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This expository article presents an overview of research, conducted mostly between the mid-1990s and late 2000s, that explores a link between commutation relations among a family of asymptotically stable vector fields and stability properties of the switched system that these vector fields generate. This topic is viewed through the lens of the author's own involvement with it, by interspersing explanations of technical developments with personal reminiscences and anecdotes.

A general upper bound for topological entropy of switched nonlinear systems is constructed, using an asymptotic average of upper limits of the matrix measures of Jacobian matrices of strongly persistent individual modes, weighted by their active rates. A general lower bound is constructed as well, using a similar weighted average of lower limits of the traces of these Jacobian matrices. In a case of interconnected structure, the general upper bound is readily applied to derive upper bounds for entropy that depend only on "network-level" information. In a case of block-diagonal structure, less conservative upper and lower bounds for entropy are constructed. In each case, upper bounds for entropy that require less information about the switching signal are also derived. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka-Volterra ecosystem model., Comment: 31 pages, 3 figures

We consider the problem of synchronizing two electric power generators, one of which (the leader) is serving a time-varying electrical load, so that they can ultimately be connected to form a single power system. Each generator is described by a second-order reduced state-space model. We assume that the generator not serving an external load initially (the follower) has access to measurements of the leader's phase angle, corrupted by some additive disturbances. By using these measurements, and leveraging results on reduced-order observers with ISS-type robustness, we propose a procedure that drives (i) the angular velocity of the follower close enough to that of the leader, and (ii) the phase angle of the follower close enough to that of the point at which both systems will be electrically connected. An explicit bound on the synchronization error in terms of the measurement disturbance and the variations in the electrical load served by the leader is computed. We illustrate the procedure via numerical simulations.

We study convergence of nonlinear systems in the presence of an `almost Lyapunov' function which, unlike the classical Lyapunov function, is allowed to be nondecreasing---and even increasing---on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed., Comment: 36 pages, 3 figures

This paper presents a library of second-order models for synchronous machines that can be utilized in power system dynamic performance analysis and control design tasks. The models have a similar structure to the classical model in that they consist of two dynamic states, the power angle and the angular speed. However, unlike the classical model, the models find applications beyond first swing stability analysis; for example, they can also be utilized in transient stability studies. The models are developed through a systematic reduction of a nineteenth-order model, using singular perturbation techniques, and they are validated by comparing their voltage, frequency, and phase profiles with that of the high-order model and that of the classical model.

We prove a novel Lyapunov-based small-gain theorem for networks of $ n \geq 2 $ hybrid systems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems., Comment: Accepted to Automatica

This article deals with input-to-state stability (ISS) of continuous-time switched nonlinear systems. Given a family of systems with exogenous inputs such that not all systems in the family are ISS, we characterize a new and general class of switching signals under which the resulting switched system is ISS. Our stabilizing switching signals allow the number of switches to grow faster than an affine function of the length of a time interval, unlike in the case of average dwell time switching. We also recast a subclass of average dwell time switching signals in our setting and establish analogs of two representative prior results., Comment: 16 pages, 2 figures