Kalman Filter of a Stochastic Vectorial Linear Differential Equation with 1 Delay

Question

I want to find the Kalman Filter of a plant with an equation with the form

$$\dot{x}(t)=Ax(t)+Bx(t-\tau)+W_{t}$$

In which $\tau > 0$, $x(t)$ is a real vector, $A$ and $B$ are fixed matrices, and $W_t$ is a Wiener process with covariance $\Sigma$. You could consider $\Sigma$ diagonal if such restriction is needed. The initial condition is deterministic

$$x(t)=\phi(t),\;t\in[-\tau,0]$$

You could set it as nice as you need.

To develop the Kalman filter you may exactly solve the equation in intervals $[t_k,t_k+\Delta t]$. For each of these solutions the solution in the previous interval will be the initial condition. Hopefully you could write this in a recursive way, and by that means you would discretize the plant. The procedure is described in Optimal and Robust Estimation With an Introduction to Stochastic Control Theory, Chapter 3. I know of an exact solution in the scalar case, given in the paper

Uwe Küchler & Beatrice Mensch (1992): Langevins stochastic differential equation extended by a time-delayed term, Stochastics and Stochastic Reports, 40:1-2, 23-42,

But I haven't found any solution in the vectorial case. However, I came up with this idea. If you make the Taylor expansion of $x(t)$

$$x(t-\tau)=x(t)-\dot{x}(t)\tau+o(\tau)$$

And consider $\tau$ is small enough, you will obtain the approximate equation

$\dot{x}(t)=(I+\tau B)^{-1}(A+Bx(t))+(I+\tau B)^{-1}W_t$

This approximate, solvable equation induce errors in the solution, but Kalman Filters are meant to deal with errors anyway. The errors induced by this approximation could be corrected in the measurement update step together with the stochastic noise errors.

So, summarizing I have two questions

1. Do you know the exact solution to the original equation?
2. Do you think the linearized approximate equation filter is an admissible approach relaying on the measurement update correction as a compensation?

Answer 1

Sorry, my reputation has not reached 50, so I give you comments here.

1. This equation you given should be a kind of functional differential equation. Even without $W_t$, I think there is no closed form for the solution. (please refer to the book "Stability Analysis and Robust Control of Time-Delay Systems" by Min Wu et al.)

2. Since the functional differential equation means it is not a Markov process (See you initial condition is in a function form. This decides that this system is a distributed system, i.e., the system state cannot be completely determined by one previous time instant). So I guess the Kalman filter might not work. But I am not sure if Wiener filter can be applied.

Hopefully, someone can give you a better answer, and I am also waiting for it.

Cheers, Ryan