  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
which are shown to be smaller, an optimal design of CRMs is proposed which
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 are also discussed.
Simulation results are presented throughout the paper to support the
theoretical derivations.
