Q1

Consider a signal whose z-transform has two complex conjugate poles. Sketch and discuss the time sequences associated with various positions of the poles in the complex plane.             [6]

Given a discrete transfer function H(z) = U(z)/E(z), demonstrate that, in the time

+

domain, uk  = Σ ej hk− j  . What is this formula commonly known as?             [6]

j=−

Consider  a  first  order  continuous  transfer  function   H (s) =  =  .

Demonstrate that the forward rectangular numerical integration rule can be

implemented through the substitution s

What is aliasing?

SECTION B

Q3       (a)        …                                                                                                                       []

Q4       (a)       ...                                                                                                                        []

SECTION C

Q5                   Consider the following transfer function:

H (s) = 10 

(a)       Find the gain (in dB) and phase (in deg) at 1  = 0.5 rad/s . Assuming a sample time T = 2 s, calculate the Nyquist frequency n . [0. 1686 dB; - 10. 1642 deg; 1.5708 rad]                [3]

(b)       Design  the discrete equivalent of H(s) using the forward rectangular rule.  [

z − 0.98

(c)       Compute the discrete equivalent of H(s) using the pole-zero matching technique

(match the steady-state gain). [1.0921                    ]                                          [9]

z − 0.9802

(d)       Find the gain (in dB) phase (in deg) at 1    of the two discrete equivalents, and compare with that of the continuous H(s). Which one is closest? [-0.6533 dB; - 10.4372 deg; 0. 1693 dB; -9.2907 deg; PZ map]                [4]

Q6       Given the following digital feedback control loop (sample time T = 0.1 s):

CL(z)

G (s) = 22(0)s(s)1(1)0         R (z) = z 10z(− 0.)1(1)2

(a)       Find the discrete equivalent G(z) of the plant G(s). [  ]                  [8]

(b)       Find the closed loop transfer function of the system CL(z). (Simplify poles and

zeros if possible) [ 0.5                   ]                                                                    [2]

z − 0.5745

(c)       Find       the       difference       equation       corresponding       to       CL(z).       [ uk  = 0.5745 uk−1 + 0.5 ek  −0.5245 ek−1]                                [4]

(d)       Estimate  the  steady-state  output  of  the  closed-loop  system   CL(z)  for  the following input:

ek  = 10sin (0.1 kT)

[ 0.585sin (0.1kT + 2.9161)]                                                                            [6]