Page 290 - Modern Control Systems

264 Chapter 4 Feedback Control System Characteristics

E(s) = R(s) - Y(s) = I R(s),
1 + G c(s)G p(s)G(s)
or
2 2
s 4 + 2ps 3 + p s
E(S) = — A r —z 7. R(s).
s 4 + 2ps 3 + (p 2 + K D)s 2 + K Ps + Kj
Using the final-value theorem, we determine that the steady-state tracking
error is
2 2
R 0(s 4 + 2/?5 3 + p s )
lim sE(s) J = lim —. 3 = 2 r 2 : = 0,
4
v
s-*Q ,-,0 s + 2ps + (p + K D)s + K Ps + Ki
where R(s) = RQ/S is a step input of magnitude R G. Therefore,
lime(0 = 0.
/—»•00
With a PID controller, we expect a zero steady-state tracking error (to a step input)
for any nonzero values of Kp, K D, and K h As we will see in Chapter 5, the integral
term, Kj/s, in the PID controller is the reason that the steady-state error to a unit
step is zero. Thus design specification DS3 is satisfied.
When considering the effect of a step disturbance input, we let R(s) = 0 and
N(s) = 0. We want the steady-state output Y(s) to be zero for a step disturbance.
The transfer function from the disturbance T d(s) to the output Y(s) is

~G(s)
1 + G c(s)G p(s)G{s)

~° 2 Us).
s 4 + 2/95 3 + (p 2 + K D)s 2 + KpS + Ki

When
A)
Us) = -f,
we find that

lim sY(s) = lim —; ^ ^ ^ = 0.

Therefore,

limy(0 = 0.
/—»00
Thus a step disturbance of magnitude D 0 will produce no output in the steady-state,
as desired.
The sensitivity of the closed-loop transfer function to changes in p is given by
cT cT cG
^p ~ °G°p-