• Ali Saberi
  • Anton A. Stoorvogel
  • Peddapullaiah Sannuti
Part of the Systems & Control: Foundations & Applications book series (SCFA)


In this chapter, we bring together the notations and acronyms used in this book as well as various definitions and facts related to matrices, linear spaces, linear operators, norms of deterministic as well as stochastic signals, norms of linear time- or shift-invariant systems, saturation functions, internal (Lyapunov) stability, and external stability.


Equilibrium Point Lyapunov Stability Output Stability Jordan Block Internal Stability 


  1. 14.
    S. Boyd, V. Balakrishnan, and P. Kabamba, “A bisection method for computing the H -norm of a transfer matrix and related problems”, Math. Contr. Sign. & Syst., 2(3), 1989, pp. 207–219.Google Scholar
  2. 15.
    S.P. Boyd and C.H. Barratt, Linear controller design: limits of performance, Information and system sciences, Prentice Hall, Englewood Cliffs, NJ, 1991.MATHGoogle Scholar
  3. 25.
    J. Choi, “Connections between local stability in Lyapunov and input / output senses”, IEEE Trans. Aut. Contr., 40(12), 1995, pp. 2139 –2143.MATHCrossRefGoogle Scholar
  4. 40.
    F.R. Gantmacher, The theory of matrices, Chelsea, New York, 1959.MATHGoogle Scholar
  5. 44.
    W. Hahn, Stability of motion, vol. 138 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer Verlag, Berlin, 1967.Google Scholar
  6. 46.
    D. Hill and P. Moylan, “Connections between finite-gain and asymptotic stability”, IEEE Trans. Aut. Contr., 25(5), 1980, pp. 931–936.MathSciNetMATHCrossRefGoogle Scholar
  7. 58.
    Hassan K. Khalil, Nonlinear systems, Prentice Hall, Upper Saddle River, 2nd Ed., 1996.Google Scholar
  8. 89.
    W. Liu, Y. Chitour, and E.D. Sontag, “On finite-gain stabilizability of linear systems subject to input saturation”, SIAM J. Contr. & Opt., 34(4), 1996, pp. 1190–1219.MathSciNetMATHCrossRefGoogle Scholar
  9. 94.
    J.L. Massera, “On Liapounoff’s Conditions of Stability”, The Annals of Mathematics, Second Series, 50(3), 1949, pp. 705–721.MathSciNetMATHCrossRefGoogle Scholar
  10. 126.
    A. Saberi, P. Hou, and A.A. Stoorvogel, “On simultaneous global external and global internal stabilization of critically unstable linear systems with saturating actuators”, IEEE Trans. Aut. Contr., 45(6), 2000, pp. 1042–1052.MathSciNetMATHCrossRefGoogle Scholar
  11. 149.
    Guoyong Shi, Control of linear systems with constraints – internal and external stabilization and output regulation, PhD thesis, School of Electrical Engineering and Computer Science, 2002.Google Scholar
  12. 151.
    G. Shi, A. Saberi, and A.A. Stoorvogel, “On the L p ( p) stabilization of open-loop neutrally stable linear plants with input subject to amplitude saturation”, Int. J. Robust & Nonlinear Control, 13(8), 2003, pp. 735–754.Google Scholar
  13. 156.
    E.D. Sontag and Y. Wang, “On characterization of the input-to-state stability”, Syst. & Contr. Letters, 24(5), 1995, pp. 351–359.MathSciNetMATHCrossRefGoogle Scholar
  14. 169.
    A.A. Stoorvogel, G. Shi, and A. Saberi, “External stability of a double integrator with saturated linear control laws”, Dynamics of Continuous Discrete and Impulsive Systems, Series B: Applications & Algorithms, 11(4–5), 2004, pp. 429–451.MathSciNetMATHGoogle Scholar
  15. 182.
    Andrew Teel, “Asymptotic convergence from p stability”, IEEE Trans. Aut. Contr., 44(11), 1999, pp. 2169–2170.Google Scholar
  16. 189.
    M. Vidyasagar, Nonlinear systems analysis, Prentice Hall, London, 2nd Ed., 1992.MATHGoogle Scholar
  17. 190.
    M. Vidyasagar and A. Vannelli, “New relationships between input-output and Lyapunov stability”, IEEE Trans. Aut. Contr., 27(2), 1982, pp. 481–483.MathSciNetMATHCrossRefGoogle Scholar
  18. 191.
    X. Wang, H.F. Grip, A. Saberi, A.A. Stoorvogel, and I. Saberi, “Remarks on the relationship between p stability and internal stability of nonlinear systems”, in American Control Conference, San Francisco, CA, 2011, pp. 1969–1970.Google Scholar
  19. 204.
    J.C. Willems, The analysis of feedback systems, MIT Press, 1971.Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Ali Saberi
    • 1
  • Anton A. Stoorvogel
    • 2
  • Peddapullaiah Sannuti
    • 3
  1. 1.School of Electrical Engineering and Computer ScienceWashington State UniversityPullmanUSA
  2. 2.Department of Electrical Engineering, Mathematics, and Computer ScienceUniversity of TwenteEnschedeThe Netherlands
  3. 3.Department of Electrical and Computer EngineeringRutgers UniversityPiscatawayUSA

Personalised recommendations