Q1       (a) Compare  and  contrast  the properties  of  open-   and  closed-loop  control structures. In your analysis, focus on the characteristics of each structure with respect to plant disturbances, changes in the plant gain, and stabilisation.      [5]

(b) Refer to the closed-loop system shown in Figure Q1. Define the sensitivity function So  and the complementary sensitivity function To  and show explicitly how  these  transfer  functions  determine  the  properties  of the  system  with respect to reference, disturbance and measurement noise signals.                   [5]

Figure Q1

(c)    Consider the linear state space system

̇(x)(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

with x(0) = x0. Show how a linear transfer function G(s) can be derived.    [5]

(d)    Given is the equation of motion of a pendulum as

I θ(̈)(t) + cθ(̇)(t) + sin θ(t) = τ(t)

Linearise this equation and present it in the standard state-space form of a differential equation,

̇(x)(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

with the angle θ as the output, and the external torque τ as the input.              [5]

Under          which           condition(s)          the           difference           equation uk = a1uk−1 + a2uk−2  + ... + b0ek + b−1ek−1 + b−2ek−2  + ... + b1ek+1 + b2ek+2  + ...             is causal? What is the physical meaning of causality?                                         [4]

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

Demonstrate that the backward rectangular 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) Describe  design  goals which  one  aims  to  achieve  when  using  feedback control.  Relate  these  to  requirements  for  the  complementary  sensitivity function  To   and  the  sensitivity  function  So ,  and  explain  how  these  can be translated into requirements of the magnitude and phase of the loop gain L. [6]

(b) Describe what is meant by controller design using loop-shaping.                   [4]

(c) Consider a PID controller in the context of frequency response design using loop-shaping.

(i)        What  are  the  three  components  of  the  PID  controller?  Give  their

responses in the Laplace domain.                                                         [3]

(ii)       Derive the transfer function of a PID controller C(s), and express the result in terms of a gain K, a time constant relating to the integral term, Ti , and a time constant relating to the derivative term, Td. What are the poles and zeros of C(s).              [4]

(iii)      Based   on  the   transfer  function  derived  in  (ii),  sketch  the  Bode

frequency response plot of an ideal PID controller.                             [3]

(b)       For the structure described in (a), describe in detail the structure of the state estimator (observer) and discuss the behaviour of the state estimation error

̃(x)(t) = x(t) − ̂(x)(t).                                                                                       [5]

(c)       Consider the plant

「- 10    1] 「0]

x(t) = |L-20    0」| x(t)+ |L2」| u(t)

y(t) = [1   0]x(t)

Calculate the observer gain vector L such that the closed loop observer poles are located at -80 and -90.                                                                                   [5]

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

「x1 ] 「-3    1]「x1 ] 「10]

|    | = |-      | |    |+ |    |u

y = [1   0]|    |

[5]

Section C

Q5       The following transfer function is a lead network:

H(s) =

(a)       Find the discrete equivalent of it, when preceded by a zero-order hold (ZOH), for sample time T = 0.5  s. Use 4  significant digits  for all numbers  in the solution.                                                                                                           [8]

(b)       Using the inverse z-transform, find the corresponding difference equation.   [4]

(c)       State  a  necessary  and sufficient condition for BIBO stability and determine whether the difference equation is BIBO stable.                                              [8]

Q6                   Consider  a continuous transfer  function of a  first-order low pass filter with steady-state gain of 20 dB and cutoff frequency (gain decays by -3 dB) at 10 rad/s.

(a)       Design  the  discrete  equivalent  of  it,  using  the  Tustin  rule,  considering  a sampling time of 0.2 s. Compute the gain (in dB) of the digital filter at the cutoff frequency, and compare it with the analogue version.                           [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.         [8]

(c)       Finally, re-design the discrete equivalent using the  forward rectangular rule, and compare the gain at the cutoff frequency.                                                  [4]