Symmetry Reduction, Contact Geometry, and Partial Feedback Linearization

Abstract

Let Pfaffian system omega define an intrinsically nonlinear control system which is invariant under a Lie group of symmetries G. Using the contact geometry of Brunovsky normal forms and symmetry reduction, this paper solves the problem of constructing subsystems alpha subset of omega such that alpha defines a static feedback linearizable control system. A method for representing the trajectories of omega from those of alpha using reduction by a distinguished class G of Lie symmetries is described. A control system will often have a number of inequivalent linearizable subsystems depending upon the subgroup structure of G. This can be used to obtain a variety of representations of the system trajectories. In particular, if G is solvable, the construction of trajectories can be reduced to quadrature. It is shown that the identification of linearizable subsystems in any given problem can be carried out algorithmically once the explicit Lie algebra of G is known. All the constructions have been automated using the Maple package DifferentialGeometry. A number of illustrative examples are given.