| The objective of our work is to analyze and control nonlinear systems along prespecified trajectories, and specifically eventually periodic ones. Such trajectories can be arbitrary for a finite amount of time, but then settle down into a periodic orbit. To that end, we introduce the concept of an eventually periodic system. Such systems contain both finite horizon and periodic systems as special cases. Due to the difficulty of dealing with nonlinear models, we opt to examine the foregoing problem using two different approaches.;In the first one, we linearize the nonlinear system equations about the trajectory, and end up with a standard discrete-time linear time-varying (LTV) model. In the case of eventually periodic trajectories, and hence eventually periodic models, we present a new linear matrix inequality (LMI) characterization of stability and performance, as well as precise conditions for closed-loop synthesis of eventually periodic controllers. We also present alternative LMI tools, stemming from semidefinite programming duality theory, that help provide new theoretical insight and new synthesis results. Moreover, we further existing results on the model reduction of stable finite horizon and periodic systems.;The second approach amounts to parametrizing the system equations about the trajectory, resulting in a nonstationary linear parameter-varying (NLPV) model, and then constructing an NLPV synthesis. Synthesis conditions are derived for NLPV systems using an operator theoretic framework with the ℓ 2-induced norm as the performance measure. These conditions are given in terms of structured operator inequalities. In general, evaluating the validity of these conditions is an infinite dimensional convex optimization problem; however, if the initial system is eventually periodic, they reduce to a finite dimensional semidefinite programming problem. Also, we give systematic approaches for the model reduction of stable as well as stabilizable NLPV models, along with means for evaluating the error resulting from the reduction process. |
