Two numerical methods for optimizing matrix stability Original Research Article

Consider the affine matrix family A(x)=A0+∑k=1mxkAk, mapping a design vector image into the space of n×n real matrices. We are interested in the question of how to choose x to optimize the stability of the matrix A(x). A typical motivation is that one wishes to control the stability of the dynamical system image . A classic example is stabilization by output feedback. We take two approaches. Let α(X) denote the spectral abscissa (the largest real part of the eigenvalues) of a matrix X; as is well known, this quantity bounds the asymptotic decay rate of the trajectories of the associated dynamical system image . Our first approach to optimizing stability is to directly minimize the function α(A(x)). The spectral abscissa α(X) is a continuous but non-smooth, in fact non-Lipschitz, function of the matrix argument X, and finding a global minimizer of α(A(x)) is hard. We introduce a novel random gradient bundle method for approximating local minimizers, motivated by recent work on non-smooth analysis of the function α(X). Our second approach is to minimize a related function αδ(A(x)), where δ is a robustness parameter in (0,1). One motivation for the definition of the “robust spectral abscissa” αδ(X) is that it bounds transient peaks as well as asymptotic decay of trajectories of image . The function αδ(X) is Lipschitz but non-convex for δset membership, variant(0,1), approaching α(X) as δ→0 and the largest eigenvalue of image as δ→1. We use a Newton barrier method to approximate local minimizers of αδ(A(x)). We compare the results of the two approaches on a number of interesting test cases.