Further Results on “Robust MPC Using Linear Matrix Inequalities”

  • M. Lazar
  • W. P. M. H. Heemels
  • D. Muñoz de la Peña
  • T. Alamo
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 384)


This paper presents a novelmethod for designing the terminal cost and the auxiliary control law (ACL) for robust MPC of uncertain linear systems, such that ISS is a priori guaranteed for the closed-loop system. The method is based on the solution of a set of LMIs. An explicit relation is established between the proposed method and \(\mathcal{H}_\infty\) control design. This relation shows that the LMI-based optimal solution of the \(\mathcal{H}_\infty\) synthesis problem solves the terminal cost and ACL problem in inf-sup MPC, for a particular choice of the stage cost. This result, which was somehow missing in the MPC literature, is of general interest as it connects well known linear control problems to robust MPC design.


robust model predictive control (MPC) linear matrix inequalities (LMIs) \(\mathcal{H}_\infty\) control input-to-state stability (ISS) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: Stability and optimality. Automatica 36(6), 789–814 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Scokaert, P.O.M., Rawlings, J.B.: Constrained linear quadratic regulation. IEEE Transactions on Automatic Control 43(8), 1163–1169 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kothare, M.V., Balakrishnan, V., Morari, M.: Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10), 1361–1379 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Magni, L., De Nicolao, G., Scattolini, R., Allgöwer, F.: Robust MPC for nonlinear discrete-time systems. International Journal of Robust and Nonlinear Control 13, 229–246 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Jiang, Z.-P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lazar, M., Heemels, W.P.M.H.: Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions. Automatica 45(1), 180–185 (2009)zbMATHCrossRefGoogle Scholar
  7. 7.
    Magni, L., Raimondo, D.M., Scattolini, R.: Regional input-to-state stability for nonlinear model predictive control. IEEE Transactions on Automatic Control 51(9), 1548–1553 (1998)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Lazar, M., Muñoz de la Peña, D., Heemels, W.P.M.H., Alamo, T.: On input-to-state stability of min-max MPC. Systems & Control Letters 57, 39–48 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kolmanovsky, I., Gilbert, E.G.: Theory and computation of disturbance invariant sets for discrete-time linear systems. Mathematical Problems in Engineering 4, 317–367 (1998)zbMATHCrossRefGoogle Scholar
  10. 10.
    Alessio, A., Lazar, M., Heemels, W.P.M.H., Bemporad, A.: Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones. Automatica 43(12), 2096–2103 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Lazar, M., Heemels, W.P.M.H., Weiland, S., Bemporad, A.: Stabilizing model predictive control of hybrid systems. IEEE Transactions on Automatic Control 51(11), 1813–1818 (2006)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Alamo, T., Muñoz de la Peña, D., Limon, D., Camacho, E.F.: Constrained minmax predictive control: Modifications of the objective function leading to polynomial complexity. IEEE Transactions on Automatic Control 50(5), 710–714 (2005)CrossRefGoogle Scholar
  13. 13.
    Kaminer, I., Khargonekar, P.P., Rotea, M.A.: Mixed \({\mathcal H}_2{\mathcal H}_\infty\) control for discrete time systems via convex optimization. Automatica 29, 57–70 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Chen, H., Scherer, C.W.: Moving horizon \({\mathcal H}_\infty\) with performance adaptation for constrained linear systems. Automatica 42, 1033–1040 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • M. Lazar
    • 1
  • W. P. M. H. Heemels
    • 1
  • D. Muñoz de la Peña
    • 2
  • T. Alamo
    • 2
  1. 1.Eindhoven Univ. of TechnologyEindhovenThe Netherlands
  2. 2.Dept. de Ingeniería de Sistemas y AutomáticaUniv. of SevilleSevilleSpain

Personalised recommendations