TY - JOUR

T1 - Recovering the initial state of an infinite-dimensional system using observers

AU - Ramdani, Karim

AU - Tucsnak, Marius

AU - Weiss, George

PY - 2010/10

Y1 - 2010/10

N2 - Let A be the generator of a strongly continuous semigroup double-struck T sign on the Hilbert space X, and let C be a linear operator from script D sign(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∈script D sign(A) (with respect to the norm of X) from the output function y(t)=Cdouble-struck T signtz0, 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 double-struck T sign to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrdinger equations, and we provide results from numerical simulations.

AB - Let A be the generator of a strongly continuous semigroup double-struck T sign on the Hilbert space X, and let C be a linear operator from script D sign(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∈script D sign(A) (with respect to the norm of X) from the output function y(t)=Cdouble-struck T signtz0, 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 double-struck T sign to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrdinger equations, and we provide results from numerical simulations.

KW - Back and forth nudging

KW - Estimatability

KW - Exact observability

KW - Observers

KW - Russell's principle

KW - Strongly continuous semigroup

KW - Time reversal focusing

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=77956397275&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2010.06.032

DO - 10.1016/j.automatica.2010.06.032

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AN - SCOPUS:77956397275

SN - 0005-1098

VL - 46

SP - 1616

EP - 1625

JO - Automatica

JF - Automatica

IS - 10

ER -