EE381 Problem Set
#2
Q1.Consider Fig. 1. Find the steady-state errors to a unit step input and to a unit ramp input
for the systems which have
1
a. G(s) = s+2
.
1
b. G(s) = s(s+2) .
c. G(s) = s2s+1
.
(s+2)
Q2. Suppose that we have a system with transfer function G(s) = N (s)/D(s) where D(s) =
s4 + Ks3 + 8s2 + 20s + 15 where K is a parameter ranging from −∞ to +∞. Find the range of
values for K for which the system is stable.
Q3. Suppose that we have a system with transfer function G(s) = N (s)/D(s) where D(s) =
s4 + 6s2 + 25. Apply the Routh stability criterion to this system and calculate the number of
poles of the system in the right half plane. Is the system stable?
U (s) +
G(s)
−
Y (s)
Figure 1: Closed loop system with unity feedback.
Q4. Consider Fig. 1. Let
G(s) =
K(s + 1)
.
+ 12s + 100)
s2 (s2
a. Sketch the locus of the closed-loop poles for K ≥ 0. On your plot show all the relevant details
such as break points, axis crossings, root directions.
Hint: The roots of the equation 3x3 + 28x2 + 136x + 200 are −3.52 ± 4.1i and −2.27.
b. What is the range of K for stability?
c. What is the range of K for stable and oscillatory step response?
Q5. Consider Fig. 1. Let
G(s) =
(s − 2)(s − 3)
.
(s − 1)(s + α)
a. Plot the root locus for the closed loop with respect to the parameter α ≥ 0. On your plot
show all the relevant details such as break points, axis crossings, root directions.
b. What is the range of α for stability?
c. What is the range of α for stable and oscillatory step response?