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                                        s.
(5.90)
(5.91)
Thus, if the steady-state error for a unit-ramp function is to vanish, the velocity
constant Kv must be very large, which impli.es that we must have q  >  2. In other
words, we must have at least two integrators in the forward path. Thus, for a type
"O" system, the steady-state error for a unit-ramp function is infinity; for a type
 "1" system, it is finite; and for systems of type lL2" or higher, it is zero.
    Similarly, the steady-state error to a unit-parabolic input function, r(t) = t2 or
r (s) = 1/S3, can be obtained as
e(oo) = ~
(5,92)
g
lr
i
.
 N
E
q
 l
d
456           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where
= lrrri_[S2G(S)]
    s~ or
(5.93)
 We observe that for a given system to have zero steady-state error to aunit-parabolic
 input, we must have at least have three integrators in the forward path. Therefore,
 the steady-state error to unit-parabolic input oftype "0" and type "1" systems is infi-
 nite; for type "2" systems,it is finite; and for systems oftype "3" or higher,itis zero.
5.6 FrequencyResponse
    In steady-state; a sinusoidal input to a linear system generates a sinusoidal
 response (output) of the same frequency. However, the magnitude and phase angles
 of the response are generally different from those of theinput and also var}r with the
 frequency of the applied input.ln the following, we will determine the steady-state
response (magnitude and phase angle) to sinusoidalinputs.2
     In general, a sinusoid input function can be represented as
r (t) = A cos cot + B sin c.ot
          - Mi cos(cot + 41)
(5.94)
(5.95)
Here, Mt and ~/ are the magrutude and phase angle of theinput sinusoid function
and are given by the following expressions:
In phasor notation,
r(t) : Mt 24t
(5.96)
(5.97)
(5.98)
Furthermore, we assume that we can represent r (t) as                         iumber, r (t) -.
A - jB so that A - jB = N.tteid and A + jB = M,ae_?4,pTfakrnug the Laplace
transform of Eq. (5.94), we get
f(s)=(AXB )
The response to a sinusoidalinput is given by
y(s) = (Al:_B )G(s)
                   As + Beo
    .     'J ))G(s)
    ~s+-]co) .s~
 ki  k2 +...
= s+ lcn + s
                   '- Jco
(5.99)
(5.100)
(5.101)
(5.102)
Because we are interested in only the steady-state response, we have ignored the
terms corresponding to G(s), which generate the transient response. Iriecall that
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW      457
the steady-state response comes from the poles because of:input function, which
in this case are at s = :+: jco. Using partial fraction method, we get
k,= A +B G(s)ls=_jto
      s- ja
 A+
= +2- jBG(_jc,o)
                   .
                   J
(5.103)
(5.104)
Because G( jc.o)is acomplex number,we can write G( jto) = M8e j48 or G(- jco) =
Mge-jb,. With A +  j B = M,e~idf , we have
Similarly,
ki - ~M,e-)*i Mge-i*8
_ Mt Mg e-j(41+48)
     2
    /
k2 = M2M  e/(4f +~)
            ~*
              ~l
where * denotes the complex conjugate. Then,
Yoo(S) = M2M [
e-j(41+4g)     ej(4f +4,)
~s+-]co) + ~s co)
(5-105)
(5.106)
(5.107)
(5.108)
(5.109)
where the suffix oo denotes the steady-state value (t -* oo) and Mg and 4g are
　
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