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In the paper, methods are presented which permit the main system variables of the Whitaker-type model-reference adaptive control system to be calculated for input signals which are essentially sinusoidal or ramp functions. The optimum adapting parameters (as the input signal varies) are calculated very accurately, by means of a simple criterion. The approximate equations for the adapting system are formed by using a parameter-perturbation technique, and the resulting time-varying coefficients are linearised for a given input signal. The equations are then used to give an estimate of the stability of the adaptive loops, and of the system responses to step changes in the adapting parameters. It is also shown that it may be necessary to limit the adaptive parameters, in order to prevent the system from driving itself unstable in its efforts to match the model at input frequencies which lie outside the designed range. The whole of the theoretical work is supported by extensive analogue-simulation studies.