EE422 - Sample EXAM II

Problem 1

Given the following continuous-time state variable model:

a) Find the eigenvalues and eigenvectors of the system
b) Find e^At

c) Use a similarity transformation to decouple the state variable model.

Problem 2

a) Find the state variable model, including initial conditions and output equation, for the following circuit:

b) For the state variable model given by:

i) Solve for x(t) if w(t) = 2e^-tu(t)
ii) Now solve for x(t) if w(t) = e^-(t-3)u(t-3) and
Problem 3

Given the following continuous time decoupled state variable model:

a) Find the eigenvalues of the system and determine which eigenvalues are: i) unstable ii) uncontrollable iii) unobservable

b) Is the overall system asymptotically stable, marginally stable, or unstable?

c) If the system is unstable, is there any way we can use state feedback (i.e., set w=-kx) to make it stable (Hint: think about the meaning of uncontrollable eigenvalues)

d) Given your answer to part a) what will the transfer function Y(s)/W(s)=H(s) be?

Problem 4

Given the following zero-input state variable model:

a) Find the eigenvectors and eigenvalues of the system

b) Is the system stable?

c) Find the state transition matrix, eAt.

d) Find x(t).

e) If we used the similarity transformation, x=Pz, to decouple the system, how quickly will the decoupled states decay to within 2% of their initial values (i.e., what is the settling time of the system).

Problem 5

Recall that in class we derive the solution to the continuous time state variable model to be .

a) Use this solution to solve the following scalar state variable model if w(t)= e-2tu(t) (you must use the above solution!):

b) Use your LTI knowledge to solve the following scalar state variable model if w(t)=5e-2(t-3)u(t-3):