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TH E MANCH ESTER M ETR O PO LITAN UNIVER SITY
FACULTY O F SCIENCE AND ENGINEER ING
D EPA R TMENT O F ENGINEER ING AND TECH NO LO GY
SESSIO N 2001/
Exam ination for th e BEng (H O NS) ELECTR ICA LAND ELECTR O NIC ENGINEER ING (FULL-TIME/SANDWICH /PA R T-TIME) YEA R TW O /TH R EE
UNIT 64EE2013: CO NTR O LENGINEER ING I & II
Monday 27 May 2002
9 .30 am to 11.30 am
Instructions to Candidates
A llquestions carry equalm ark s.
Th is paper contains tw o sections A and B.
Candidates for ControlEngineering I (one h our paper) answ er TW O questions from Section A.
Candidates for ControlEngineering I and II (tw o h our paper) answ er FO UR questions, selecting TW O questions from each section.
Som e usefulrelationsh ips and Laplace Tab les are attach ed to th e b ack of th e question paper.
- Th e tab le (of m ass M k g) illustrated in Figure Q 2 is used for precision m easurem ent and is supported by a suspension system as sh ow n. Each spring h as a spring constant of K Nm -1. Th e dam per h as a viscous dam ping constant B Nsm -1.
(a) D e rive th e differentialequation th at relates th e m otion of th e tab le, y(t), to th e displacem ent, x(t), w h ich is caused by vib rations of th e floor. [12]
(b ) Clearly state th ree m odelling assum ptions th at you h ave used in h elping you to derive th is equation. [3]
(c) Use th e differentialequation to find th e transfer function ( )
X s
Y s. [5]
(d) D e scrib e , w ith th e aid of a sim ple sk etch , th e construction and principles of operation of a dam per. [5]
x
M kg
y
FLOOR
TABLE
Figure Q
- (a) A system is describ e d b y th e transfer function ( 2 )( 4 )
Us s s
Y s
. Use
th e Laplace Transform tab les provided to find th e tim e dom ain expression for y(t), to a unit step input, u(t) = 1 , t ≥ 0. You m ay assum e zero initial conditions. [9 ]
(b ) Evaluate th e system output, y(t), at tim e t = 0.5. [3]
(c) Sh ow th at th e system 2 12 50
( )^100
G s = (^) s (^2) + s + h as a dam ping ratio ζ = 0.6.
[6]
(c) Evaluate th e percentage oversh oot and 5% settling tim e for a step input. [3]
(d) Sk etch th e unit step input response and m ark clearly on your graph th e oversh oot, settling tim e and steady state system output. [4]
SECTIO N B
- A n open-loop system is describ e d b y th e transfer function (^2) ( 1 )
ss
G s.
(a) Sh ow th at th e gain and ph ase angle of th e process is 0.5 and – 180 o respectively if a sinusoidalsignalof frequency ω = 1 rad/s is applied. [6]
(b ) A sub set of data tak en from a frequency response test of th is system is sh ow n in Tab le T5. Find th e ph ase m argin and gain m argin of th e process. Com m ent on:
(i) th e relative stab ility of th e system in unity fe e d b ack closed-loop control;and (ii) th e nature of th e closed-loop response a unit step input. [7]
(c) If a proportionalcontroller is introduced into th e forw ard path of th e closed-loop system , determ ine th e controller gain such th at a ph ase m argin of 40o^ is ob tained. W h at w illb e th e new gain m argin?Com m ent on th e closed-loop stab ility w ith th e proportionalcontroller included. [8]
(d) Sk etch th e Nyquist diagram of th e open-loop system w ith and w ith out th e proportionalcontroller sh ow ing low and h igh frequency points. [4]
Frequency (rad/s)
G ( jω ) 2.^43 1.^74 1.^33 1.^00 0.^72 0.^50 ∠ G ( jω ) deg -130^ -140^ -150^ -160^ -170^ -
Table T
- (a) Th e frequency response of an open-loop plant is sh ow n in Figure Q 6.
(i) D e term ine th e order of th e system and w rite dow n th e form of th e transfer function of th e system. Justify your answ ers. [5]
(ii) D e term ine th e ph ase m argin, gain m argin, ph ase and gain crossover frequencies, and com m ent on th e relative stab ility of th e system. [7]
(b ) Th e open-loop transfer function of a plastic extrusion process is given by
( 1 )( 0. 2 )^2
( )^10
s s
G s.
(i) Estim ate th e ph ase and gain of th e system at low and h igh frequencies. [4]
(ii) Sk etch th e Bode plot sh ow ing th e frequency response of th e system using asym ptotic approxim ations. Sh ow allyour construction clearly indicating th e b reak frequencies. [6]
(iii) If th e gain is now ch anged to 20, sh ow h ow th is w illalter th e Bode plots. [3]
10 -2 10 -1 100 101 102
0
20
40
magnitude dB
Bode gain diagram
10 -2 10 -1 100 101 102
Frequency rad/s
Phase deg.
Bode phase diagram
Figure Q6: Bode plot of an open-loop system
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- Figure Q 8 sh ow s a h eating tank controlled using a Proportional+ Integral(PI) controller. Th e controller adjusts th e opening of th e controlvalve to regulate th e am ount of steam flow ing th rough th e h eating coil to ach ieve th e required tem perature, To , of liquid in th e tank. Th e tem perature To is m easured by th e transducer TT and is com pared w ith th e required set point Ts to generate th e error signal e.
It is k now n th at th e open loop transfer function relating th e valve controlsignal U
and th e tem perature To is given by
T (s) U(s)
3s 1
o (^) =
(a) D raw a b lock diagram of th e closed loop system sh ow ing clearly allth e signals. [5]
(b ) Th e controller proportionalgain is set to 1 and th e integraltim e is set to 30 seconds. W rite dow n th e controller transfer function Gc(s), th en find th e
closed loop transfer function
T (s) T (s)
o s
. [6]
(c) Using th e finalvalue th eorem , or oth erw ise, sh ow th at th e steady state error is zero w h en a step ch ange in th e set point Ts is introduced to th e system. [4]
(d) Briefly discuss th e possib le disturb ances th at can act on th e system. [5]
(e) State one m ain feature th at distinguish es a process system from a servo system and sh ow h ow th e transfer function of th e h eating tank system can b e m odified to reflect a practicalsystem. [5]
ControllerG c
Flow in
Flow out steam
U Control valve
e To
Ts
TT
Figure Q
END
