Convergence of discrete time Kalman filter estimate to continuous time estimate

This article is concerned with the convergence of the state estimate obtained from the discrete time Kalman filter to the continuous time estimate as the temporal discretization is refined. We derive convergence rate estimates for different systems, first finite dimensional and then infinite dimensional with bounded or unbounded observation operators. Finally, we derive the convergence rate in the case where the system dynamics is governed by an analytic semigroup. The proofs are based on applying the discrete time Kalman filter on a dense numerable subset of a certain time interval $[0,T]$.Comment: Author's version of the manuscript accepted for publication in International Journal of Contro

Output error minimizing back and forth nudging method for initial state recovery

We show that for linear dynamical systems with skew-adjoint generators, the initial state estimate given by the back and forth nudging method with colocated feedback, converges to the minimizer of the discrepancy between the measured and simulated outputs - given that the observer gains are chosen suitably and the system is exactly observable. If the system's generator A is essentially skew-adjoint and dissipative (with not too much dissipation), the colocated feedback has to be corrected by the operator e^{At}e^{A*t} in order to obtain such convergence. In some special cases, a feasible approximation for this operator can be found analytically. The case with wave equation with constant dissipation will be demonstrated.Comment: This is the preprint version of the article. The final, published version is available on the journal's websit

Sense and Substance in Wittgenstein"s Tractatus

In the early pages of his Tractatus, Wittgenstein says that the substance of the world consists of unalterable, simple objects (Gegenstände) (2.021, 2.027). Substance is connected to the Sinn of a proposition in the following cryptic way: "If the world had no substance, then whether a proposition had sense [Sinn] would depend on whether another proposition was true.â€? (2.0211)â€? "In that case we could not sketch any picture of the world (true or false).â€? (2.0212) How can the sense of a proposition depend on the existence of simple, unalterable objects? No connection between Sinn and substance is evident on the basis of these short remarks. Furthermore, the remarks sound strange because elsewhere in the Tractatus Wittgenstein makes it clear that we can understand the sense of a proposition without knowing anything about how things actually stand in the world (see e.g. 4.024)