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Problem 3.53PP
Suppose that unity feedback is to be applied around the listed open-loop systems. Use Routh’s 
stability criterion to determine whether the resulting closed-loop systems will be stable.
(b) = ^
W «” =
Step-by-step solution
Step-by-step solution
(a)
The transfer function of open-loop system is.
' ' +25^+35+4)
First, find the characteristic equation.
I + ^ G ( s ) = 0
5 ( i ’ + 2 j*+ 3 » + 4 ) 
j * + 2 j *+ 3 j ’ +8 s +8 = 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial. 
1 3 8 
2 8i * :
i * : a b
c
d
Evaluate the variables a,b,cnoAd - 
2 x 3 - 8 x l
_6-8
2
= - l
, 2 x 8 - lx 0
16 
° 2
Sa-2b
-1
- 8 -1 6
-1
- 2 4
d = b = ^
According to Routh array a system is stable If and only If all the elements in the first column of 
the Routh array are positive.
From the obtained Routh array, it is clear that the sign changes twice in the first column, so there 
are two poles not in the left hand side of the plane, hence the given closed-loop system Is 
unstable.
Step 5 of 11
(b)
The transfer function of open-loop system is.
XG(s)
S* ( j + l)
First, find the characteristic equation.
I + ̂ G ( j ) = 0
5*(j + I )+ 2 ( 5 + 4 ) = 0
5’ + 5 *+ 2 j + 8 » 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial. 
1 2 
1 8 
- 6 
8
>7 of 11
According to Routh array a system is stable If and only If all the elements in the first column of 
the Routh array are positive.
From the obtained Routh array, it is clear that the sign changes twice in the first column, so there 
are two poles not in the left hand side of the plane, hence the given closed-loop system Is 
unstable.
(c)
The transfer function of open-loop system is.
First, find the characteristic equation.
I-flC G (s) = 0
4 (j ’ + 2 j * + j +1)
1 + - ; V ; ----- ;----------{ = 0
s‘ (s’ +2s‘ - s - l )
s‘ (s’ +2s‘ -s - l)+ 4 (s ’ +2s‘ +s+ l) = 0 
j ’ + 2s*+ 3 j ’ + 7 j ’ + 4 j + 4 - 0
To determine the Routh array, first arrange the coefficients of the characteristic polynomial in two 
rows, beginning with first and second coefficients and followed by the even numbered and odd- 
numbered coefficients.
Write the Routh array for the polynomial. 
1 3 4 
2 7 4
“ i
* :
i * :