Current algebras and the identification problem

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Abstract
In this paper, we investigate the identification problem of linear system theory from the viewpoint of nonlinear filtering. Following the work of Brockett and Mitter, one associates in a natural way a certain (infinite dimensional) Lie algebra of differential operators known as the estimation algebra of the problem. For the identification problem the estimation algebra is a subalgebra of a current algebra. In this paper we study questions of representation and integrability of current algebras as they impinge upon the identification problem. A Wei-Norman type procedure for the associated Cauchy problem is developed which reveals a sequence of functionals of the observations that play the role of joint sufficient statistics for the identification problem.