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input all the time. In other words, it is desired that thevsteady-state value of the
output be equal to the value of the reference input as closely as possible. However,
it may not always be possible to achieve this goal and, in reality, the steady-state
value of the output differs from the value of the reference input.
   The steady-state error is the difference between the steady-state value of the
output and the reference input. Usually, unit~step, unit-ramp, or parabolic functions
are used as test inputs to determine the steady-state error.ln the following, we will
derive expressions for steady-state error for unity feedback systems as shown in
Fig. 5.6. It may be noted that any given nonunity feedback syjstem (Fig. 5.7) can
be expressed as an equivalent unity feedback system by adding and subtracting
a unity feedback loop as shown in Fig. 5.8a and obta:lning an equivalent unity
feedback system as shown in Fig. 5.8b.
   Let e(t) be the error signal that is the difference between the output and the
input. For steady-state error to be zero, e(t) -* O as t + oo.
We have
Taking Laplace transforms,
e(t) : r (t) - y(t)
(5.79)
(5.80)
(5.81)
(5.82)
Using the final value theorem in Eq. (5.30), we can obtain the steady-state error
e(t) as follows:
e(oo) = shmo[se(s)]
          A'
= sl-,[-+~i)]
Fig.5.6   Uruty feedback system.
(5.83)
(5.84)
?.!
I
i
[
;j
a
~
i
t.;
$
ll
~
*:
~
dl
~
        .
..
e(s) = r(s) - y(s)
     - r(s) - G(s)e(s)
   r(s)
           =  1+ G(s)
454             PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
r
Fig.5.7   Nonunity feedback system.
a) Addition and subtraction of \uuty feedback
b) Equiyalent unity feedback system
F/g. 5.8    EquiYalent unity feedback system for a pven nonunity feedback system.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       455
The steady-state error to a unit-step function r (s) = lls is given by
e(oo) -.
1 +lims_*0 G(s)
(5.85)
In other words, for steady-state error to a unit-step function to be zero,litris_*0 G (s)
_. oo-
   Generally, we have
                    (s + Zl)(S + z, ) .. . (s + Znt)
G(s) ~ Sq~S + Pi)(s + p2) - - - (S + pn)
(5.86)
For rationaltransfer functions, n  >  m,i.e.,the number ofpoles exceeds the number
of zeros. The value of the index q designates the type of the system. For example,
if q = 0, the system is said to be a type "0" system and, for a type ";0" system,
thrrio G(s) =
2122 . . . Zm
pl P2 ' . . pn
(5.87)
which is finite. Hence, the steady-state error for a type "O" system to a unit-step
input is nonzero and is given by
e(oo)
1+Kp
Here, K p is called the position constant and is given by
Kp
2122 ' .  Zm
Pi P2 ' . ' Pn
(5.88)
(5.89)
For type "1" or higher systems (q > 1), Kp - oo, and the steady-state error to a
unit-step input approaches zero. A type sil" system is said to have one integrator in
the forward path.ln other words, the integer value of q corresponds to the number
ofintegrators in the forward path.
     It can be shown that the steady-state error to a unit-ramp function r (t) = t or
r (s) = 1/S2 jS given by
e(oo) ~ ~
where the velocity error coefficient Kv is given by
                                                          Kv = .,hmo[s G(s)]
                                        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
　
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