Contributions to the theory of robustness of systems with multilinear uncertainty structure

This dissertation addresses a number of robustness analysis problems for systems whose description involves multilinear functions of uncertain parameters. For many common systems which possess this multilinear uncertainty structure, standard robustness problems are NP-hard. One of our approaches to this fundamental difficulty is to soften the problem formulation to allow violation of the robustness specification over some set of acceptably small volume in parameter space. To this end, in some cases, we view the uncertain parameters as random variables and consider probabilistic performance measures which are estimated by Monte Carlo techniques; in other cases exact volume bounds are obtained. The first problem addressed involves the solution of a linear system of equations depending on uncertain parameters which enter the system matrix as rank-one perturbations. Subsequently, we obtain a multilinear factorization of the partial derivatives of the solution components, leading to an extreme point test for convex, concave, or monotonic dependence of the solutions; this facilitates the computation of a probabilistic performance measure. The second problem considered is that of robust negativity of a multilinear function over a hypercube. Since this problem is NP-hard, we instead determine whether the function is practically robust; i.e., the function is negative everywhere except on an acceptably small subset of the hypercube. The third problem addressed involves dynamic systems which depend multilinearly on random complex gains with probability distribution known only to belong to a given class. In this setting, we obtain distributionally robust results describing the maximum and minimum achievable value of a measure of the overall system gain. Our fourth result is a distributionally robust worst-case estimate of the probability of stability for polynomials with multilinear dependence of the coefficients on the uncertain parameters. While this estimate recovers the deterministic result provided by the Mapping Theorem, it does not require extreme point tests. The final problem addressed involves random resistive networks with normally distributed resistances or conductances. In this case, we exploit the homogeneous multilinear uncertainty structure of the equivalent resistance and conductance to obtain explicit formulae for the probability distributions of these quantities.