Magnitude and phase envelopes of systems with affine linear uncertainty

In this paper, the magnitude and phase extremums of a family of polynomials of the form P(s, q)=a₀(q)+a₁(q)s+...+a _n(q)sⁿ whose coefficients depend linearly on q=[p ₁, p₂, …, p_q]^T and Q={q: p_i∈[p_i, p_i],i=1, 2, …, q} are first obtained by using the geometric structure of the value set. The magnitude and phase extremums of this polynomial family multiplied with a fixed polynomial are then investigated. Finally, a procedure is presented for computing the Bode envelopes of a control system with parametric uncertainty. The distinguishing feature of the results given in this paper is the efficient procedure introduced for constructing the 2q-convex parpolygon of P(s, q)