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Math Help - System identification using Extended Kalman Filter

  1. #1
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    Thumbs down 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.

    Could you please kindly advise my?

    Thank you very much for your time and attention.
    Best regards,
    tvauce
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by tvauce View Post
    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.

    Could you please kindly advise my?

    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(<br />
\begin{array}{cc}<br />
Q & 0\\<br />
0 &q<br />
\end{array}<br />
\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
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  3. #3
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    Joined
    Nov 2007
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    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
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