Ultimate Bounds and Robust Invariant Sets for Linear Systems with State-Dependent Disturbances

  • Sorin OlaruEmail author
  • Vasso Reppa
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 464)


The objective of this chapter is to present a methodology for computing robust positively invariant sets for linear, discrete time-invariant systems that are affected by additive disturbances, with the particularity that these disturbances are subject to state-dependent bounds. The proposed methodology requires less restrictive assumptions compared to similar established techniques, while it provides the framework for determining the state-dependent (parameterized) ultimate bounds for several classes of disturbances. The added value of the proposed approach is illustrated by an optimization-based problem for detecting the mode of functioning of a switching system.


Invariant sets Ultimate bounds State dependent disturbances 



The results presented in this chapter were initiated in the framework of the French-Italian collaborative research project Galileo 2014. The authors would like to thank Prof. F. Blanchini, Prof. D. Casagrane, Prof. S. Miani, and G. Giordano for the fruitful discussion on the topic. This work was supported by the People Programme (Marie Curie Actions) of the European FP7 (2007–2013) under REA grant agreement n\(^{\circ }\) 626891 (FUTuRISM).


  1. 1.
    J.-P. Aubin, Viability Theory (Springer Science & Business Media, Heidelberg, 2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    J.-P. Aubin, H. Frankowska, Set-Valued Analysis. (Springer, Heidelberg, 2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    B.R. Barmish, G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions. IEEE Trans. Autom. Control 27(1), 153–158 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B.R. Barmish, J. Sankaran, The propagation of parametric uncertainty via polytopes. IEEE Trans. Autom. Control 24(2), 346–349 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D.P. Bertsekas, I.B. Rhodes, Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Autom. Control 16(2), 117–128 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Bitsoris, Positively invariant polyhedral sets of discrete-time linear systems. Int. J. Control 47(6), 1713–1726 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced lyapunov functions, in Proceedings of the 30th IEEE Conference on Decision and Control (1991), pp. 1755–1760Google Scholar
  8. 8.
    F. Blanchini, S. Miani, Set-Theoretic Methods in Control (Springer, Heidelberg, 2007)zbMATHGoogle Scholar
  9. 9.
    E. De Santis, Invariant sets: a generalization to constrained systems with state dependent disturbances, in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 1 (1998), pp. 622–623Google Scholar
  10. 10.
    E. De Santis, On positively invariant sets for discrete-time linear systems with disturbance: an application of maximal disturbance sets. IEEE Trans. Autom. Control 39(1), 245–249 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    X. Feng, V. Puig, C. Ocampo-Martinez, S. Olaru, F. Stoican, Set-theoretic methods in robust detection and isolation of sensor faults. Int. J. Syst. Sci. 46(13), 2317–2334 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J.D. Glover, F.C. Schweppe, Control of linear dynamic systems with set constrained disturbances. IEEE Trans. Autom. Control 16(5), 411–423 (1971)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.C. Kerrigan, Robust constraint satisfaction: invariant sets and predictive control, Ph.D. thesis. University of Cambridge, 2001Google Scholar
  14. 14.
    E. Kofman, H. Haimovich, M.M. Seron, A systematic method to obtain ultimate bounds for perturbed systems. Int. J. Control 80(2), 167–178 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    I. Kolmanovsky, E.G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems. Math. Probl. Eng. 4(4), 317–367 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    A.V. Kuntsevich, V.M. Kuntsevich, Invariant sets for families of linear and nonlinear discrete systems with bounded disturbances. Autom. Remote Control 73(1), 83–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V.M. Kuntsevich, B.N. Pshenichnyi, Minimal invariant sets of dynamic systems with bounded disturbances. Cybern. Syst. Anal. 32(1), 58–64 (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    G. Leitmann, Guaranteed asymptotic stability for a class of uncertain linear dynamical systems. J. Optim. Theory Appl. 27(1), 99–106 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. Leitmann, On the efficacy of nonlinear control in uncertain linear systems. J. Dyn. Syst. Meas. Control 103(2), 95–102 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. McLane, Optimal stochastic control of linear systems with state- and control-dependent disturbances. IEEE Trans. Autom. Control 16(6), 793–798 (1971)CrossRefGoogle Scholar
  21. 21.
    S. Olaru, J.A. De Doná, M.M. Seron, F. Stoican, Positive invariant sets for fault tolerant multisensor control schemes. Int. J. Control 83(12), 2622–2640 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S.V. Raković, E.C. Kerrigan, D.Q. Mayne, Reachability computations for constrained discrete-time systems with state-and input-dependent disturbances, in Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 4 (2003), pp. 3905–3910Google Scholar
  23. 23.
    S.V. Raković, E.C. Kerrigan, K.I. Kouramas, D.Q. Mayne, Invariant approximations of the minimal robust positively invariant set. IEEE Trans. Autom. Control 50(3), 406–410 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. Reppa, S. Olaru, M.M. Polycarpou, Structural detectability analysis of a distributed sensor fault diagnosis scheme for a class of nonlinear systems, in Proceedings of the 9th IFAC SAFEPROCESS (Paris, France, 2015), pp. 1485–1490Google Scholar
  25. 25.
    R.M. Schaich, M. Cannon, Robust positively invariant sets for state dependent and scaled disturbances, in Proceedings of the 54th IEEE Conference on Decision and Control (2015)Google Scholar
  26. 26.
    M.M. Seron, X.W. Zhuo, J.A. De Doná, J.J. Martínez, Multisensor switching control strategy with fault tolerance guarantees. Automatica 44(1), 88–97 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M.M. Seron, J.A. De Doná, Robust fault estimation and compensation for LPV systems under actuator and sensor faults. Automatica 52, 294–301 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    F. Stoican, S. Olaru, Set-theoretic Fault-tolerant Control in Multisensor Systems Wiley-ISTE, (2013)Google Scholar
  29. 29.
    J.L. Willems, J.C. Willems, Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica 12(3), 277–283 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Laboratory of Signals and Systems (UMR CNRS 8506)CentraleSupélec-CNRS-Univ. Paris-Sud, Université Paris-SaclayGif-sur-YvetteFrance

Personalised recommendations