  Our recently developed "unbiased" extremum seeking (uES) algorithms ensure
perfect convergence to the optimum at a user-assigned exponential rate or, more
powerfully, within a user-prescribed time. Unlike classical approach, these
algorithms use time-varying adaptation and controller gains, along with
constant or time-varying probing frequencies (chirp signals). This paper
advances our earlier uES designs from strongly convex maps with static optima
to a broader class of convex cost functions with time-varying optima diverging
at arbitrary rates, even in finite time. This advancement first motivates the
use of Lie bracket averaging instead of classical averaging due to the average
system system, which doesn't necessarily need to be exponentially convergent,
and the existence of non-periodic time-varying parameters; second, it
necessitates the formulation of non-trivial and key feasibility conditions for
the choice of time-varying design parameters and their decay/growth rates in
relation to the convexity of the map and the divergence rate of optima. These
conditions indicate that, for constant-frequency probing, the user-defined
asymptotic rate of unbiasing is limited by the convexity of the map. However,
this rate can be made arbitrarily fast (including asymptotic, exponential, and
prescribed time) using chirpy probing, which requires sufficiently rapid
frequency and adaptation growth to enable tracking of faster-diverging optima.
In addition to numerical simulations of the designs, we experimentally test the
feasibility of exponential uES for tuning the angular velocity of a unicycle to
seek a static light source.
