Recovering the initial state of an infinite-dimensional system using observers

Abstract

. Recovering the initial state of an infinitedimensional system using observers. Automatica, Elsevier, 2010, 46 (10) 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 z 0 ∈ D(A) (with respect to the norm of X) from the output function y(t) = CT t z 0 , 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 z 0 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.

Improving Convergence in Numerical Analysis Using Observers — The Wave-Like Equation Case

“…These methods were initially proposed for systems of rather limited sizes, and indeed only much more recently have they been considered for infinite-dimensional systems modeled by PDEs -as such in Refs. 39,18,15,34 and under the impetus of data assimilation in Refs. 28, 1 -as an effective alternative to sequential methods based on optimization principles -in particular the Kalman filter or various methods derived thereof -which are not easily suited to such systems due to the "curse of dimensionality" coined by R.E.…”

Data assimilation of time under-sampled measurements using observers, the wave-like equation example

“…Combining (29), (31) and the fact that P n L(X ) ≤ 1 and R n L(X ) ≤ 1, we obtain the following estimate…”

“…In order to benefit from the available data z(t) and considering only the available a priori x 0 that we have on the initial condition, we consider the Luenberger observerx(t) [7] -see also similar formulations in [13,31] -estimating x • (t) from the dynamics…”

“…As an alternative for this "curse of dimensionality", numerous strategies have been proposed for example the Ensemble Kalman Filter [15] or Reduced Order Kalman-like filters [34]. In the specific context of the conservative wave-like equation, however, several works [7,23,27,31] have rather proposed simplified but effective feedback laws directly based on the physical properties of the system at hand which stabilize at a certain rate -potentially sub-optimal -any errors. This idea follows the path proposed by Luenberger's work [25] for finite dimensional systems and is popularized for PDEs with the nudging appellation as initiated in [1,19] -a complete historical perspective can be found in [21].…”

Data assimilation experiments using diffusive back-and-forth nudging for the NEMO ocean model

“…The BFN convergence was proved by Auroux and Blum (2005) for linear systems of ordinary differential equations and full observations, by Ramdani et al (2010) for reversible linear partial differential equations (wave and Schrödinger equations), and by Donovan et al (2010) and Leghtas et al (2011) for the reconstruction of quantum states, and was studied by Auroux and Nodet (2012) for non-linear transport equations. The BFN performance in numerical applications using a variety of models, including non-reversible models such as a shallow water (SW) model (Auroux, 2009) and a multi-layer quasi-geostrophic (LQG) model (Auroux and Blum, 2008), are very encouraging.…”