Transfer Function of an Observer designed by Kalman filter

Hi to all,

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

Once your Kalman gain has settled out, your system is a linear time invariant system. If u is zero then you can eliminate it from the equation. Rearranging you system, you get:

xdot = (A - LC)x + Ly

If you let Cn be a selector output gain that chooses the nth element out of x (i.e. if you want x1 then C1 = [1 0 0 0 0...], if you want x4 then C4 = [0 0 0 1 0 ...]). Then your transfer function from y to xn is just

Hn = Cn * (Is - (A - LC))^-1 * L

Assuming you're in s -- 'Is' in this case is the identity matrix times s.