# System identification using Extended Kalman Filter

• December 26th 2007, 05:22 PM
tvauce
System identification using Extended Kalman Filter
Dear friends,

I am trying to solve a system identification problem using EKF. The state which I need to estimate include displacement (d), velocity (v) and system stiffness (k), i.e. X=[d v k]'.
Because only d & v change with time, time derivative of state vector X will produce zero value for the third term (since k=const), i.e. dX/dt=[v a 0]' (in which a is acceleration).
Time update equation using EKF:
X(k+1)=X(k)+dX/dt*T
where T is time step.
Following this equation, my stiffness component k of the state vector X can not be updated meaning that the initial guess value of k is kept unchanged through time update process.
In the observation equation update
X(k+1)=X(k+1)+K*(z(k+1)-h(X))
My observation is displacement d, the Kalman gain K has the size of 3x1 with the last component having zero value at every time instant. My stiffness component of the state vector is thus can not be updated either.
Hence, following EKF procedure, my last component of state vector k can not be updated meanwhile d & v is updated correctly.

Thank you very much for your time and attention.
Best regards,
tvauce
• December 26th 2007, 06:09 PM
CaptainBlack
Quote:

Originally Posted by tvauce
Dear friends,

I am trying to solve a system identification problem using EKF. The state which I need to estimate include displacement (d), velocity (v) and system stiffness (k), i.e. X=[d v k]'.
Because only d & v change with time, time derivative of state vector X will produce zero value for the third term (since k=const), i.e. dX/dt=[v a 0]' (in which a is acceleration).
Time update equation using EKF:
X(k+1)=X(k)+dX/dt*T
where T is time step.
Following this equation, my stiffness component k of the state vector X can not be updated meaning that the initial guess value of k is kept unchanged through time update process.
In the observation equation update
X(k+1)=X(k+1)+K*(z(k+1)-h(X))
My observation is displacement d, the Kalman gain K has the size of 3x1 with the last component having zero value at every time instant. My stiffness component of the state vector is thus can not be updated either.
Hence, following EKF procedure, my last component of state vector k can not be updated meanwhile d & v is updated correctly.

Thank you very much for your time and attention.
Best regards,
tvauce

You are telling us that there is no relationship between the stiffness and
displacement and displacement rate (and no mixing between $k$ and $d$ and $v$
during time propagation). Also that the initial covariance is of
the form:

$\left(
\begin{array}{cc}
Q & 0\\
0 &q
\end{array}
\right)$

where $Q$ is the covariance matrix of the initial estimate of ${d \brack v}$, and $q$ the
variance of the initial estimate of $k$.

If this is so then $k$ is unobservable in this model.

What exactly is $k$ doing?

RonL
• December 26th 2007, 06:20 PM
tvauce
Dear RonL,

All of my variable have the relationship through a second order differential equation:
m*a+0*v+k*d=u
in which m is known constant, u is observed time history input and
dd/dt=v; dv/dt=a (time derivative of d and v, respectively)
with that relationship, can I observe k?

Thank you very much.

Best regards,
tvauce