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Problem 4.20PP
Suppose you are given the system depicted in Fig.(a) where the plant parameter a is subject to 
variations.
(a) Find G(s) so that the system shown in Fig.(b) has the same transfer function from r to y as the 
system in Fig.(a).
(b) Assume that a = 1 is the nominal value of the plant parameter. What are the system type and 
the error constant in this case?
(c) Now assume that a = ̂ + 5a, where 6a is some perturbation to the plant parameter. What are 
the system type and the error constant for the perturbed system?
Figure Control system
the system type and the error constant for the perturbed system?
Figure Control system
Step-by-step solution
step 1 of 8
(a)
Refer to the system in Figure 4.35(a) in the textbook. 
Consider the expression for left side loop.
1
J(s+a) 
4 R * X -4 Y = {s+ a)X
= X
x ^ : ^ ^ ...... (1,
Step 2 of 8
Consider the expression for right side loop. 
[ j r + ; i r ( s + o ) ] = iy 
X {s+ a+ l) = sY 
„ sY
( s + a + I )
(2)
Step 3 of 8
Equate both the equations (1) and (2).
4 {R -Y ) sY
( i+ a + 1 )
4 ( / e - y ) ( s + o + l ) = j K ( j + a - l ) 
4 / i ( « + a + l) - 4 r ( 5 + a + I ) = 5 y ( « + a - l ) 
4 / t ( j + a + l ) = y [ s ( j + f l - l ) + 4 ( j + a + l ) ]
Y 4 (g 4 -g 4 -l)
R ^ ( ^ + f l - l ) + 4 ( i + g + l )
The closed-loop transfer function of the system is. 
r ( j ) 4 (5 -t-g + i)
R(s) 4 (* + f l - l ) + 4 ( 5 + g + l )
(3)
step 4 of 8
The general fonri of the closed-loop transfer function is.
y ( 4 g ( ^ )
« ( s ) 1 + G ( j )
(4)
Here C {4 ) is the open-loop transfer function. 
Find the open-loop transfer function