Minimum residual error in estimation of deterministic system using Kalman filter

Question

Let the process equation for a state vector $\mathbf{x}_t$ at time $t$ be: \begin{equation} \bf{x}_{t+1} = \bf{f}\left(\bf{x}_t\right) \end{equation} where, $\mathbf{f}\left(.\right)$ is a nonlinear function. There is no process noise. So, the $\mathbf{x}_t$ is deterministic at every given time. Similarly, the measurement equation is given as: \begin{equation} \bf{y}_{t} = \bf{h}\left(\bf{x}_t\right) \end{equation} So, there is no noise in the measurement, as well. Now we use a Kalman filter (KF )framework to estimate the states $\mathbf{x}_t$ at every given time, using the measurements. However, we assume some non-zero process noise $\mathbf{Q}\left(t\right)$ and measurement noise $\mathbf{R}\left(t\right)$ as required by the KF framework. We also assume initial error in the states as $\mathbf{P}_0$. My question is : What is the minimum error in the estimation of $\mathbf{x}_t$ using this KF framework as a function of $\mathbf{Q}\left(t\right)$, $\mathbf{R}\left(t\right)$ and $\mathbf{P}_0$?

Answer 1

I doubt that there is an answer to this question in general, but there are a couple of ways that you might approach the problem.

For linear systems, there are known ways of determining the steady-state covariance, which would give you an idea of the minimum error that you would expect. I know that Dan Simon's Optimal State Estimation book covers this in Section 5.4.1. If you don't have the book, I can update my answer to include the method.

For nonlinear systems that are periodic (i.e. the dynamics and measurement matrices repeat every $N$ timesteps) I wrote a paper a couple of years ago that finds the steady-state covariance. I can share the paper with you if that sounds like it would be helpful for you.