Simultaneous stabilization and optimization of unknown, time-varying systems

Extremum Seeking (ES) has a long history as a tool for locating the extremum points of unknown functions. Recently, by using the ES method to seek the minimum of Lyapunov functions, the ES approach has been demonstrated as a tool for stabilization of unknown systems, removing the distinction between optimization and stabilization. This work combines the dual aspects of the ES method for simultaneous stabilization and optimization. The resulting control scheme optimizes the output of unknown maps, of unknown and possibly unstable, time-varying dynamic systems. Intuitively the approach can be thought of as achieving trajectory tracking of an unknown trajectory, which is the time-varying optimal point of a time-varying map, despite the influence of unstable system dynamics. The motivation for this is the control of an autonomous vehicle, which is chasing a source whose trajectory is unknown, but which is somehow detectable with the detection strength dropping off as a function of distance, physical examples of which may be electric charge, temperature, or chemical concentration. The destabilizing terms may be thought of as unknown environmental variables, such as wind or fluid flow, which may push the autonomous vehicle away from its desired path. Obviously, the multi-dimensional results presented here can be applied to more abstract systems, with many dimensions, such as the states (voltages, currents...) of multiple components of complicated circuits operating in unknown conditions. The results presented guarantee that the system's state will converge to and remain ultimately bounded within some local max/min of the unknown, time-varying output function. In the case of the output having a global max/min ultimate boundedness to within a neighborhood of the global extremum is guaranteed.