082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

21.12 Poles, zeros and thesplane 669
Example21.31 Find the poles of
s −2
(s 2 +2s +5)(s +1) .
Solution The denominator isfactorized into linear factors:
(s 2 +2s+5)(s+1)= (s−p 1
)(s−p 2
)(s−p 3
)
where p 1
= −1 +2j, p 2
= −1 −2j, p 3
= −1.The poles are −1 +2j,−1 −2j,−1.
IfX(s)= 1
s−p 1
where the pole p 1
isgiven bya +bj,then
x(t) =e p 1 t =e at e btj = e at (cosbt +jsinbt)
Hence the real part of the pole,a, gives rise to an exponential term and the imaginary
part,b,givesrisetoanoscillatoryterm.Ifa < 0theresponse,x(t),willdecreasetozero
ast→∞.
ConsidertheLaplacetransforminExample21.25.Therearethreepoles: −1, −1+j
and −1 − j. The real pole is negative, and the real parts of the complex poles are also
negative.Thisensurestheresponse,x(t),decreaseswithtime.Theimaginarypartofthe
complex poles gives rise to the oscillatory term, cost. The characteristics of poles and
the corresponding responses arenow discussed.
GivenasystemwithtransferfunctionG(s),inputsignalR(s)andoutputsignalC(s),
then
C(s)
R(s) =G(s)
thatis,
and so
C(s) =G(s)R(s)
C(s) = K(s−z 1 )(s−z 2 )...(s−z m )R(s)
(s−p 1
)(s−p 2
)...(s−p n
)
The poles and zeros of the system are independent of the input that is applied. All that
R(s)contributes tothe expression forC(s)isextra poles and zeros.
Consider the case whereR(s) = 1 , corresponding toaunit stepinput:
s
C(s) = K(s−z 1 )(s−z 2 )...(s−z m )
(s−p 1
)(s−p 2
)...(s−p n
)s
= A 1
+ A 2
+···+
A n
+ B 1
s−p 1
s−p 2
s−p n
s
whereA 1
,A 2
,...,A n
andB 1
areconstants. Taking inverse Laplace transformsyields
c(t) =A 1
e p 1 t +A 2
e p 2 t +···+A n
e p n t +B 1
If the system is stable then p 1
,p 2
,...,p n
will have negative real parts and their contribution
toc(t) will vanish ast → ∞.
Theresponsecausedbythesystempolesisoftencalledatransientresponsebecause
itdecreaseswithtimeforastablesystem.Thecomponent ofthetransientresponsedue
to a particular pole is often termed its transient. Notice that the form of the transient
responseisindependentofthesysteminputandisdeterminedbythenatureofthesystem
poles. It is now possible to derive a series of rules relating the transient response of the
systemtothe positions of the systempoles inthesplane.