This preview shows pages 1–2. Sign up to view the full content.
CHAPTER 5
CONCEPTS OF STABILITY AND THE ROUTH
STABILITY CRITERION
5.1
(a) All the elements in the first column of the Routh array are +ve.
Therefore, all the roots are in the lefthalf plane.
(b) Two sign changes are found in the first column of the Routh array.
Therefore, two roots are in the righthalf plane and the rest in the left
half plane.
(c)
s
3
row of the Routh array has zero pivot element, but the entire row is
not all zeros. We replace the pivot element by
e
and then proceed with
the construction of the Routh array. As
e
ﬁ
0, two sign changes are
found in the first column of the Routh array. Therefore, two roots are
in righthalf plane and the rest in the lefthalf plane.
(d)
s
1
row of the Routh array is an allzero row. Auxiliary polynomial
formed using the elements of
s
2
row is given by
A
(
s
) =
s
2
+ 1
We replace the elements of s
1
row with the coefficients of
dA s
ds
()
= 2
s
+ 0
and proceed with the construction of the Routh array. There are no
sign changes in the resulting Routh array; the characteristic
polynomial does not have any root in the righthalf plane. The roots of
the 2ndorder auxiliary polynomial are therefore purely imaginary.
The given characteristic equation has two roots on the imaginary axis
and the rest in the lefthalf plane.
(e) Since all the coefficients of the given characteristic polynomial are
not of the same sign, the system is unstable. The Routh array
formation is required only if the number of roots in the righthalf plane
are to be determined.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '12
 M.lee

Click to edit the document details