Page 411 - Mechanical Engineers' Handbook (Volume 2)

402 Basic Control Systems Design

the change in the control signal is proportional to the integral of the error. In the terminology
of Fig. 7, this gives

K I
F(s) E(s) (16)
s
is the integral gain. In the time
where F(s) is the deviation in the control signal and K I
domain, the relation is
t
ƒ(t) K e(t) dt (17)
I
0
if ƒ(0) 0. In this form, it can be seen that the integration cannot continue indefinitely
because it would theoretically produce an infinite value of ƒ(t)if e(t) does not change sign.
This implies that special care must be taken to reinitialize a controller that uses integral
action.

Integral Control of a First-Order System
Integral control of the velocity in the system of Fig. 20 has the block diagram shown in Fig.
22, where G(s) K/s, K K K K /R. The integrating action of the amplifier is physically
1 I T
obtained by the techniques to be presented in Section 6 or by the digital methods presented
in Section 10. The control system is stable if I, c, and K are positive. For a unit step command
input, 1; so the offset error is zero. For a unit step disturbance, the steady-state
ss
deviation is zero if the system is stable. Thus, the steady-state performance using integral
control is excellent for this plant with step inputs. The damping ratio is c/2 IK. For
slight damping, the response will be oscillatory rather than exponential as with proportional
control. Improved steady-state performance has thus been obtained at the expense of de-
graded transient performance. The conflict between steady-state and transient specifications
is a common theme in control system design. As long as the system is underdamped, the
time constant is 2I/c and is not affected by the gain K, which only influences the
oscillation frequency in this case. It might by physically possible to make K small enough
so that 1, and the nonoscillatory feature of proportional control recovered, but the
response would tend to be sluggish. Transient specifications for fast response generally re-
quire that 1. The difficulty with using 1 is that is fixed by c and I.If c and I are
such that 1, then is large if I c.

Integral Control of a Second-Order System
Proportional control of the position servomechanism in Fig. 23 gives a nonzero steady-state
deviation due to the disturbance. Integral control [G(s) K/s] applied to this system results
in the command transfer function

(s) K
(18)
3
2
(s) Is cs K
r
With the Routh criterion, we immediately see that the system is not stable because of the
missing s term. Integral control is useful in improving steady-state performance, but in gen-
eral it does not improve and may even degrade transient performance. Improperly applied,
it can produce an unstable control system. It is best used in conjunction with other control
modes.