This paper considers the model reference adaptive control problem for the system (1) with unmodelled dynamics eta(n) dominated by a small constant epsilon multiplied by a quantity dependent on the input, output and noise. Using bounded external excitation and randomly varying truncation techniques, we give design method of a model reference adaptive controller which is similar to that in Meyn and Brown (1992). It is shown that the closed-loop system is globally stable, the estimation error for parameter contained in the modelled part is of order epsilon, and the closed-loop system under the model reference adaptive control law is suboptimal in the sense of [GRAPHICS] while the SPR condition usually used in other papers is replaced by a stability condition.