Q1       …

Discuss the features of a digital signal, as opposed to an analogue one.          [4]

With the help of sketches as necessary, highlight the main differences of a

digital feedback controller, with respect to an analogue one.                           [6]

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

Demonstrate that the trapezoid numerical integration rule can be implemented

through the substitution s

Derive how the stability region of a continuous transfer function maps into the z-plane, using that rule, and use sketches to explain your results. What are the consequences of your result, when applying this rule to a real system.           [4]

SECTION B

Q3       (a)        …                                                                                                                       []

Q4       (a)       ...                                                                                                                        []

SECTION C

Q5       Given the following digital feedback control loop CL(z) (sample time T = 0.1 s):

CL(z)

G (s) =

R (z) =

z − 0.995

z −1

Find the discrete equivalent G(z) of the plant G(s). [ G (z) = 0.05 z − 0(z − 0.)995(9002) ] [8]

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

zeros if possible) [ CL  = 0.047619z(z)  0(0).(.)9952(9002) ]                        [2]

Find        the        difference        equation         corresponding        to        CL(z). [ uk  = 0.9952uk−1 + 0.047619ek  − 0.042866ek−1 ]                        [4]

By verifying a suitable condition, demonstrate whether the difference equation is BIBO stable. [yes because …]         [6]

Q6

Consider a continuous transfer function of a first-order low pass filter with cutoff frequency (-3 dB) of 15 rad/s and steady-state gain of 0 dB.

(a)       Design  the  discrete  equivalent  of  it,  using  the  Tustin  rule,  considering  a

sampling time of 0.1 s. Compute the gain (in dB) of the digital filter at the cutoff frequency,       and       compare       it       with       the       analogue       version. [ G(z) =  ; −4.0533dB < -3 dB]                                           [8]

(b)       Re-design the  same, but this time apply a pre-warping such that the gain is preserved  at the  original  cutoff frequency.  Once  designed,  verify the  gain numerically. [ G(z) = 0.4823 ; −3.0103dB]                                [8]

(c)       Finally, re-design the discrete equivalent using the backward rectangular rule, and        compare        the        gain        at        the        cutoff        frequency. [ G(z) =  ; −4.8644 dB ]                                                     [4]