MECH0023 Example Control Exam Questions
1. The position of a robot arm in an automated assembly line is given by:
Φ()() = 1 !* +". +
Where U(s) is the required position in radians,
KA = 3000 is the amplifier gain,
Total friction, f = 600 N/rad/s,
Robot arm mass, m = 30 kg,
and Mp is the mass of the object being handled.
a) Assuming unity feedback, find the closed loop transfer function in the two
cases below. Write your answers using the standard forms for transfer
function components.
i) With an empty gripper, i.e. Mp = 0.
ii) When holding a heavy item: Mp = 90 kg.
b) Sketch the closed loop step response in both cases (when Mp = 0 and 90
kg) and comment on the differences in damping and speed of response of
the robot.
c) Which of the following controllers would be most effective at improving the
performance of this robot? Provide reasoning.
Proportional controller (Gc = KP)
P I controller (Gc = KP + KI /s)
P D controller (Gc = KP + KDs)
2. The directional response of a large plane is approximated by the following
transfer function, whose frequency response is shown in Figure 3:
#() = Φ()() = 45 × 10$( + 29.8)(% + 34 + 612)
a) Measuring from Figure 3, identify the phase margin in degrees, the gain margin
in dB, and the frequencies at which these are measured.
b) Sketch the root locus of Gm(s) indicating the position of any asymptotes.
c) i) Write a controller transfer function, Gc(s), which would add zeros to cancel
the oscillatory poles of Gm(s) and which would add a pole at the origin.
ii) Sketch the root locus of the system including the controller: GmGc(s).
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3. The position of an accelerator pedal in a car is measured using a potentiometer
whose output ranges between 0.4 V and 2.8 V over the full range of the pedal’s
motion. This potentiometer voltage is measured with an 8-bit analogue to digital
converter (ADC) with an input range of -5V to +5V and a sample rate of 1,000 Hz.
a) Regarding the ADC, answer the following:
i) The range of motion is divided into how many discrete position steps?
ii) Will the sample rate of the ADC add a significant phase lag to the pedal angle
measurement? Provide reasoning.
b) The overall car engine model has the following equations of motion where x1, x2
and x3 are the state variables and u1 and u2 are the two inputs. ̇! = −50" − 13! + 130" ̇" = ! + 24" + 0.5! ̇# = 0.2! + 5" − #
Write these equations in the conventional state-space matrix format: ̇ = +
c) Select one of the following automatic feedback control systems on a modern car.
• Cruise control (maintaining a constant speed).
• Active suspension (reducing body roll when cornering).
• Antilock braking system, “ABS”.
• Auto-lane-following (staying within the white-line lane markings on a highway)
Briefly describe how the system works, specifically describing the following
components:
i) Input or set-point
ii) Output
iii) Sensor
iv) Disturbance (choose one example)
4. The behaviour of a magnetic levitation system has been approximated to have the
following open loop transfer function between control voltage, V(s), in volts and
vertical position, X(s), in mm.
()() = ( + 3)( − 1)( + 5.6)
a) How can we tell the system is unstable and what would that mean in
practice?
b) Would the system be stable in a unity-feedback closed-loop arrangement for
any values of k? Show justification using a root locus plot or algebra if you
prefer.
c) Choose any other inherently unstable system and describe how a controller is
used to stabilise it. Ensure you also specify:
i) Input or set-point
ii) Output
iii) Sensor
iv) Disturbance (choose one example)
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5. One of the position servo motors of a large production-line assembly machine
has the following open-loop transfer function:
&() = ()() = 2( + 57) 8.7 × 10$(% + 14 + 170)
Where V(s) is the linear speed in m/s and E(s) is the difference between desired
(“set-point”) speed and actual speed.
a) If a proportional gain controller, K, is used:
i) Sketch a root locus plot, specifying any asymptotes.
ii) Describe how the performance of the closed loop system would change
depending on the value of K.
b) A controller unit is inserted into the control loop:
'() = 0.1 × (% + 12 + 136)
i) Given that this is a PID controller, find the values of (, ) and *.
ii) Explain how this PID controller improves the performance in this specific
situation: sketch the new root locus and compare with the root-locus plot
from part a).
iii) The PID controller could be implemented via analogue electrical circuitry
or by using a digital controller. Describe three differences between these
methods.
6. An actuator in an automatic vehicle steering system has the measured frequency
response Gx(jw) shown in Figure 5.
Figure 5. Bode plot for Question 6.
a) Describe the low-frequency regions of the magnitude and phase plots and
therefore determine whether the actuator is likely to include one or more
integrators (also known as a pole at the origin, or “1/s”).
b) If the actuator receives a sinusoidal input: u(t) = 0.2 sin(200t), refer to Figure
5 to identify the magnitude and relative phase of the output sinusoid, and
sketch the input and output sinusoidal waveforms (both on one set of axes).
c) Measuring from Figure 5, identify the gain margin in dB, the phase margin in
degrees and state the frequencies where these occur.
d) Describe how proportional control could increase these stability margins,
being specific to the Bode plot shown. Then, also describe any other valid
control approach to increase stability in this system.
Frequency (rad/s)