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Adaptive Tracking for Simple Linear Plants with Unknown Control Direction

Examining adaptive trackers for a class of the first-order linear continuous-time plants, with different estimators, and the modified certainty equivalence pole-placement control law, we overcome the difficulty of a loss of stabilizability (or controllability) when the estimate of the plant gain is zero and point out a few substantive mistakes of the results from Middleton and Kokotovic's paper (1992). In the constructed adaptive tracking systems, the explicit expression for the phase-plane trajectories or the explicit solution completely describing the nonlinear behavior of the resulting closed-loop systems are obtained, and some problems of these adaptive tracking systems, due to a loss of stabilizability of the estimated model probably, are analyzed