On the stability of a differential Riccati equation for continuous-discrete observers

Abstract

This article deals with the stability properties of a continuous-discrete Riccati differential equation. The main motivation comes from the theory of Kalman filters for continuous-time nonlinear systems with sampled measurements, where this type of equation often arises. Stability is to be understood in the following sense: under appropriate hypotheses, the solution matrix is always symmetric positive definite and bounded from above and below. No uniformity is expected from the sampling procedure, only an upper bound on the elapsed time between consecutive samples is needed. The exposition consists of two parts. First, the stability properties are proven and second, by means of three examples, it is shown how the article's main result applies.

A Kalman filter for linear continuous-discrete systems with asynchronous measurements

“…The present section is dedicated to the proof of Lemma 2. It follows the ideas exposed in [4] for synchronous continuous-discrete systems, and expands them to the multi-rate setting. For the sake of brevity, this quite long proof is not reproduced here.…”

“…As it is done in [4] -but also in [11] for continuous systems-the existence of bounds for the Riccati equation is demonstrated in two parts:…”

“…Now, we want to combine Lemma 3 with the fact that G c (T ) − G cda (T ) can be made small enough. This is done with the help of the following lemma, which can be proved using similar arguments as the proof of Lemma 3.11 in [4].…”

“…The main difficulties are, on the one hand, dealing with several subdivisions of time in order to represent the asynchronous outputs, and on the other hand, proving that the observer's Riccati equation is bounded over time. This latter issue is handled by following the ideas developed in [4], where only the synchronous setting is considered. Moreover, the present work is done with the purpose of expanding the results to the nonlinear setting (see e.g.…”

“…The proof of this result is quite technical and long, and is therefore not detailed in this article. It basically follows the ideas developed in [4], with an increased complexity coming from the asynchronicity of the measurements. However, this paper adresses the important issue of the preservation of observability under multi-rate sampling, which is a necessary condition in order to bound the Riccati matrix.…”