We present results for the analysis of input/output properties of a general class of hybrid systems given by a flow set, a flow map, a jump set, a jump map, and an output map. For this class of systems, the notion of input-output-to-state stability is introduced in the first part of the technical note. Under mild assumptions on the functions and sets defining a hybrid system, sufficient conditions for this notion in terms of Lyapunov functions are derived. Equivalences between Lyapunov functions for input-output-to-state stability for asymptotic and exponential decay rates are established. The sufficient conditions and equivalences are linked to the existence of norm observers for hybrid systems. These results are used in the second part of the technical note to study interconnections of hybrid systems. An interconnection result in terms of a Lyapunov-based small gain theorem is also presented. Examples illustrate the results. © 2013 IEEE.

This chapter presents a unified overview of recent results on controlling hybrid dynamical systems. The first section of the chapter introduces basic notation and definitions, as well as the modeling framework. The second section pertains to stabilization of sets for hybrid systems, stabilizability and control Lyapunov functions. The third section presents conditions guaranteeing the existence of continuous static state feedback controllers. The fourth section introduces the notion of passivity, links it to asymptotic stability and presents sufficient conditions useful in the design of passivity-based controllers. The final section states a tracking control problem for hybrid systems and presents design conditions for tracking controllers. © 2013 by John Wiley & Sons, Inc.

Dynamical systems with trajectories given by sequences of sets are studied. For this class of generalized systems, notions of solution, invariance, and omega limit sets are defined. The structural properties of omega limit sets are revealed. In particular, it is shown that for complete and bounded solutions, the omega limit set of a bounded and complete solution is nonempty, compact, and invariant. Lyapunov-like conditions to locate omega limit sets are also derived. Tools from the theory of set convergence are conveniently used to prove the results. The findings are illustrated in several examples and applications, including the computation of reachable sets and forward invariant sets, as well as in propagation of uncertainty.

We propose a computationally tractable implementation of state-feedback laws for hybrid systems given by differential equations capturing the continuous dynamics or flows, and by difference equations capturing the discrete dynamics or jumps. By exploiting the availability of a control Lyapunov function, along with other properties of the system, we show that pointwise minimum norm control laws can be implemented in a sample-and-hold fashion, with events triggered by timers, to render a desired compact set semiglobally and practically asymptotically stable. Examples illustrate the results.

Minimum-norm control laws for hybrid dynamical systems are proposed. Hybrid systems are given by differential equations capturing the continuous dynamics or flows, and by difference equations capturing the discrete dynamics or jumps. The proposed control laws are defined as the pointwise minimum norm selection from the set of inputs guaranteeing a decrease of a control Lyapunov function. The cases of individual and common inputs during flows and jumps, as well as when inputs enter through one of the system dynamics, are considered. Examples illustrate the results. ©2013 IEEE.

For a class of hybrid systems given in terms of constrained differential and difference equations/inclusions, we study the existence of control Lyapunov functions when compact sets are asymptotically stable as well as the stabilizability properties guaranteed when control Lyapunov functions exist. An existence result asserting that asymptotic stabilizability of a compact set implies the existence of a smooth control Lyapunov function is established. When control Lyapunov functions are available, conditions guaranteeing the existence of stabilizing continuous state-feedback control laws are provided. © 1963-2013 IEEE.

Notions and tools for finite time stability of closed sets for hybrid dynamical systems modeled as hybrid inclusions are introduced. Finite time stability of a closed set is defined as the following two properties: Lyapunov stability, namely, the property that solutions that start close to the set stay close to it, and finite time convergence. In the latter property, the amount of time required to converge to the set of interest is captured by a settling-time function that depends on hybrid time, namely, the time elapsed during flows and the number of jumps (or events) of the hybrid system. Various sufficient conditions for such properties to hold for a given closed set are established. Conditions involving Lyapunov-like functions that strictly decrease during flows, that strictly decrease during jumps, and that strictly decrease along both regimes are proposed — these functions are only required to be locally Lipschitz. A link between (non-finite time) asymptotic stability and the proposed notion is also established. In addition, sufficient conditions for finite time attractivity involving properties at jumps of the system, the second derivative of a Lyapunov-like function, and a property of nested sets are provided. Throughout the paper, examples exercise the results. In addition, an application to finite time parameter estimation is provided and results are applied to it.

Hybrid systems are inherently fragile with respect to perturbations when their state experiences jumps on surfaces. Zero-crossing detection algorithms are capable of robustly detecting the crossing of such surfaces, but, up to now, the effects of adding such algorithms to the system being simulated are unknown. In this paper, we propose a mathematical model for hybrid systems that incorporates zero-crossing detection as well as a hybrid simulator for it. First, we discuss adverse effects that measurement noise and discretization can have on hybrid systems jumping on surfaces and prove that, under mild regularity conditions, zero-crossing detection algorithms can robustify the original system. Then, we show that integration schemes with zero-crossing detection actually compute a robustified version of the fragile nominal model. In this way, we rigorously characterize their effect on solutions to the simulated system. Finally, we show that both the model and simulator are not only robust, but also that the hybrid simulator preserves asymptotic stability properties, semiglobally and practically (on the step size), of the original system. Several examples throughout the paper illustrate these ideas and results.

State-feedback optimal control and cost evaluation problems for constrained difference inclusions are considered. Sufficient conditions, in the form of Lyapunov-like inequalities, are provided to derive an upper bound on the cost associated with the solution to a constrained difference inclusion with respect to a given cost functional. Under additional sufficient conditions, we determine the cost exactly without computing solutions. The proposed approach is extended to study an optimal control problem for discrete-time systems with constraints. In this setting, sufficient conditions for closed-loop optimality are given in terms of a constrained steady-state-like Hamilton-Jacobi-Bellman equation. Applications and examples of the proposed results are presented.

The property of synchronization between two identical linear time-invariant (LTI) systems connected through a network with stochastically-driven isolated communication events is studied. More precisely, the goal is to design feedback controllers that, using information obtained over such networks, asymptotically drive the values of their state to synchronization and render the synchronization condition Lyapunov stable. To solve this problem, we propose a dynamic controller with hybrid dynamics, namely, the controller exhibits continuous dynamics between communication events while it has variables that jump at such events. Due to the additional continuous and discrete dynamics inherent in the networked systems and communication structure, we utilize a hybrid systems framework to model the closed-loop system. The problem of synchronization is then recast as a set stabilization problem and, by utilizing recent Lyapunov stability tools for hybrid systems, sufficient conditions for asymptotic stability of the synchronization set are provided for two network topologies: a cascade (unidirectional) network and a feedback (bidirectional) communication network with independent transmission instances. Furthermore, we study the robustness of synchronization by considering a class of perturbations on the transmitted data. Numerical examples are provided.