Advertisement

The Small-Gain Theorem for Linear Systems and Its Applications to Robust Stability

  • Alberto IsidoriEmail author
Chapter
Part of the Advanced Textbooks in Control and Signal Processing book series (C&SP)

Abstract

In a system consisting of the interconnection of several component subsystems, some of which could be only poorly modeled, stability analysis and feedback design might not be easy tasks. Thus, methods allowing to understand the influence of interconnections on stability and asymptotic behavior are important. The methods in question are based on the use of a concept of gain, which can take alternative forms and can be evaluated by means of a number of alternative methods. This chapter describes the various alternative forms of such concept of gain, and shows why this is useful in the analysis of stability of interconnected systems. A major consequence is the development of a systematic method for stabilization in the presence of (general class of) model uncertainties.

References

  1. 1.
    B.D.O. Anderson, J.B. Moore, Optimal Control: Linear Quadratic Methods (Prentice Hall, Englewoof Cliffs, 1990)zbMATHGoogle Scholar
  2. 2.
    T. Basar, P. Bernhard, \(H_\infty \) Optimal Control and Related Minimaz Design Problems (Birkäuser, Boston, 1990)Google Scholar
  3. 3.
    S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Studies in Applied and Numerical Mathematics (SIAM, Philadelphia, 1994)Google Scholar
  4. 4.
    C.A. Desoer, M. Vidyasagar, Feedback Systems: Input-Output Properties (Academic Press, New York, 1975)zbMATHGoogle Scholar
  5. 5.
    J. Doyle, B. Francis, A. Tannenbaum, Feedback Control Theory (MacMillan, New York, 1992)Google Scholar
  6. 6.
    B. Francis, A Course in \(H_\infty \) Control Theory (Springer, Berlin, 1987)Google Scholar
  7. 7.
    P. Gahinet, P. Apkarian, A linear matrix inequality approach to \(H_\infty \) control. Int. J. Robust Nonlinear Control 4, 421–448 (1994)Google Scholar
  8. 8.
    P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox: For Use with MATLAB (The Mathworks Inc., Massachusetts, 1995)Google Scholar
  9. 9.
    C. Scherer, S. Weiland, Linear matrix inequalities in control, in The Control Handbook: Control Systems Advanced Methods, Chap. 24, ed. by W.S. Levine (CRC Press, Boca Raton, 2011)Google Scholar
  10. 10.
    C.W. Scherer, P. Gahinet, M. Chilali, Multiobjective output-feedback control via LMI optimization. IEEE Trans. Autom. Control 42, 896–911 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A.J. Van der Schaft, \({L}_2\) Gain and Passivity Techniques in Nonlinear Control (Springer, London, 1996)Google Scholar
  12. 12.
    J.G. VanAntwerp, R.D. Braatz, A tutorial on linear and bilinear matrix inequalities. J. Process Control 10, 363–385 (2000)CrossRefGoogle Scholar
  13. 13.
    J. Willems, Dissipative dynamical systems. Arch. Ration. Mech. Anal. 45, 321–393 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Informatica, Automatica e GestionaleUniversità degli Studi di Roma “La Sapienza”RomeItaly

Personalised recommendations