EE611
Homework #24
Due Wednesday November 5
1. Given the continuous-time state variable model:
You have learned two methods to discretize this model and obtain
a) Is the original system stable? Find values for 
if we use a first order Euler approximation on the derivative
of the state if our sampling time is T.
b) Find 
for your answer
in part a) (leave your answer in terms of T) (Hint:
)
c) Evaluate the zero-input discrete state response, xk,
for k=9 and a sampling time of T=1 second.
d) Evaluate the zero-input solution to the original system for
t=t0 + kT and compare to your answer to part c).
2. a) Repeat 1a-1c using the discretization method based upon
a piecewise constant input (i.e., 
).
b) Prove that the eigenvectors of A and 
are the same and that the eigenvalues of 
are 
are the eigenvalues
of A.
c) Prove that the eigenvectors of A and 
are the same and that the eigenvalues of 
are 
are the eigenvalues
of A.
d) Prove the following properties of the discrete state transition matrix, (k,j):
e) Property 4 in problem 2d) assumes that 
is invertible. Is this necessarily true for the two discretization
methods we learned in class? (i.e., 
and 
)