  We present multivariable extremum seeking (ES) designs that achieve unbiased
convergence to the optimum. Two designs are introduced: one with exponential
unbiased convergence (unbiased extremum seeker, uES) and the other with
user-assignable prescribed-time unbiased convergence (unbiased PT extremum
seeker, uPT-ES). In contrast to the conventional ES, which uses persistent
sinusoids and results in steady-state oscillations around the optimum, the
exponential uES employs an exponentially decaying amplitude in the perturbation
signal (for achieving convergence) and an exponentially growing demodulation
signal (for making the convergence unbiased). The achievement of unbiased
convergence also entails employing an adaptation gain that is sufficiently
large in relation to the decay rate of the perturbation amplitude. Stated
concisely, the bias is eliminated by having the learning process outpace the
waning of the perturbation. The other algorithm, uPT-ES, employs
prescribed-time convergent/blow-up functions in place of constant amplitudes of
sinusoids, and it also replaces constant-frequency sinusoids with chirp signals
whose frequency grows over time. Among the convergence results in the ES
literature, uPT-ES may be the strongest yet in terms of the convergence rate
(prescribed-time) and accuracy (unbiased). To enhance the robustness of uES to
a time-varying optimum, exponential functions are modified to keep oscillations
at steady state. Stability analysis of the designs is based on a state
transformation, averaging, local exponential/PT stability of the averaged
system, local stability of the transformed system, and local exponential/PT
stability of the original system. For numerical implementation of the developed
ES schemes and comparison with previous ES designs, the problem of source
seeking by a two-dimensional velocity-actuated point mass is considered.
