Page 77 - Understanding Automotive Electronics

2735 | CH 2 Page 64 Tuesday, March 10, 1998 10:55 AM

2 THE SYSTEMS APPROACH TO CONTROL AND INSTRUMENTATION

PROPORTIONAL INTEGRAL CONTROLLER

One way of avoiding the steady disturbance response error of a
proportional (P) controller is to use a controller that is called a proportional
integral (PI) controller. The proportional integral control corrects for a serious
defect in a purely proportional controller. In the latter type, any steady
disturbance leads to a steady error in the output regulation. The configuration
for a PI controller is shown in Figure 2.26. This controller combines a term
proportional to the error with the integral of the error.
The error signal for a PI controller is generated the same way as for the P
controller. That is, the error e is the difference between the desired and actual values:
e = x – x o

The controller then generates the integral of the error,

∫ et
d

electronically. The control signal that is applied to the plant includes a part that
is proportional to the error as well as a part that is proportional to the integral
of the error. The resulting control signal is the sum of these two parts:

∫
v = Ge + k e t
d
c
where G is the proportional gain and k is the integral gain. The proportional
part of the control signal acts the same as in a P controller, that is, it drives the
plant so as to reduce the error to zero.
However, consider the case of a disturbance to the system that tends
to produce a steady error in a P controller. In a PI controller, the integral
part continues to increase (since the integral of a steady error continuously
increases until the error itself is zero). This integral part of the control
signal drives the plant in a direction to reduce the error due to disturbance
to zero.
The disturbance rejection of a PI and P controller are illustrated in Figure
2.24. Here a steady disturbance occurs at time t = 0. The P controller response
is shown in the dotted curve, which depicts the steady error. The PI controller
response is shown by the solid curve, for which the error eventually is reduced
to zero. This ability to reduce errors due to disturbance to zero is a fundamental
property of PI controllers.
In principle a PI controller could be implemented in a purely mechanical
control system. However, in practice the only feasible implementation of an
integral is electronic (where integration can be implemented by means of a
capacitor in a simple electronic circuit). Integrals can also be implemented with

64 UNDERSTANDING AUTOMOTIVE ELECTRONICS