Robust Stability of Multilinear Affine Polynomials

This paper deals with the robust stability problem of multilinear affine polynomials. By multilinear affine polynomials, we mean an uncertain polynomial family consisting of multiples of independent uncertain polynomials of the form P(s, q) = l0(q)+l1(q)s+···+ln(q)sn whose coefficients depend linearly on q = [q1, q2, ..., qq]T and the uncertainty box is Q = {q : qi¿[q_i_, q~i~], i = 1, 2, ...., q}. Using the geometric structure of the value set of P(s, q), a powerful edge elimination procedure is proposed for computing the value set of multilinear affine polynomials. In order to construct the value set of a multilinear affine polynomial, the mapping theorem can be used. However, in this case, it is necessary to find the images of all vertex polynomials and then taking the convex hull of the images of the vertex polynomials in the complex plane which is a computationally expensive procedure. On the other hand, the approach of the present paper greatly reduces the number of the images of vertex polynomials which are crucial for the construction of the value set. Using the proposed approach for construction of the value set of multilinear affine polynomials together with the zero exclusion principle, a robust stability result is given. The proposed stability result is important for the robust stability of control systems with multilinear affine transfer functions.