We study the convergence properties of a direct model reference adaptive control system by applying techniques from numerical analysis. In particular, a first-order discrete system coupled to a minimal control synthesis algorithm discretized by the one-step one-stage zero-order-hold sampling is studied. This results in a strongly non-linear dynamic system owing to the adaptive mechanism where stability at steady state, i.e. at the operating point, equates to successful control. This paper focuses on the convergence analysis of the overall dynamical system for understanding accuracy, stability and performance at steadystate. The local stability of the steady state solution is considered by linearizing the system in the neighbourhood of an operating point when the input is a step function. This analysis allows us to specify two gain space domains which define the region of local stability. Moreover, both the accuracy and the frequency-domain analyses give insight into the range of adaptive control weightings that results in optimal performance of the minimal control synthesis algorithm and also highlights a possible approach to a priori selection of the time step and adaptive weighting values.The effectiveness of the proposed analysis is further demonstrated by simulations and experiments on a first-order plant.Adaptive control methods are generally divided into (i) direct and (ii) indirect methods. In the first case the adjustment rules provide directly how to update the controller parameters; in the indirect methods, the parameters of the unknown plant are estimated on-line, and the controller parameters are calculated on the base of these estimates. In the context of control theory some recent books and papers relevant to direct methods can be found [2,3]. In the context of structural control several applications have been made by several researches as well [4][5][6].When a direct reference model controller is used, it can be also applied to systems where the details of the plant cannot be fully known a priori or are varying with time. Using this type of algorithms without the knowledge of plant parameters, such that we assume zero initial conditions for the controller gains, has become known as the minimal control synthesis (MCS) approach [7]. Basing adaptive control schemes on a reference model enables the system to be controlled to behave like the model itself. This type of approach [8,9] has been applied to a wide range of systems including non-linear and chaotic systems [3,10]. As the approach is based primarily on linear control theory being the reference model usually a linear one, the effect of non-linearities and/or disturbances in non-linear systems is compensated for by the adaptive nature of the controller. Very recently in the field of dynamic and seismic testing [11], the MCS method was used in real-time substructuring, in order to minimize the error between displacements imposed by the actuators on the substructure and displacements at the interfaces of the numerical model [12,13]; and it was adop...