Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Notation and Background Kapitel lesen Erstes Kapitel lesen

2011 | OriginalPaper | Buchkapitel

3. Analysis of Continuous-Time Nonlinear Systems

verfasst von : Prof. Laura Menini, Prof. Antonio Tornambè

Erschienen in: Symmetries and Semi-invariants in the Analysis of Nonlinear Systems

Verlag: Springer London

Einloggen

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

search-config
loading …

Abstract

In this chapter, the concepts of semi-invariants (an extension of first integrals) and symmetries (and orbital symmetries) are introduced for continuous-time systems, and their relations with known results in nonlinear systems theory are explored. Such two concepts are strongly related for planar systems, and their relation can be extended to higher order systems through the concept of the inverse Jacobi last multiplier. Before exploring such relations, the concepts of homogeneity and homogeneous systems (with their characteristic solutions) have been introduced. With all such a machinery available, it is possible to detail many properties of two very well known normal forms for autonomous systems: the Poincaré–Dulac and the Belitskii normal forms. Dual semi-invariants are strictly related with semi-invariants and, in turn, they are related with invariant distributions. Both symmetries and invariant distributions can be used to obtain a reduction of the order of a given system, and, in particular, the reduction based on invariant distributions is useful to study structural properties of the system, possibly endowed with inputs and outputs. The last sections of this chapter propose a brief review of other uses of the concept of symmetry; such ideas, although almost out of the scope of the book, are inherently related with the topics studied here and are therefore useful to deepen the comprehension.

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!

Jetzt informieren

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!

Jetzt informieren
Vorheriges Kapitel Analysis of Linear Systems
Nächstes Kapitel Analysis of Discrete-Time Nonlinear Systems
Literatur
5.
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1982)
9.
Bacciotti, A.: Local Stabilizability of Nonlinear Control Systems. Advances in Mathematics for Applied Sciences, vol. 8. World Scientific, Singapore (1991)
11.
Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory, 2nd edn. Communications and Control Engineering. Springer, Berlin (2005) MATH
14.
Bambusi, D., Cicogna, G., Gaeta, G., Marmo, G.: Normal forms, symmetry and linearization of dynamical systems. J. Phys. A, Math. Gen. 31, 5065–5082 (1998) MathSciNetMATHCrossRef
15.
Barenblatt, G.I.: Dimensional Analysis. Gordon & Breach, Routledge (1987)
16.
20.
Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002) MATH
21.
Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations, vol. 2. Springer, New York (1974) MATH
22.
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, 2nd edn. Springer, New York (1989) MATH
23.
Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, Orlando (1986) MATH
24.
25.
Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Springer, New York (1989) MATH
26.
Buckingham, E.: On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4(4), 345–376 (1914) CrossRef
28.
Cheb-Terrab, E.S., Roche, A.D.: Symmetries and first order ODE patterns. Comput. Phys. Commun. 113(2–3), 239–260 (1998) MATHCrossRef
30.
Christopher, C.J.: Invariant algebraic curves and conditions for a centre. Proc. R. Soc. Edinb., Sect. A, Math. 124(6), 1209–1229 (1994) MathSciNetMATH
31.
Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields. I. Theory and algorithm. IEEE Trans. Circuits Syst. 35(7), 863–880 (2002) MathSciNetCrossRef
32.
Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields. II. Applications. IEEE Trans. Circuits Syst. 36(1), 51–70 (2002) MathSciNetCrossRef
34.
Cicogna, G., Gaeta, G.: Symmetry and Perturbation Theory in Nonlinear Dynamics. Lecture Notes in Physics Monographs, vol. 57. Springer, Berlin (1999) MATH
37.
Culver, W.J.: On the existence and uniqueness of the real logarithm of a matrix. In: Proceedings of the American Mathematical Society, pp. 1146–1151 (1966)
38.
Darboux, G.: Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (mélanges). Bull. Sci. Math 2(2) (1878)
45.
Elphick, C., Tirapegui, E., Brachet, M.E., Coullet, P., Iooss, G.: A simple global characterization for normal forms of singular vector fields. Physica D 29(1–2), 95–127 (1987) MathSciNetMATHCrossRef
46.
Evans, G., Blackledge, J.M., Yardley, P.: Analytic Methods for Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Berlin (2000) MATH
49.
50.
51.
53.
Giné, J.: On some open problems in planar differential systems and Hilbert’s 16th problem. Chaos Solitons Fractals 31(5), 1118–1134 (2007) MathSciNetMATHCrossRef
55.
Goodman, R.W.: Nilpotent Lie Groups: Structure and Applications to Analysis. Lecture Notes in Mathematics, vol. 562. Springer, Berlin (1976) MATH
56.
Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 19. World Scientific, Singapore (2001) MATHCrossRef
57.
58.
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 7th edn. Applied Mathematical Sciences, vol. 42. Springer, New York (2002)
60.
Hahn, W.: Stability of Motion. Springer, Berlin (1967) MATH
62.
Hartman, P.: Ordinary Differential Equations. Classics in Applied Mathematics, vol. 18. SIAM, Philadelphia (2002) MATHCrossRef
64.
67.
Hydon, P.E.: Symmetry Methods for Differential Equations. Cambridge University Press, Cambridge (2000) MATHCrossRef
69.
Isidori, A.: Nonlinear Control Systems, 3rd edn. Communications and Control Engineering. Springer, Berlin (1995) MATH
73.
Kamke, E.: Differetialgleichungen: Losungsmethoden und Losungen. Chelsea, New York (1959)
74.
Kawski, M.: Homogeneous stabilizing feedback laws. Control Theory Adv. Technol. 6, 497–516 (1990) MathSciNet
75.
Kawski, M.: Geometric homogeneity and stabilization. In: Proceedings of the IFAC NOLCOS, Lake Tahoe, USA (1995)
80.
Lie, S.: Zur theorie des integrabilitetsfaktors. Christiania Forh. 242–254 (1874)
81.
Lie, S.: Differentialgleichungen. Chelsea, New York (1967)
87.
Menini, L., Tornambè, A.: Linearization of Hamiltonian systems through state immersion. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 1261–1266 (2008)
88.
Menini, L., Tornambè, A.: On the use of semi-invariants for the stability analysis of planar systems. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 634–639 (2008)
89.
Menini, L., Tornambè, A.: Linearization through state immersion of nonlinear systems admitting Lie symmetries. Automatica 45(8), 1873–1878 (2009) MATHCrossRef
90.
Menini, L., Tornambe, A.: On the generation of classes of planar systems with given orbital symmetries. In: Proceedings of the 48th IEEE Conference on Decision and Control, pp. 7442–7447 (2009)
91.
Menini, L., Tornambe, A.: A procedure for the computation of semi-invariants. In: Proceedings of the 48th IEEE Conference on Decision and Control, pp. 7460–7465 (2009)
92.
Menini, L., Tornambè, A.: Computation of a linearizing diffeomorphism by quadrature. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 6281–6286 (2010)
93.
Menini, L., Tornambè, A.: Computation of the real logarithm for a discrete-time nonlinear system. Syst. Control Lett. 59(1), 33–41 (2010) MATHCrossRef
94.
Menini, L., Tornambè, A.: Generalized Lax pairs for the computation of semi-invariants. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 5384–5389 (2010)
96.
Menini, L., Tornambè, A.: Semi-invariants and their use for stability analysis of planar systems. Int. J. Control 83(1), 154–181 (2010) MATHCrossRef
100.
Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990) MATH
102.
Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107. Springer, New York (1986) MATH
105.
Robert, E.K., Christopher, C.J.: Algebraic invariant curves and the integrability of polynomial systems. Appl. Math. Lett. 6(4), 51–53 (1993) MathSciNetMATHCrossRef
106.
Rosier, L.: Homogeneous Lyapunov functions for homogeneous continuous vector fields. Syst. Control Lett. 19, 467–473 (1992) MathSciNetMATHCrossRef
111.
Stephani, H.: Differential Equations: Their Solutions Using Symmetries. Cambridge University Press, Cambridge (1989) MATH
113.
Takens, F.: Forced oscillations and bifurcations. In: Applications of Global Analysis, Communications of the Mathematical Institute Rijksuniversiteit, Utrecht, vol. 3. pp. 1–59 (1974). (Reprinted in Broer, H.W., Krauskopf, B., Vegter Gert (eds.) Global Analysis of Dynamical Systems. IOP Publishing, 2001)
117.
118.
Walcher, S.: Plane polynomial vector fields with prescribed invariant curves. Proc. R. Soc. Edinb., Sect. A, Math. 130, 633–649 (2000) MathSciNetMATH
Metadaten
Titel
Analysis of Continuous-Time Nonlinear Systems
verfasst von
Prof. Laura Menini
Prof. Antonio Tornambè
Copyright-Jahr
2011
Verlag
Springer London
Buch
Symmetries and Semi-invariants in the Analysis of Nonlinear Systems

Print ISBN: 978-0-85729-611-5

Electronic ISBN: 978-0-85729-612-2

Copyright-Jahr: 2011

DOI
https://doi.org/10.1007/978-0-85729-612-2_3

Neuer Inhalt

    Bildnachweise