Anantharaman Subbaraman

Stability analysis for stochastic hybrid systems

This survey addresses stability analysis for stochastic hybrid systems (SHS), which are dynamical systems that combine continuous change and instantaneous change and that also include random effects. We re-emphasize the common features found in most of the models that have appeared in the literature, which include stochastic switched systems, Markov jump systems, impulsive stochastic systems, switching diffusions, stochastic impulsive systems driven by renewal processes, diffusions driven by Levy processes, piecewise-deterministic Markov processes, general stochastic hybrid systems, and stochastic hybrid inclusions. Then we review many of the stability concepts that have been studied, including Lyapunov stability, Lagrange stability, asymptotic stability, and recurrence. Next, we detail Lyapunov-based sufficient conditions for these properties, and additional relaxations of Lyapunov conditions. Many other aspects of stability theory for SHS, like converse Lyapunov theorems and robustness theory, are not fully developed; hence, we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces.

A converse Lyapunov theorem for strong global recurrence

Abstract A converse Lyapunov theorem is established for discrete-time stochastic systems modeled by a set-valued mapping under mild regularity conditions. For this class of systems, it is shown that strong global recurrence is a necessary and sufficient condition for the existence of a smooth Lyapunov function that decreases in expected value along solutions outside of the set that is strongly globally recurrent. Robustness of strong global recurrence to sufficiently small perturbations is also established.

Equivalent Characterizations of Input-to-State Stability for Stochastic Discrete-Time Systems

Input-to-state stability (ISS) for stochastic difference inclusions is studied. First, ISS in probability relative to a compact set is defined. Subsequently, several equivalent characterizations are given. For example, ISS in probability is shown to be equivalent to global asymptotic stability in probability when the disturbance takes values in a ball whose radius is determined by a sufficiently small, but unbounded, function of the distance of the state to the compact set. In turn, a recent converse Lyapunov theorem for global asymptotic stability in probability provides an equivalent Lyapunov characterization. Finally, robust ISS in probability is defined and is shown to give another equivalent characterization.

A Converse Lyapunov Theorem and Robustness for Asymptotic Stability in Probability

A converse Lyapunov theorem is established for discrete-time stochastic systems with non-unique solutions. In particular, it is shown that global asymptotic stability in probability implies the existence of a continuous Lyapunov function, smooth outside of the attractor, that decreases in expected value along solutions. The keys to this result are mild regularity conditions imposed on the set-valued mapping that characterizes the update of the system state, and the ensuing robustness of global asymptotic stability in probability to sufficiently small state-dependent perturbations.

Discrete-time stochastic control systems: a continuous Lyapunov function implies robustness to strictly causal perturbations

Abstract Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.

GPS-optimal micro air vehicle navigation in degraded environments

We investigate a computationally and memory efficient algorithm for radio frequency (RF) source-seeking with a single-wing rotating micro air vehicle (MAV) operating in an urban canyon environment. We present an algorithm that overcomes two significant difficulties of operating in an urban canyon environment. First, Global Positioning System (GPS) localization quality can be degraded due to the lack of clear line of sight to a sufficient number of GPS satellites. Second, the spatial RF field is complex due to multipath reflections leading to multiple maxima and minima in received signal strength (RSS). High quality GPS localization is maintained by observing the GPS signal to noise ratio (SNR) to each satellite and making inferences about directions of high GPS visibility (allowable) and directions of low GPS visibility (forbidden). To avoid local maxima in RSS due to multipath reflections we exploit the rotation of the MAV and the directionality of its RF antenna to derive estimates of the angle of arrival (AOA) at each rotation. Under mild assumptions on the noise associated with the AOA measurements, a greedy algorithm is shown to exhibit a global recurrence property. Simulations supplied with actual GPS SNR measurements indicate that this algorithm reliably finds the RF source while maintaining an acceptable level of GPS visibility. Additionally, outdoor experiments using Lockheed Martins Samarai MAV demonstrate the efficacy of this approach for static source-seeking in an urban canyon environment.

A Matrosov theorem for strong global recurrence

A theorem on nested Matrosov functions is presented for time-varying stochastic, set-valued discrete-time systems satisfying mild regularity conditions. It establishes sufficient conditions for uniform strong global recurrence of an open, bounded set. In general Matrosov functions are required to satisfy less rigid requirements than typical Lyapunov functions that satisfy a strict decrease condition along trajectories.

On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

Abstract We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness.

A Krasovskii-LaSalle function based recurrence principle for a class of stochastic hybrid systems

We characterize the sets to which bounded random solutions generated by a class of stochastic hybrid systems converge under the existence of a Lyapunov-like function that is non-increasing almost surely during flows and on average during jumps. In particular, we establish that we get almost sure convergence to the largest weakly totally recurrent in probability set that is contained in a level set of this function. We also apply this result to establish weak sufficient conditions for uniform global asymptotic stability in probability of compact sets and uniform global recurrence of open, bounded sets for a class of stochastic hybrid systems.

A stochastic hybrid algorithm for robust global almost sure synchronization on the circle: All-to-all communication

A stochastic hybrid algorithm for guaranteeing global almost sure synchronization of a finite number of agents evolving on the circle is presented. We assume all-to-all communication where every agent has access to the same aggregate quantity corresponding to an average of the position of the agents. The lack of robustness of deterministic algorithms to adversaries and slow convergence properties of stochastic gossip algorithms are overcome by using a stochastic hybrid feedback.