Omega-limit sets of nonlinear systems that are semiglobally practically stabilized.

摘要

In nonlinear control theory, the equilibrium of a system is semiglobally practically stabilizable if, given two balls centered at the equilibrium, one of arbitrarily large radius and one of arbitrarily small radius, we are able to design a feedback so that the resulting closed-loop system has the following property: all the trajectories originating in the large ball enter, within a fixed finite time, into the small ball and stay inside thereafter.; In this work, given a nonlinear system that is semiglobally practically stabilized, we focus on the problem of characterizing the asymptotic behavior of its trajectories that start inside the large ball. It turns out that inside the small ball where these trajectories enter, there is a compact, invariant, connected, stable set that attracts them; such set is the omega-limit set of the large ball. Here, we address the problem of studying the structure of this omega-limit set. Specifically, we carry out the characterization for closed-loop systems obtained applying a semiglobally practically stabilizing feedback law to a nonlinear minimum-phase system belonging to a certain class. The characterization is carried out when a memory-less state feedback is employed and when a dynamic output feedback is employed. It is then found that using output feedback rather than state feedback does not affect the structure of the omega-limit set.