Given an observable z, r and unobservable x, I have a process of the form: x k + 1 = x k + w k z k = a 1 z k − 1 + a 2 z k − 2 − s 2 ( r k − 1 − x k − 1 + r k − 2 − x k − 2 ) + v k which in my understanding are transit and measurement equations.I am a beginner in Kalman-filter. In all places I researched, the measurement equation is in the form of z k = H x k + v k , but how should I apply Kalman filter to the above system?

Question

Given an observable   z,  r and unobservable   x, I have a process of the form:      x          k      +      1        =      x    k    +      w    k        z    k    =      a    1        z          k      −      1        +      a    2        z          k      −      2        −      s    2    (      r          k      −      1        −      x          k      −      1        +      r          k      −      2        −      x          k      −      2        )  +      v    k  which in my understanding are transit and measurement equations.I am a beginner in Kalman-filter. In all places I researched, the measurement equation is in the form of       z    k    =  H      x    k    +      v    k  , but how should I apply Kalman filter to the above system?

Marco Ford · Accepted Answer

For the measurement equation, the general equation is      z    k    =  h  (      x    k    )  +      v    k  where   h() is a function. For the vector case,   h is replaced with an operator   H(). For a linear measurement,   h is linear in       x    k   (  H is hence a matrix multiplication). The optimal tracker is a Kalman one. For non-linear measurements, an Extended Kalman Filter is used to linearize   h in the neighborhood of       x    k  .In your case,   h depends on previous measurements, since      z    k    −      a    1        z          k      −      1        −      a    2        z          k      −      2        =  −      s    2    (      r          k      −      1        +      r          k      −      2        −      x          k      −      1        −      x          k      −      2        )  +      v    k  that is      z    k    (  1  −      a    1        z          −      1        −      a    2        z          −      2        )  =  −      s    2    (      r          k        (  z  +      z    2    )  −      x          k        (  z  +      z    2    )  )  +      v    k  which is      z    k    (  1  −      a    1        z          −      1        −      a    2        z          −      2        )  =  −      s    2    (      r          k        −      x          k        )  (  z  +      z    2    )  +      v    k  This looks like an infinite impulse response. I&#039;m not sure whether Kalman filters are of usage here.

Given an observable z , r and unobservable x , I have a process of the form:

Answered question

Answer & Explanation

New Questions in Pre-Algebra