One of the main features of adaptive systems is an oscillatory convergence that exacerbates with the speed of adaptation. Recently, it has been shown that closed-loop reference models (CRMs) can result in improved transient performance over their open-loop counterparts in model reference adaptive control. In this paper, we quantify both the transient performance in the classical adaptive systems and their improvement with CRMs. In addition to deriving bounds on L-2 norms of the derivatives of the adaptive parameters that are shown to be smaller, an optimal design of CRMs is proposed that minimizes an underlying peaking phenomenon. The analytical tools proposed are shown to be applicable for a range of adaptive control problems including direct control and composite control with observer feedback. The presence of CRMs in adaptive backstepping and adaptive robot control is also discussed. Simulation results are presented throughout this paper to support the theoretical derivations.
Closed-loop reference model adaptive control removes high frequency oscillations.