The Stability of Linear Feedback Systems

5.2 The Routh·HuTWitz Stability Criterion
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. ' desired to determine the gain K that results in borderline stability, The
.1. 15 . h
aurwitz array IS t en
s' 11K
s' 110
•
s' ,K 0
s' clOD
I' K 0 0
t - K -K
CI =-----
, ,
for any value of K greater than zero, the system is unstable. Also,
me last term in the first column is equal to K, a negative value of K will
in an unstable system, Therefore the system is unstable for all values of
L
3. Zeros in the first column, and the other elements of the row
g the zero are also zero.
3 occurs when all the elements in one row are zero or when the row
of a single element which is zero. This condition occurs when the
iaI contains singularities that are symmetrically located about the on·
of the s·plane. Therefore Case 3 occurs when factors such as
.Xs - If) or (s + jw)(s - jw) occur. This problem is circumvented by utithe
auxiliary equation, which immediately precedes the zero entry in the
amy. The order ofthe auxiliary equation is always even and indjcates the
ofsymmetrical root pairs.
order to illustrate this approach, let us consider a third-order system with
lIl""'ri"stic equation:
q(5) - s' + 2s' + 45 + K,
K is an adjustable loop gain. The Routh array is then
s' 1 4
s' 2 K
8 - K
Sl 2 0
I' K O.
re, for a stable system, we require that
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