The Stability of Linear Feedback Systems

5.2 The Routh·Hurwitz Stability Criterion 211
der to ascertain the stability of a feedback control system, one could
In ~re the roots of the characteristic equation q(s). However, we are first
deIt..nu~ in determining the answer to the question: "'s the system stable" If
iD1t::t late the roots of the characteristjc equation in order to answer this ques­
~ Cll have determined much more information than is necessary. Therefore,
IiCJIl. ~methods have been developed which provide th@ required "yes" or "no"
JIVCI'I.. 10 the stability question. The three approaches to the Question ofstability
~) The s-plane approach, (2) the frequency plane Uw) approach, and (3) the
~omain app~oach. The ~eal frequ~ncy (jw) ap~roach is oullin.ed in Chapter
..aad the discussion of the ttme·domalO approach IS deferred uniii Chapter 9.
• The Routh-Hurwitz Stability Criterion
']"be discussion and determination of stability has occupied the interest of many
_neers. Maxwell and Vishnegradsky first considered the question of stability
ofdynamic systems. In the late 1800s, A. Hurwitz and E. J. Routh published
iDdePendently a method of investigating the stability of a linear system [2, 3].
The Routh-Hurwitz stability method provides an answer to the Question of stability
by considering the characteristic equation of the system. The characteristic
equation in the laplace variable is written as
6(s) = q(s) = a"sn + on_Is"-1 + ... + ajs + au ::: O. (5.3)
III order to ascertain the stability ofthe system, it is necessary to determine ifany
of the roots of q(s) lie in thc right half of the s-plane. If Eq. (5.3) is written in
6ctored form, we have __
an(s - ,\)(s - (2) ..• (s - 'n) = 0, (5.4)
Where" ... ilh root ofthe characteristic cquation. Multiplying the factors togcther
we find that
'
q(s) = a"s" -
a"(rl + 'z + ... + 'n)s"-I
+ a.('I'2 + rZr J + 'I') +. .)s"-2
- a"(rl,Z') + 'Ir z ,.' •)sn-J +
L + an( -IY",z') . 'n = O.
All other Wo d l'.
r S, lor an nth-degree equation, we obtain
q(s) '"' a"s" -
an(sum of all the roots)s"-I
(5.5)
+ an(sum oflhe products of the roots taken 2 at a time)s"-z
- a"(sum ofthe products of the roots taken 3 at a time)s"-J
(5.6)
Eta + ... + a"(-!)"(product ofall II roolS) = O.
mining Eq (5 5) . .
e the same . '.' ,we note that .a11 the coeffiCients of the polynomial must
sign Ifall the roots are In the left-hand plane. Also, it is necessary