ECE 722 Spring 2010
Homework 7, due April 8, 2010
Given is the system (the same as in the test):
x(t ) Fx(t ) Gw(t )
y(t ) Hx(t ) v(t ) x(0) x0
0 1 1
F G H 0 1
2 3 0
The noise v(t) is “white”, has zero mean, and has normal distribution with power r.
The input w(t) is a “white noise” with zero mean, normal distribution, and power q.
Assume initially q=1, r=1
a. Solve numerically Matrix Algebraic Riccati Equation for P(∞) using
“care” algorithm in Matlab.
b. Form the Hamiltonian matrix (page 164)
c. Find the eigenvalues of the Hamiltonian matrix and make a plot displaying
their location in the complex plane.
d. Find the eigenvectors of the Hamiltonian matrix and use Lemma 1 on page
165 to find
The problem here is to find which particular eigenvectors of the
Hamiltonian matrix will provide appropriate matrices A and B such that P
calculated in this way matches the P(∞) from point a. above. Please describe
2. Given the system above use the MacFarlane-Potter-Fath method to compute the
steady state solution of the Riccati equation, the Kalman Filter gain K, and the
closed loop eigenvalues of the filter for the following cases:
a. q=1, r=1
b. q=1, r=10
c. q=1, r=100
d. q=10, r =1
e. q=100, r=1
f. q=1, r=0.01
Please make a plot of the closed loop filter eigenvalues corresponding to these six
After you have done this, please interpret the results. The interpretation should
address the effect of the input noise power q and measurement noise power r on the
the closed loop eigenvalues of the filter.
3. Given is a discrete-time system:
, and both noises are white
H 0 1 R=r
Assume q=1, r=1
a. Form the discrete time version of the Hamiltonian matrix
shown in line 5 on page 170
b. Find the steady state covariance P∞, steady state gain K∞
and the location of the closed-loop eigenvalues of the filter.
c. Again, vary q and r as in problem 2 above and observe the
changes in the closed loop eigenvalues. Report your