| In this dissertation the development of a nonequilibrium theory for local robust output regulation of finite-dimensional, smooth, continuous-time, autonomous, nonlinear systems is presented. By nonequilibrium it is meant that asymptotic tracking and/or disturbance rejection is sought in a neighborhood of a dynamic invariant set as opposed to an isolated equilibrium. Results are obtained for locally, exponentially stable (stabilizable), one-dimensional, compact, invariant sets, although the generalization to locally, exponentially stable, k-dimensional, compact, invariant sets follows in a very natural way.; For the first time, to the best of the author's knowledge, the conditions for the solvability of the regulator are formulated in terms of ω-limit sets. This formulation is shown to be consistent with the manifold theory approach used for the isolated equilibrium case. However, the ω-limit set story leads to a theory of local robust output regulation which applies to a more general class of nonlinear plants and exogenous systems. Important novel methodologies include: (1) asymptotic immersion, which is an extension of the application of system immersion employed by Fliess, Byrnes, and Isidori, (2) asymptotic internal modeling, which is a generalization of the asymptotic dynamics of any controller used to achieve a desired steady-state; this includes adaptive controllers, stabilizers, or any design which yields the “right” internal model, asymptotically, (3) a more general internal model principle which is an extension of the celebrated internal model principle of Hepburn and Wonham, and, in addition, (4) the discovery of a “universal nonlinear oscillator”, which is a finite-dimensional nonlinear system that can produce a sinusoidal trajectory having any desired frequency, is presented.; Similarities between robust regulation and robust generalized unidirectional synchronization are pointed out and a counter-example to a necessary condition in the literature is illustrated. Finally, an auxiliary system method for achieving identical synchronization with the electric field of an circulary polarized plane wave having uncertain amplitude, phase, and frequency is presented. |
