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The paper is concerned with recent developments in the application of harmonic approximants to predict limit-cycling conditions in multivariable feedback systems. Emphasis is placed on computational procedures giving graphical interpretations in the frequency domain which will be useful in the subsequent design of suitable linear compensator elements. Early work based on the single-harmonic approximant is reviewed, and it is shown how this can be subsequently extended and refined to generate more accurate solutions based on higher-order-harmonic approximants. Some of the numerical problems involved are described, and solutions are proposed which make efficient use of the computational effort available. An example is given which allows results obtained from the various methods described to be directly compared.