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Differential Geometric Methods in Nonlinear Control

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Encyclopedia of Systems and Control

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Abstract

In the early 1970s, concepts from differential geometry were introduced to study nonlinear control systems. The leading researchers in this effort were Roger Brockett, Robert Hermann, Henry Hermes, Alberto Isidori, Velimir Jurdjevic, Arthur Krener, Claude Lobry, and Hector Sussmann. These concepts revolutionized our knowledge of the analytic properties of control systems, e.g., controllability, observability, minimality, and decoupling. With these concepts, a theory of nonlinear control systems emerged that generalized the linear theory. This theory of nonlinear systems is largely parallel to the linear theory, but of course it is considerably more complicated.

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References

  • Arnol’d VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, Berlin

    Book  Google Scholar 

  • Brockett RW (1972) Systems theory on group manifods and coset spaces. SIAM J Control 10: 265–284

    Article  MATH  MathSciNet  Google Scholar 

  • Chow WL (1939) Uber Systeme von Linearen Partiellen Differentialgleichungen Erster Ordnung. Math Ann 117:98–105

    MathSciNet  Google Scholar 

  • Gauthier JP, Hammouri H, Othman S (1992) A simple observer for nonlinear systems with applications to bioreactors. IEEE Trans Autom Control 37:875–880

    Article  MATH  MathSciNet  Google Scholar 

  • Griffith EW, Kumar KSP (1971) On the observability of nonlinear systems, I. J Math Anal Appl 35:135–147

    Article  MATH  MathSciNet  Google Scholar 

  • Haynes GW, Hermes H (1970) Non-linear controllability via Lie theory SIAM J Control 8: 450–460

    MATH  MathSciNet  Google Scholar 

  • Hermann R, Krener AJ (1977) Nonlinear controllability and observability. IEEE Trans Autom Control 22:728–740

    Article  MATH  MathSciNet  Google Scholar 

  • Hermes H (1994) Large and small time local controllability. In: Proceedings of the 33rd IEEE conference on decision and control, vol 2, pp 1280–1281

    Google Scholar 

  • Hirschorn RM (1981) (A,B)-invariant distributions and the disturbance decoupling of nonlinear systems. SIAM J Control Optim 19:1–19

    Article  MATH  MathSciNet  Google Scholar 

  • Isidori A, Krener AJ, Gori Giorgi C, Monaco S (1981a) Nonlinear decoupling via feedback: a differential geometric approach. IEEE Trans Autom Control 26:331–345

    Article  MATH  MathSciNet  Google Scholar 

  • Isidori A, Krener AJ, Gori Giorgi C, Monaco S (1981b) Locally (f,g) invariant distributions. Syst Control Lett 1:12–15

    Article  MATH  MathSciNet  Google Scholar 

  • Kostyukovskii YML (1968a) Observability of nonlinear controlled systems. Autom Remote Control 9:1384–1396

    MathSciNet  Google Scholar 

  • Kostyukovskii YML (1968b) Simple conditions for observability of nonlinear controlled systems. Autom Remote Control 10:1575-1584-1396

    Google Scholar 

  • Kou SR, Elliot DL, Tarn TJ (1973) Observability of nonlinear systems. Inf Control 22:89–99

    Article  MATH  Google Scholar 

  • Krener AJ (1971) A generalization of the Pontryagin maximal principle and the bang-bang principle. PhD dissertation, University of California, Berkeley

    Google Scholar 

  • Krener AJ (1974) A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control problems. SIAM J Control 12:43–52

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ (1975) Local approximation of control systems. J Differ Equ 19:125–133

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ (2002a) The convergence of the extended Kalman filter. In: Rantzer A, Byrnes CI (eds) Directions in mathematical systems theory and optimization. Springer, Berlin, pp 173–182. Corrected version available at arXiv:math.OC/0212255 v. 1

    Google Scholar 

  • Krener AJ (2002b) The convergence of the minimum energy estimator. I. In: Kang W, Xiao M, Borges C (eds) New trends in nonlinear dynamics and control, and their applications. Springer, Heidelberg, pp 187–208

    Google Scholar 

  • Krener AJ (2010a) The accessible sets of linear free nilpotent control systems. In: Proceeding of NOLCOS 2010, Bologna

    Google Scholar 

  • Krener AJ (2010b) The accessible sets of quadratic free nilpotent control systems. Commun Inf Syst 11:35–46

    MathSciNet  Google Scholar 

  • Krener AJ (2013) Feedback linearization of nonlinear systems. Baillieul J, Samad T (eds) Encyclopedia of systems and control. Springer

    Google Scholar 

  • Krener AJ, Schaettler H (1988) The structure of small time reachable sets in low dimensions. SIAM J Control Optim 27:120–147

    Article  Google Scholar 

  • Lobry C (1970) Cotrollabilite des Systemes Non Lineaires. SIAM J Control 8:573–605

    Article  MATH  MathSciNet  Google Scholar 

  • Sussmann HJ (1973) Minimal realizations of nonlinear systems. In: Mayne DQ, Brockett RW (eds) Geometric methods in systems theory. D. Ridel, Dordrecht

    Google Scholar 

  • Sussmann HJ (1975) A generalization of the closed subgroup theorem to quotients of arbitrary manifolds. J Differ Geom 10:151–166

    MATH  MathSciNet  Google Scholar 

  • Sussmann HJ (1977) Existence and uniqueness of minimal realizations of nonlinear systems. Math Syst Theory 10:263–284

    Article  MathSciNet  Google Scholar 

  • Sussmann HJ, Jurdjevic VJ (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116

    Article  MATH  MathSciNet  Google Scholar 

  • Wonham WM, Morse AS (1970) Decoupling an pole assignment in linear multivariable systems: a geometric approach. SIAM J Control 8:1–18

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Naval Postgrauate School, 93943, Monterey, CA, USA

    A. J. Krener

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  1. A. J. Krener
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Correspondence to A. J. Krener .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

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Krener, A.J. (2013). Differential Geometric Methods in Nonlinear Control. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_80-1

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_80-1

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  • Online ISBN: 978-1-4471-5102-9

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