Sensitivities of stability constraints and their applications

Abstract

Let the real polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(a(s) = a_{0} + a_{1}s + ... + a_{n}s^{n}</tex> with the coefficients being known differentiable functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a_{k}(x)</tex> be given and let the constraints <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_{i}(x) &gt; 0</tex> determine the strictly Hurwitz property of the polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(s)</tex> . A simple and efficient method to calculate the derivatives <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\partial g_{i}(x)/\partial x_{j}</tex> is proposed. Then, the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed. Also, a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given.