Page 632 - Mechanical Engineers' Handbook (Volume 2)

2 Fundamentals of Closed-Loop Performance 623

Figure 3 Simplified position servo.

where e command signal V
c
e error signal, V
e
e velocity feedback signal, V
v
e position feedback signal, V
x
F disturbance force applied to the load, N
d
K integrating servoamplifier gain, (V/s)/V
1
K proportional servoamplifier gain, V/V
2
K actuator gain, (mm/s)/V
3
K actuator velocity droop due to force disturbance, (mm/s)/N
4
K velocity transducer gain, volts/(mm/s)
v
K position transducer gain, volts/mm
x
K vv open-loop gain of velocity servo K K K ,s 1
v
3
1
K open-loop gain of position servo K K K ,s 1
vx 2 3 x
X load position, mm
V X ˙ , mm/s
As shown in Fig. 4, the velocity and position responses to commands are both char-
acterized by a first-order lag having a break frequency equal to the open-loop gain. However,
the responses to disturbance forces are quite different in the two cases (Fig. 5). When the
disturbance is downstream of the integrator (velocity servo), the servo error is K F at high
4 d
frequencies but rolls off at frequencies (in radians per second) below K and is zero statically.
vv
When the disturbance is upstream of the integrator (position servo), there is a static error
inversely proportional to the open-loop gain, which rolls off at frequencies above K . Note
vx
that Eqs. (1)–(4) remain reasonably valid when the dynamic response characteristics of the
various open-loop elements are considered, for those cases in which K vv (or K ) is well
vx
below the lowest break frequencies of those elements. At higher loop gains, the closed-loop
dynamics can change considerably, as shown in the following discussion.
The conclusions regarding the effects of disturbance forces on servo accuracy can be
generalized to any forward-loop offset or uncertainty. Referring again to Fig. 2, it can be
seen that the integrating amplifier will compensate for any forward-loop offset downstream
of the integrator, so that the static errors are zero. From Fig. 3, it is apparent that static errors
due to offsets upstream of the integration can be quantified as
X 1
e
(5)
V 0 K vx