Advertisement

Discrete Event Dynamic Systems

, Volume 17, Issue 3, pp 329–354 | Cite as

Stable Model Predictive Control for Constrained Max-Plus-Linear Systems

  • Ion Necoara
  • Bart De Schutter
  • Ton J. J. van den Boom
  • Hans Hellendoorn
Article

Abstract

Discrete-event systems with synchronization but no concurrency can be described by models that are “linear” in the max-plus algebra, and they are called max-plus-linear (MPL) systems. Examples of MPL systems often arise in the context of manufacturing systems, telecommunication networks, railway networks, parallel computing, etc. In this paper we provide a solution to a finite-horizon model predictive control (MPC) problem for MPL systems where it is required that the closed-loop input and state sequence satisfy a given set of linear inequality constraints. Although the controlled system is nonlinear, by employing results from max-plus theory, we give sufficient conditions such that the optimization problem that is performed at each step is a linear program and such that the MPC controller guarantees a priori stability and satisfaction of the constraints. We also show how one can use the results in this paper to compute a time-optimal controller for linearly constrained MPL systems.

Keywords

Discrete-event systems Max-plus-linear systems Input-state constraints Model predictive control Stability Positively invariant sets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baccelli F, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity. Wiley, New YorkzbMATHGoogle Scholar
  2. Cofer DD, Garg VK (1996) Supervisory control of real-time discrete-event systems using lattice theory. IEEE Trans Automat Contr 41(2):199–209, FebruaryzbMATHCrossRefGoogle Scholar
  3. Cottenceau B, Hardouin L, Boimond J, Ferrier J (2001) Model reference control for timed event graphs in dioid. Automatica 37(8):1451–1458zbMATHCrossRefGoogle Scholar
  4. Cuninghame-Green RA (1979) Minimax algebra, Lecture notes in economics and mathematical systems, vol 16. Springer, Berlin, GermanyGoogle Scholar
  5. De Schutter B (1996) Max-algebraic system theory for discrete event systems. PhD thesis, Faculty of Applied Sciences, K.U.Leuven, Leuven, BelgiumGoogle Scholar
  6. De Schutter B, van den Boom T (2001) Model predictive control for max-plus-linear discrete event systems. Automatica 37(7):1049–1056, JulyzbMATHCrossRefGoogle Scholar
  7. Elsner L, Johnson CR, Dias da Silva JA (1988) The Perron root of a weighted geometric mean of nonnegative matrices. Linear Multilinear Algebra 24(1):1–13zbMATHCrossRefGoogle Scholar
  8. Gaubert S (1996) On the Burnside problem for semigroups of matrices in the (max,+) algebra. Semigroup Forum 52:271–292zbMATHCrossRefGoogle Scholar
  9. Gazarik MJ, Kamen BEW (1999) Reachability and observability of linear systems over max-plus. Kybernetika 35(1):2–12, JanuaryGoogle Scholar
  10. Gilbert EG, Tan KT (1991) Linear systems with state and control constraints: the theory and applications of maximal output admissible sets. IEEE Trans Automat Contr 36(9):1008–1020, SeptemberzbMATHCrossRefGoogle Scholar
  11. Heemels WPMH, De Schutter B, Bemporad A (2001) Equivalence of hybrid dynamical models. Automatica 37(7):1085–1091, JulyzbMATHCrossRefGoogle Scholar
  12. Heidergott B, Olsder GJ, Woude J (2005) Max plus at work. Princeton University Press, PrincetonGoogle Scholar
  13. Kumar R, Garg VK (1994) Extremal solutions of inequations over lattices with applications to supervisory control. In: Proceedings of the 33rd IEEE conference on decision and control. Orlando, Florida, pp 3636–3641, DecemberGoogle Scholar
  14. Libeaut L, Loiseau JJ (1995) Admissible initial conditions and control of timed event graphs. In: Proceedings of the 34th IEEE conference on decision and control. New Orleans, Louisiana, pp 2011–2016, DecemberGoogle Scholar
  15. Maciejowski JM (2002) Predictive control with constraints. Prentice Hall, Harlow, EnglandGoogle Scholar
  16. Maia CA, Hardouin L, Santos-Mendes R, Cottenceau B (2003) Optimal closed-loop control of timed event graphs in dioids. IEEE Trans Automat Contr 48(12):2284–2287, DecemberCrossRefGoogle Scholar
  17. Mairesse J (1995) A graphical approach to the spectral theory in the (max,+) algebra. IEEE Trans Automat Contr 40(10):1783–1789, OctoberzbMATHCrossRefGoogle Scholar
  18. Mayne DQ, Rawlings JB, Rao CV, Scokaert POM (2000) Constrained model predictive control: stability and optimality. Automatica 36(7):789–814, JunezbMATHCrossRefGoogle Scholar
  19. Menguy E, Boimond JL, Hardouin L (1997) A feedback control in max-algebra. In: Proceedings of the european control conference (ECC’97), Brussels, Belgium, paper 487, JulyGoogle Scholar
  20. Menguy E, Boimond JL, Hardouin L, Ferrier JL (2000) A first step towards adaptive control for linear systems in max algebra. Discret Event Dyn Syst Theor Appl 10(4):347–367zbMATHCrossRefGoogle Scholar
  21. Necoara I (2006) Model predictive control for max-plus-linear and piecewise affine systems. PhD thesis, Delft Center for Systems and Control, Delft University of Technology, The Netherlands, OctoberGoogle Scholar
  22. Necoara I, van den Boom TJJ, De Schutter B, Hellendoorn J (2006) Stabilization of MPL systems using model predictive control: the unconstrained case. Technical Report 06-006, Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands, Revised version. Provisionally accepted for Automatica, SeptemberGoogle Scholar
  23. Passino KM, Burgess KL (1998) Stability analysis of discrete event systems. Wiley, New YorkGoogle Scholar
  24. La Salle JP (1976) The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia, PAGoogle Scholar
  25. van den Boom TJJ, De Schutter B, Necoara I (2005) On MPC for max-plus-linear systems: analytic solution and stability. In: Proceedings of the 44th IEEE conference on decision and control, and the european control conference, (CDC-ECC’05). Seville, Spain, pp 7816–7821, DecemberGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Ion Necoara
    • 1
    • 2
  • Bart De Schutter
    • 1
  • Ton J. J. van den Boom
    • 1
  • Hans Hellendoorn
    • 1
  1. 1.Delft Center for Systems and ControlDelft University of TechnologyDelftNetherlands
  2. 2.Department of Electrical Engineering (ESAT)Katholieke Universiteit LeuvenLeuven-HeverleeBelgium

Personalised recommendations