Stability analysis of indirect adaptive regulating for simple linear control systems

The paper deals with the adaptive regulators for a class of first-order linear continuous-time plants, with different estimators and modified certainty-equivalence control law. We overcame the difficulty of the loss of stabilizability (or controllability) as the estimate of the plant gain is zero and corrected a few mistakes in the paper (Middleton and Kokotovic, 1992). In the constructed adaptive regulating systems, the explicit expression for the phase plane trajectories or the explicit solution completely describing the nonlinear behavior of the resulting closed-loop systems is obtained, and some problems of the resultant adaptive systems, which due to a loss of stabilizability of the estimated model probably are analyzed. Implications of these results for the indirect adaptive regulating case of higher order linear continuous-time plants, with the unknown control direction and model parameters are discussed.