Page 488 - Mechanical Engineers' Handbook (Volume 2)

6 Stability 479

Special Cases
1. Zero in the first column, while some other elements of the row containing a zero in
the first column are nonzero
2. Zero in the first column, and the other elements of the row containing the zero are
also zero

CASE 1: In this case the zero is replaced with a small positive constant
0, and the array
is completed as before. The stability criterion is then applied by taking the limits of entries
of the first column as
→ 0. For example, consider the following characteristic equation:
s 2s 2s 4s 11s 10 0
3
4
2
5
The Routh array is then
1 2 11
2 4 10
0 6 0

6 0 first column zero replaced by
c 1 10
d 1 0
where
4
12 6c 10
c and d 1
1

1 c 1
As
→ 0, we get c
12/
, and d
6. There are two sign changes due to the large
1
1
negative number in the first column. Therefore the system is unstable, and two roots lie in
the right-half plane. As a final example consider the characteristic polynomial
2
3
4
P(s) s 5s 7s 5s 6
The Routh array is
17 6
55
66

← Zero replaced by
0
6
If
0, there are no sign changes. If
0, there are two sign changes. Thus if
0, it
indicates that there are two roots on the imaginary axis, and a slight perturbation would
drive the roots into the right-half plane or the left-half plane. An alternative procedure is to
define the auxiliary variable
1
z
s
and convert the characteristic polynomial so that it is in terms of z. This usually produces a
Routh array with nonzero elements in the first column. The stability properties can then be
deduced from this array.