Question 1: Find E*(s) for the following functions using the direct method, i.e. from the time sequence. Express E*(s) in closed form.

a) e(t) = exp(at)

b) E(s)= e^-2Ts/s - a

c) E(s)= 1/s - a

Question 2: For each of the transfer functions below,

1) Find the z-transform using the residue method.

2) Find the starred transform using the residue method.

3) Compare the pole-zero locations of E(z) in the z-plane with those of E(s) and E*(s) in the s- plane. Let T=0.1s.

a) E(s) = 20/(s + 2)(s + 5)

b) E(s) = (s + 2)/s²(s +1)

Question 3: Find E*(s), for T=0.1s, for the two functions below. Explain why the two transforms are equal.

a) e₁(t) = cos(4Πt)  b)e₂(t) = cos(16Πt)

Question 4: If possible, find C(z)/R(z) for each of the following block diagrams,

Question 5: For the block diagram given below,

a) If r_d(t) = 0, derive the transfer function C(z)/R(z)

b) If r(t) = 0, derive the transfer function C(z)/R_d(z)

c) Write the complete expression of the output C(z) using superposition

Question 6: For the block diagram given below,

Given that: D(z)= K_iz/z-1 H_k = 0.02; T = 1sec; K=2; J = 0.01.

Derive the transfer function Φ(z)/R(z)

Question 7: For the block diagram given below,

Given that: D(z)= K_p + K_iz/z-1

a) Derive the transfer function C(z)/R(z)

b) Compute and plot C(kT) when the input is a step input with magnitude 5.

Question 8: Consider the following continuous time systems:

x.(t) = A_cx(t) + B_cu(t) , y(t) = C_cx(t), where

a) Find a state space representation of the discrete time system when T=1.

b) Find the transfer function of the discretized system.

c) Find the transfer function Gp(s) of the continuous time system.

d) Compute the Z-transform of Gp(s) with the addition of ZOH.

e) Determine a state space representation of the transfer function G(z) obtained in d).

f) Compare the answers in a) and in e).