New approach to assessing the effects of parametric variations in feedback loops

A method for the computation of the magnitude and phase envelopes of uncertain transfer functions is presented. The idea is to factor the transfer function into its real and complex pair roots and find the maximum and minimum magnitudes of the gain and phase of each factor. The Bode envelopes of the given uncertain system are then found from those of the individual factors. This approach, which is different from those based on the interval polynomial method of Kharitonov, has the major advantage that the representation is more applicable to practical situations where typically the coefficients of the various factored terms relate to physical parameters of a mathematical model. Further, the method results in the narrowest Bode envelopes and therefore can yield improved controller designs. The describing function analysis and absolute stability problem of nonlinear systems with variable plant parameters are also studied. An approach which enables one to predict the existence of limit cycles in a control system which simultaneously contains nonlinearities and parametric uncertainties is given. The proposed method makes use of the popular describing function technique and these narrowest possible Bode envelopes of linear uncertain transfer functions in factored form. The technique can be used to cover the cases of linear elements, which have a multilinear or nonlinear uncertainty structure, and a nonlinear element with or without memory. New formulations of the circle and off-axis circle criteria are given for use with Bode diagrams so that the absolute stability of nonlinear systems with variable plant parameters can be studied. Examples are given to show how the proposed method can be used to assess the effects of parametric variations in feedback loops.