Control System Radii and Robustness Under Approximation

Abstract

Summary. The purpose of this paper is twofold. First, we provide a short review and summarize results on the robustness of controllability and stabilizability for finite dimensional control problems. We discuss the computation of system radii which provide a measure of robustness. Second, we consider systems which arise as finite difference and finite element approximations to control systems defined by partial differential equations. In particular, we derive controllability criteria for approximations of the controlled heat equation which are easy to check numerically. For a particular example we establish tight theoretical upper and lower bounds on the controllability radii for the finite difference and finite element models and compare these bounds with numerical results. Finally, we present numerical results on stabilizability radii which suggests that conditioning of the LQR control problem may be measured by this radii.

Partial Observability and Its Consistency for Linear PDEs

“…In this section, some examples are used to illustrate Definition 1. In the literature, a measure of controllability for linear systems is defined based upon the radius of matrices [1,4,14]. Using duality, we can define the observability radius, γ o , as the distance between the system and the set of unobservable systems.…”

Mesh Independence of Kleinman–Newton Iterations for Riccati Equations in Hilbert Space

“…It is important to note that even standard numerical schemes may not preserve these important control system properties. However, for the delay systems considered below it is known that all of the schemes discussed in Kappel's survey [35] satisfy these conditions (see [15], [18], [19], and [28]). Moreover, as we see below, although these conditions are sufficient for the finite dimensional Newton iterates to converge, they do not guarantee that the limit of the Newton iterates X N ∞ converges to X ∞ as N → +∞.…”