Frequency response of uncertain systems: a 2q-convex parpolygonal approach

Abstract

The paper deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are independent uncertain polynomials of the form P(s, q)=a0( q)+a1( q)s+…+an( q)sn, whose coefficients depend linearly on q=[p1 , p2 ,…, pq]T, and the uncertainty box is Q={q:pi∈[ pi, pi], i=1, 2,…, q}. Using the geometric structure of the value set of P(s, q), powerful procedures are developed for computing the frequency response of these uncertain systems. A feature of the approach is the use of the 2q-convex parpolygonal value set of P(s, q) and transition frequency concept. Thus, the approach eliminates some exposed edges of the corresponding polytopes of the numerator and denominator polynomials which are not useful for construction of the Bode, Nyquist, and Nichols envelopes. These results are used for computing robust gain and phase margins, and to design robust controllers for systems with affine linear uncertainty. Examples illustrate the benefit of the presented method.