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Problem 5.17PP
For the system shown in Fig., determine the characteristic equation and sketch the root locus of it 
with respect to positive values of the parameter c. Give L(s), a(s), and b(s), and be sure to show 
with arrows the direction in which c increases on the locus.
Figure Control system
Step-by-step solution
step 1 of 4
Step 1 of 4
Refer to Figure 5.57 in the textbook.
From the block diagram, the process transfer function is.
/ 'C + I65V 9 ''G (,)
V C+S
Calculate the characteristic equation.
1 + G (»)H (») = 0
( c + j) i* + 9 c + I4 4 5
( c + j ) j*
cy* + 5* + 9 c + 144s * 0
c(s* + 9) + s ( s* + 144) = 0
l + c - i- = 0 (1)
j ( j " + 1 4 4 )
Therefore, the characteristic equation is
(s*+9) 
s(s* + 144)
Step 2 of 4
From the equation {1), the loop transfer function is.
s (s^ + 1 4 4 )
From the characteristic equation.
a(s) = s* + 144s
And.
6(s)=s*+9
Therefore, the loop transfer function, L {s ) i: 
are |a(j)sj^+144s and ^(s) = s*
p(s* + 144) e values of 0 (5) andi(s)
Step 3 of 4
Write a MATLAB code to plot the root locus. 
» num=[1 0 9];
» den=[1 0 144 0]:
» sys=tf(num,den) 
sys = 
s^2 + 9
s^3 + 144 s
Continuous-time transfer function. 
» rlocus(sys)
Step 4 of 4
Draw the root locus for all positive values of c .