An estimator is presented which generates sequential estimates for nonlinear, time-variable discrete-time dynamic systems in which the system state estimates are subject to an instantaneous constraint. That is, at each sample time the state estimate is constrained to lie in a given region of the state space. This nonlinear sequential estimator is an extended version of an optimal sequential estimator for linear, time-variable discrete-time systems with state estimates constrained to a given region of the state space. The linear estimator was developed from a non-probabilistic weighted linear least squares basis with the constraints added through the mechanism of Lagrange multipliers; therefore, the estimator produces "hard" constraints on the state estimate. The solution of the constrained estimation problem, at each instant of time, requires only the unconstrained state estimate at that time instant and the instantaneous constraints which define the constraint region. If the unconstrained sequential estimate satisfies the constraints, then that solution is also the constrained solution. On the other hand, if the unconstrained estimate does not satisfy the constraints, then the constrained solution is generated from the solution of a set of static equations. The constrained estimation problem is thus reduced to a sequence of nonlinear programming problems. The estimator for the state of a nonlinear system was developed by quasi-linearization of the optimal constrained linear estimator. The estimates resulting from this estimator are "optimal in the small" for nonlinear systems and are optimal for linear systems.