# Norm of Covariance Matrix

Hi to all,
I have been also confused about the covariance matrix of the Kalman filter. I have a Kalman filter which has 9 states and therefore 9 X 9
error covariance matrix which is updated at the every time step.
My question is;
How can i be sure that the Kalman filter works properly by using the error covariance matrices. Should the norm of the error covariance matrix converge to zero or any constant value?
The same problem is valid for the Kalman gains. Should the Kalman gains converge to zero or to any constant value?
Are there any other metrics for evaluating the Kalman filter performance?
Best regards from Germany
Volkan Ozturk
Robert Bosch GmbH
Stuttgart/Germany
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The covariance array will converge to a constant value as long as the process and measurement noise doesn't change.

The Kalman gains better not converge on zero because then there would be no correction between the estimated and actual feedback. The Kalman gains will converge on a non zero values.

I make simulations where I know what the process is doing because I generate the transfer function. I use the transfer function to simulate the process but add process noise and measurement error to generate feed back that resemble reality. I then use that data as the input to the Kalman filter and compare the Kalman filter results with the process without the noise. There isn't much you can do to tweak the output of the Kalman filter except to change the Q and R arrays but if you do that you might as well use a steady state filterr and just adjust the bandwidth to get the desired results. Another thing you should consider is calculating the Kalman gains ahead of time ( steady state ). It will take a lot of processing time to calcalute 9 Kalman gains real time.
Peter Nachtwey
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Hi Peter,
Another question of mine is about the transfer function of an observer designed using a Kalman filter. How can we convert the observer equation to a good form to analyze the frequency response of the oberserver?
The observer equation is:
xdot=Ax+Bu+L(y-Cx);
where L is the Kalman gain, y is the measurement and the input u=0.
I want to get the transfer function between some states inside x and y. I have acceleration estimates inside x, and y measurements are speed measurements. If I take the input as y, and choose the acceleration states from x, I am supposed to get a differentiator characteristic.
I have 9 states which means my Kalman gain is a vector with 9 elements. My Kalman gain is updated in everytime step and converges to constant values for all of the states.
Regards
Volkan