appl. Methods 32 (2011) 668–680; doi:10.1002/oca.967.CrossRefGoogle Scholar
Teo, K. L., Lee, W. R., Jennings, L. S., Wang, S. and Liu, Y., “Numerical solution of an optimal control problem with variable time points in the objective function”, ANZIAM J. 43 (2002) 463–478.Google Scholar
Trélat, E., “Optimal control and applications to aerospace: some results and challenges”, J. Optim. Theory Appl. 154 (2012) 713–758; doi:10.1007/s10957-012-0050-5.Google Scholar
Wang, L. Y., Gui, W. H., Teo, K. L., Loxton, R. C. and Yang, C. H., “Time delayed optimal control problems with multiple characteristic time points: computation and industrial applications”, J. Ind. Manag. Optim. 5 (2009) 705–718; doi:10.3934/jimo.2009.5.705.Google Scholar
Wong, K. H. and Tang, W. M., “Optimal control of switched impulsive systems with time delay”, ANZIAM J. 53 (2012) 292–307; doi:10.1017/S1446181112000284.Google Scholar
Wu, C., Teo, K. L. and Wu, S., “Min–max optimal control of linear systems with uncertainty and terminal state constraints”, Automatica 49 (2013) 1809–1815; doi:10.1016/j.automatica.2013.02.052.Google Scholar
Yu, C., Li, B., Loxton, R. and Teo, K. L., “Optimal discrete-valued control computation”, J. Global Optim. 56 (2013) 503–518; doi:10.1007/s10898-012-9858-7.CrossRefGoogle Scholar