Mathematical Problems in Engineering
Volume 4 (1998), Issue 2, Pages 135-163
Nonnegativity of uncertain polynomials
1Department of Electrical Engineering, Santa Clara University, Santa Clara 95053, CA, USA
2Department of Electrical Engineering, Stanford University, Stanford 94305, CA, USA
Received 8 November 1996
Copyright © 1998 Dragoslav D. Šiljak and Matija D. Šiljak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.