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Measured Frequency - Linear Control Systems I - Past Exam Paper, Exams of Linear Control Systems

National Center For Biological Sciences Linear Control Systems

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

Typology: Exams

2012/2013

Uploaded on 03/26/2013

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Autumn Examinations 2008/ 2009

Exam Code(s) 3BN1

Exam(s) Third Year Electronic Engineering

Module Code(s) EE310

Module(s) Linear Control Systems

Paper No. 1

Repeat Paper

External Examiner(s) Prof. G. Irwin

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) Electrical & Electronic Engineering

Course Co-ordinator(s)

Requirements:

MCQ

Handout

Statistical Tables

Graph Paper Yes

Log Graph Paper Yes

Other Material Nichols Chart paper

Partial preview of the text

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

Autumn Examinations 2008/ 2009

Exam Code(s) 3BN

Exam(s) Third Year Electronic Engineering

Module Code(s) EE

Module(s) Linear Control Systems

Paper No. 1

Repeat Paper

External Examiner(s) Prof. G. Irwin

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) Electrical & 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 Mo

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

Tr ( 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 −

Ts (±2%) ≤

ζ ω n

^ ( ζ < 1)

Ts (±5%) ≤

ζ ω n

^ ( ζ < 1)

Ziegler-Nichols Rules : Proportional : K = 0.5 Kc P+I control : K = 0.45 Kc , Ti = 0.83 Tc PID : K = 0.6 Kc , Ti = 0.5 Tc , Td = 0.125 Tc

1. Measured frequency response results for the plant, Gp(s), in the system of Fig. 1 are given in

Table 1, where

s(s 1 )(s 2 )

1 G (^) p (s)

Table 1

ωωωω (rad/s) 0.4^ 0.7^1 1.4^2

Gp(j ωωωω ) –0.62 – j0.95^ –0.45 – j0.32^ –0.17 –j0^ –0.075 +j0.025^ 0 + j

Fig. 1

(a) Complete Table 1 by calculating Gp(s) at ω = 1 rad/s. [ 4 marks ]

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

gain, Kc, at which the system becomes unstable. Specify the period of the unstable

oscillations. [ 7 marks ]

(c) For what value of K is a gain margin of 6 dB achieved? Draw a new Polar Plot for this value

of K, and determine the resulting value for the phase margin (degrees). [ 9 marks ]

2. A pure integral controller is used in the system of Fig. 2; i.e.

K Gc (s)= , while the plant

transfer function, Gp(s), is given as:

s 2 s 5

1 G (^) p (s) 2

(a) Write an expression for the closed-loop transfer function of the system of Fig. 2, and

determine the locations of poles and zeros of the root locus for K. [ 5 marks ]

(b) Sketch a root locus for K. Include a calculation of the angle of departure from the complex

poles and show asymptotes for all branches. [ 12 marks ]

(c) Given that the root locus cross the imaginary axis at s = ±j2.24, apply the magnitude

condition to determine the corresponding value of Kc. [ 3 marks ]

Fig. 2

_

R(s) C(s)

K Gp(s)

_

R(s) C(s)

Gc(s) Gp(s)

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

given in Table 2.

Table 2

ω (rad/s) 1 2 2.6^ 3.4^ 4.2^ 5.2^6 7

G( jω ) (dB) 12 8 4.5^ 2.5^ 0.5^ – 0.5^ – 3^ – 6^ – 9.

Arg(G(ω)) (o) – 107^ – 115^ – 130^ – 140^ – 147^ – 156^ – 167^ – 175^ – 187

(a) Plot the data of Table 2 onto a Nichols Chart and specify the gain margin (dB) and phase

margin (degrees). [ 7 marks ]

(b) Given that Mo = 0 dB, estimate the percentage overshoot and the frequency of dominant

oscillations produced at the output for a unit step input. [ 9 marks ]

(c) If the system can withstand a minimum phase margin of 20o, what is the maximum time-

delay that can be introduced in the controller? [ 4 marks ]

4. A DC motor with transfer function:

s(s 4s 4.2)

1 G (^) p (s) 2

is connected into the tachometric feedback loop shown in Fig. 3, and the gain of the power

amplifier is set at K = 3.6.

(a) Derive an expression for the closed loop transfer function of the system in terms of the

parameter T. [ 2 marks ]

(b) Confirm that without tachometric feedback, i.e. with T = 0, the transfer function has one

pole at s = −3 and find the locations of the other two poles. Proceed to calculate the

percentage overshoot and settling time to within 5% of the step response. [ 12 marks ]

(c) With tachometric feedback in operation, find the value of T which will locate one of the

closed-loop poles at s = −2 and determine the locations of the other two poles. Calculate

the new value of percentage overshoot, and comment on the effectiveness of tachometric

feedback. [ 6 marks ]

Fig. 3

C(s)

K

R(s)^ +

1+sT

Gp(s)

7. Design specifications for a digital controller for a voltage regulator specify a step-response

overshoot of less than 20% and a ±2% settling time of ~2 s.

(a) Assuming that the closed-loop behaviour may be approximated by that of a second-order

system, specify pole locations in the s-plane corresponding to the design specifications.

Suggest a suitable sampling interval, T, for a digital controller design. [ 9 marks ]

(b) An analogue controller with a transfer function,

s 3 Gc(s)

= , is found to meet the

Measured Frequency - Linear Control Systems I - Past Exam Paper, Exams of Linear Control Systems

Related documents

Partial preview of the text

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

Autumn Examinations 2008/ 2009

Exam Code(s) 3BN

Exam(s) Third Year Electronic Engineering

Module Code(s) EE

Module(s) Linear Control Systems

Paper No. 1

Repeat Paper

External Examiner(s) Prof. G. Irwin

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) Electrical & Electronic Engineering

Course Co-ordinator(s)

Requirements :

MCQ

Handout

Statistical Tables

Graph Paper Yes

Log Graph Paper Yes

Other Material Nichols Chart paper

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

Tr ( 0 − 95 %) ≅ 3 / ω b ( ζ > 0. 4)

π − sin−^1 1 − ζ 2

ω n 1 − ζ 2 ( ζ < 1)

ζ ω n

^ ( ζ < 1)

ζ ω n

^ ( ζ < 1)

1. Measured frequency response results for the plant, Gp(s), in the system of Fig. 1 are given in

Table 1, where

Table 1

ωωωω (rad/s) 0.4^ 0.7^1 1.4^2

Gp(j ωωωω ) –0.62 – j0.95^ –0.45 – j0.32^ –0.17 –j0^ –0.075 +j0.025^ 0 + j

Fig. 1

(a) Complete Table 1 by calculating Gp(s) at ω = 1 rad/s. [ 4 marks ]

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

gain, Kc, at which the system becomes unstable. Specify the period of the unstable

oscillations. [ 7 marks ]

(c) For what value of K is a gain margin of 6 dB achieved? Draw a new Polar Plot for this value

of K, and determine the resulting value for the phase margin (degrees). [ 9 marks ]

2. A pure integral controller is used in the system of Fig. 2; i.e.

transfer function, Gp(s), is given as:

(a) Write an expression for the closed-loop transfer function of the system of Fig. 2, and

determine the locations of poles and zeros of the root locus for K. [ 5 marks ]

(b) Sketch a root locus for K. Include a calculation of the angle of departure from the complex

poles and show asymptotes for all branches. [ 12 marks ]

(c) Given that the root locus cross the imaginary axis at s = ±j2.24, apply the magnitude

condition to determine the corresponding value of Kc. [ 3 marks ]

Fig. 2

_

R(s) C(s)

K Gp(s)

_

R(s) C(s)

Gc(s) Gp(s)

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

given in Table 2.

Table 2

ω (rad/s) 1 2 2.6^ 3.4^ 4.2^ 5.2^6 7

Arg(G(ω)) (o) – 107^ – 115^ – 130^ – 140^ – 147^ – 156^ – 167^ – 175^ – 187

(a) Plot the data of Table 2 onto a Nichols Chart and specify the gain margin (dB) and phase

margin (degrees). [ 7 marks ]

(b) Given that Mo = 0 dB, estimate the percentage overshoot and the frequency of dominant

oscillations produced at the output for a unit step input. [ 9 marks ]

(c) If the system can withstand a minimum phase margin of 20o, what is the maximum time-

delay that can be introduced in the controller? [ 4 marks ]

is connected into the tachometric feedback loop shown in Fig. 3, and the gain of the power

amplifier is set at K = 3.6.

(a) Derive an expression for the closed loop transfer function of the system in terms of the

parameter T. [ 2 marks ]

(b) Confirm that without tachometric feedback, i.e. with T = 0, the transfer function has one