ANZIAM J. 49 (2007), no. 2, pp. 187–203.
Tracking control of linear switched systems
R. Li Z. G. Feng
Center for Control Theory and Guidance Technology
Harbin Institute of Technology
Harbin
P.R. China
hitlirui@gmail.com		 College of Mathematics and Computer Science
Chongqing Normal University
Chongqing
P.R. China
K.L. Teo† G. R. Duan
Department of Mathematics & Statistics
Curtin University of Technology
Perth
WA 6845
Australia
K.L.Teo@curtin.edu.au		 Center for Control Theory and Guidance Technology
Harbin Institute of Technology
Harbin
P.R. China
Received 9 May, 2007

Abstract

This paper deals with the optimal tracking problem for switched systems, where the control input, the switching times and the switching index are all design variables. We propose a three-stage method for solving this problem. First, we fix the switching times and switching index sequence, which leads to a linear tracking problem, except different subsystems are defined in their respective time intervals. The optimal control and the corresponding cost function obtained depend on the switching signal. This gives rise to an optimal parameter selection problem for which the switching instants and the switching index are to be chosen optimally. In the second stage, the switching index is fixed. A reverse time transformation followed by a time scaling transform are introduced to convert this subproblem into an equivalent standard optimal parameter selection problem. The gradient formula of the cost function is derived. Then the discrete filled function is used in the third stage to search for the optimal switching index. On this basis, a computational method, which combines a gradient-based method, a local search algorithm and a filled function method, is developed for solving this problem. A numerical example is solved, showing the effectiveness of the proposed approach.

Download the article in PDF format (size 168 Kb)

2000 Mathematics Subject Classification: primary 49N90; secondary 90C11, 93C95
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2376???
†indicates author for correspondence

References

  1. B. D. Anderson and J. B. Moore, Optimal control: Linear quadratic methods (Prentice Hall, Englewood Cliffs, NJ, 1990).
  2. S. Bengea and R. DeCarlo, “Optimal control of switching systems”, Automatica 41 (2005) 11–27. MR2157228
  3. S. Devasia, B. Paden and C. Rossi, “Exact-output tracking theory for systems with parameter jumps”, Int. J. Control 67 (1997) 11–131. MR1685845
  4. Z. G. Feng, K. L. Teo and V. Rehbock, “A discrete filled function method for the optimal control of switched systems in discrete time”, Submitted.
  5. R. P. Ge, “A filled function method for finding a global minimizer of a function of several variables”, Math. Program. 46 (1990) 191–204. MR1047374
  6. S. Hedlund and A. Rantzer, “Convex dynamic programming for hybrid systems”, IEEE Trans. Automat. Contr. 47 (2002) 1536–1540. MR1924325
  7. A. Isidori, Nonlinear control systems, 2nd ed. (Springer-Verlag, Berlin, 1989). MR1015932
  8. L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3.3-Optimal control software: Theory and user manual, 2004.
  9. H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, “Control parametrization enhancing technique for time optimal control problems”, Dyn. Syst. Appl. 6 (1997) 243–261. MR1461441
  10. F. L. Lewis, Optimal control, 2nd ed. (John Wiley, New York, 1995). MR833285
  11. C. K. Ng, L. S. Zhang, D. Li and W. W. Tian, “Discrete filled function method for discrete global optimization”, Comput. Optim. Appl. 31 (2005) 87–115. MR2143506
  12. B. Piccoli, “Necessary conditions for hybrid optimization”, in Proc. 38th IEEE CDC, (1999) 410–415.
  13. P. Riedinger, C. Iung and F. Kratz, “An optimal control approach for hybrid systems”, European J. Contr. 49 (2003) 449–458.
  14. Z. Sun and S. S. Ge, Switched linear systems– control and design (Springer, Berlin, 2004).
  15. H. Sussman, “A maximum principle for hybrid optimal control problems”, in Proc. 38th IEEE CDC, Phoenix, USA, (1999) 425–430.
  16. K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems (Longman Scientific and Technical, United Kingdom, 1991). MR1153024
  17. C. Z. Wu and K. L. Teo, “Global impulsive optimal control computation”, J. Ind. Manag. Optim. 2 (2006) 435–450. MR2247961
  18. X. Xu and P. Antsaklis, “Optimal control of switched systems based on parameterization of the switching instants”, IEEE Trans. Automat.Contr. 49 (2004) 2–16. MR2028538
  19. Y. Yin and S. Hosoe, “Tracking control of discrete and continuous hybrid systems: modeling and servoing problem of dextrous hand manipulation”, in Proc. 2004 IEEE Int. Conf. Control Applications, 2–4 Sept. 2004, (2004) 860–865.