A Distance Measure for Perspective Observability and Observability of Riccati Systems

Systems governed by Riccati differential equations arise in several areas of control system theory. In combination with a linear fractional output, observability of such systems is relevant in the context of robotics and computer vision, for example, when studying the reconstruction of point locations from their perspective projections. The so-called perspective observability criteria exist to verify this observability property algebraically, but they provide only a binary answer. The present contribution studies the assessment of perspective and Riccati observability in a quantitative way, in terms of the distance to the closest nonobservable system. For this purpose, a distance measure is proposed. An optimization problem for determining it is derived, which features a quadratic cost function and an orthogonality constraint. The solution of this optimization problem by means of a descent algorithm is discussed and demonstrated in the course of a practically motivated numerical example.