Skip to main content

2011 | OriginalPaper | Buchkapitel

3. Converse Lyapunov Results

verfasst von : Iasson Karafyllis, Zhong-Ping Jiang

Erschienen in: Stability and Stabilization of Nonlinear Systems

Verlag: Springer London

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Chapter 3 is devoted to answering the following question: do Lyapunov functionals always exist for a robustly globally asymptotically output stable system? The previous Chap. 2 showed that one of the most important ways of proving stability is the derivation of estimates which guarantee appropriate stability properties by means of Lyapunov functionals. The converse Lyapunov results obtained in this chapter show that such Lyapunov functionals always exist.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Angeli, D., Sontag, E.D.: Forward completeness, unbounded observability and their Lyapunov characterizations. Systems and Control Letters 38, 209–217 (1999) MathSciNetMATHCrossRef Angeli, D., Sontag, E.D.: Forward completeness, unbounded observability and their Lyapunov characterizations. Systems and Control Letters 38, 209–217 (1999) MathSciNetMATHCrossRef
2.
Zurück zum Zitat Bacciotti, A., Rosier, L.: Lyapunov stability and Lagrange stability: Inverse theorems for discontinuous systems. Mathematics of Control, Signals and Systems 11, 101–125 (1998) MathSciNetMATHCrossRef Bacciotti, A., Rosier, L.: Lyapunov stability and Lagrange stability: Inverse theorems for discontinuous systems. Mathematics of Control, Signals and Systems 11, 101–125 (1998) MathSciNetMATHCrossRef
3.
Zurück zum Zitat Bacciotti, A., Rosier, L.: On the converse of first Lyapunov theorem: The regularity issue. Systems and Control Letters 41, 265–270 (2000) MathSciNetMATHCrossRef Bacciotti, A., Rosier, L.: On the converse of first Lyapunov theorem: The regularity issue. Systems and Control Letters 41, 265–270 (2000) MathSciNetMATHCrossRef
4.
Zurück zum Zitat Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences, vol. 267. Springer, London (2001) MATH Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences, vol. 267. Springer, London (2001) MATH
5.
Zurück zum Zitat Bernfeld, S.A., Corduneanu, C., Ignatyev, A.O.: On the stability of invariant sets of functional differential equations. Nonlinear Analysis: Theory Methods and Applications 55, 641–656 (2003) MathSciNetMATHCrossRef Bernfeld, S.A., Corduneanu, C., Ignatyev, A.O.: On the stability of invariant sets of functional differential equations. Nonlinear Analysis: Theory Methods and Applications 55, 641–656 (2003) MathSciNetMATHCrossRef
6.
Zurück zum Zitat Burton, T.A.: Uniform asymptotic stability in functional differential equations. Proceedings of the American Mathematical Society 68, 195–199 (1978) MathSciNetMATHCrossRef Burton, T.A.: Uniform asymptotic stability in functional differential equations. Proceedings of the American Mathematical Society 68, 195–199 (1978) MathSciNetMATHCrossRef
7.
Zurück zum Zitat Grüne, L.: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Transactions on Automatic Control 47, 1499–1504 (2002) CrossRef Grüne, L.: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Transactions on Automatic Control 47, 1499–1504 (2002) CrossRef
8.
Zurück zum Zitat Hahn, W.: Stability of Motion. Springer, Berlin (1967) MATH Hahn, W.: Stability of Motion. Springer, Berlin (1967) MATH
9.
Zurück zum Zitat Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATH Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATH
10.
Zurück zum Zitat Ignatyev, A.O.: On the partial equiasymptotic stability in functional differential equations. Journal of Mathematical Analysis and Applications 268, 615–628 (2002) MathSciNetMATHCrossRef Ignatyev, A.O.: On the partial equiasymptotic stability in functional differential equations. Journal of Mathematical Analysis and Applications 268, 615–628 (2002) MathSciNetMATHCrossRef
11.
Zurück zum Zitat Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002) MathSciNetMATHCrossRef Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002) MathSciNetMATHCrossRef
12.
Zurück zum Zitat Karafyllis, I.: Non-uniform in time robust global asymptotic output stability. Systems and Control Letters 54, 181–193 (2005) MathSciNetMATHCrossRef Karafyllis, I.: Non-uniform in time robust global asymptotic output stability. Systems and Control Letters 54, 181–193 (2005) MathSciNetMATHCrossRef
13.
Zurück zum Zitat Karafyllis, I.: Non-uniform in time robust global asymptotic output stability for discrete-time systems. International Journal of Robust and Nonlinear Control 16, 191–214 (2006) MathSciNetMATHCrossRef Karafyllis, I.: Non-uniform in time robust global asymptotic output stability for discrete-time systems. International Journal of Robust and Nonlinear Control 16, 191–214 (2006) MathSciNetMATHCrossRef
14.
Zurück zum Zitat Karafyllis, I.: A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. Journal of Mathematical Analysis and Applications 328, 876–899 (2007) MathSciNetMATHCrossRef Karafyllis, I.: A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. Journal of Mathematical Analysis and Applications 328, 876–899 (2007) MathSciNetMATHCrossRef
15.
Zurück zum Zitat Karafyllis, I., Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM Journal Control and Optimization 42, 936–965 (2003) MathSciNetMATHCrossRef Karafyllis, I., Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM Journal Control and Optimization 42, 936–965 (2003) MathSciNetMATHCrossRef
16.
Zurück zum Zitat Karafyllis, I., Pepe, P., Jiang, Z.-P.: Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. European Journal of Control 14, 516–536 (2008) MathSciNetCrossRef Karafyllis, I., Pepe, P., Jiang, Z.-P.: Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. European Journal of Control 14, 516–536 (2008) MathSciNetCrossRef
17.
Zurück zum Zitat Kellett, C.M., Teel, A.R.: Smooth Lyapunov functions and robustness of stability for difference inclusions. Systems and Control Letters 52, 395–405 (2004) MathSciNetMATHCrossRef Kellett, C.M., Teel, A.R.: Smooth Lyapunov functions and robustness of stability for difference inclusions. Systems and Control Letters 52, 395–405 (2004) MathSciNetMATHCrossRef
18.
Zurück zum Zitat Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice-Hall, New York (1996) Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice-Hall, New York (1996)
19.
Zurück zum Zitat Kharitonov, V.L.: Lyapunov–Krasovskii functionals for scalar time delay equations. Systems and Control Letters 51, 133–149 (2004) MathSciNetMATHCrossRef Kharitonov, V.L.: Lyapunov–Krasovskii functionals for scalar time delay equations. Systems and Control Letters 51, 133–149 (2004) MathSciNetMATHCrossRef
20.
Zurück zum Zitat Kharitonov, V.L., Melchor-Aguilar, D.: On delay dependent stability conditions for time-varying systems. Systems and Control Letters 46, 173–180 (2002) MathSciNetMATHCrossRef Kharitonov, V.L., Melchor-Aguilar, D.: On delay dependent stability conditions for time-varying systems. Systems and Control Letters 46, 173–180 (2002) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963) MATH Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963) MATH
22.
Zurück zum Zitat Kurzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. Amer. Math. Soc. Trans. 2(24), 19–77 (1956) Kurzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. Amer. Math. Soc. Trans. 2(24), 19–77 (1956)
23.
Zurück zum Zitat Lin, Y., Sontag, E.D., Wang, Y.: A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization 34, 124–160 (1996) MathSciNetMATHCrossRef Lin, Y., Sontag, E.D., Wang, Y.: A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization 34, 124–160 (1996) MathSciNetMATHCrossRef
26.
Zurück zum Zitat Sontag, E.D., Wang, Y.: Lyapunov characterizations of input-to-output stability. SIAM Journal on Control and Optimization 39, 226–249 (2001) CrossRef Sontag, E.D., Wang, Y.: Lyapunov characterizations of input-to-output stability. SIAM Journal on Control and Optimization 39, 226–249 (2001) CrossRef
27.
Zurück zum Zitat Teel, A.R., Praly, L.: A Smooth Lyapunov function from a class-KL estimate involving two positive functions. ESAIM Control Optimisation and Calculus of Variations 5, 313–367 (2000) MathSciNetMATHCrossRef Teel, A.R., Praly, L.: A Smooth Lyapunov function from a class-KL estimate involving two positive functions. ESAIM Control Optimisation and Calculus of Variations 5, 313–367 (2000) MathSciNetMATHCrossRef
28.
Zurück zum Zitat Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global exponential robust stability. Systems and Control Letters 44, 373–384 (2001) MathSciNetMATHCrossRef Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global exponential robust stability. Systems and Control Letters 44, 373–384 (2001) MathSciNetMATHCrossRef
29.
Zurück zum Zitat Yoshizawa, T.: Asymptotic behavior of solutions in nonautonomous systems. In: Lakshmikantham, V. (ed.) Trends in Theory and Practice of Nonlinear Differential Equations, pp. 553–562. Dekker, New York (1984) Yoshizawa, T.: Asymptotic behavior of solutions in nonautonomous systems. In: Lakshmikantham, V. (ed.) Trends in Theory and Practice of Nonlinear Differential Equations, pp. 553–562. Dekker, New York (1984)
Metadaten
Titel
Converse Lyapunov Results
verfasst von
Iasson Karafyllis
Zhong-Ping Jiang
Copyright-Jahr
2011
Verlag
Springer London
DOI
https://doi.org/10.1007/978-0-85729-513-2_3

Neuer Inhalt