Dear All,
I have a question about Kalman filter. I am using kalman filter for a state space model as following:
X(k+1)=A(k)x(k)+B(k)u(k)+w(k), w(k) ∼ N(0,Q)
Y(k)=C(K)x(k)+D(k)u(k)+v(k), v(k) ∼ N(0,R)
Which the state space matrixes (A(k),B(k),C(k),D(k)) are updated in each sampling time but Q and R matrixes are considered to be constant. The equations which calculate the kalman gain (K(k)) and covariance P matrix (P(k)) are as following:
K(k) = (P(k)*C(k)' )/(R + C(k)*P(k)*C(k)');
Pxx(k)=(eye(n)-K(k)*C(k))*P(k)*(eye(n)-K(k)*C (k))'+K(k)*R*K(k)';%Joseph form
P(k)= A(k)*Pxx(k)*A(k)' + Q;
The problem that I face is related to stability of (A(k)-K(k)*C(k)). In some sampling times, the calculated kalman gain can not stabilize the (A(k)-K(k)*C(k)) matrix and the eigenvalues of (A(k)-K(k)*C(k)) are outside of unit circle. Could you please help me to figure out the reason for this problem? I am expecting that the kalman filter gives me the gain which make the (A(k)-K(k)*C(k)) matrix stable with eigenvalues inside the unit circle.
"Just to clarify, the state space matrixes are updated based on a subspace identification technique in each sampling time. So, I am not using an extended kalman filter. Furthermore, In each sampling time which the state space matrixes are updated, I also check the observabilty matrix of the system which is full rank all the time. I also check the controlability matrix based on (A, Q^1/2) which is full rank. Another point, in subspace identification technique which I am using, input and output data are considered from a real process without any preprocessing such as filtering."
You need to check detectablity and stabilizability of your system! Since your system is timevarying, these concepts are not so straightforward and easy to check as in the time-invariant case: checking the rank of the controlability / observability matrices might not be enough in the timevarying case. You should look at the Gramians. I suggest checking textbooks on timevariant control.
You told that the state matrices are uptated in each sampling period. Does it mean that the plant model is nonstationary? If yes, you should use a Extended Kalman Filter.
For the calculation of a stabilizing Kalman gain, we need detectability of the system dynamics, i.e,. all unstable (including states corresponding to poles on the imaginary axis) states are observable.
For an extended Kalman filter, which is what you appear to be implementing, this depends upon the local linearization, and the measurements you have. Instability can occur simply because of a burst of noise in your measurement for example.
To avoid instability, you need to use projection methods that bound your estimator based on your physical knowledge.
Dear all
First of all, thank you. I really appreciate your answers. Just to clarify, the state space matrixes are updated based on a subspace identification technique in each sampling time. So, I am not using an extended kalman filter. Furthermore, In each sampling time which the state space matrixes are updated, I also check the observabilty matrix of the system which is full rank all the time. I also check the controlability matrix based on (A, Q^1/2) which is full rank. Another point, in subspace identification technique which I am using, input and output data are considered from a real process without any preprocessing such as filtering.
I think it is the sampling time. use high sampling time. To figure out the stable sampling time minimum, find the discrete model matrices and check the sampling time effect on the stability of the z-domain poles of the system.
Best!
You need to check detectablity and stabilizability of your system! Since your system is timevarying, these concepts are not so straightforward and easy to check as in the time-invariant case: checking the rank of the controlability / observability matrices might not be enough in the timevarying case. You should look at the Gramians. I suggest checking textbooks on timevariant control.
Although you are using the Joseph form of the covariance update equation, you still might be facing a numerical stability issue, in which case square-root filtering is the solution. A thought to consider.
I agree with Patrick and suggest to study textbooks on the stability of linear time-varying (LTV) systems. The question is why you want (A(k)-K(k)*C(k)) to be stable at every sampling time? Of course, in the time-invariant (LTI) case, this implies asymptotic stability of the estimation error. However, this is no longer the case in the LTV situation, as the following example shows:
Define H(k) = (A(k)-K(k)*C(k)). Then, the estimation error evolves as e(k+1) = H(k)*e(k). Now assume that H(0) = [.6 .3; .1 .9] with eigenvalues 0.52, 0.97. Assume now that H(1) = (H(0))^T and therefore has the same (stable) eigenvalues. However, the eigenvalues of the product H(1)*H(0) are 0.26 and 1.01, i.e. they are unstable! If you assume that for even k, H(k) = H(0) and for odd k H(k) = H(1), then you can see that the estimation error will eventually blow up, even if all individual H(k) matrices have stable eigenvalues!
The conclusion is to be aware that many LTI results become inconclusive in the LTV case.
When using the Kalman filter, there are tunable parameters that should be suitably adjusted in order to get a stable Kalman gain. Among the more influencing parameters, the process covariance matrix Q, the sampling time step and the noise covariance matrix R should be adjusted while observing the behavior of the Kalman filter.
https://www.researchgate.net/publication/267326130_Multi-site_modeling_and_prediction_of_annual_and_monthly_precipitation_in_the_watershed_of_Cheliff_Algeria
I suggest taking a look at this very interesting publication on the subject.
Article Detectability Analysis and Observer Design for Linear Time V...
http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf
Dear Iman Hajizadeh
This can happen when the system is unobservable. Check the rank of the observability matrix
Best regards
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