Q1       (a)       Refer to the closed-loop system shown in Figure Q1(a)

Derive the closed-loop equations relating the plant output y to the signals r, d, and n. Discuss how should the feedback system be designed in order to respond appropriately to  each  of these  signals  and  what  the  associated  limitations are.                                                                                                                   [5]

Figure Q1(a)

(b)       Derive an expression linking the vector marginsm  of the closed-loop system to the  peak  magnitude  of  the  sensitivity  function  So.  Derive  an  equivalent expression linking the complementary vector margin Tm   of the  closed-loop system to the peak magnitude of the complementary sensitivity function To. Briefly  discuss  the  relevance  of  these  expressions  in  relation  to  stability robustness of the feedback system.                                                                   [5]

(c)       Consider the linear state space system

̇(x)(t) = Ax(t) + Bu(t)    x(0)=x0

y(t) = Cx(t) + Du(t)

Show how  a  linear  transfer  function  G(s)  can  be  derived.  Explain  which element of the linear state space representation cannot be represented in the linear transfer function G(s).         [5]

(d)       In your own words, explain what is meant by controllability in the context of state feedback control. Describe a test for controllability of a state space system. [5]

Q2       (a)       The formula:

r (t ) = kΣ=+伪-伪r (kT)sinc

can be used for reconstructing a continuous signal r (t ) from its samples r (kT )

(i)        State  under  which  condition(s)  exact  reconstruction  is  theoretically possible.                   [3]

(ii)       Explain why this formula cannot be implemented in a realistic scenario, and state arealizable, approximated version of the formula.               [4]

(iii)      Explain what can be done to improve the reconstruction of the signal in quasi-real-time, and what is the potential impact on system control and signal broadcasting. Use sketches of time-signal plots as required.    [8]

(b)       Given the following system of two transfer functions in series:

R (z) =  ;        G (z) = 

Discuss  the   system’s   asymptotic   and   BIBO   stabilities,   after   zero-pole cancellations, and justify your answers.                                                           [5]

SECTION B

Attempt ONE question

Q3       (a)   Consider  a  feedback  control  system  with  loop  gain  Lo (s).  Discuss  design targets of the closed loop system in terms of the sensitivity function So (s) and the complementary sensitivity function To (s). How can these design targets be translated into requirements for the frequency response of Lo?                      [6]

(b)  Consider a PID controller.

(i)        State the control law in the time domain and in the Laplace domain. [3]

(ii)       Derive the transfer function of the PID controller in terms an overall controller gain K, a time-constant associated with the integral term, TI , and a time constant associated with the derivative term, TD . What are the poles and zeros of C(s).         [4]

(iii)      Sketch the Bode frequency response of a PID controller with K  =  100,

TI  = 1 and TD  = 0.05. Clearly marks the corner frequencies and the corresponding asymptotes of the magnitude and phase components of the frequency response.                                                                        [4]

(iv)      Describe how the PID controller can be extended to make it realisable.

Based in the numerical values in Q3(b)(iii), choose a suitable value for the extra component and explain your choice. Amend the Bode plot of the PID controller accordingly.                                                             [3]

(b)       For  the  structure  described  in  (a),  describe  in  detail  the   state  estimator

 = x(t) 一 根(x)(t) and discuss its behaviour.                                                                        [5]

(c)       Consider the plant

l2(1)) = l一一 15(11)   0(1)] lx2(x1)] + l23(2)] u

y = [1   0] lx2(x1)]

Derive the observer gain vector L such that the closed loop observer poles are located at - 100 and - 110.                                                                                 [5]

(d)       Describe  a test for observability of a  state space system. Show whether the following system is observable:

l2(1)) = l 3    一01] lx2(x1)] + l2(5)] u

y = [一2    1] lx2(x1)]

[5]

SECTION C

Attempt ONE question

Q5       Consider the following digital plant with sample time T = 0.1s:

G (z) = 

(a)       Design a digital PD controller that cancels the unstable pole of the plant, and find the open-loop transfer function L(三).                                                           [4]

(b)       Select  the  open-loop gain K such that the time constant of the closed-loop system is approximately τ = 0.2 s .                [6]

(c)       Find the values of the damping and natural frequency of the closed-loop system. [5]

(d)       Estimate the steady-state error (of the closed-loop system) in response to a unit step input.                                                                                                        [5]

Q6       Consider the following low-pass filter:

G (s) = (s - 1)(s - 2)

(a)       Find the sampling time that corresponds to 10 times the bandwidth of the filter (frequency at 0 dB).                            [4]

(b)       Now set a sample time  T = 0.1s . Calculate the Nyquist frequency.               [2]

(c)       Find the discrete  equivalent of the  filter, using the pole-zero matching rule; calculate the gain at 负c   = 4.19 rad / s , and compare it to that of the continuous filter at the same frequency.                           [8]

(d)       Now design another discrete equivalent, using the Tustin rule with pre-warping

at frequency 负c   = 4.19 rad / s . Show that the gain is now preserved.            [6]