  In this work we find the capacity of a compound finite-state channel with
time-invariant deterministic feedback. The model we consider involves the use
of fixed length block codes. Our achievability result includes a proof of the
existence of a universal decoder for the family of finite-state channels with
feedback. As a consequence of our capacity result, we show that feedback does
not increase the capacity of the compound Gilbert-Elliot channel. Additionally,
we show that for a stationary and uniformly ergodic Markovian channel, if the
compound channel capacity is zero without feedback then it is zero with
feedback. Finally, we use our result on the finite-state channel to show that
the feedback capacity of the memoryless compound channel is given by
$\inf_{\theta} \max_{Q_X} I(X;Y|\theta)$.
