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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Bibliography
Arnol’d VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, Berlin
Brockett RW (1972) Systems theory on group manifods and coset spaces. SIAM J Control 10: 265–284
Chow WL (1939) Uber Systeme von Linearen Partiellen Differentialgleichungen Erster Ordnung. Math Ann 117:98–105
Gauthier JP, Hammouri H, Othman S (1992) A simple observer for nonlinear systems with applications to bioreactors. IEEE Trans Autom Control 37:875–880
Griffith EW, Kumar KSP (1971) On the observability of nonlinear systems, I. J Math Anal Appl 35:135–147
Haynes GW, Hermes H (1970) Non-linear controllability via Lie theory SIAM J Control 8: 450–460
Hermann R, Krener AJ (1977) Nonlinear controllability and observability. IEEE Trans Autom Control 22:728–740
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
Hirschorn RM (1981) (A,B)-invariant distributions and the disturbance decoupling of nonlinear systems. SIAM J Control Optim 19:1–19
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
Isidori A, Krener AJ, Gori Giorgi C, Monaco S (1981b) Locally (f,g) invariant distributions. Syst Control Lett 1:12–15
Kostyukovskii YML (1968a) Observability of nonlinear controlled systems. Autom Remote Control 9:1384–1396
Kostyukovskii YML (1968b) Simple conditions for observability of nonlinear controlled systems. Autom Remote Control 10:1575-1584-1396
Kou SR, Elliot DL, Tarn TJ (1973) Observability of nonlinear systems. Inf Control 22: 89–99
Krener AJ (1971) A generalization of the Pontryagin maximal principle and the bang-bang principle. PhD dissertation, University of California, Berkeley
Krener AJ (1974) A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control problems. SIAM J Control 12:43–52
Krener AJ (1975) Local approximation of control systems. J Differ Equ 19:125–133
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 ar**v:math.OC/0212255 v. 1
Krener AJ (2002b) The convergence of the minimum energy estimator. I. In: Kang W, **ao M, Borges C (eds) New trends in nonlinear dynamics and control, and their applications. Springer, Heidelberg, pp 187–208
Krener AJ (2010a) The accessible sets of linear free nilpotent control systems. In: Proceeding of NOLCOS 2010, Bologna
Krener AJ (2010b) The accessible sets of quadratic free nilpotent control systems. Commun Inf Syst 11:35–46
Krener AJ (2013) Feedback linearization of nonlinear systems. Baillieul J, Samad T (eds) Encyclopedia of systems and control. Springer
Krener AJ, Schaettler H (1988) The structure of small time reachable sets in low dimensions. SIAM J Control Optim 27:120–147
Lobry C (1970) Cotrollabilite des Systemes Non Lineaires. SIAM J Control 8:573–605
Sussmann HJ (1973) Minimal realizations of nonlinear systems. In: Mayne DQ, Brockett RW (eds) Geometric methods in systems theory. D. Ridel, Dordrecht
Sussmann HJ (1975) A generalization of the closed subgroup theorem to quotients of arbitrary manifolds. J Differ Geom 10:151–166
Sussmann HJ (1977) Existence and uniqueness of minimal realizations of nonlinear systems. Math Syst Theory 10:263–284
Sussmann HJ, Jurdjevic VJ (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116
Wonham WM, Morse AS (1970) Decoupling an pole assignment in linear multivariable systems: a geometric approach. SIAM J Control 8:1–18
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J. Krener, A. (2021). Differential Geometric Methods in Nonlinear Control. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, Cham. https://doi.org/10.1007/978-3-030-44184-5_80
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DOI: https://doi.org/10.1007/978-3-030-44184-5_80
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