EECS 564________________________PROBLEM SET #10________________________Winter 1999
ASSIGNED: April 05, 1999 (Monday) READ: Catch up Chapter 6 of Srinath (Kalman filtering).
DUE DATE: April 12, 1999 (Monday) THIS WEEK: Simple applications of Kalman filtering.
  1. Joe observes y(t)=x(t)+v(t) where v(t) is 0-mean white noise with Sv(w)=½
    and x(t) is 0-mean WSS with E[x(t)v(s)]=0 and autocorrelation Rx(t)=e-|t|.

    Joe is too lazy to compute the causal Wiener filter, so he simply uses a suboptimal filter
    h(t)=ke-at,t> 0, and chooses k and a to minimize the steady-state mean square error.

    1. Show that x(t) and Joe's estimate z(t) of x(t) satisfy the vector state equation
      d_|x(t)| = |-1 0||x(t)| + |1 0||w(t)|
dt|z(t)|   |k -a||z(t)|   |0 k||v(t)|
      where w(t) is a 0-mean white process which when filtered gives x(t).
    2. Compute the steady-state error covariance matrix for this,
      and from that read off the mean square error E[(x(t)-z(t))2] in terms of k and a.
    3. Show that this is minimized for a=2.236 and k=1.24 and compute it.
    4. Compute the mean square error for the infinite smoothing filter for estimating x(t).
      Compare this to your answer to (c). Why is it smaller?
  2. A satellite is spinning at constant angular rate w radians/second (w is very small).
    We observe y(n)=x(n)+v(n), n=1,2... where v(n) is 0-mean WGN with variance r,
    x(n) is angular position of the satellite, x(0)=0, and w is N(m,s2).
    1. Formulate this as a Kalman filtering problem by augmenting the state x(n) with w.
    2. Write out the Kalman filtering equations for the augmented-state problem.
    There are other ways of doing this, but use state augmentation here.
  3. A dynamic system is described by x(n+1)=x(n)+u(n),n=0,1,2...where in fact
    u(n)=x(n-1)+w(n-1) where w(n) is iid and N(0,1) and x(-1)=0 and x(0)=10.
    We observe y(n)=x(n)+v(n) where v(n) is iid and N(0,3) and E[w(i)v(j)]=0.
    1. Augment the state x(n) with u(n) and write out the Kalman filter equations.
      These should be specifically the ones for estimating x(n) from {y(0)...y(n)}.
    2. Write out the algebraic Riccati equation for the steady-state error covariance matrix.
    3. Solve these equations. You should get P=[6 3;3 3] (this is a 2× 2 matrix).