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

Abstract

This is the preprint version of the article. The final, published version is available on the journal's websiteInternational audienceWe 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

Iterative observer-based state and parameter estimation for linear systems

“…This strategy is generally known as back and forth nudging (BFN), and it was originally proposed by Auroux and Blum in [3] and [4]. A more rigorous treatment was carried out by Ramdani et al in [33] and further studies include [19] by Haine, [20] by Haine and Ramdani, [18] by Fridman, and [1] by Aalto. In [1] it is shown that the initial state estimate given by the BFN method converges to the minimizer of the L 2 -norm of the output discrepancy if the observer gains are taken to zero with a suitable rate.…”

“…The main result of [33] is that if there are no noises, that is, η, ν = 0 in (2.1), and if the system is both forward and backward stabilizable, that is, there exist forward and backward feedback operators K ± such that A − K + C and −A − K − C are exponentially stable, then the initial state estimate given by the BFN method converges exponentially to the true initial state of (2.1). The main result of [1] is concerned with colocated feedback K ± j = κ j C * , and it states that if A is skew-adjoint and the system is exactly observable at time T , then assuming that the observer gains satisfy ∞ j=1 κ j = ∞ and ∞ j=1 κ 2 j < ∞, then the initial state estimate converges to the minimizer of the cost function…”

“…This is an important result from the point of view of this article, since we wish to use the BFN method as a minimization scheme. We do not use the results of [1] directly, but the techniques used in this paper are similar to the techniques used there.…”

“…In this method, the initial state is estimated by the BFN method, and the parameters by taking one Gauss-Newton step between the BFN iterations. As in [1], we intend to use variable gain in the BFN iterations (in general). Therefore also the cost function to which the Gauss-Newton method is applied, changes between iterations.…”

“…This is motivated by the fact that the extended Kalman filter is equivalent to a Gauss-Newton optimization step on a suitably chosen cost function. Using the results of [1] on the optimality of the BFN method, in case the system dynamics are governed by a skew-adjoint main operator, the proposed strategy can be regarded as a hybrid optimization method with separate optimization schemes for the initial state and the system parameters. The method can be used also in the case the main operator is not skew-adjoint as long as the BFN method is stable.…”