The problem of estimation of states in nonlinear dynamical systems containing time delays is formally studied. The plant is specified by a set of nonlinear differential-difference equations. Observations are a nonlinear function of current and/or delayed states. Both contain deterministic additive disturbances. The criterion used for the optimal estimates is the integral of the weighted squared error. Using the theory of the calculus of variations, equations are developed for the estimation. They are first expressed in the form of a split boundary value problem, which is then converted to an (approximate) initial value problem for online estimation. The result yields an estimation scheme in which filtered and smoothed estimates are computed in a sequential manner. The applicability of the procedure is demonstrated by estimating variables of a nonlinear model describing the behavior of a stirred tank reactor.

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