CONTROL M (ENG5022) 2017

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CONTROL M (ENG5022)

15th December 2017

SECTION A

Q1       (a)    Draw aNyquist plot for astable closed loop system and for an unstable system. For the stable system, mark the gain margin, phase margin and the vector margin in the plot. Explain the meaning of gain margin, phase margin and vector margin.                   [5]

(b)    For a control system with a nominal plant P0 , the Nyquist plot of the loop-gain L shows a phase margin of 60deg at a frequency of 4rad/sec. The plot crosses the negative real axis at -0.8.

(i)        What additional delay can be added to the closed loop before stability is lost?                                    [3]

(ii)       By  how  much  can  the  plant  gain  increase  before  the  closed  loop becomes unstable?              [2]

(c)    Given the transfer function

G(s) = 

derive the state space description in observer canonical form.                         [5]

(d)   What is meant by the Separation Theorem in the context of state-estimator feedback control?                [5]

Q2       (a)   Discuss the features of a digital signal, as opposed to an analogue one.          [4]

(b)   With the help of sketches as necessary, highlight the main differences of a digital feedback controller, compared to an analogue one.        [6]

(c)   Consider a first order continuous transfer function H(s) = U (s) =    a   . Show that the trapezoid numerical integration rule can be implemented through the

2 z - 1

T z +1

(d)    Derive how the stability region of a continuous transfer function maps into the z-plane, using the trapezoid rule from (c), 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 potential benefits and drawbacks of a closed-loop control strategy and contrast  these  to   an   open-loop  strategy.  In  your   analysis,  focus  on  the characteristics of each structure with respect to plant disturbances, changes in the plant gain, and stabilisation.                                                                        [5]

(i)        Derive the closed-loop equation of this system, and define the loop gain L, the sensitivity function S and the complementary sensitivity function T.                                                                                                          [5]

(ii)      Sketch the typical shapes of the frequency responses of the magnitudes of L, S, and T. Explain how this is related to characteristics of the closed loop  system,  in  particular  how  the  closed  loop  behaviour  at  low frequencies and at high frequencies is defined by this.                        [5]

(b)   Consider the closed loop system shown in Figure Q3.

(c) Show  that  the   sensitivity  function   S0     is  equal  to  the   sensitivity  of  the complementary  sensitivity  T0   to  changes  in  the  plant  P0.  By  a  symmetry argument, briefly describe the effect of T0   on the sensitivity of S0   to plant changes.                [5]

 

Figure Q3

Describe the concept of state feedback control using a block diagram and derive the differential equation of the closed loop system.                               [5]

For the  state  feedback  system  from  (a),  what  is  controller  design  by pole assignment and how can it be used to design a state feedback controller?      [5]

What is controllability in the context of state feedback control and what is meant by stabilisability?                [3]

Consider the system

[2(1)] = 

y = [0     2] [x2(x1)]

Show whether this system is controllable or not.                                              [2]

Design a state feedback controller K  for the state space system given in (d) using the pole placement method in such a way that the closed system has an overshoot  of Mp=1% and a rise time  of tr=0.1  seconds. You  can use the following equations to derive the natural frequency and the damping of the desired closed loop system:

ξ = _ 

[5]

Section C

Q5       Given the following digital feedback control loop CL(z) (sample time T = 0.01 s):

CL(z)

G (s) = 105(s)1(0)0           R (z) = z10(-)z(0) .-91(9)8

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

(b)       Find the transfer function of the closed loop system CL(z). (Simplify poles and zeros if possible)         [2]

(c)       Find the difference equation corresponding to CL(z).                                       [4]

(d)       By verifying a suitable condition, demonstrate whether the difference equation is BIBO stable.               [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.                                      [8]

(b)       Re-design the system in (a), 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 backward rectangular rule, and compare the gain at the cutoff frequency.                                                  [4]

2023-09-05