ELL 225: Control Engineering
HW #8
Due on April 5, 2026
1. Discussion is allowed but all students are expected to turn in their original work. Offenders
will get an F and their names will be forwarded to the Academic Section.
2. You can type or handwrite legibly. Please check the quality of your PDF file before submission.
If we cannot read it, we cannot grade it!
3. Only Problems 1 and 2 will be graded.
1. Sketch the root locus for the systems with poles and zeros shown below and then use MATLAB
to verify your answer.
2. Use the root locus method to determine feedback controllers that can stabilize the systems
with poles and zeros shown above. The choice of the controllers need to be properly justified.
No marks for coming up with the controller expression magically!
3. You have shown in HW#2 that the ordinary differential equation
Ke Kt Kt
J ω̇ + b +
ω=
va ,
Ra
Ra
is a simplified description of the motion of the rotor of a DC motor.
ELL225
Semester 2, 2025-2026
(a) Let the voltage va be the control input and the rotor angular velocity ω be the measured
output. Let J = 227 × 10−6 kg m2 , Kt = 0.02 N m/A, Ke = 0.02 V s/rad, b = 289.4 ×
10−6 kg m2 /s, and Ra = 7 Ω. Use the root locus method to design a dynamic feedback
controller so that the closed-loop system is capable of asymptotically tracking a constant
reference input ω(t) = ω̄, t ≥ 0.
(b) Repeat (a) to design a position controller that uses a measurement of the angular position
Z t
θ(t) = θ(0) +
ω(τ )dτ,
0
as output and is capable of asymptotically tracking a constant angular reference θ(t) = θ̄,
t ≥ 0.
(c) In HW#5, you would have shown that the torque of a DC motor, τ, is related to the
armature voltage, va , through the transfer function
Kt
(s + b/J)
T (s)
=
.
Va (s)
Ra Ra s + b/J + Ke Kt /(Ra J)
Use the data from (a) and the root locus method to design a controller that uses the
voltage va as the control input and the torque τ as the measured output so that the
closed-loop system is capable of asymptotically tracking a constant reference input torque
τ(t) = τ̄, t ≥ 0.
2
0
