ELL 225: Control Engineering
HW #5
Due on February 26, 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!
1. The purpose of this problem is to examine the effect of feedback on sensitivity. In particular,
we would like to compare the topologies shown below for connecting three amplifier stages
with a gain of −K into a single amplifier with a gain of −10.
Figure for Problem 1
(a) For each topology above, compute βi if K = 10 and Y = −10R.
Y
H
(b) For each topology, compute SK
, where H =
(Use the respective βi values found in
R
part (a)). Which case is the least sensitive?
(c) Compute the sensitivities of the systems in Figures (b) and (c) to β2 and β3 . Using your
results, comment on the relative need for precision in sensors and actuators.
ELL225
Semester 2, 2025-2026
2. Consider the system shown below where
Dc (s) =
K(s2 + αs + 1)
.
(s2 + ωo2 )
Figure for Problem 2
(a) Prove that if the system is stable, it is capable of tracking a sinusoidal reference input
r = sin ωo t with a zero steady-state error.
(b) Use the Routh’s criterion to find the range of K such that the closed loop system remains
stable if ωo = 1 and α = 0.3.
3. Consider the system (a) shown below
Figure for Problem 3
(a) What is the system type? Compute the steady-state tracking error due to a ramp input
r(t) = µt, t ≥ 0.
(b) For the modified system with a feed forward path shown in (b), give the value of Hf so
the system is Type 2 for reference inputs and compute the Ka in this case.
(c) Is the resulting Type 2 property of this system robust with respect to changes in Hf , that
is, will the system remain Type 2 if Hf changes slightly?
4. Suppose you are given the system depicted in (a), where the plant parameter a is subject to
variations
2
ELL225
Semester 2, 2025-2026
Figure for Problem 4
(a) Find G(s) so that the system shown in (b) has the same transfer function from r to y as
the system in (a).
(b) Assume that a = 1 is the nominal value of the plant parameter. What is the system type
and the error constant in this case?
(c) Now assume that a = 1 + δa, where δa is some perturbation to the plant parameter.
What is the system type and the error constant for the perturbed system?
5. 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. 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.
(a) Design a feedback controller
va = K(ω̄ − ω),
and select K such that the closed-loop system is stable. Can the closed-loop system
asymptotically track a constant-angular-velocity reference ω(t) = ω̄, t ≥ 0? Assuming
zero as the initial condition, sketch or use MATLAB to plot the closed-loop response
when ω̄ = 900 RPM.
(b) Repeat (a) with the controller
Z t
va (t) = Ki
e(τ )dτ, e = ω̄ − ω.
0
(c) Repeat (a) with the controller
va = Ki (θ̄ − θ),
Rτ
where θ(t) = θ(0) + 0 ω(τ )dτ, and the angular reference θ(t) = θ̄ = π, t ≥ 0.
(d) Explain the difference between the controllers in (b) and (c) for ω̄ = θ̄ = 0, if any.
(e) Repeat (a) with the controller
va = K(τ̄ − τ),
where τ is the rotor torque and the reference torque τ̄ = 0.5 N m.
(f) Repeat (e) with the controller
Z t
va (t) = Ki
e(τ )dτ, e = τ̄ − τ.
0
3