Recovering the initial state of an infinite-dimensional system using observers |
| |
Authors: | Karim Ramdani [Author Vitae] Marius Tucsnak [Author Vitae] George Weiss [Author Vitae] |
| |
Affiliation: | a INRIA Nancy Grand-Est (CORIDA), 615 rue du Jardin Botanique, 54600 Villers-les-Nancy, Franceb Université Henri Poincaré (IECN), B.P. 70239, 54506 Vandoeuvre-les-Nancy, Francec Department of EE-Systems, Tel Aviv University, Ramat Aviv 69978, Israel |
| |
Abstract: | Let A be the generator of a strongly continuous semigroup T on the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X, not necessarily admissible). We consider the problem of estimating the initial state z0∈D(A) (with respect to the norm of X) from the output function y(t)=CTtz0, given for all t in a bounded interval [0,τ]. We introduce the concepts of estimatability and backward estimatability for (A,C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z0 from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell’s principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations. |
| |
Keywords: | Strongly continuous semigroup Observers Estimatability Exact observability Russell&rsquo s principle Back and forth nudging Time reversal focusing Wave equation |
本文献已被 ScienceDirect 等数据库收录! |
|