EE422

Homework #11

Due Tuesday, March 1

0. Be sure to check your tests and your posted scores and tell me if there are any grade discrepancies!

1. Given the state variable model,

a) Find the transfer function, H(s)=C[sI-A]^-1B + D

b) Find the eigenvectors and eigenvalues of the matrix A (you may use the Matlab eig() command to check your answers, but remember that the choice of eigenvectors is NOT unique and Matlab will normalize the length of each eigenvector to one. So to compare, divide each of your eigenvectors by the length of each eigenvector)

c) If eivenvalues are analogous to transfer function poles, would you say that the system in part 1a) is asymptotically stable, marginally stable, or unstable?

d) Form P = [P₁ P₂ ... P₃] (the matrix of your eigenvectors) and (the diagonal matrix of eigenvalues, then verify that A^k=PS^kP^-1 for k=0,1,2, and 3.

e) Now find e^At = Pe^StP^-1

f) Next, in the state model let w=0 solve for x(t)= e^Atx(0).

2a) Check your answer to 1f) by using Laplace transforms to solve for x(t)

b) Use Matlab's eig() funtion to find the eigenvalues and eigenvectors for the following 4th-order circuit model:

c) Recall that we defined settling time for a system modelled by a transfer function as, t_s = -4/Re[p_i]_max where Re[p_i]_max is the real part of the transfer function pole closest to the jw axis. Let us define settling time for a state variable model as t_s = -4/Re[s_i]_max where Re[s_i]_max is the real part of the eigenvalue closest to the jw -axis. What is the settling time for the system in 2b)?

d) If eivenvalues are analogous to transfer function poles, would you say that the system in part 2b) is asymptotically stable, marginally stable, or unstable?

e) Use Matlab's expm() function to find e^At for t=0,1,2,3, and 4 seconds. Does the system seem stable?

f) Again use Matlab to find x(t) at t=0,1,2,3, and 4 seconds (hint: x at t seconds is expm(A*t)*x0 where x0 is the initial state). Has x(t) "settled" by the settling time?