Control-Affine Extremum Seeking Control with Attenuating Oscillations

Abstract

Control-affine Extremum Seeking Control (ESC) systems have been increasingly studied and applied in the last decade. Similar to classic ESC related structures, control-affine ESC systems are operable by assuming access to measurements of the objective function, and not necessarily its expression. In contrast to classic ESC related structures, in a control-affine ESC, the objective function -- or a map of it -- is incorporated within the system's vector fields themselves. This has invoked the use of tools from geometric control theory, namely Lie Bracket Systems (LBSs). Said LBSs play a crucial role in stability and performance characterization of ESCs. In a recent effort, many control-affine ESC structures have been generalized in a unifying class and analyzed through LBSs. In addition, this generalized class converge asymptotically to the extremum point; however, the extremum point has to be known a priori and guaranteeing vanishing control input at the extremum point requires the application of strong conditions. In this paper, we introduce a LBS-based ESC structure that: (1) does not require the extremum point a priori, (2) its oscillations attenuate structurally via a novel application of a geometric-based Kalman filter estimating LBSs; and (3) its stability is characterized by a time-dependent (one bound) condition that is verifiable via simulations and relaxed when compared to the generalized approach mentioned earlier. We provide numerical simulations of three problems to demonstrate the ability of our proposed ESC; these problems cannot be solved with vanishing oscillations using the above-mentioned generalized approach in literature