  Classical discrete-time adaptive controllers provide asymptotic
stabilization. While the original adaptive controllers did not handle noise or
unmodelled dynamics well, redesigned versions were proven to have some
tolerance; however, exponential stabilization and a bounded gain on the noise
was rarely proven. Here we consider a classical pole placement adaptive
controller using the original projection algorithm rather than the commonly
modifed version; we impose the assumption that the plant parameters lie in a
convex, compact set and that the parameter estimates are projected onto that
set at every step. We demonstrate that the closed-loop system exhibits very
desireable closed-loop behaviour: there are linear-like convolution bounds on
the closed loop behaviour, which implies exponential stability and a bounded
noise gain, as well an easily proven tolerance to unmodelled dynamics and plant
parameter variation. We emphasize that there is no persistent excitation
requirement of any sort.
