Convergence analysis for reduced-order adaptive controller design of uncertain SISO linear systems with noisy output measurements

We consider in this article a class of uncertain SISO linear systems that are subject to system and measurement noises. Reduced-order adaptive controller designs have been proposed before for such systems by the authors and stability analysis of the closed-loop systems has been established. Here we analyse, further, the robustness properties for these reduced-order adaptive control systems by providing detailed convergence analysis results for the key closed-loop signals and parameter estimates. We rigorously prove that, whenever the exogenous disturbance input is of finite energy and bounded, and the reference trajectory and its derivatives up to rth order are bounded, r being the relative degree of the transfer function of the true system, a set of signals, including the tracking error, the estimation error between the system output and its estimate, the projection signal, are of finite energy and converge to zero; and the system states and their estimates exhibit asymptotic behaviours with certain formats. With an additional persistency of excitation condition, it is also proved that the estimate and the worst-case estimate of the state vector asymptotically track the actual state vector; and the estimate and the worst-case estimate of the unknown parameter vector converge to the true value. A numerical example is given to illustrate the theoretical findings.