Optimal Control of Piece-Wise Polynomial Hybrid Systems Using Cylindrical Algebraic Decomposition

  • Ioannis A. Fotiou
  • A. Giovanni Beccuti
  • Georgios Papafotiou
  • Manfred Morari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


We present a new method to solve the constrained finite-time optimal control (CFTOC) problem for piece-wise polynomial (PWP) hybrid systems, based on Cylindrical Algebraic Decomposition (CAD). The computational approach consists of two parts. The off-line, where the method re-formulates the original CFTOC optimization problem in algebraic form, decomposes it into smaller subproblems and then independently pre-processes each subproblem to obtain certain structural information, and the on-line, where this available precomputed information is used to efficiently compute the optimal solution of the original problem in real time. The method is illustrated through its application to the control of a boost dc-dc converter.


Duty Cycle Model Predictive Control Prediction Horizon Boost Converter Model Predictive Control Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Maciejowski, J.: Predictive control with Constraints. Pearson Education, London (2001)MATHGoogle Scholar
  2. 2.
    Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bemporad, A., Borrelli, F., Morari, M.: Model predictive control based on linear programming — the explicit solution. IEEE Trans. Automat. Contr. 47(12) (2002)Google Scholar
  4. 4.
    Borrelli, F., Baotić, M., Bemporad, A., Morari, M.: k An efficient algorithm for computing the state feedback optimal control law for discrete time hybrid systems. In: Proc. American Control Conf., Denver, Colorado, pp. 4717–4722 (2003)Google Scholar
  5. 5.
    Kerrigan, E.C., Mayne, D.Q.: Optimal control of constrained, piecewise affine systems with bounded disturbances. In: Proc. 41st IEEE Conf. on Decision and Control, Las Vegas, Nevada, USA (2002)Google Scholar
  6. 6.
    Borrelli, F.: Constrained Optimal Control of Linear and Hybrid Systems. LNCIS, vol. 290. Springer, Heidelberg (2003)MATHGoogle Scholar
  7. 7.
    Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40(2), 293–300 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fotiou, I.A., Parrilo, P.A., Morari, M.: Nonlinear parametric optimization using cylindrical algebraic decomposition. In: Proc. of the Conf. on Decision & Control, Seville, Spain, pp. 3735–3740 (2005)Google Scholar
  9. 9.
    Collins, G.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition, London, UK. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  10. 10.
    Lafferriere, G., Pappas, G., Yovine, S.: Reach set computation for linear vector fields using quantifier elimination. In: Electronic Proceedings of the IMAC Conference on Applications of Computer Algebra, El Escorial, Spain (1999)Google Scholar
  11. 11.
    Alur, R., Henzinger, T., Lafferriere, G., Pappas, G.: Discrete abstractions of hybrid systems. Proceedings of the IEEE 88(7), 971–984 (2000)CrossRefGoogle Scholar
  12. 12.
    Vidal, R., Soatto, S., Ma, Y., Sastry, S.: An Algebraic Geometric Approach to the Identification of a Class of Linear Hybrid Systems. In: Proc. of the Conf. on Decision & Control, Maui, HI (2003)Google Scholar
  13. 13.
    Beccuti, A.G., Papafotiou, G., Morari, M.: Optimal Control of the Boost dc-dc Converter. In: Proc. of the Conf. on Decision & Control, Seville, Spain (2005)Google Scholar
  14. 14.
    Baotić, M., Christophersen, F.J., Morari, M.: A new Algorithm for Constrained Finite Time Optimal Control of Hybrid Systems with a Linear Performance Index. In: Proc. of the European Control Conference, Cambridge, UK (2003)Google Scholar
  15. 15.
    Brown, C.W.: QEPCAD B: a program for computing with semialgebraic sets using CADs. ACM SIGSAM Bulletin 37, 97–108 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Mohan, N., Undeland, T.M., Robbins, W.P.: Power Electronics: Converters, Applications and Design. Wiley, London (1989)Google Scholar
  17. 17.
    Qin, S.J., Badgwell, T.A.: A survey of industrial model predictive control technology. Control Engineering Practice 11, 733–764 (2003)CrossRefGoogle Scholar
  18. 18.
    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982)MATHGoogle Scholar
  19. 19.
    Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004), Available from http://control.ee.ethz.ch/~joloef/yalmip.php
  20. 20.
    Mishra, B.: Computational Real Algebraic Geometry. Handbook of discrete and computational geometry, 537–556 (1997)Google Scholar
  21. 21.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry. Springer, New York (2003)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ioannis A. Fotiou
    • 1
  • A. Giovanni Beccuti
    • 1
  • Georgios Papafotiou
    • 1
  • Manfred Morari
    • 1
  1. 1.Automatic Control LaboratorySwiss Federal Institute of Technology (ETH)ZürichSwitzerland

Personalised recommendations