Deterministic Kalman Filtering on Semi-infinite Interval

We relate a deterministic Kalman filter on semi-infinite interval to linear-quadratic tracking control model with unfixed initial condition.


Introduction
In [4], E. Sontag considered the deterministic analogue of Kalman filtering problem on finite interval.The deterministic model allows a natural extension to semi-infinite interval.It is of a special interest because for the standard linearquadratic stochastic control problem extension to semi-infinite interval leads to complications with the standard quadratic objective function (see e.g.[1]).According to [4], the model which we are going to consider has the following form: x(0) = x 0 . ( Here we assume that the pair (x, u) ∈ a(x 0 ) + Z, where Z is a vector subspace of the Hilbert space ) a Hilbert space of R n −value square integrable functions) defined as follows: by n matrix; ȳ ∈ L r 2 [0, +∞).Notice that in (1) -(3) x 0 is not fixed and we minimize over all triple (x, u, x 0 ) ∈ L n 2 [0, +∞) × L m 2 [0, +∞) × R n satisfying our assumption.
Notice also that we interpret (1) -( 3) as an estimation problem of the form ẋ = Ax + Bu, where we try to estimate x with the help of observation ȳ by minimizing perturbations u, v and choosing an appropriate initial condition x 0 .

Solution of the deterministic problem
Consider the algebraic Riccati equation where L = BR −1 B T .Assuming that the pair (A, B) is stabilizable and the pair (C, A) is detectable, there exists a negative definite symmetric solution K st to (4) such that the matrix A+LK st is stable (see e.g.Theorem 12.3 in [5]).Using the result of [2], we can describe the optimal solution to ( 1) -( 3) with fixed x 0 as follows.
There exists a unique solution Moreover, ρ 0 can be explicitly described as follows: The optimal solution (x, u) to ( 1) -( 3) has the form For details see [2].Notice that ρ 0 does not depend on x 0 .To solve the original problem ( 1) -(3) we need to express the minimal value of the functional (1) in term of x 0 .Theorem 1.Let (x, u) be an optimal solution of ( 1) -( 3) with fixed x 0 given by ( 5) -( 8).Then Remark.Notice that J(x, u, x 0 ) is a strictly convex function of x 0 and hence minimum of J as a function of x 0 is attained at Hence ( 5) -( 8) gives a complete solution of the original problem (1) -(3).

Steady state deterministic Kalman filtering
In light of (10), it is natural to consider the process as a natural estimate for the optimal solution to problem (1) -( 3).Let us find the differential equation for z.
Remark: Notice that K −1 st is a solution to the algebraic equation In other words, the differential equation ( 12) is a precise deterministic analogue for the stochastic differential equation describing the optimal (steady-state) estimation in Kalman filtering problem.See e.g.[1].
Proof.Using ( 5) and (11), we obtain: Since K st is a solution to (4), we have

Concluding remarks
The differential equation (12) allows one to recursively estimate of z based on observation ȳ provided we know z(0) = x opt 0 .Notice that according to ( 6) requires the knowledge of the whole ȳ.However, due to the integral nature of this relationship the infinite integral can be very well approximated by the integral over the finite interval [0, T ] (since the matrix A + LK st ) T is stable) and hence the recursive nature of ( 12) is recovered for t ≥ T for some finite T .Notice, further, that similar problem with discrete time can be solved using the formalism developed in [3].