  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.
