On certain performance issues arising in adaptive control

In this note we address the problem of improving the performance of a class of Model Reference. Adaptive Controllers (MRAC) under insufficient excitation and in the presence of perturbations, e.g., non-parametric uncertainty and bounded disturbances. In such an environment, the global boundedness of a MRAC closed-loop (BIBO stability) has been established, without any excitation requirements, for both bounded disturbances and 'small' non-parametric uncertainty (e.g., see [1,2] and references therein). In addition to this form of robust stability, the adaptive controller also guarantees robust tracking performance in a root-mean-square (RMS) sense. That is, the RMS value of the normalized tracking error is of order of the size of the non-parametric uncertainty and/or the RMS value of the external disturbances. Unfortunately, however, this weak performance measure is rarely reliable in assessing the effectiveness of the adaptive controller. Indeed, in the absence of sufficient excitation, even arbitrarily small perturbations may cause the adjustable parameters to drift away from their desired or actual values. Such a parameter drift can then produce large instantaneous tracking errors due to either a change in the excitation signal or the (local) destabilization of the closed loop system. This type of undesirable behavior, often referred to as burst phenomena, has been the subject of several studies (e.g., see [3, 4, 5, 6]). A typical remedy of the bursting problem has been recognized in the form of absolute or relative dead-zones [7,8]. The main idea of this approach is to cease adaptation when the estimation error becomes smaller than a threshold, derived as a conservative estimate of the worst-case upper bound of the perturbation entering the estimation algorithm of the adaptive controller. The net result of this approach is that if the threshold is chosen appropriately, the estimated parameters converge and the tracking error enters a residual set where its magnitude is proportional to the dead-zone threshold. We refer to this type of performance characterization as performance in a lim sup sense. Thus, for adaptive laws with dead-zone both the RMS and lim sup values of the tracking error are of the order of the threshold. (In the relative dead-zone case, all quantities are suitably normalized while the threshold should be sufficiently small to guarantee the BIBO stability of the closed-loop). The need for a conservative selection of the dead-zone threshold, however, constitutes the main drawback of this approach in the sense that if the threshold is grossly overestimated, the RMS tracking performance may deteriorate unnecessarily.