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Feedback control method using Haar wavelet operational matrices for solving optimal control problems. (English) Zbl 1291.93253

Summary: Most of the direct methods solve optimal control problems with nonlinear programming solver. In this paper we propose a novel feedback control method for solving for solving affine control system, with quadratic cost functional, which makes use of only linear systems. This method is a numerical technique, which is based on the combination of Haar wavelet collocation method and successive Generalized Hamilton-Jacobi-Bellman equation. We formulate some new Haar wavelet operational matrices in order to manipulate Haar wavelet series. The proposed method has been applied to solve linear and nonlinear optimal control problems with infinite time horizon. The simulation results indicate that the accuracy of the control and cost can be improved by increasing the wavelet resolution.

MSC:

93D15 Stabilization of systems by feedback
93B40 Computational methods in systems theory (MSC2010)
49N35 Optimal feedback synthesis
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[1] Beard, R. W.; McLain, T. W., Successive Galerkin approximation algorithms for nonlinear optimal and robust control, International Journal of Control, 71, 5, 717-743 (1998) · Zbl 0945.49021
[2] Feng, Y.; Anderson, B. D. O.; Rotkowitz, M., A game theoretic algorithm to compute local stabilizing solutions to HJBI equations in nonlinear \(H_\infty\) control, Automatica, 45, 4, 881-888 (2009) · Zbl 1162.93021 · doi:10.1016/j.automatica.2008.11.006
[3] Huang, J.; Lin, C.-F., Numerical approach to computing nonlinear \(H_\infty\) control laws, Journal of Guidance, Control, and Dynamics, 18, 5, 989-996 (1995) · Zbl 0841.93018
[4] Aliyu, M. D. S., An approach for solving the Hamilton-Jacobi-Isaacs equation (HJIE) in nonlinear \(ℋ_∞\) control, Automatica, 39, 5, 877-884 (2003) · Zbl 1030.93013 · doi:10.1016/S0005-1098(03)00025-6
[5] Abu-Khalaf, M.; Lewis, F. L.; Huang, J., Policy iterations on the Hamilton-Jacobi-Isaacs equation for \(H_\infty\) state feedback control with input saturation, IEEE Transactions on Automatic Control, 51, 12, 1989-1995 (2006) · Zbl 1366.93147 · doi:10.1109/TAC.2006.884959
[6] Glad, S. T., Robustness of nonlinear state feedback—a survey, Automatica, 23, 4, 425-435 (1987) · Zbl 0633.93051
[7] von Stryk, O.; Bulirsch, R., Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37, 1, 357-373 (1992) · Zbl 0784.49023 · doi:10.1007/BF02071065
[8] Beard, R. W.; Saridis, G. N.; Wen, J. T., Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation, Automatica, 33, 12, 2159-2176 (1997) · Zbl 0949.93022
[9] Jaddu, H. M., Numerical methods for solving optimal control problems using Chebyshev polynomials [Ph.D. thesis] (1998), School of Information Science, Japan Advanced Institute of Science and Technology
[10] Beeler, S. C.; Tran, H. T.; Banks, H. T., Feedback control methodologies for nonlinear systems, Journal of Optimization Theory and Applications, 107, 1, 1-33 (2000) · Zbl 0971.49023
[11] Park, C.; Tsiotras, P., Approximations to optimal feedback control using a successive wavelet collocation algorithm, Proceedings of the American Control Conference
[12] Chen, C. F.; Hsiao, C. H., Haar wavelet method for solving lumped and distributed parameter systems, IEE Proceeding on Control Theory and Application, 144, 1, 87-94 (1997) · Zbl 0880.93014 · doi:10.1049/ip-cta:19970702
[13] Hsiao, C. H.; Wang, W. J., Optimal control of linear time-varying systems via Haar wavelets, Journal of Optimization Theory and Applications, 103, 3, 641-655 (1999) · Zbl 0941.49018
[14] Dai, R.; Cochran, J. E., Wavelet collocation method for optimal control problems, Journal of Optimization Theory and Applications, 143, 2, 265-278 (2009) · Zbl 1176.49035 · doi:10.1007/s10957-009-9565-9
[15] Curtis, J. W.; Beard, R. W., Successive collocation: an approximation to optimal nonlinear control, Proceeding of the American Control Conference
[16] Hsiao, C. H.; Wu, S. P., Numerical solution of time-varying functional differential equations via Haar wavelets, Applied Mathematics and Computation, 188, 1, 1049-1058 (2007) · Zbl 1118.65077 · doi:10.1016/j.amc.2006.10.070
[17] Brewer, J. W., Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25, 9, 772-781 (1978) · Zbl 0397.93009 · doi:10.1109/TCS.1978.1084534
[18] Courrieu, P., Fast computation of Moore-Penrose inverse matrices, Neural Information Processing-Letters and Reviews, 8, 2, 25-29 (2005)
[19] Slotine, J.-J.; Li, W., Applied Nonlinear Control (1991), Englewood Cliffs, NJ, USA: Prentice-Hall, Englewood Cliffs, NJ, USA · Zbl 0753.93036
[20] Isidori, A., Nonlinear Control Systems. Nonlinear Control Systems, Communication and Control Engineering (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0714.93021
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