On approximating the nearest Ω‐stable matrix. Issue 3 (28th February 2020)

• Record Type: Journal Article
• Authors: Choudhary, Neelam; Gillis, Nicolas; Sharma, Punit
• Is Part Of: Numerical linear algebra with applications. Volume 27:Issue 3(2020)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-02-28
• Subjects: Ω‐stability -- convex optimization -- linear time‐invariant systems -- stability radius; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2282
• Languages: English
• ISSNs: 1070-5325

Abstract

Summary: In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω of the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips, and disks. We refer to this problem as the nearest Ω‐stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time‐invariant systems. In order to achieve this goal, we parameterize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples. Abstract : Illustration of Ω = { x + i y | sin ( θ ) x < cos ( θ ) y < − sin ( θ ) x, x < − h < 0, | x + i y | < r } =Ω C (0, θ )∩Ω V ( h, + ∞ )∩Ω D (0, r ).