Feedback-Equivalence of Nonlinear Systems with Applications to Power System Equations.

The key concept of the dissertation is feedback equivalence among systems affine in control. Feedback equivalence to linear systems in Brunovsky canonical form and the construction of the corresponding feedback transformation are used to: (i) design a nonlinear regulator for a detailed nonlinear model of a synchronous generator connected to an infinite bus; (ii) establish which power system network structures enjoy the feedback linearizability property and design a stabilizing control law for these networks with a constraint on the control space which comes from the use of d.c. lines. It is also shown that the feedback linearizability property allows the use of state feedback to contruct a linear controllable system with a positive definite linear Hamiltonian structure for the uncontrolled part if the state space is even; a stabilizing control law is derived for such systems. Feedback linearizability property is characterized by the involutivity of certain nested distributions for strongly accessible analytic systems; if the system is defined on a manifold M diffeomorphic to the Euclidean space, it is established that the set where the property holds is a submanifold open and dense in M. If an analytic output map is defined, a set of nested involutive distributions can be always defined and that allows the introduction of an observability property which is the dual concept, in some sense, to feedback linearizability: the goal is to investigate when a nonlinear system affine in control with an analytic output map is feedback equivalent to a linear controllable and observable system. Finally a nested involutive structure of distributions is shown to guarantee the existence of a state feedback that takes a nonlinear system affine in control to a single input one, both feedback equivalent to linear controllable systems, preserving one controlled vector field.