Absolute stability problem of systems with parametric uncertainties

The paper deals with the problem of robust absolute stability of systems with parametric uncertainties. By using a 2q-convex parpolygonal value set 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 = [q₁, q₂,..., q_q]^T and the uncertainty box is Q = {q : q_i ϵ [q_i, q_i], i = 1,2,....q}, an extremal boundary result for a transfer function, whose numerator and denominator polynomials are in the form of P(s, q), is first given. Then, using this boundary result, the robust versions of the classical absolute stability criteria of Lur´e, Popov and the Circle criterion are obtained. An example is included to illustrate the benefits of the method presented.