Page 160 - Modern Control Systems

134 Chapter 2 Mathematical Models of Systems

14. Consider the closed-loop system in Figure 2.79 with
1000
G c(s) = 15, H(s) = 1, and G(s) =
s 3 + 50s 2 + 45005 + 1000'
Compute the closed-loop transfer function and the closed-loop zeros and poles.

15000
a. T(s) = - 3 2 ,5! = -3.70,5 2 3 = -23.15 ± 61.59/
5 + 505 + 45005 + 16000 ' '
. ^, N 15000 „„„ n, nn
b. 7/(5) = — , Si = -3.70,5, = -86.29
2
505 + 45005 + 16000
1
c. T(5) = -r 5 ,5 X = -3.70,5 2 , = -23.2 ± 63.2/
3
' 5 + 505 2 + 45005 + 16000
d. 7/(5) = 1_292 s = _ 3>70 s = -23.2, s 3 = -63.2
5 3 + 505 2 + 45005 + 16000
15. Consider the feedback system in Figure 2.79 with
K(s + 0.3) 1
G c(s) = — -, H{s) = 2s, and G(s) = 2
(5 - 2)(5 + 10s + 45)'
Assuming R(s) = 0 and N(s) = 0, the closed-loop transfer function from the distur-
bance 7/ rf(5) to the output Y(s) is:
Y(s) 1
a.
(
T d(s) 53 + 8^ + 2^ + 25)5 + (0.6K - 90)
Y(s) 100
b. 3 2
T d(s) 5 + 85 + (2K + 25)5 + (0.6K - 90)
Y(s) 1
T d(s) 85 2 + (2K + 25)5 + (0.6K - 90)
Y(s) K(s + 0.3)
d. 4 3 2
T d(s) s + 85 + {2K + 25)5 + {0.6K - 90)5

In the following Word Match problems, match the term with the definition by writing the
correct letter in the space provided.

a. Actuator An oscillation in which the amplitude decreases with
time.
b. Block diagrams A system that satisfies the properties of superposition
and homogeneity.
c Characteristic The case where damping is on the boundary between
equation underdamped and overdamped.
d. Critical damping A transformation of a function f(t) from the time
domain into the complex frequency domain
yielding F(s).
e. Damped oscillation The device that provides the motive power to the
process.
f. Damping ratio A measure of damping. A dimensionless number
for the second-order characteristic equation.
g. DC motor The relation formed by equating to zero the
denominator of a transfer function.