Convergence Analysis for Reduced-Order Adaptive Controller Design: Disturbance Attenuation and Asymptotic Tracking for SISO Linear Systems with Noisy Output Measurements

To analyze the robustness of adaptive control system, stability analysis of the closed-loop system and convergence analysis of parameters are two issues of most importance. In this paper, we study the convergence properties for the reduced-order adaptive control system achieved by Zhao and Pan (2004), where we established the stability properties for the closed-loop system. We explore the conditions under which various closed-loop signals converge. In particular, the convergence properties of the parameter estimates to their true values and the asymptotic behavior of the state variables of the unknown system are of interest. We rigorously prove that, whenever the exogenous disturbance inputs 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, then a set of closed-loop signals are of finite energy and converge to zero; the system states and their estimates exhibit asymptotic behaviors with certain formats. With an additional persistency of excitation condition, the parameter estimates converge to their true values and the state estimates asymptotically track the true state.