Extremum Seeking for Single-Variable Static Map with Measurement Noise via A Time-Delay Approach

In this article, we introduce a time-delay approach to extremum seeking (ES) for scalar static quadratic maps with measurement noise. The approach involves transforming the ES system into a time-delay neutral type system with stochastic perturbations, which guarantees the stability of the original ES system. Using a Lyapunov-Krasovskii (L-K) method, explicit conditions are established for the mean-square ultimate boundedness of the ES control systems. These conditions are expressed in terms of LMIs and depend on the intensity of measurement noise, tuning parameters, and a known arbitrarily large constant
$L$
that bounds the 6th moment of the estimation error. Compared to existing results for ES with measurement noise obtained via qualitative analysis, our approach provides a quantitative analysis via a time-delay approach to averaging. A numerical example is provided to illustrate the efficiency of the proposed method.