Global linearization by feedback and state transformation

Differential geometric conditions equivalent to the existence of a solution to the global feedback linearization problem are given. If global feedback for linearization is obtained with an atlas of local state space transformations, the resulting closed loop system still has almost all the important features of a linear system. Existence of a compact leaf in any of the standard foliations arising in the local feedback linearization problem is shown to represent nontrivial linear holonomy and hence an obstruction to the existence of global feedback. We prove that in the analytic case, if the state space is simply-connected, this obstruction does not occur. We show that in the two dimensional (C¿) case, if the manifold is simply connected, then the local conditions and controllability are sufficient for global feedback linearization.

[1]  L. Conlon,et al.  Transversally parallelizable foliations of codimension two , 1974 .

[2]  Christopher I. Byrnes,et al.  Remarks on nonlinear planar control systems which are linearizable by feedback , 1985 .

[3]  Loring W. Tu,et al.  Differential forms in algebraic topology , 1982, Graduate texts in mathematics.

[4]  William M. Boothby.  Global Feedback Linearizability of Locally Linearizable Systems , 1986 .

[5]  A. Isidori.  The matching of a prescribed linear input-output behavior in a nonlinear system , 1985 .

[6]  A. Haefliger.  Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes , 1958 .

[7]  R. Su.  On the linear equivalents of nonlinear systems , 1982 .

[8]  Raoul Bott,et al.  Lectures on characteristic classes and foliations , 1972 .

[9]  L. Hunt,et al.  Global transformations of nonlinear systems , 1983 .

[10]  R. Palais.  A Global Formulation of the Lie Theory of Transformation Groups , 1957 .

[11]  W. Respondek.  Global Aspects of Linearization, Equivalence to Polynomial Forms and Decomposition of Nonlinear Control Systems , 1986 .

[12]  Gilbert Hector,et al.  Introduction to the geometry of foliations , 1986 .

[13]  Roger W. Brockett,et al.  Feedback Invariants for Nonlinear Systems , 1978 .

[14]  Daizhan Cheng,et al.  Global feedback linearization of nonlinear systems , 1984, The 23rd IEEE Conference on Decision and Control.

[15]  A. Krener.  On the Equivalence of Control Systems and the Linearization of Nonlinear Systems , 1973 .

[16]  R. Brockett.  The global description of locally linear systems , 1982 .

[17]  W. Boothby,et al.  Global state and feedback equivalence of nonlinear systems , 1985 .

[18]  W. Boothby.  Some comments on global linearization of nonlinear systems , 1984 .

[19]  Witold Respondek.  Geometric methods in linearization of control systems , 1985 .

[20]  Bruce L. Reinhart.  Differential Geometry of Foliations , 1983 .

[21]  J. Pasternack.  Foliations and compact lie group actions , 1971 .