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Spring Examinations 2007/2008: Linear Control Systems Exam, Exams of Linear Control Systems

National Center For Biological SciencesLinear Control Systems

Information about a spring examinations for the module linear control systems in the electronic engineering department. It includes details about the exam code, modules covered, exam duration, instructions, and required materials. It also provides formulas and rules for calculating various control system parameters.

Typology: Exams

2012/2013

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Spring Examinations 2007/ 2008
Exam Code(s) 3BN1
Exam(s) Third Year Electronic Engineering
Module Code(s) EE310
Module(s) Linear Control Systems
Paper No. 1
Repeat Paper No
External Examiner(s) Prof. P. Cheung
Internal Examiner(s) Prof. G. Ó Laighin
Dr. M. Duffy
Instructions:
Answer five questions from seven
All questions carry equal marks (20 marks)
Duration
3 hours
No. of Pages 7 (including cover)
Department(s) Electronic Engineering
Course Co-ordinator(s)
Requirements:
MCQ
Handout
Statistical Tables
Graph Paper Yes
Log Graph Paper Yes
Other Material Nichols Chart paper
pf3
pf4
pf5

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Spring Examinations 2007/ 2008

Exam Code(s) 3BN

Exam(s) Third Year Electronic Engineering

Module Code(s) EE

Module(s) Linear Control Systems

Paper No. 1

Repeat Paper No

External Examiner(s) Prof. P. Cheung

Internal Examiner(s) Prof. G. Ó Laighin

Dr. M. Duffy

Instructions: Answer^ five^ questions from seven

All questions carry equal marks (20 marks)

Duration 3 hours

No. of Pages 7 (including cover)

Department(s) Electronic Engineering

Course Co-ordinator(s)

Requirements :

MCQ

Handout

Statistical Tables

Graph Paper Yes

Log Graph Paper Yes

Other Material Nichols Chart paper

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

Mp M (^) o

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)

T (^) r (0 − 100%) =

π − sin−^1 1 −ζ 2

ω n 1 −ζ 2 ( ζ < 1)

Overshoot = 100 exp − πζ 1 −ζ 2

( ζ < 1)

T (^) s (±2%) ≤

ζ ω (^) n ln

1 −ζ 2

⎭⎪^ ( ζ < 1)

T (^) s (±5%) ≤

ζ ω (^) n ln

1 −ζ 2

⎭⎪^ ( ζ < 1)

Ziegler-Nichols Rules : Proportional : K = 0.5 Kc P+I control : K = 0.45 Kc , Ti = 0.83 T (^) c PID : K = 0.6 Kc , T (^) i = 0.5 T (^) c , T (^) d = 0.125 Tc

1. Open-loop frequency response data measured on a robotic arm, Gp (s), is given in Table

1. Note that the frequency values are given in Hz.

Table 1

f (Hz) 1 5 7 10

Gp(j2 π f) – 1.7 – j 2.5 – 1.3 – j 1.0 – 0.9 – j 0.35 – 0.5 – j 0

The position of the arm is to be controlled by the controller G c (s) shown in Fig. 1.

R(s) + C(s)

Gc (s) Gp (s)

_

Fig. 1

(a) Using cm graph paper, plot the data of Table 1 on a Polar Plot, and determine the

gain margin and the phase margin of the closed-loop system for a simple

proportional controller with K = 1. [ 7 marks ]

(b) Derivative action is introduced so that Gc (s) = (1+0.01s). Calculate the new open-

loop frequency response values of the system for the same frequency values as

given in Table 1. [ 5 marks ]

(c) Plot the frequency response calculated in part (b) on a Polar Plot and determine the

new phase margin. [ 4 marks ]

(d) How much time delay can the system tolerate before it becomes unstable? [ 4

marks ]

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 )

(s 2 ) G (^) p (s) 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)

3. The open-loop frequency response of a speed-control system was measured to produce

the results in Table 2.

Table 2

ω (rad/s) 0.5^ 0.7^ 1.0^ 1.

G (jω ) (dB) 6.0^0 – 6.0^ – 12.

Arg(G(jω) ( o^ ) – 140^ – 160^ – 180^ – 200

(a) Plot the data of Table 2 onto a Nichols Chart and estimate values of the resonance

peak, M p and the resonant frequency, ωr. Given that Mo = 1 (0 dB), predict the step

response overshoot of the closed-loop system for a unit step input. [ 9 marks ]

(b) In order to improve the system stability, a phase-lead compensator with

(s 1. 5 )

  1. 14 (s 0. 326 ) G (^) c (s)

= is introduced into the system in the forward path. Calculate

values for the gain and phase of the compensator at the same frequency values as

given in Table 2. [ 6 marks ]

(c) Modify the dB-angle plot of your system on the Nichols Chart to include the effect

of the phase-lead compensator. Use the results of the plot to predict the new value of

overshoot achieved for a unit step input. [ 5 marks ]

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

Fig. 3. 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 at 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. 3

_

R(s) C(s)

s(s 2 )

K +

1 + sT

7. It is required to design a digital controller for a certain chemical process. Root locus

analysis of the system found that the gain at which the system becomes unstable is 18

dB, and the frequency of unstable oscillations is ωc = 0.628 rad/s.

(a) Design an analogue PID controller for the process according to the Ziegler-Nichols

Rule. [ 7 marks ]

(b) Choosing a sampling interval of T = 1.5 s, directly apply the bilinear transformation

to emulate the analogue controller of part (a). Write the computer algorithm for the

resulting digital controller. [ 8 marks ]

(c) Discuss how the performance of the digital controller is likely to compare with that

of the analogue controller, and suggest one method for overcoming the differences

expected. [ 5 marks ]