Extremum Seeking (ES) is Practically Stable Whenever Model-Based ES is Stable
Extremum seeking controllers (ESCs) are optimization algorithms in continuous time, with model-based ESCs using true derivative information of the cost function and model-free ESCs utilizing perturbation-based estimates instead. Stability analysis of model-free ESCs often employs the associated average system, whose stability is dependent on the selection of the dither signal. We first demonstrate the challenge of this analysis approach by showing selections of relative dither amplitudes and rates at different ESC inputs which result in the average system always having an unstable equilibrium. We then show that, under averaging hypotheses that hold for the standard ESC constructions, if the model-based ESC is a continuous (but not necessarily locally Lipschitz) vector field that is globally asymptotically stable (GAS), then the average system is semiglobally practically asymptotically stable (sGPAS), and the model-free ESC is semiglobally practically uniformly asymptotically stable (sGPUAS). Thus, the system analyst does not need to analyze the stability of each average system that corresponds to the choice of dither signals, as the stability of the model-based ESC is sufficient to show that all the average systems are sGPAS for appropriately selected dither signals.