EE422
Solution to HW#22
1a) First order Euler approximation with T=2 seconds:
From Matlab, the eigenvalues of 
are {5, -11} which are both outside the unit circle!! (recall the original continuous time model had the eigenvalues {2, -6} which had one eigenvalue in the LHP and one in the RHP. Thus, this method of discretization made the system more unstable! (both eigenvalues are now unstable, not just one!)
b)
Preferred method of discretization:
From Matlab, the eigenvalues of are { 54.5982, 6.1442e-006} which are consistent with the original eigenvalues (i.e., one is inside the unit circle, the other is outside the unit circle). Thus, using the preferred method of discretization, the relative stability of the original model is preserved (one stable, one not stable)!
c) From class, the solution to the state-variable model is 
Applying this solution to our discrete model in part 1b) we find:
If we evaluate this solution for x(4)=x(kT)=x(k2) we find that k=2 and
To compare this to the solution obtained in HW#11, we can look at the zero input solution and evaluate at t=2T=4 seconds:
We obtain the exact same answer as x(kT) at k=2.
d) Recall that the transfer function is given by 
2. a) Prove the Z-transform convolution property: 
b) Use Z-transforms together with the above result and the fact that 
to prove that the solution to the discrete state variable model, 
is
Taking the Z-transform of both sides: 
c) Given the discrete state variable model: 
Use the solution you derived in part b) to solve this model for both x(kT) and y(kT).
We can find 
one of three ways: , 
, and Cayley-Hamilton. Looking ahead to part d), let's use 
:
d) Find the eigenvalues and eigenvectors of 
in part c)
e) Form the matrix of eigenvectors, P, then use this as a similarity transformation x(kT)=Pz(kT) to decouple the discrete model in part c)
f) Block diagram:
i) stable {1/2} ii) uncontrollable {3} iii) unobservable {none}
g) Since the uncontrollable eigenvalue is unstable as well, we cannot make this system stable using state feedback for the input, w(kT).