While the original classical parameter 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 modified version; we impose the assumption that the plant parameters lie in a convex, compact set. 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 confers exponential stability and a bounded noise gain, and can be leveraged to prove tolerance to unmodelled dynamics and plant parameter variation. We emphasize that there is no persistent excitation requirement of any sort; the improved performance arises from the vigilant nature of the parameter estimator.
In this paper, we consider the problem of tracking for a discrete-time plant with unknown plant parameters; we assume knowledge of an upper bound on the plant order, and for each admissible order we assume knowledge of a compact set in which the plant parameters lie. We carry out parameter estimation of an associated auxiliary model; indeed, for each admissible dimension, we cover the set of admissible parameters by a finite number of compact and convex sets and use an originalprojection-algorithm-based estimator for each set. At each point in time, we employ a switching algorithm to determine which model and parameter estimates are used in the poleplacement-based control law. We prove that this adaptive controller guarantees desirable linear-like closed-loop behavior: exponential stability, a bounded noise gain in every p-norm, a convolution bound on the effect of the exogenous inputs, as well as exponential tracking for certain classes of reference and noise signals; this linear-like behavior is leveraged to immediately show tolerance to a degree of plant time-variations and unmodelled dynamics.
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