Feasibility, Stability, Convergence and Markov Chains

  • Basil KouvaritakisEmail author
  • Mark Cannon
Part of the Advanced Textbooks in Control and Signal Processing book series (C&SP)


This chapter considers the closed-loop properties of stochastic MPC strategies based on the predicted costs and probabilistic constraints formulated in Chap.  6. To make the analysis of closed-loop stability and performance possible, it must first be ensured that the MPC law is well-defined at all times and the most natural way to approach this is to ensure that the associated receding horizon optimization problem remains feasible whenever it is initially feasible . We therefore begin by discussing the conditions for recursive feasibility.


  1. 1.
    D. Bernardini, A. Bemporad, Stabilizing model predictive control of stochastic constrained linear systems. IEEE Trans. Autom. Control 57(6), 1468–1480 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G.C. Calafiore, L. Fagiano, Stochastic model predictive control of LPV systems via scenario optimization. Automatica 49(6), 1861–1866 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Cannon, B. Kouvaritakis, P. Couchman, Mean-variance receding horizon control for discrete-time linear stochastic systems, In Proceedings of the 17th IFAC World Congress, Seoul, Korea, (2008), pp. 15321–15326Google Scholar
  4. 4.
    C. Wang, C.J. Ong, M. Sim, Linear systems with chance constraints: Constraint-admissible set and applications in predictive control, Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China, (2009), pp. 2875–2880Google Scholar
  5. 5.
    A. Prékopa, Stochastic Programming, Mathematics and Its Applications, vol. 324 (Kluwer Academic Publishers, Dordrecht, 1995)zbMATHGoogle Scholar
  6. 6.
    W. Feller, Introduction to Probability Theory and Its Applications, vol. 2 (Wiley, New York, 1971)zbMATHGoogle Scholar
  7. 7.
    J.L. Doob, Stochastic Processes (Wiley, New York, 1953)zbMATHGoogle Scholar
  8. 8.
    H.J. Kushner, Introduction to Stochastic Control (Holt, Rinehart and Winston, New York, 1971)zbMATHGoogle Scholar
  9. 9.
    M. Cannon, B. Kouvaritakis, X. Wu, Probabilistic constrained MPC for multiplicative and additive stochastic uncertainty. IEEE Trans. Autom. Control 54(7), 1626–1632 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Evans, M. Cannon, B. Kouvaritakis, Robust MPC tower damping for variable speed wind turbines. IEEE Trans. Control Syst. Technol. 23(1), 290–296 (2014)CrossRefGoogle Scholar
  11. 11.
    S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics (1994)Google Scholar
  12. 12.
    W. Feller, Introduction to Probability Theory and Its Applications, vol. 1 (Wiley, New York, 1968)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of OxfordOxfordUK

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