Page 55 - Servo Motors and Industrial Control Theory -

2.8 Steady State Error 47

The above equation can be used to calculate the steady state error. The steady state
error is defined as the deference between the input (desired value) and the output
variables.
Example 9 Error for unit step, ramp, and acceleration inputs for the system dis-
cussed in Example 8.
In Example 8, the root locus of a servo system was discussed. The roots of the
characteristic equation determine the transient response behavior of the system. The
transient response can be calculated by various computer programs. The final value
theorem can be used to evaluate the steady state error for various standard inputs.
By definition,

e = θ − θ (2.39)
i o
It is clear that when the variable is function of time or s, the above equation can be
considered, as the variables are function of time or function of s.
Substituting for θ from Eq. (2.37) and with some algebraic manipulation gives
o

3
0.5s + s + K Ks
2
e = d θ (2.40)
0.5s + s + K Ks + K i
2
3
d
The final value theorem can now be used to find the steady state error for various
input function.
For unit step input, 1
θ = s (2.41)
i
Substituting Eq. (2.41) in Eq. (2.40) and using the final value theorem gives

c Lim(f(t)) : Lim(sF(s))=
t →∞ → 0
s
Solving Eq. (2.40) gives
e = 0

It shows that the steady state error for step input is zero. Although unit step input is
used, similar results can be obtained by multiplying the result by constant of the step
input, which in this example is unity. The same way for unit ramp input of θ = t the
i
Laplace Transform by referring to the table of Laplace Transform gives
1
θ =
i
s 2