SECTION A

(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]

Show that for an open-loop structure the ideal compensator should cancel the plant dynamics, and state the drawbacks of this approach. Describe a feedback structure which can provide approximate plant inversion, and show therefore

that high-gain closed-loop control implicitly inverts plant dynamics.            [5]

What  are  the  three  components  of a  PID  controller?  Briefly  analyse  the advantages and disadvantages of each. In your analysis you should focus on how each component affects the shape of the loop gain, and how this in turn

influences characteristics of the closed loop system.                                      [5]

Explain the concept of “internal stability” and relate this to BIBO stability, using a block-diagram similar to that shown in Figure 1. Why is it important to

Describe a test for controllability of a state space system.                              [5]

Section B

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]

Consider the closed loop system shown in Figure Q3

(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

ofL, 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]

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)] = [ ] [2(1)] + [0(1)]