Skip to main content

Advertisement

SpringerLink
Log in

Numerical algorithm for a class of constrained optimal control problems of switched systems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we consider an optimal control problem of switched systems with continuous-time inequality constraints. Because of the complexity of such constraints and switching laws, it is difficult to solve this problem by standard optimization techniques. To overcome the difficulty, we adopt a bi-level algorithm to divide the problem into two nonlinear constrained optimization problems: one continuous and the other discrete. To solve the problem, we transform the inequality constraints into equality constraints which is smoothed using a twice continuously differentiable function and treated as a penalty function. On this basis, the smoothed problem can be solved by any second-order gradient algorithm, e.g., Newton’s Method. Finally, numerical examples show that our method is effective compared to existing algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zhang, W., Abate, A., Hu, J.H., Vitus, M.: Exponential stabilization of discrete–time switched linear systems. Autom. 45, 2526–2536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Kirches, C., Sager, S., Bock, H.G., Schloder, J.P.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Meth. 31, 137–153 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ghosh, R., Tomlin, C.: Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling. Syst. Biol. 1, 170–183 (2004)

    Article  Google Scholar 

  4. Cassandras, C.G., Pepyne, D.L., Wardi, Y.Y.: Optimal control of a class of hybrid systems. IEEE Trans. Autom. Control. 46, 398–415 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Rehbinder, H., Sanfridson, M.: Scheduling of a limited communication channel for optimal control. Autom. 40, 491–500 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Autom. 35, 407–428 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bengea, S.C., DeCarlo, R.A.: Optimal control of switching systems. Autom. 41, 11–27 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Xu, X.P., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Autom. Control. 49, 2–16 (2004)

    Article  MathSciNet  Google Scholar 

  9. Boccadoro, M., Wardi, Y.Y., Egerstedt, M., Verriest, E.: Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn. Syst.-Theory Appl. 15, 433–448 (2005)

    MATH  MathSciNet  Google Scholar 

  10. Hwang, I., Li, J., Du, D.: A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach. Int. J. Control. 81, 277–293 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Axelsson, H., Wardi, Y.Y., Egerstedt, M., Verriest, E.: Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. J. Optim. Theory Appl. 136, 167–186 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kamgarpour, M., Tomlin, C.: On optimal control of non-autonomous switched systems with a fixed mode sequence. Autom. 48, 1177–1181 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gorges, D., Izak, M., Liu, S.: Optimal control and scheduling of switched systems. IEEE Trans. Autom. Control. 56, 135–140 (2011)

    Article  MathSciNet  Google Scholar 

  14. Cao, N., Zhang, H.G., Luo, Y.H.: Infinite horizon optimal control of affine nonlinear discrete switched systems using two-stage approximate dynamic programming. Int. J. Syst. Sci. 43, 1673–1682 (2012)

    Article  MathSciNet  Google Scholar 

  15. Balandat, M., Zhang, W., Abate, A.: On infinite horizon switched LQR problems with state and control constraints. Syst. Control. Lett. 61, 464–471 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gao, J.J., Li, D.: Linear-quadratic switching control with switching cost. Autom. 48, 1138–1143 (2012)

    Article  MATH  Google Scholar 

  17. Zhang, W., Hu, J.H., Abate, A.: On the value functions of the discrete-time switched LQR problem. IEEE Trans. Autom. Control. 54, 462–471 (2009)

    Article  MathSciNet  Google Scholar 

  18. Zhang, W., Hu, J.H., Abate, A.: Infinite-horizon switched LQR problems in discrete time: a suboptimal algorithm with performance analysis. IEEE Trans. Autom. Control. 57, 1815–1821 (2012)

    Article  MathSciNet  Google Scholar 

  19. Wu, X., Zhang, K.J., Sun, C.Y.: Constrained optimal control of switched systems and its application. Optim. (2013). doi:10.1080/02331934.2013.784764

  20. Wu, X., Zhang, K.J., Sun, C.Y.: Parameter tuning of multi-proportional-integral-derivative controllers based on optimal switching algorithms. J. Optim. Theory. Appl. 159, 454–472 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Azhmyakov, V., Basin, M.V., Raisch, J.: A proximal point based approach to optimal control of affine switched systems. Discrete Event Dyn. Syst.-Theory. Appl. 22, 61–81 (2012)

    MATH  MathSciNet  Google Scholar 

  22. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control. 43, 31–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Xu, X.P., Antsaklis, P.J.: On time optimal control of integrator switched systems with state constraints. Nonlinear Anal.–Theory Meth. Appl. 62, 1453–1465 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Caldwell, T.M., Murphey, T.D.: Switching mode generation and optimal estimation with application to skid-steering. Autom. 47, 50–64 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Caldwell, T.M., Murphey, T.D.: Single integration optimization of linear time-varying switched systems. IEEE Trans. Autom. Control. 57, 1592–1597 (2012)

    Article  MathSciNet  Google Scholar 

  26. Egerstedt, M., Wardi, Y.Y., Axelsson, H.: Transition-time optimization for switched-mode dynamical systems. IEEE Trans. Autom. Control. 51, 110–115 (2006)

    Article  MathSciNet  Google Scholar 

  27. Howlett, P., Pudney, P., Vu, X.: Local energy minimization in optimal train control. Autom. 45, 2692–2698 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jiang, C.H., Teo, K.L., Loxton, R., Duan, G.R.: A neighboring extremal solution for an optimal switched impulsive control problem. J. Ind. Manage. Optim. 8, 591–609 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Loxton, R., Teo, K.L., Rehbock, V.: Computational method for a class of switched system optimal control problems. IEEE Trans. Autom. Control. 54, 2455–2460 (2009)

    Article  MathSciNet  Google Scholar 

  30. Loxton, R., Teo, K.L., Rehbock, V., Ling, W.: Optimal switching instants for a switched-capacitor DC/DC power converter. Autom. 45, 973–980 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shaikh, M., Caines, P.: On the hybrid optimal control problem: Theory and algorithms. IEEE Trans. Autom. Control. 52, 1587–1603 (2007)

    Article  MathSciNet  Google Scholar 

  32. Passenberg, B.: Theory and algorithms for indirect methods in optimal control of hybrid systems. PhD thesis, Technical University of Munich (2012)

    Google Scholar 

  33. Graichen, K., Petit, N.: Incorporating a class of constraints into the dynamics of optimal control problems. Optim. Control. Appl. Meth. 30, 537–561 (2009)

    Article  MathSciNet  Google Scholar 

  34. Sakawa, Y., Shindo, Y.: Optimal control of container cranes. Autom. 18, 257–266 (1982)

    Article  MATH  Google Scholar 

  35. Soler, M., Olivares, A., Staffetti, E.: Hybrid optimal control approach to commercial aircraft trajectory planning. J. Guid. Control. Dyn. 33, 985–991 (2010)

    Article  Google Scholar 

  36. Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  37. Gonzalez, H., Vasudevan, R., Kamgarpour, M., Sastry, S., Bajcsy, R., Tomlin, C.: A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems. In: Johansson K., Yi W. (eds.) 13th International Conference On Hybrid Systems: Computation And Control, 12–16 April 2010. Lecture Notes in Computer Science, pp. 5160. Stockholm (2010)

  38. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program 66, 327–349 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automation, Southeast University, Nanjing, 210096, China

    Xiang Wu, Kanjian Zhang & Changyin Sun

  2. School of Electrical and Information Engineering, Hunan Institute of Technology, Hengyang, 421002, China

    Xiang Wu

Authors
  1. Xiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kanjian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Changyin Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kanjian Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, X., Zhang, K. & Sun, C. Numerical algorithm for a class of constrained optimal control problems of switched systems. Numer Algor 67, 771–792 (2014). https://doi.org/10.1007/s11075-013-9822-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9822-8

Keywords

Search

Navigation