Questions on Kalman filter...

Sep 26, 2005 3 Replies
B

I have the following problem to which I am thinking of applying Kalman filter:



x1(k+1) = x1(k) + u1(k)-u2(k)+w1(k) x2(k+1) = x2(k) + u2(k)-u3(k)+w2(k) y(k) = x1(k) + x2(k) + v(k)



w1,w2,v are noises. u1, u2 are control actions. u3 is a disturbance. u1, u2 and u3 are known to only upto a certain level and they are corrupted by noises which I am thinking of modeling as being included in w1 and w2. y is the observation and I need to estimate the two states x1 and x2.



I have read thro' some introductory material on Kalman filters but have the following unanswered questions:


  1. Is the above formulation such that Kalman filter can be applied? Specifically, something bothers me about estimating two variables (x1, x2) from one measurement y which is the sum of the two variables.


  1. If I can apply KF, then what is the structure of the covariance matrix Q of the state noises w1, w2? Note that the presence of u2 in both x1 and x2 equations will correlate w1 and w2 and hence I expect Q to have nonzero off-diagonal elements? Any suggesstions on how to estimate this matrix from the noise characterizations of u1,u2 and u3?

  2. If I can apply KF, an additional requirement I have is that state estimates should be >= 0 at all times. Any idea on how to satisfy this constraint?

TIA



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A

Hello there, I am also new to kalman, As i see from your problem, I believe ur modal is ideal for kalman filter.

1)estimating two variables (x1,x2) from one measurement y which is the sum of the two variables. For this u need to structure ur input-output state transition matrix appropriately 2) for the rest of your questions i would like to read more about ur system. Abhijit
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B

Hi Abhijit, you wrote "...For this u need to structure ur input-output state transition matrix appropriately"

I am not sure what you mean by this. The model I presented is the final form I am thinking of. Mapping it to the state-space representation, one would get

X(k+1) = AX(k)+BU(k)+W(k) y(k) = Hx(k)

where

X = [x1,x2]', U = [u1,u2,u3]', W=[w1,w2]' A = [[1,0],[0,1]] B = [[1,-1,0],[0,1,-1]] H = [1,1]

This is my model.

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R

What do you mean by disturbance ( measurable input, but should be zero?).

If the system is observable you can estimate the state with the Kalman filter. But of course the estimation might be better with additional sensors.

I'd suggest just choosing a Q and doing some simulations. You can regard Q as a tuning means, so being off impacts performance but not functionality. Just take care that Q is positive definite.

That is an interesting question, but I don't think that is possible with a Kalman Filter framework. That would mean, that your system of transition equations is nonlinear. You might want to look into the unscented kalman filter, where you can have even discrete transition equations.

Regards

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