ECE 486 Control Systems
Review Sheet for Exam # 2
Exam # 2 will be in class, room 163 Everitt Lab on Friday, April 4th, 10:00-10:50 AM.
The exam is closed book. You will be allowed a single 8 12 × 11 sheet of notes. Calculators
will be allowed.
Basic Concepts
The exam will test the material covered since the first exam.
1. Routh-Hurwitz Criterion: You should be able to compute a Routh-Hurwitz array
for a polynomial and know what the array tells you about the location of the roots of
the polynomial. You should be able to compute the Routh-Hurwitz array in terms of
a single unknown parameter and determine the range of that parameter for which a
system will be stable.
2. Root Locus: You should be able to roughly sketch a root locus given the open-loop
transfer function. You should understand the magnitude and phase angle conditions
that define the root locus. You should know the effect of added poles and zeros on the
root locus. You should know how to apply the root locus to design compensators (PD,
Lead, Lag).
3. Frequency Response: You should know what the frequency response function tells
you and how to compute it.
4. Bode Plots: You should be able to sketch the asymptotic Bode plot – not the actual
Bode plot. You should understand the terms bandwidth, corner frequency, crossover
frequency.
5. Nyquist Plot: You should be able to sketch a Nyquist plot for a simple system. You
should know the Nyquist Stability Criterion. You should know how to get the Nyquist
Plot from the Bode plot. You should understand Gain Margin and Phase Margin.
Practice Problems
1. Compute the Routh-Hurwitz array for each of the polynomials below and determine
how poles each has in the left and right half planes.
(a) p(s) = s4 + s3 + 5s2 + 2s + 4
(b) p(s) = s4 + s3 + s + 0.5
2. Suppose p(s) = s3 +(3a−1)s2 +2as+4 is the characteristic polynomial for a closed-loop
system. Find the range of the parameter a for which the system is stable.
3. Suppose p(s) = s4 + 3s3 + 2s2 + as + 3. Show that p(s) has two unstable roots for all
values of the parameter a.
4. Sketch the root locus 1 + KG(s) for each of the systems shown.
s−1
s(s2 + s + 10)
s+6
(b) g(s) =
s(s + 1)(s + 3)
(a) G(s) =
5. Sketch the asymptotic Bode Plot ( both magnitude and phase plots) for each of the
systems shown.
(a) G(s) = s 10s
+ 10
s
(b) G(s) = + 1002
(s + 10)
6. For the Nyquist plot shown, determine if the closed loop system is stable if
(a) G(s) has one right half plane pole
(b) G(s) has zero right half plane pole
7. Consider the unity feedback system shown below:
R(s)
Gc (s)
Gp (s)
C(s)
1
design a phase lead compensator so that the closed loop poles are
s(s + 1)
at s = −2 ± 2j.
If Gp (s) =
8. Consider the unity feedback system shown below:
R(s)
and suppose Gp (s) =
Gc (s)
Gp (s)
C(s)
0.2 . For G (s) = 10 the closed loop poles of the system are
c
s(s + 2)
at s = −1 ± j.
(a) What is the steady state error to a step input?
(b) What is the steady state error to a ramp input?
(c) Design a phase lag compensator so that the steady state error to a ramp is reduced
by a factor of 10 without moving the closed loop poles significantly.
(d) What are the closed loop poles resulting from your design?