Extremum seeking in the presence of large delays via time-delay approach to averaging

In this paper, we study gradient-based classical extremum seeking (ES) for uncertain n-dimensional (nD) static quadratic maps in the presence of known large constant distinct input delays and large output constant delay with a small time-varying uncertainty. This uncertainty may appear due to network-based measurements. We present a quantitative analysis via a time-delay approach to averaging. We assume that the Hessian has a nominal known part and norm-bounded uncertainty, the extremum point belongs to a known box, whereas the extremum value to a known interval. By using the orthogonal transformation, we first transform the original static quadratic map into a new one with the Hessian containing a nominal diagonal part. We apply further a time-delay transformation to the resulting ES system and arrive at a time-delay system, which is a perturbation of a linear time-delay system with constant coefficients. Given large delays, we choose appropriate gains to guarantee stability of this linear system. To find a lower bound on the dither frequency for practical stability, we employ variation of constants formula and exploit the delay-dependent positivity of the fundamental solutions of the linear system with their tight exponential bounds. Sampled-data ES in the presence of large distinct input delays is also presented. Explicit conditions in terms of simple scalar inequalities depending on tuning parameters and delay bounds are established to guarantee the practical stability of the ES control systems. We show that given any large delays and initial box, by choosing appropriate gains we can achieve practical stability for fast enough dithers and small enough uncertainties.