Rafal Goebel

Hybrid dynamical systems

Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability

This paper shows several versions of the (LaSalles) invariance principle for general hybrid systems. The broad framework allows for nonuniqueness of solutions, Zeno behaviors, and does not insist on continuous dependence of solutions on initial conditions. Instead, only a mild structural property involving graphical convergence of solutions is posed. The general invariance results are then specified to hybrid systems given by set-valued data. Further results involving invariance as well as observability, detectability, and asymptotic stability are given.

Hybrid systems: Generalized solutions and robust stability 1

Abstract Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used. Then a generalized solution definition is given that leads to continuity with respect to initial conditions and perturbations of the system data. This property enables new results on necessary conditions for asymptotic stability in hybrid systems.

Smooth Lyapunov Functions for Hybrid Systems Part II: (Pre)Asymptotically Stable Compact Sets

It is shown that (pre)asymptotic stability, which generalizes asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. This result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small, state-dependent perturbations. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The converse Lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, to the insertion of jumps according to an ldquoaverage dwell-timerdquo rule, and to the insertion of flow according to a ldquoreverse average dwell-timerdquo rule.

Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness

Hybrid systems are dynamical systems where the state is allowed to either evolve continuously (flow) on certain subsets of the state space or evolve discontinuously (jump) from other subsets of the state space. For a broad class of such systems, characterized by mild regularity conditions on the data, we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth Lyapunov function. This result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right-hand sides. Furthermore, we give a description of forward completeness of a hybrid system in terms of a smooth Lyapunov-like function.

Hybrid Feedback Control and Robust Stabilization of Nonlinear Systems

In this paper, we show, for any nonlinear system that is asymptotically controllable to a compact set, that a logic-based, hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise, actuator error, and external disturbance. The construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state, while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed-loop hybrid system. The robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties. Auxiliary results give uniformity of convergence and of overshoots for hybrid systems, and give a characterization of asymptotic stability of compact sets.

Notions of Relative Interior in Banach Spaces

Extensions to a Banach space of the equivalent notions of relatively absorbing, non-support, and relative interior points of a convex set in

The Proximal Average: Basic Theory

\reals^n

Conjugate convex Lyapunov functions for dual linear differential inclusions

are presented. The relations between these extensions are studied, and their basic calculus rules are developed. Several explicit examples and counterexamples in general Banach spaces are given; and the tools for development of further examples are explained. Various implications for infinite dimensional optimization are highlighted.

Invariance principles for switching systems via hybrid systems techniques

The recently introduced proximal average of two convex functions is a convex function with many useful properties. In this paper, we introduce and systematically study the proximal average for finitely many convex functions. The basic properties of the proximal average with respect to the standard convex-analytical notions (domain, Fenchel conjugate, subdifferential, proximal mapping, epi-continuity, and others) are provided and illustrated by several examples.