Frequency Response Envelopes of a Family of Uncertain Discrete-Time Systems

Abstract

An efficient technique is presented for the calculation of the exact amplitude and phase frequency response envelopes (Bode envelopes) of an infinite impulse response filter (i.e., a real rational function in a complex variable) with interval uncertainty in the coefficients. In the first stage, the order of complexity is reduced from 2n to n2, using the structure of the value set. In the second stage, the problem is completely solved by calculating the frequency response of a finite number of pertinent transfer functions with fixed coefficients. The computations involved in this stage consist of finding a number of zeros of certain polynomials plus some combinatorics. The algorithm extends easily to complex coefficient filters with box uncertainties. Also, the problem trivializes for complex filters with disk uncertainties. However, this latter case leads to extremely simple and completely analytic bounds for the frequency response envelopes of both real and complex coefficient filters.