Modern Control Engineering

400 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

whereaand the bi(wherei=1, 2,p,n)are constants and is the complex conjugate

ofa. The inverse Laplace transform of Equation (7–2) gives

(7–3)

For a stable system,–s 1 ,–s 2 ,p,–snhave negative real parts. Therefore, as tapproaches

infinity, the terms and approach zero. Thus, all the terms on the right-

hand side of Equation (7–3), except the first two, drop out at steady state.

IfY(s)involves multiple poles sjof multiplicity mj,theny(t)will involve terms such

as For a stable system, the terms approach zero

astapproaches infinity.

Thus, regardless of whether the system is of the distinct-pole type or multiple-pole

type, the steady-state response becomes

(7–4)

where the constant acan be evaluated from Equation (7–2) as follows:

Note that

SinceG(jv)is a complex quantity, it can be written in the following form:

where@G(jv)@represents the magnitude and frepresents the angle of G(jv); that is,

The angle fmay be negative, positive, or zero. Similarly, we obtain the following

expression for G(–jv):

Then, noting that

Equation (7–4) can be written

=Ysin( vt+f) (7–5)

=X@G(jv)@sin (vt+f)

yss(t)=X@G(jv)@

ej(vt+f)-e-j(vt+f)

2j

a=-

X@G(jv)@e-jf

2 j

, a–=

X@G(jv)@ejf

2 j

G(-jv)=@G(-jv)@e-jf= @G(jv)@e-jf

f=/G(jv)=tan-^1 c

imaginary part of G(jv)

real part of G(jv)

d

G(jv)=@G(jv)@ejf

a–=G(s)

vX

s^2 +v^2

(s-jv)^2

s=jv

=

XG(jv)

2j

a=G(s)

vX

s^2 +v^2

(s+jv)^2

s=-jv

=-

XG(-jv)

2j

yss(t)=ae-jvt+a–ejvt

thje-sj^ tAhj=0, 1, 2,p,mj- 1 B. thje-sjt^

e-s^1 t^ ,e-s^2 t,p, e-snt^

y(t)=ae-jvt+a–ejvt+b 1 e-s^1 t+b 2 e-s^2 t+p+bn e-sn^ t (t 0 )

a–

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