Advertisement

Analysis via the Regions of Saturation Model

  • Sophie TarbouriechEmail author
  • Germain Garcia
  • João Manoel Gomes da SilvaJr.
  • Isabelle Queinnec
Chapter
  • 1.5k Downloads

Abstract

This chapter deals with the problem of stability analysis of linear control systems with input saturations using the regions of saturation model. It is shown that considering such a representation for the closed-loop system it is possible to derive necessary and sufficient conditions to ensure that a polyhedral region in the state space is a region of asymptotic stability. These conditions can be tested by linear programming algorithms. The determination of ellipsoidal regions of stability is also presented. In this case only sufficient conditions can be derived. From these conditions, LMI-based optimization problems are proposed to generate regions of asymptotic stability.

References

  1. 12.
    Aubin, J.P.: A survey of viability theory. SIAM J. Control Optim. 28(4), 749–788 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 31.
    Bitsoris, G.: Existence of positively invariant polyhedral sets for continuous-time linear systems. Control Theory Adv. Technol. 7(3), 407–427 (1991) MathSciNetGoogle Scholar
  3. 35.
    Blanchini, F.: Nonquadratic Lyapunov functions for robust control. Automatica 31(3), 451–461 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 38.
    Blanchini, F., Miani, S.: Constrained stabilization of continuous-time linear systems. Syst. Control Lett. 28, 95–102 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 39.
    Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Birkhäuser, Basel (2008) zbMATHGoogle Scholar
  6. 44.
    Boulingand, G.: Introduction À la Géométrie Infinitésimale. Gauthiers-Villars, Paris (1932) Google Scholar
  7. 45.
    Boyd, S., El Ghaoui, L., Féron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1994) zbMATHGoogle Scholar
  8. 58.
    Castelan, E.B., Hennet, J.C.: Eigenstructure assignment for state constrained linear continuous time systems. Automatica 28(3), 605–611 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 59.
    Castelan, E.B., Hennet, J.C.: On invariant polyhedra of continuous-time linear systems. IEEE Trans. Autom. Control 38(11), 1680–1685 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 61.
    Castelan, E.B., Gomes da Silva Jr., J.M., Cury, J.E.R.: A reduced order framework applied to linear systems with constrained controls. IEEE Trans. Autom. Control 41(2), 249–255 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 71.
    Clarke, F.H.: Optimization and Non Smooth Analysis. Wiley, New York (1983) Google Scholar
  12. 96.
    Fong, I.-K., Hsu, C.-C.: State feedback stabilization of single input systems through actuators with saturation and dead zone characteristics. In: Conference on Decision and Control, Sydney, Australia (2000) Google Scholar
  13. 132.
    Gomes da Silva Jr., J.M., Tarbouriech, S.: Polyhedral regions of local asymptotic stability for discrete-time linear systems with saturating controls. In: Conference on Decision and Control, San Diego, USA, pp. 925–930 (1997) Google Scholar
  14. 136.
    Gomes da Silva Jr., J.M., Tarbouriech, S.: Polyhedral regions of local stability for linear discrete-time systems with saturating controls. IEEE Trans. Autom. Control 44(11), 2081–2085 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 169.
    Hennet, J.C., Castelan, E.B.: Constrained control of unstable multivariable linear systems. In: European Control Conference, Groningen,The Netherlands, pp. 2039–2043 (1993) Google Scholar
  16. 202.
    Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43(4), 555–559 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 217.
    Kiendl, H., Adamy, J., Stelzner, P.: Vector norms as Lyapunov functions for linear systems. IEEE Trans. Autom. Control 37(6), 839–842 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 218.
    Kim, J., Bien, Z.: Robust stability of uncertain linear systems with saturating actuators. IEEE Trans. Autom. Control 39(1), 202–207 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 232.
    Laloy, M., Rouche, N., Habets, P.: Stability Theory by Lyapunov’s Direct Method. Springer, New York (1977) Google Scholar
  20. 238.
    Limon, D., Gomes da Silva Jr., J.M., Alamo, T., Camacho, E.F.: Improved MPC design based on saturating control laws. Eur. J. Control 11, 112–122 (2005) MathSciNetCrossRefGoogle Scholar
  21. 252.
    Luenberger, D.G.: Linear and Nonlinear Programming. Addison-Wesley, Reading (1984) zbMATHGoogle Scholar
  22. 258.
    Milani, B.E.A.: Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls. Automatica 38, 2177–2184 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 270.
    Nagumo, M.: Uber die lage der integralkurven gewöhnlicher differential-gleichungen. Proc. Phys. Math. Soc. Jpn. 24(3), 272–559 (1942) Google Scholar
  24. 340.
    Sznaier, M.: A set induced norm approach to the robust control of constrained systems. SIAM J. Control Optim. 31(3), 733–746 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 383.
    Vassilaki, M., Bitsoris, G.: Constrained regulation of linear continuous-time dynamical systems. Syst. Control Lett. 13, 247–252 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 403.
    Yang, Y., Sussmann, H.J., Sontag, E.: Stabilization of linear systems with bounded controls. In: IFAC Nonlinear Control Systems Design Symposium (NOLCOS), Bordeaux, France, pp. 15–20 (1992) Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Sophie Tarbouriech
    • 1
    Email author
  • Germain Garcia
    • 2
  • João Manoel Gomes da SilvaJr.
    • 3
  • Isabelle Queinnec
    • 1
  1. 1.Laboratoire Analyse et Architecture des Systèmes (LAAS)CNRSToulouse CX 4France
  2. 2.Laboratoire Analyse et Architecture des Systèmes (LAAS)CNRSToulouse CX 4France
  3. 3.Departamento de Engenharia ElétricaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil

Personalised recommendations