Optimal control of switched systems by a modified pseudo spectral method

Document Type: Research Paper


Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan


In the present paper, we develop a modified pseudospectral scheme for solving an optimal control problem which is governed by a switched dynamical system. Many real-world processes such as chemical processes, automotive systems and manufacturing processes can be modeled as such systems. For this purpose, we replace the problem with an alternative optimal control problem in which the switching times appear as unknown parameters. Using the Legendre-Gauss-Lobatto quadrature and the corresponding differentiation matrix, the alternative problem is discretized to a nonlinear programming problem. At last, we examine three examples in order to illustrate the efficiency of the proposed method.


  1. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer–Verlag, Berlin, 2006.
  2. B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1996.
  3. L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
  4. D. E. Kirk, Optimal Control Theory, Prentice–Hall, Englewood Cliffs, New Jersey, 1970.
  5. L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, New York, 1962.
  6. C. P. Neuman and A. Sen, A suboptimal control algorithm for constrained problems using cubic splines, Automatica 9 (1973) 601–613.
  7. J. Vlassenbroeck, A Chebyshev polynomial method for optimal control with state constraint, Automatica 24 (1988) 499–506.
  8. J. Vlassenbroeck and R. Van Doreen, A Chebyshev technique for solving nonlinear optimal control problems, IEEE Trans. Automat. Control 33 (1988) 333–340.
  9. C. J. Goh and K. L. Teo, Control parametrization: a unified approach to optimal control problems with general constraint, Automatica 24 (1988) 3–18.
  10. O. Rosen and R. Luus, Evaluation of gradients for piecewise constraint optimal control, Computers and Chemical Engineering 15 (1991) 273–281.
  11. W. W. Hager, Multiplier methods for nonlinear optimal control, SIAM J. Numer. Anal. 27 (1990) 1061–1080.
  12. D. H. Jacobson and M. M. Lele, A Transformation technique for optimal control problems with a state variable inequality constraints, IEEE Trans. Automat. Control AC–14 (1969) 457–464.
  13. E. Polak, T. H. Yang and D. Q. Mayne, A method of centers based on barrier function methods for solving optimal control problems with continuum state and control constraints, SIAM J. Control Optim. 31 (1993) 159–179.
  14. G. N. Elnagar, M. A. Kazemi and M. Razzaghi, The pseudospectral Legendre method for discretizing optimal control problems, IEEE Trans. Automat. Control 40 (1995) 1793–1796.
  15. G. N. Elnagar and M. A. Kazemi, Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems, Comput. Optim. Appl. 11 (1998) 195–217.
  16. F. Fahroo and I. M. Ross, Costate estimation by a Legendre pseudospectral method, J. Guid. Control Dyn. 24 (2001) 270–277.
  17. G. T. Huntington, D. A. Benson and A. V. Rao, Post–optimality evaluation and analysis of a formation flying problem via a Gauss pseudospectral method, 2005, AAS paper no. 05–339.
  18. Q. Gong, W. Kang and I. M. Ross, A pseudospectral method for the optimal control of constrained feedback linearizable systems, IEEE Trans. Automat. Control 51 (2006) 1115–1129.
  19. M. Shamsi, A modified pseudospectral scheme for accurate solution of Bang–Bang optimal control problems, Optimal Control Appl. Methods 32 (2011) 668–680.
  20. R. Fletcher, Practical Methods of Optimization, John Wiely, Chichester, 1987.
  21. J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer, New York, 1999.
  22. I. M. Ross and F. Fahroo, Pseudospectral knotting methods for solving optimal control problems, J. Guid. Control Dyn. 27 (2004) 397–405.
  23. I. E. Grossmann, S. A. Van Den Heever and I. Harjukoski, Discrete optimization methods and their role in the integration of planning and scheduling, AIChE Symposium Series 98 (2002) 150–168.
  24. N. H. El–Farra, P. Mhaskar and P. D. Christofides, Feedback control of switched nonlinear systems using multiple Lyapunov functions, in Proceedings of American Control Conference, pages 3496–3502, Arlington, VA, 2001.
  25. R. A. Decarlo, M. S. Branicky, S. Petterson and B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88 (2000) 1069–1082.
  26. A. Bemporad and M. Morari, Control of systems integrating logic, dynamics and constraints, Automatica 35 (1999) 407–427.
  27. B. Hu, X. Xu, P. J. Antsaklis and A. N. Michel, Robust stabilizing control law for a class of second–order switched systems, Systems Control Lett. 38 (1999) 197–207.
  28. N. H. El–Farra, P. Mhaskar and P. D. Christofides, Output feedback control of switched nonlinear systems using multiple Lyapunov functions, Systems Control Lett. 54 (2005) 1163–1182.
  29. R. Ghosh and C. Tomlin, Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta–Notch protein signalling, Syst. Biol. 1 (2004) 170–183.
  30. P. G. Howlett, P. J. Pudney and X. Vu, Local energy minimization in optimal train control, Automatica 45 (2009) 2692–2698.
  31. S. Engell, S. Kowalewski, C. Schulz and O. Stursberg, Continuous–discrete interactions in chemical processing plants, Proceedings of the IEEE 88 (2000) 1050–1068.
  32. C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer–Verlag, Berlin Heidelberg, 2014.
  33. X. Xu and P.J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. control 49 (2004) 2–16.
  34. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1984.