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Gain Margin - Linear Control Systems I - Past Exam Paper, Exams of Linear Control Systems

National Center For Biological SciencesLinear Control Systems

Main points of this exam paper are: Gain Margin, Phase Margin, Proportional Plus Integral, Respective Transfer Functions, Locations, Poles, Zeroes, Damping Ratio, Measurements, Open-Loop Conditions

Typology: Exams

2012/2013

Uploaded on 03/26/2013

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EE318 Linear Control Systems I Page 1 of 7
Autumn Examinations 2011/2012
Exam Code(s) 3BEI, 3BN, 3BM, 3BSE
Exam(s) Third Engineering Innovation – Electronic
Third Electronic Engineering
Third Mechanical Engineering
Third Energy Systems Engineering
Module Code(s) EE318
Module(s) Linear Control Systems I
Paper No. 1
Repeat Paper No
External Examiner(s) Prof. G. W. Irwin
Internal Examiner(s) Prof. G. Ó Laighin
Dr. M. Duffy
Instructions:
Answer any three questions from four.
All questions carry equal marks (20 marks).
Duration 2hrs
No. of Pages 7
Discipline Electrical & Electronic Engineering
Course Co-ordinator(s) Dr. Maeve Duffy
Requirements:
MCQ
Handout
Statistical Tables
Graph Paper Yes: mm graph paper
Log Graph Paper
Other Material Nichols Chart Paper
pf3
pf4
pf5

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Download Gain Margin - Linear Control Systems I - Past Exam Paper and more Exams Linear Control Systems in PDF only on Docsity!

Autumn Examinations 2011/

Exam Code(s) 3BEI, 3BN, 3BM, 3BSE

Exam(s) Third Engineering Innovation – Electronic

Third Electronic Engineering

Third Mechanical Engineering

Third Energy Systems Engineering

Module Code(s) EE

Module(s) Linear Control Systems I

Paper No. 1

Repeat Paper No

External Examiner(s) Prof. G. W. Irwin

Internal Examiner(s) Prof. G. Ó Laighin

Dr. M. Duffy

Instructions: Answer any^ three^ questions from four.

All questions carry equal marks (20 marks).

Duration 2hrs

No. of Pages 7

Discipline Electrical & Electronic Engineering

Course Co-ordinator(s) Dr. Maeve Duffy

Requirements :

MCQ

Handout

Statistical Tables

Graph Paper Yes: mm graph paper

Log Graph Paper

Other Material Nichols Chart Paper

The following standard formulas are given and may be freely used:

M (^) p M (^) o^ =^

1 2 ζ 1 − ζ (^2) ( ζ ≤ 0. 707 )

ω (^) r = ω (^) n 1 − 2 ζ (^2) ( ζ ≤ 0. 707 )

ω (^) d = ω (^) n 1 − ζ 2

ω (^) b = ω (^) n (1 − 2 ζ 2 ) + (1 − 2 ζ 2 ) + 1

T (^) r ( 0 − 95 % ) ≅ 3 / ω (^) b ( ζ > 0. 4 )

Tr (0 − 100%) = π − sin−^1 1 − ζ 2 ω (^) n 1 − ζ^2 ( ζ < 1 )

Overshoot = 100 exp − π^ ζ 1 − ζ^2

( ζ < 1 )

Ts (±2%) ≤ (^) ζω^1 n

ln 50 1 − ζ^2

^ ( ζ < 1 )

Ts (±5%) ≤ (^) ζω^1 n

ln 20 1 − ζ^2

^ ( ζ < 1 )

Ziegler-Nichols Rules : Proportional control : K = 0.5 Kc

P+I control : K = 0.45 Kc , Ti = 0.83 Tc

PID control: K = 0.6 Kc , Ti = 0.5 Tc , Td = 0.125 Tc

1. Results of the open-loop frequency response measured on the system shown in Fig. 1 with

K = 1, are given in Table 1.

Table 1

ω (rad/s) 0.25^ 0.37^ 0.48^ 0.61^ ∞ Gp(jω) −1.48 − j1.48 −1.07 − j0.55 −0.76 − j0.19 −0.5 + j0 0 − j

In order to increase the accuracy of frequency response plots, an additional measurement was

taken at ω = 0.85 rad/s, with the following results:

r(t) = 0.2 Sin(0.85t)

c(t) = 0.08 Sin(0.85t + 162.5o)

(a) Complete the frequency response of Gp(jω) at ω = 0.85 rad/s, and sketch the polar

plot for the plant on mm graph paper. [ 6 marks ]

(b) Using the polar plot of part (a), determine the maximum gain, Kc, that can be

applied to the system of Fig. 1, before it becomes unstable. Then, choosing the gain

according to the Ziegler-Nichols rules for proportional control, calculate the

resulting open-loop frequency response, KGp(jω) (use the same frequency values as

in Table 1) and sketch your results on a polar plot. [ 6 marks ]

(c) For the polar plot of part (b), calculate the gain margin (dB) and phase margin (o).

Using the approximation ξ = 0.01 φPM(o), estimate the unit step-response overshoot

of the closed-loop system. [ 6 marks ]

(d) Explain how derivation action would act so as to reduce the percentage overshoot in

this case. [ 2 marks ]

Fig. 1

_

R(s) C(s)

K^ Gp(s)

2. The block diagram of a system for controlling the tilt of a high-speed train is given in Fig.

2. When K = 1, the open-loop transfer function of the system is given as:

(s 1 )(s 4 )(s 4 s 8 )

G (s) (s^2 )

p + + 2 + +

=^ +

(a) Determine the locations of the poles and zeroes of the root locus for the system of

Fig. 2 as K increases from zero to infinity. Mark in portions of the real axis that are

on the root locus, and show the asymptotes for the poles. [ 9 marks ]

(b) Determine the angle of departure of the root locus from the complex conjugate

poles. [ 5 marks ]

(c) Apply the angle condition to confirm that the point s = – 1 + j 2.8 is on the root

locus, and then apply the magnitude condition to find the value of K that will place

the pole at this point. [ 6 marks ]

Fig. 2

_

R(s) C(s)

K^ Gp(s)

4. The block diagram of a position-control system with tachometric feedback is shown in

Fig. 4. It is required to design the system so as to provide a damping factor of ζ = 0.

and an undamped natural frequency of ωn = 2.235 rad/s.

(a) Confirm that the required location of the system’s closed-loop poles is s = –2 ± j1.

[ 4 marks ]

(b) Write an expression for the closed-loop transfer function, and use this to

determine design values for K and T that will satisfy the design requirements.

[ 5 marks ]

(c) For the resulting controller design: [ 7 marks ]

(i) Calculate the percentage overshoot and 0 – 100 % rise time of the closed-

loop system.

(ii) Calculate the steady-state error of the system for a unit ramp input, r(t) = t.

(d) Describe how the system response will change if the tachometric feedback

constant, T, is made (i) smaller and (ii) larger than the design value. [ 4 marks ]

Fig. 4

_

R(s) C(s)

s(s 2 )

K +

1 + sT