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 rd(t) = 0, derive the transfer function C(z)/R(z)
b) If r(t) = 0, derive the transfer function C(z)/Rd(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)= Kiz/z-1 Hk = 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)= Kp + Kiz/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) = Acx(t) + Bcu(t) , y(t) = Ccx(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).