Page 634 - Mechanical Engineers' Handbook (Volume 2)

2 Fundamentals of Closed-Loop Performance 625

e e V e 1 (7)
˙
˙ e c V c K vv
where V output error V V, mm/s
c
e
V velocity command e /K , mm/s
c
v
c
Note that the following error for the position servo is the position error resulting from
a steady rate of change of position command. For the velocity servo, the following error is
the velocity error resulting from a steady rate of change of velocity command.
The servo errors discussed thus far have been those that can be minimized by a tight
servoloop (high loop gain). To these must be added errors in the transducer mechanism.
Even if infinite loop gain were achievable, the servo can be no more accurate than the
transducer itself. The most important types of transducer inaccuracies are repeatability, res-
olution, and linearity. Errors due to transducer location and mounting geometry must also
be taken into account.
Many of the foregoing concepts can be applied to servos in general. For example, a
force or pressure servo working against a spring load is similar to a position servo in the
sense that output force is proportional to actuator position. Also, temperature control servos
tend to behave like position servos since the controlling device tends to provide heat flow
proportional to temperature error, and thermal loads tend to produce temperature rate of
change proportional to heat flow.

2.2 Dynamic Response and Stability

As discussed in Section 2.1, open-loop gain has a strong influence on servo accuracy. High
loop gains also provide fast dynamic response in most cases. However, stability considera-
tions will limit the maximum useful loop gain. The dynamic response and stability of a servo
are determined by the dynamic characteristics of the various loop components. In many
situations, the forward-loop dynamics are dominated by a relatively small number of low-
frequency lag elements, and the transducer dynamics are negligible. In these cases, it is often
possible to obtain an adequate estimate of servo performance by approximating the combined
forward-loop characteristics with an integrator plus a first-order or second-order lag. The
adequacy of this approximation can be determined by the match of the frequency response
gain and phase for the frequency range in which the phase lag is less than 180 . Using these
two rather basic dynamic forms, the relationships among loop gain, stability, and dynamic
response are easily seen.
A block diagram using the basic dynamic forms is shown in Fig. 6,
where U generalized controlled variable
U generalized command input
c
D generalized disturbance input
G open-loop response of U to D
d
The first-order lag is typical of simple temperature control systems and dc servos having
short electrical time constants. The second-order lag is often representative of electrohy-
draulic servos and dc servos having long electrical time constants. The closed-loop responses
to command inputs are
U 1 1 (8)
2
U ( /K )s (1/K )s 1
c 1 u1 u1
and