Solutions to hybrid inclusions via set and graphical convergence with stability theory applications

Direct design of robustly asymptotically stabilizing hybrid feedback

“…If the feedback is regular and f is continuous (in both variables) then the resulting closed-loop hybrid system (2.2) has the properties required by [5] in developing the robust stability theory for hybrid systems. In particular, for such hybrid systems, appropriately understood limits of solutions are still solutions, and solutions from points near a reference point are close (again, in appropriate sense) to some solution from the reference point.…”

“…In particular, for such hybrid systems, appropriately understood limits of solutions are still solutions, and solutions from points near a reference point are close (again, in appropriate sense) to some solution from the reference point. Such properties guarantee that asymptotic stability of compact sets is always robust; see [5]. Technically, the robustness results of [5] is not applicable here, since we are talking about stability of A×Q, which need not be compact, and f is not continuous in (x, u).…”

“…In [5], various regularity properties of general hybrid systems were established and robustness of stability was tied to some basic properties of the system's data. The work in [9] adapted such results to nonlinear systems in closed loop with hybrid patchy feedbacks, and in particular, showed that any hybrid patchy feedback that is strongly regular (in the terminology of the current paper, see Def.…”

Reset passivation of nonlinear controllers via a suitable time-regular reset map

“…implies that the uniformly non-Zeno definition in Goebel and Teel (2006) (see also Collins (2004)) is satisfied with T = ρ and J = 2.…”

“…Controller (2) will be dealt with in this paper following the framework of Cai andTeel (2009), Goebel et al (2009) and Goebel and Teel (2006). In particular, by Assumption 1, controller (2) satisfies the hybrid basic assumptions (see, e.g., Cai and Teel (2009)) which, under the hypothesis that v and θ are measurable signals, ensure desirable regularity properties of the solutions, such as existence, and robustness to arbitrarily small perturbations (see Goebel et al (2009) for details).…”