A nonlinear filtering theory from a deterministic point of view is presented and an application to attitude determination is considered. The approach that is taken in this paper is motivated largely by the H∞ control and estimation theory for linear systems which has been evolved within the last decade. Rather than formulating the estimation problem as a game played by two adversaries, as has been done in the linear case, we employ in this work some notions from the theory of dissipative systems as a vehicle for arriving at certain Hamilton-Jacobi inequality, which in turn, provides a solution to the filtering problem, whenever it is satisfied. Application of this method to a linear estimation problem and to the problem of estimating a spacecraft attitude quaternion and gyro drift bias vector are presented. In limiting cases these give the Kalman filter and the extended Kalman filter respectively. The main advantages of this approach over its probabilistic counterpart are that this approach does not require a prior knowledge of any statistics, and that in general it is more amenable to a quantitative assessments regarding approximations such a linearization, and that in certain cases this approach yields an exact solution to the nonlinear filtering.