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Derivation of equivalent linear models of nonlinear systems

Derivation of equivalent linear models of nonlinear systems

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The design of feedback controllers for uncertain nonlinear plants is greatly simplified if the vagaries of the nonlinear behaviour of the plant can be accounted for by the variation of the parameters of an equivalent linear model of the plant, whereby the latter is able to predict the span of all possible responses generated by the plant. This change of philosophy permits the application of all existing linear control theory. In particular, familiar frequency-domain techniques may be employed in the design of controllers to constrain the operation of the feedback system to within toleranced bounds enveloping an acceptable time response.Thus, a necessary stage in the application of such sensitivity concepts to system design, where the plant may be both nonlinear and uncertain, is the determination of the complete range of equivalent linear models to represent the plant. This requires the identification of an appropriate model structure and the estimation of the range of variation of its parameters.Two methods, which previously have been employed in the time and frequency-domain synthesis of signals, are compared in this new role with a modification of the standard recursive least-squares routine which allows a direct estimation of the Laplacian system function. It is seen that, in general and not unexpectedly, the equivalent linear model may have to be of comparatively high order and, for certain regimes of operation, may call for ‘operationally-localized’ right-half plane poles. However, these attributes are of no disadvantage when the model is to be used for sensitivity reduction based upon the synthesis of the loop frequency response.

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