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A linear time-invariant dynamical system is robustly stable if the system and all of its nearby systems in a neighborhood of interest are stable. An important property of robustly stable systems is that they decay asymptotically without exhibiting significant transient behavior. The first part of this thesis work focuses on measures revealing the degree of(More)
Two useful measures of the robust stability of the discrete-time dynamical system xk+1 = Axk are the -pseudospectral radius and the numerical radius of A. The -pseudospectral radius of A is the largest of the moduli of the points in the -pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We(More)
The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke–Lewis–Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement(More)
The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix(More)
The eigenvalues of a Hermitian matrix function that depends on one parameter analytically can be ordered so that each eigenvalue is an analytic function of the parameter. Ordering these analytic eigenvalues from the largest to the smallest yields continuous and piece-wise analytic functions. For multi-variate Hermitian matrix functions that depend on d(More)
This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. While the usual resolvent norm addresses this question for a single eigenvalue(More)
Structured eigenvalue problems feature a prominent role in many algorithms for the computation of robust measures for the stability or controllability of a linear control system. Structures that typically arise are Hamiltonian, skewHamiltonian, and symplectic. The use of eigenvalue solvers that preserve such structures can enhance the reliability and(More)
We are concerned with the computation of the H∞-norm for H∞-functions of the form H(s) = C(s)D(s)−1B(s), where the middle factor is the inverse of an analytic matrixvalued function, and C(s), B(s) are analytic functions mapping to short-and-fat and talland-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay(More)
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions, and of practical interest because of wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the(More)