Question 1

[75 marks]

A system is described by the ordinary differential equation given

below with zero initial conditions, where y(t) is the output of the

system and x(t) is the input of the system;

x(t) =

d³ y(t)

dt³

d²y(t) dy(t) + 4y(t)

+5.

dt²

dt

+5.

a. Use Simulink to numerically solve the system and sketch the

output y(t) for a unit step input. Indicate the steady-state

value and the maximum overshoot of the system response.

[9 marks]

b. Calculate the Laplace transform of the ordinary differential

equation and express the system transfer function.

[6 marks]

c. Use the final value theorem to calculate the steady-state

value of the system response and compare it to that

obtained in part a.

[6 Marks]

d. Write down the characteristic equation and use Matlab to

find the position of any system poles and zeros. Plot the

pole-zero map for this system.

[9 marks]

e. Considering the pole-zero map obtained above, explain how

the system can be approximated to a second order system

and find out the transfer function of the equivalent second-

order system that offers similar transient and steady-state

characteristics.

[9 Marks]/n[ENG2026]

f. Calculate the damping factor and natural frequency for this

second order system obtained in e. and comment on the

system dynamics and stability.

[7 marks]

g. Use Matlab commands to plot the system response of the

second order system obtained in e. to a unit step input, find

the percentage overshoot and the settling time to within 2%

of the final value and compare them to the response of the

original system obtained in a.

[8 marks]

h. Verify the values of the percentage overshoot and settling

time of the second order system obtained in g. using the

damping factor and natural frequency values calculated in

part f.

[7 marks]

i. If the simplified system is connected to a controller with gain

K and a unity gain negative feedback loop imposed, sketch a

block diagram of the closed-loop system and derive the

closed-loop transfer function.

[5 marks]

j. Use Matlab to sketch the root locus of the closed loop

system as the controller gain K varies.

[3 marks]

k. What effect on the system dynamics does the increase in

controller gain K have?

[3 marks]

1. State if there is any value of gain K for which the system

becomes unstable

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