On Absolute Equivalence and Linearization I

In this paper, we study the absolute equivalence between Pfaffian systems with a degree 1 independence condition and obtain structural results, particularly for systems of corank 3. We apply these results to understanding dynamic feedback linearization of control systems with 2 inputs.Comment: 32 page

Dynamic Feedback Linearization of Control Systems with Symmetry

Control systems of interest are often invariant under Lie groups of transformations. Given such a control system, assumed to not be static feedback linearizable, a verifiable geometric condition is described and proven to guarantee its dynamic feedback linearizability. Additionally, a systematic procedure for obtaining all the system trajectories is shown to follow from this condition. Besides smoothness and the existence of symmetry, no further assumption is made on the local form of a control system, which is therefore permitted to be fully nonlinear and time varying. Likewise, no constraints are imposed on the local form of the dynamic compensator. Particular attention is given to those systems requiring non-trivial dynamic extensions; that is, beyond augmentation by chains of integrators. Nevertheless, the results are illustrated by an example of each type. Firstly, a control system that can be dynamically linearized by a chain of integrators, and secondly, one which does not possess any linearizing chains of integrators and for which a dynamic feedback linearization is nevertheless derived. These systems are discussed in some detail. The constructions have been automated in the Maple package DifferentialGeometry.Comment: 41 pages, minor revisions and error correction

Symmetry Reduction, Contact Geometry, and Partial Feedback Linearization

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