Solution to HW#12
1. Given the following state variable model:
a) Use Laplace Transforms to find 
b) Repeat part a) using the Cayley-Hamilton method
c) Again using Cayley-Hamilton, find coefficients c0 and c1 such that 
. Verify your answers for k=0,1, and 2.
d) Lastly, find 
and evaluate at t=1, 2, and 3 seconds
e) Is the system stable, marginally stable, or unstable?
Although the system is marginally stable, by choosing the initial state, x(0)=[0 2]T, the marginally stable eigenvalue (s1 = 0) does not appear in x(t) and thus x(t) appears asymptotically stable.
2. Given the following state variable model with repeated eigenvalues:
a) Use Laplace Transforms to find 
b) Repeat part a) using the Cayley-Hamilton method
To obtain 2 linearly independent equations, we must take d/dsi:
c) Again using Cayley-Hamilton, find coefficients c0 and c1 such that .
Again, to obtain 2 linearly independent equations, we must take d/dsi:
d) Verify your answers for k=0, 1, and 2. What happens if you let k=-1?
Note that if we let k=-1, we get an undefined answer. This is consistent because A-1 does NOT exist ( det[A]=0)!!!
e) Lastly, find and evaluate at t=0, 1, 2, and 3 seconds
f) Is the system stable, marginally stable, or unstable?
Although the system is unstable (repeated eigenvalues on jw -axis), by choosing the initial state, x(0)=[0 4]T, the unstable term in eAt (i.e., 2t), never appears in x(t) and thus x(t) appears marginally stable. Your favorite EE422 professor should do a better job of picking the initial states as he is 0 for 2 in this HW!