Feasibility and Robustness

Part of the Communications and Control Engineering book series (CCE)


In this chapter we consider two different but related issues. In the first part we discuss the feasibility problem, i.e., that the nominal NMPC closed loop solutions remain inside a set on which the finite horizon optimal control problems defining the NMPC feedback law are feasible. We formally define the property of recursive feasibility and explain why the assumptions of the previous chapters, i.e., viability of the state constraint set or of the terminal constraint set ensure this property. Then we present two ways to relax the viability assumption on the state constraint set in the case that no terminal constraints are used. After a comparative discussion on NMPC schemes with and without stabilizing terminal constraints, we start with the second part of the chapter in which robustness of the closed loop under additive perturbations and measurement errors is investigated. Here robustness concerns both feasibility and admissibility as well as stability of the closed loop. We provide different assumptions and resulting NMPC schemes for which we can rigorously prove such robustness results and also discuss examples which show that in general robustness may fail to hold.


Optimal Control Problem State Constraint Feasibility Problem Input Constraint Terminal Constraint 


  1. 1.
    Artstein, Z.: Stabilization with relaxed controls. Nonlinear Anal.7(11), 1163–1173 (1983) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Camilli, F., Grüne, L., Wirth, F.: Control Lyapunov functions and Zubov’s method. SIAM J. Control Optim.47, 301–326 (2008) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Curtain, R.F., Zwart, H.: An Introduction to Infinite-dimensional Linear Systems Theory. Texts in Applied Mathematics, vol. 21. Springer, New York (1995) MATHCrossRefGoogle Scholar
  4. 4.
    De Nicolao, G., Magni, L., Scattolini, R.: On the robustness of receding-horizon control with terminal constraints. IEEE Trans. Automat. Control41(3), 451–453 (1996) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Examples when nonlinear model predictive control is nonrobust. Automatica40(10), 1729–1738 (2004) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Model predictive control: for want of a local control Lyapunov function, all is not lost. IEEE Trans. Automat. Control50(5), 546–558 (2005) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Nominally robust model predictive control with state constraints. IEEE Trans. Automat. Control52(10), 1856–1870 (2007) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grüne, L., Nešić, D.: Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models. SIAM J. Control Optim.42, 98–122 (2003) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kellett, C.M., Shim, H., Teel, A.R.: Further results on robustness of (possibly discontinuous) sample and hold feedback. IEEE Trans. Automat. Control49(7), 1081–1089 (2004) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Langson, W., Chryssochoos, I., Raković, S.V., Mayne, D.Q.: Robust model predictive control using tubes. Automatica40(1), 125–133 (2004) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lazar, M., Heemels, W.P.M.H.: Predictive control of hybrid systems: input-to-state stability results for sub-optimal solutions. Automatica45(1), 180–185 (2009) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Limón, D., Alamo, T., Camacho, E.: Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties. In: Proceedings of the 41st IEEE Conference on Decision and Control – CDC 2002, Las Vegas, Nevada, pp. 4619–4624 (2002) Google Scholar
  13. 13.
    Limón, D., Alamo, T., Raimondo, D.M., Muñoz de la Peña, D., Bravo, J.M., Ferramosca, A., Camacho, E.F.: Input-to-state stability: a unifying framework for robust model predictive control. In: Magni, L., Raimondo, D.M., Allgöwer, F. (eds.) Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol. 384, pp. 1–26. Springer, Berlin (2009) CrossRefGoogle Scholar
  14. 14.
    Magni, L., Scattolini, R.: Robustness and robust design of MPC for nonlinear discrete-time systems. In: Findeisen, R., Allgöwer, F., Biegler, L.T. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol. 358, pp. 239–254. Springer, Berlin (2007) CrossRefGoogle Scholar
  15. 15.
    Michalska, H., Mayne, D.Q.: Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Automat. Control38(11), 1623–1633 (1993) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Nešić, D., Teel, A.R., Kokotović, P.V.: Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. Systems Control Lett.38(4–5), 259–270 (1999) MathSciNetMATHGoogle Scholar
  17. 17.
    Primbs, J.A., Nevistić, V.: Feasibility and stability of constrained finite receding horizon control. Automatica36(7), 965–971 (2000) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Quincampoix, M.: Differential inclusions and target problems. SIAM J. Control Optim.30(2), 324–335 (1992) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Raimondo, D.M., Limón, D., Lazar, M., Magni, L., Camacho, E.F.: Min–max model predictive control of nonlinear systems: a unifying overview on stability. Eur. J. Control15(1), 5–21 (2009) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison (2009) Google Scholar
  21. 21.
    Shim, D.H., Jin Kim, H., Sastry, S.: Decentralized nonlinear model predictive control of multiple flying robots. In: Proceedings of the 42nd IEEE Conference on Decision and Control – CDC 2003, Maui, Hawaii, USA, pp. 3621–3626 (2003) Google Scholar
  22. 22.
    Soner, H.M.: Optimal control with state-space constraint. I. SIAM J. Control Optim.24(3), 552–561 (1986) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim.24(3), 1110–1122 (1986) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Wills, A.G., Heath W.P.: Barrier function based model predictive control. Automatica40(8), 1415–1422 (2004) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Yu, S., Böhm, C., Chen, H., Allgöwer, F.: Robust model predictive control with disturbance invariant sets. In: Proceedings of the American Control Conference – ACC 2010, Baltimore, Maryland, USA, pp. 6262–6267 (2010) Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

There are no affiliations available

Personalised recommendations