I am working on designing an interconnected distributed control system. In this system, I need to use Kalman filter because the measurements are noisy. I have two questions:
1 - What is the relationship between the infinity norm estimation error and the square root of trace of the estimation error covariance matrix in Kalman filter?
I have found a paper that has an inequality relationship between the two mentioned terms. This relationship can be found on Page 4 below Equation 14. The relationship is:
$E(\|\mathbf{\hat{s}}(k)-\mathbf{s}(k)\|_{\infty})\leq\sqrt{tr(\Sigma(k))}$,
where $\mathbf{\hat{s}}(k)$ is the estimated state vector at time step $k$, $\mathbf{s}(k)$ is the actual state vector, and $\Sigma(k)$ is the estimation error covariance matrix. The Kalman filter equations are taken from the original paper by Kalman.
I can not find a reference where this property of Kalman falter is mentioned. If anyone can introduce a reference for this property, I would be grateful.
2 - I want to use the same idea to apply it to a distributed Kalman filter from paper I found with the title "Distributed Kalman Filtering for Interconnected Dynamic Systems". The paper can be found in ieee Xplore. In this paper, they have found an upper bound for the trace of estimation error covariance matrix to show that the proposed distributed Kalman filter is asymptotically stable.
My main question is if the inequality I mentioned in part 1 is a purely mathematical relationship between a variable and its covariance matrix or it is a property of Kalman filter.
Thank you so much in advance!
