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CONTROL M (ENG5022)

Monday 16th December 2019

SECTION A

Q1 (a) Draw the structure of an open-loop control system and of a two-degree-of- freedom  feedback  control  system.  Show  the  different  signals,  including disturbances, and explain their physical meaning. Explain which additional components are required in the feedback control structure. [5]

(b) Show that for an open-loop structure the ideal compensator should cancel the plant dynamics, and state the drawbacks of this approach. Demonstrate that a feedback control structure can provide approximate plant inversion and show therefore that high-gain closed-loop control implicitly inverts plant dynamics. [5]

(c) What is observability in the context of state feedback control? Describe a test for observability of a state space system.         [5]

(d) What is meant by state estimator feedback control? Use a block diagram to illustrate   your   explanations   and   mark   the    elements   which   form   the compensator. Explain what is meant by the Separation Theorem in the context. [5]

Q2 (a) Consider a signal whose z-transform has two complex conjugate poles. Sketch and discuss the time sequences associated with various positions of the poles in the complex plane.                                                                                             [6]

(b) Given a discrete transfer function H(z) = U(z)/E(z), demonstrate that, in the time

+

domain, uk = Σ ej hk−j . What is this formula commonly known as?             [6]

j=−

(c)

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

Demonstrate that the forward rectangular numerical integration rule can be

where T is the sample time. [6]

(d) What is aliasing in the context of sampled system?                                         [2]

SECTION B

Q3 (a) Explain why ‘peaking’ in the sensitivity function S0  and the complementary sensitivity function T0  should be avoided. The explanation should be based on performing the following analysis.

i.          Derive the closed-loop equations of a feedback control system in terms of So    and  To ,  considering  the  output  responses  to  the  reference, disturbance and noise. Discuss how peaking would affect the system’s response to these signals.        [4]

ii.         Show that the complementary vector margin for the inverse loop gain is equal to the inverse of the peak value of |T0 |. By asymmetry argument,

briefly describe the effect of |S0 | on the vector margin for the loop gain. Based on this, explain what a strongly oscillatory response of a closed- loop system means for stability-robustness.      [5]

iii.        Show that T0  is equal to the sensitivity of S0    to changes in the plant P0 .

By  a  symmetry  argument,  briefly  describe  the  effect  of  S0    on  the sensitivity of T0  to plant changes.                [5]

(b)       Refer to a closed loop control structure. Assume that the compensator C(s) is chosen in such a way that the nominal loop gain Lo (s) gives a stable closed- loop system. Assume also that the nominal loop gain Lo (s) is perturbed to the actual loop gain L(s), i.e. Lo (s) → L(s)  (or, equivalently, that the nominal inverse  loop  gain  is  perturbed  as  1⁄Lo (s) →  1⁄L(s)).  Derive  a  sufficient condition for closed-loop stability which combines two tests, one involving |So | and the other involving |To |. What is the name of this criterion?                     [6]