A parameter adaptive controller which provides exponential stability: The first order case

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, neither exponential stabilization nor a bounded noise gain is typically proven. Here we consider the first order case and prove that if the original, ideal, projection algorithm is used in the estimation process (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set), then it guarantees 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.