A redesign of model reference adaptive control using a non-quadratic Lyapunov function for deriving a new adaptive law leads to a tracking errore(t) belonging to theL1+αsignal space for plants with relative degree one, where α > 0 is a design parameter. A choice of 0 < α < 1 leads to the energy of |e(t;)|(1+α)/2smaller than that with the standard adaptive law for α = 1. Simulation results indicate that the new adaptive control scheme with0 < α < 1improves system tracking performance. Preliminary analysis shows that the new scheme with a projection modification is also robust with respect to external disturbances.