A framework for a class of hybrid extremum seeking controllers with dynamic inclusions

This paper presents a prescriptive framework for the design and analysis of a class of deterministic extremum seeking controllers (ESCs) based on hybrid dynamic inclusions. This type of ESCs combines continuous and discrete-time dynamics during the seeking process, and its evolution in time is characterized by differential and difference inclusions rather than standard difference and differential equations. Examples of hybrid ESCs include, but are not limited to, purely continuous ESC with optimizers described by set-valued mappings, ESC with arbitrarily fast and slow switching modes, ESC with weakly-jumping parameters, as well as distributed ESCs for multi-agent systems with time-varying graphs. Since solutions of systems described by set-valued mappings are usually not unique, we do not insist on this property, but rather we characterize the behavior of all possible solutions generated by the closed-loop system. Making use of recent results in singular perturbations, averaging, and Ω-limit sets of sets for hybrid dynamical systems, we establish a practical asymptotic stability result. Some examples of common classes of seeking dynamics described by hybrid systems are presented, as well as some numerical simulations illustrating their application in standard maximization problems and Nash seeking problems in game theoretical scenarios.