Abstract and Applied Analysis

Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems

Waleeda Swaidan and Amran Hussin

Full-text: Open access


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.

Article information

Abstr. Appl. Anal., Volume 2013 (2013), Article ID 240352, 8 pages.

First available in Project Euclid: 27 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Swaidan, Waleeda; Hussin, Amran. Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems. Abstr. Appl. Anal. 2013 (2013), Article ID 240352, 8 pages. doi:10.1155/2013/240352. https://projecteuclid.org/euclid.aaa/1393512007

Export citation


  • R. W. Beard and T. W. McLain, “Successive Galerkin approximation algorithms for nonlinear optimal and robust control,” International Journal of Control, vol. 71, no. 5, pp. 717–743, 1998.
  • Y. Feng, B. D. O. Anderson, and M. Rotkowitz, “A game theoretic algorithm to compute local stabilizing solutions to HJBI equations in nonlinear ${H}_{\infty }$ control,” Automatica, vol. 45, no. 4, pp. 881–888, 2009.
  • J. Huang and C.-F. Lin, “Numerical approach to computing nonlinear ${H}_{\infty }$ control laws,” Journal of Guidance, Control, and Dynamics, vol. 18, no. 5, pp. 989–996, 1995.
  • M. D. S. Aliyu, “An approach for solving the Hamilton-Jacobi-Isaacs equation (HJIE) in nonlinear $\mathcal{H}_{\infty }$ control,” Automatica, vol. 39, no. 5, pp. 877–884, 2003.
  • M. Abu-Khalaf, F. L. Lewis, and J. Huang, “Policy iterations on the Hamilton-Jacobi-Isaacs equation for ${H}_{\infty }$ state feedback control with input saturation,” IEEE Transactions on Automatic Control, vol. 51, no. 12, pp. 1989–1995, 2006.
  • S. T. Glad, “Robustness of nonlinear state feedback–-a survey,” Automatica, vol. 23, no. 4, pp. 425–435, 1987.
  • O. von Stryk and R. Bulirsch, “Direct and indirect methods for trajectory optimization,” Annals of Operations Research, vol. 37, no. 1, pp. 357–373, 1992.
  • R. W. Beard, G. N. Saridis, and J. T. Wen, “Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation,” Automatica, vol. 33, no. 12, pp. 2159–2176, 1997.
  • H. M. Jaddu, Numerical methods for solving optimal control problems using Chebyshev polynomials [Ph.D. thesis], School of Information Science, Japan Advanced Institute of Science and Technology, 1998.
  • S. C. Beeler, H. T. Tran, and H. T. Banks, “Feedback control methodologies for nonlinear systems,” Journal of Optimization Theory and Applications, vol. 107, no. 1, pp. 1–33, 2000.
  • C. Park and P. Tsiotras, “Approximations to optimal feedback control using a successive wavelet collocation algorithm,” in Proceedings of the American Control Conference, vol. 3, pp. 1950–1955, June 2003.
  • C. F. Chen and C. H. Hsiao, “Haar wavelet method for solving lumped and distributed parameter systems,” IEE Proceeding on Control Theory and Application, vol. 144, no. 1, pp. 87–94, 1997.
  • C. H. Hsiao and W. J. Wang, “Optimal control of linear time-varying systems via Haar wavelets,” Journal of Optimization Theory and Applications, vol. 103, no. 3, pp. 641–655, 1999.
  • R. Dai and J. E. Cochran Jr., “Wavelet collocation method for optimal control problems,” Journal of Optimization Theory and Applications, vol. 143, no. 2, pp. 265–278, 2009.
  • J. W. Curtis and R. W. Beard, “Successive collocation: an approximation to optimal nonlinear control,” in Proceeding of the American Control Conference, vol. 5, pp. 3481–3485, June 2001.
  • C. H. Hsiao and S. P. Wu, “Numerical solution of time-varying functional differential equations via Haar wavelets,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 1049–1058, 2007.
  • J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Transactions on Circuits and Systems, vol. 25, no. 9, pp. 772–781, 1978.
  • P. Courrieu, “Fast computation of Moore-Penrose inverse matrices,” Neural Information Processing-Letters and Reviews, vol. 8, no. 2, pp. 25–29, 2005.
  • J.-J. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1991.
  • A. Isidori, Nonlinear Control Systems, Communication and Control Engineering, Springer, New York, NY, USA, 2nd edition, 1989.