In this paper, we formulate state estimation problem for usually non-time invertible evolutionary systems into a nonstandard finite horizon linear quadratic output tracking problem. This problem directly leads to the classical filtering problem with disturbance constraint. For a given reference trajectory, we give the explicit expressions of optimal initial state and optimal control. In addition, based on some well-known results for classical finite horizon linear quadratic optimal control problems, we obtain, in terms of measured output, the pointwise linear state feedbacks to optimal initial state and optimal control, and the representation of optimal cost is also given. As an important consequence, the differential equation satisfied by the optimal state estimate for filtering problem is developed. Some examples are given as applications.