Uncertain Systems and Robustness

  • Rosario Toscano
Part of the Advances in Industrial Control book series (AIC)


The ability of dealing with uncertainty (uncertain system) is an essential part when designing a robust feedback controller. The objective is indeed to determine the controller parameters ensuring acceptable performance of the closed-loop system despite the unknown disturbances affecting the system as well as the uncertainties about the plant dynamics. To this end, it is necessary to be able to take into account the model uncertainties during the design phase of the controller. In this chapter, we briefly describe some basic concepts regarding uncertain systems and robustness analysis. The last part of this chapter is devoted to structured robust control, for which a specific stochastic algorithm is developed.


Uncertain System Structure Controller Structure Uncertainty Dynamic Uncertainty Small Gain Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 17.
    Bhattacharyya SP, Chapellat H, Keel LH (1995) Robust control. The parametric approach. Prentice-Hall, Upper Saddle River MATHGoogle Scholar
  2. 21.
    Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Society for Industrial and Applied Mathematics, Philadephia MATHCrossRefGoogle Scholar
  3. 26.
    Braatz RP, Young PM, Doyle JC, Morari M (1994) Computational complexity of μ calculation. IEEE Trans Autom Control AC-39:1000–1002 MathSciNetCrossRefGoogle Scholar
  4. 35.
    Chesi G (2008) On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems. Automatica 44(11):2973–2976 MathSciNetMATHCrossRefGoogle Scholar
  5. 36.
    Chesi G (2011) LMI conditions for time-varying uncertain systems can be non-conservative. Automatica 47(3):621–624 MathSciNetMATHCrossRefGoogle Scholar
  6. 44.
    Desoer CA, Vidyasagar M (1975) Feedback systems: input–output properties. Academic Press, New York MATHGoogle Scholar
  7. 45.
    Doyle J, Packard A, Zhou K (1991) Review of LFTs, LMIs, and μ. In: Proceedings of the 30th conference on decision and control, Brighton, England, pp 1227–1232 CrossRefGoogle Scholar
  8. 47.
    Doyle JC (1982) Analysis of feedback systems with structured uncertainty. IEE Proc, Part D, Control Theory Appl 129:242–250 MathSciNetCrossRefGoogle Scholar
  9. 74.
    Kharitonov VL (1978) Asymptotic stability of an equilibrium position of a family of systems of differential equations. Differ Equ 14(11):2086–2088. Translated from Russian MathSciNetMATHGoogle Scholar
  10. 96.
    Megretski A (1993) On the gap between structured singular values and their upper bounds. In: Proceedings of the IEEE conference on decision and control, pp 3461–3462 Google Scholar
  11. 118.
    Safonov MG (1982) Stability margins for diagonally perturbed multivariable feedback systems. IEE Proc, Part D, Control Theory Appl 129:251–256 MathSciNetCrossRefGoogle Scholar
  12. 139.
    Treil S. The gap between complex structured singular value μ and its upper bound is infinite.
  13. 150.
    Zames G (1966) On the input-output stability on nonlinear time-varying feedback systems, part I: conditions derived using concepts of loop gain, conicity, and positivity. IEEE Trans Autom Control AC-11:228–238 CrossRefGoogle Scholar
  14. 154.
    Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, New York MATHGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Rosario Toscano
    • 1
  1. 1.Laboratoire de Tribologie et DynamiqueEcole Nat. d’Ingenieurs de Saint-EtienneSaint-EtienneFrance

Personalised recommendations