Control System Design Exam questions PDF

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CONTROL SYSTEM DESIGN (14ELB004) June 2015

Two Hours

Answer ALL 4 questions. Each question carries 25 marks (total of 100). Any University approved calculator is permitted. For Q1 candidates are expected to refer to their laboratory notebooks used to record design methodology and results during the coursework assignment in Semester 2 Important Notes: Appendices: Laplace transform tables and theorems Blank Bode diagrams and Nichols charts are provided which should be removed and used as required. Candidates must ensure their completed chars clearly identify to which question they apply and are securely fastened to the answer book.

1.

This question relates directly to Coursework 2 undertaken in Semester 2. Candidates are expected to refer to their laboratory notebooks used to record results from MATLAB/Simulink and associated laboratory implementation. a)

In tasks 3 and 4 you investigated two different approaches to designing a position controller for the motor. Sketch the two Simulink block diagrams showing the DC motor with its position controllers. On the two diagrams indicate clearly which part of the diagram is the motor model and which part the controller. Clearly mark and explain signals. [10 marks]

b)

Comment on the advantages and disadvantages of the two position control approaches (tasks 3 and 4). Which do you favour and why?

c)

d)

[4 marks]

Sketch, on a single graph, the velocity step responses obtained for task 2 (MATLAB/Simulink) and task 5 (the motor lab). Compare the MATLAB/Simulink prediction with the actual result obtained from the motor. (Hint: What exactly are you comparing? Highlight and explain similarities/differences)

[6 marks]

Explain why absolute stability is not a particularly useful concept for practical engineering systems. In your answer introduce the use of relative stability making reference to the DC motor controller design process.

[5 marks]

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/continued

2.

Answer the following short questions a)

The standard transfer function for a second order system is written as

G (s )=

b)

c)

Write down the transfer function G(s) for a second order system with a natural frequency of 5rad/s and a damping ratio of 0.5.

[5 marks]

16 and state whether this s 2 + 4 + 16 system is stable, marginally stable or unstable.

[5 marks]

Calculate the poles of the system G ( s )=

Use the final value theorem to find the value of the steady state of the system

d)

e)

ω n2 . s 2 + 2ζω n + ω n2

X ( s) 16 in response to a step U (s ) of amplitude 2. = 2 U ( s) s + 4 + 16

Given the system, G( s) =

[5 marks]

6 , calculate the frequency when the gain is 10s + 4

unity

[5 marks]

Figure Q2d shows the experimental time response of a system to an input sinusoid. Determine: the frequency in Hz and the gain and phase response at that frequency. (Show your methodology by reproducing a sketch of the response and clearly showing where measurements were taken to get the results)

[5 marks]

Figure Q2d Frequency response results for part e)

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/continued

3.

Newtons 2nd Law can be used to describe the dynamic behaviour a car of mass, m, with a linear drag co-efficient, b, as follows:

mv (t ) = F (t) − bv( t) . Where, v(t ) is the velocity of the car and F (t ) is the force applied via the drive train and wheels. a)

Draw and label a block diagram model relating the input, F (t ) , to the output, v(t ) .

b)

[5 marks]

Show (by block diagram algebra or otherwise) that the transfer function model for the system is given by:

P( s) = c)

1 v( s) = F (s ) ms + b

[5 marks]

Reproduce the diagram below with a controller inserting the car model P(s) and a P+I controller G(s). On the diagram, clearly label the command signal, vin(s), the error E(s), the force, F(s) and the velocity, v(s).

[6 marks] Figure Q3 d)

Determine the closed-loop transfer function,

v(s) , for the system from (c). vin( s) [5 marks]

e)

Comment on the likely size of the steady-state error for the above system. Explain how you arrive at your conclusion. [4 marks]

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/continued

4.

Preliminary designs for a car cruise control have been undertaken. The speed of the vehicle (in km/hr) is controlled by the application of force at the wheel (in kN). A simplified transfer function model of the dynamics has been created as

G(s) =

9 3 s + 0.3

Design requirements have been established as • Gain margin of greater than 6 dB • Phase margin of greater than 60o • Overshoot of less than 20% • Steady state error of zero An initial proportional and integral controller has been selected in the form

K( s) =

K p (τ i s + 1)

τi s

where K p = 0.07 and τ i = 2 You are to analyse the design to see if it matches the requirements. a)

b)

c)

d)

e)

Sketch the approximate frequency response of the system, G(s), on the Bode diagram provided, Figure Q4.a-d

[5 marks]

Sketch the approximate frequency response of the controller, K(s), on the same Bode diagram, Figure Q4.a-d

[5 marks]

Combine the Bode sketches of G(s) and K(s) to generate the open loop frequency response of the controlled system, KG(s)

[5 marks]

From the Bode diagram find the gain and phase margins of KG(s) {Hint: show your method on the chart}

[4 marks]

The closed loop time response to a 100 km/hr step demand of speed at 0 seconds is shown in Figure Q4.e. Calculate the overshoot and the steady state error (show working on the Figure). How do these time response metrics and the gain and phase margins from d) compare to the design requirements? What could be changed in the design to better reach the requirements?

[6 marks]

R. Dixon C. P. Ward

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/continued

APPENDICES

Table 2: Useful Laplace transform theorems Homogeneity

1.

ℒ{𝑘𝑓 (𝑡)} = 𝑘𝐹(𝑠)

2.

ℒ{𝑓(𝑡 ) + 𝑔(𝑡 )} = 𝐹(𝑠) + 𝐺(𝑠)

Superposition

3.

ℒ{𝑒 −𝑎𝑡 𝑓(𝑡)} = 𝐹(𝑠 + 𝑎)

Frequency shift

4(a). ℒ � 4(b). ℒ � 4(c). ℒ � 5.

𝑑𝑓(𝑡) 𝑑𝑡

� = 𝑠𝐹(𝑠) − 𝑓(0)

Differentiation

𝑑2 𝑓(𝑡) 𝑑𝑡 2

� = 𝑠 2 𝐹(𝑠) − 𝑠𝑓(0) − 𝑓󰇗(0)

Differentiation (second order)

𝑑3 𝑓(𝑡) 𝑑𝑡 3

� = 𝑠 3 𝐹(𝑠) − 𝑠2 𝑓(0) − 𝑠𝑓 󰇗 (0) − 𝑓󰇘 (0)

Differentiation (third order)

𝑡

1

ℒ �∫0 𝑓(𝜏) 𝑑𝜏� = 𝐹(𝑠)

Integration

𝑠

𝑓(∞) = lim 𝑓(𝑡) ≡ lim 𝑠𝐹(𝑠)

Final value Theorem

𝑓(0+ ) = lim 𝑓(𝑡) ≡ lim 𝑠𝐹(𝑠)

Initial value Theorem

𝑡→∞ 𝑡→0

𝑠→0

𝑠→∞

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/continued

Figure Q4.a-d Bode diagram

Figure Q4.e - Closed loop step response

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/continued

Figure Q4.a-d Bode diagram

Figure Q4.e - Closed loop step response

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