Nontrivial closed-loop analysis for an extremely simple one-step-guess adaptive controller

In this paper, we investigate a class of the scalar discrete-time adaptive control system based on an extremely simple one-step-guess (OSG) estimator, whose closed loop is rather complex and nontrivial despite that it has ever been ignored for a long time partially due to its quite simple form and idea. The OSG adaptive controller is based on the most intuitive idea to estimate the unknown parameter with only the information available from one step, and its form is much simpler than and different from that of other widely used adaptive controllers based on least-squars algorithm or gradient-type algorithms. We show that the closed-loop system of the OSG adaptive controller is indeed governed by a time-varying nonlinear difference equation, which was not thoroughly investigated in previous studies of difference equations. The primordial OSG adaptive controller without dead zone and the extended OSG adaptive controller with time-varying dead zone are analyzed in this contribution, and their closed-loop properties are rigorously established, which show that the simple OSG adaptive controller is stable under mild conditions. Extensive numerical simulations also illustrate the effectiveness of the proposed method.