EE422

HW#22

Due Thursday, April 15 (Tax Day! )

1. Recall in HW#11 we solve the following continuous time state variable model:

a) Use the first order Euler approximation method learned in class to discretize the above state variable model using a sampling time of T=2 seconds. Is the resulting discrete state model stable (i.e., are the eigenvalues inside the unit circle)?

b) Repeat part a) but now use the preferred method of discretization where we use the approximation that w(t) is constant over the period kT < t < (k+1)T (you may use the solution to HW#11 to find e^AT ). Is the resulting discrete state variable model stable?

c) Use Z-transforms to solve the discrete state variable model in part b) for x(kT) if w(kT)=u(kT). Evaluate your answer for x(4 seconds) and compare your answer to the answer in the solution to HW#11 evaluated at t=4 seconds. Is there any difference?

d) For the discrete state variable model in part b), let x(0)=0 and find the transfer function H(z)=Y(z)/W(z).

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

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).

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) Draw a block diagram of the decoupled system and determine which eigenvalues are: i) stable ii) uncontrollable iii) unobservable

g) Given your answer to part f), is there anyway we can make this system stable using state feedback for the input, w(kT)?