Extremum Seeking of Nonlinear Static Maps with Constant Delays via a Time-delay Approach∗

Extremum seeking (ES) is a real-time optimization strategy, thus transmission delays in the feedback loop of ES have big impact on its stability. How big delay that ES control systems are able to withstand? This paper tries to provide a potential answer to this problem. We extend a newly developed time-delay approach of ES from quadratic maps to general nonlinear maps under known constant transmission delays. The gradient-based ES in the case of single-variable and multivariable nonlinear static maps are considered. Different from the recent literature, the time-delay method in this paper is applicable to a big family of nonlinear static maps, and the robustness of ES control systems to transmission delays is investigated. By transforming the original ES system into a kind of time-delay system of neutral type, and further transforming the neutral system into a perturbed retarded system with constant delays, we provide simple Lyapunov-based sufficient conditions in the form of linear matrix inequalities (LMIs) to guarantee practical stability of the ES closed-loop system.