Chaohong Cai

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.

Characterizations of input-to-state stability for hybrid systems☆

Abstract We study input-to-state stability (ISS) for a broad class of hybrid systems, which are combinations of a differential equation on a constraint set and a difference equation on another constraint set. For this class of hybrid systems, we establish the equivalence of ISS, nonuniform ISS, and the existence of a smooth ISS-Lyapunov function by “additionally” assuming that the right-hand side of the differential equation has a convex property with respect to inputs. Moreover, we demonstrate by examples that the equivalence may fail when such a convexity assumption is removed.

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.

Results on input-to-state stability for hybrid systems

We show that, like continuous-time systems, zero-input locally asymptotically stable hybrid systems are locally input-to-state-stable (LISS). We demonstrate by examples that, unlike continuous-time systems, zero-input locally exponentially stable hybrid systems may not be LISS with linear gain, input-to-state stable (ISS) hybrid systems may not admit any ISS Lyapunov function, and nonuniform ISS hybrid systems may not be (uniformly) ISS. We then provide a strengthened ISS condition as an equivalence to the existence of an ISS Lyapunov function for hybrid systems. This strengthened condition reduces to standard ISS for continuous-time and discrete-time systems. Finally under some other assumptions we establish the equivalence among ISS, several asymptotic characterizations of ISS, and the existence of an ISS Lyapunov function for hybrid systems.

Converse Lyapunov theorems and robust asymptotic stability for hybrid systems

We state results on the existence of smooth Lyapunov functions for hybrid systems whose solutions satisfy a class-KLL estimate with respect to two measures. The class-KLL estimate, a natural extension of class-KL estimate, is in terms of the elapsed time and the number of jumps that have occurred. The main result is that a smooth Lyapunov function exists if and only if the class-KLL estimate is robust. In turn, sufficient conditions for robustness are given. Special cases include systems with compact attractors. Most of the results parallel, and unify, what has been developed previously for differential inclusions and difference inclusions.

Hybrid Systems: Limit Sets and Zero Dynamics with a View Toward Output Regulation

Summary. We present results on omega-limit sets and minimum phase zero dynamics for hybrid dynamical systems. Moreover, we give pointers to how these results may be useful in the future for solving the output regulation problem for hybrid systems. We highlight the main attributes of omega-limit sets and we show, under mild conditions, that they are asymptotically stable. We define a minimum phase notion in terms of omega-limit sets and establish an equivalent Lyapunov characterization. Then we study the feedback stabilization problem for a class of minimum phase, relative degree one hybrid systems. Finally, we discuss output regulation for this class of hybrid systems. We illustrate the concepts with examples throughout the paper.

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

Abstract We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].

Robust Input-to-State Stability for Hybrid Systems

This work addresses input-to-state stability (ISS) for hybrid dynamical systems, which combine continuous-time dynamics on a flow set and discrete-time dynamics on a jump set. The main result entails equivalent characterizations of ISS when the right-hand side of the differential equation for the continuous-time dynamics is locally Lipschitz but the set of admissible derivative values, generated by considering all possible input values, is not necessarily convex. Under some mild assumptions, we show that existence of an ISS-Lyapunov function is equivalent to various types of robust ISS for these hybrid systems. As an application, we demonstrate how a hybrid system framework can be used to study some stability properties for continuous-time systems; in particular, we use the derived equivalent characterization results of ISS for hybrid systems to recover and generalize Lyapunov and asymptotic characterizations of input-output-to-state stability for continuous-time systems.

Output-to-state stability for hybrid systems

Output-to-state stability (OSS) is a dual notion of input-to-state stability for dynamical systems. This paper presents Lyapunov and asymptotic characterizations of OSS for hybrid dynamical systems, emphasizing that a globally detectable (i.e. nonuniformly OSS) hybrid system admits a smooth OSS-Lyapunov function.