A new class of adaptive control schemes for minimum-phase linear time invariant (LTI) systems has recently been developed using nonlinear design techniques which guarantee improved transient performance in addition to closed-loop stability and asymptotic tracking. In this paper we establish the parameter convergence properties of this new class of schemes in the presence of persistently exciting signals and compare them with the properties of the traditional adaptive controllers, We show that the new class of adaptive controllers has stronger parameter convergence properties in the presence of overparameterization.