ECE 382 Quiz # 3 Solution
Problem 1. (5 pt) Consider a system with the following transfer function
H(s) = s2+ 1
(s2+ 6s+ 8)(s2+ 5s+ 10)(s+ 3).
Find its dominant pole/poles.
Solution: The poles of H(s) are −2,−3,−4,−2.5±j√3.75. The dominant pole is −2.
Problem 2. (15 pt)
(a) (7 pt) Given the transfer function
H(s) = s−2
(s+ 2)(s4+s3+s2+s+ 3),
is H(s) stable? If not, how many unstable poles does it have?
Solution: In the denominator, the first factor yields a pole at −2 which is stable. For the
second factor, its Routh array is
s4: 1 1 3
s3: 1 1
s2:ǫ3
s1:ǫ−3
ǫ
s0: 3
By looking at the first column and letting ǫgo to zero from above and below, we have
N+=N−= 2. Therefore, the second factor, hence H(s), has exactly two unstable poles.
(b) (8 pt) In the above feedback control system, C(s) = Kwhere K > 0 is a positive, adjustable
parameter, and G(s) = 1
s(s+1)(s+3) . Find the range of Kso that the closed-loop system is
stable, i.e., the transfer function Y(s)
U(s)is stable.
Solution: The closed loop transfer function is
Y(s)
U(s)=KG(s)
1 + KG(s)=K
s3+ 4s2+ 3s+K.
The Routh array for s3+ 4s2+ 3s+Kis:
s3: 1 3
s2: 4 K
s1:12−K
4
s0: K
For the close-loop system to be stable, the first column needs to have the same sign. Hence,
Kshould lies in the range
0< K < 12.
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