such as the level in a surge tank, the cruise control in a car, or a thermostat in a

house, this may not be of any major consequence. For processes in which the process

variable can be controlled within a band from set point, proportional controllers are

suf¬cient. However, when the process variable must be controlled at the set point,

not away from it, proportional controllers do not provide the required control.

3-2.2 Proportional“Integral Controller

Most processes cannot be controlled with an offset; that is, they must be controlled

at the set point. In these instances an extra amount of “intelligence” must be added

to the proportional controller to remove the offset. This new intelligence, or new

mode of control, is the integral, or reset, action; consequently, the controller

becomes a proportional“integral (PI) controller. The describing equation is

KC

m(t ) = m + KC e (t ) + Ú e(t ) dt (3-2.6)

tI

where tI is the integral (or reset) time with units of time. Most often, the time unit

used is minutes; less often, seconds are used. The unit used depends on the manu-

facturer. Therefore, the PI controller has two parameters, KC and tI, both of which

must be adjusted (tuned) to obtain satisfactory control.

To understand the physical signi¬cance of the reset time, consider the hypothet-

ical example shown in Fig. 3-2.4. At some time t = 0, a constant error of 1% in

magnitude is introduced in the controller. At this moment the PI controller solves

the following equation:

45

TYPES OF FEEDBACK CONTROLLERS

c(t), 1%

set point

% TO

t=0 time

m(t),

% CO

52

KC

51

KC

50

time

tI

Figure 3-2.4 Response of a PI controller (direct acting) to a step change in error.

KC t

m(t ) = 50% + KC e (t ) + e (t ) dt

t I Ú0

KC t

= 50% + KC (1) + (1) dt

t I Ú0

KC

= 50% + KC + t

tI

When the error is introduced at t = 0, the controller output changes immediately by

an amount equal to KC; this is the response due to the proportional mode.

KC

m(t = 0) = 50% + KC + (0) = 50% + KC

tI

As time increases the output also increases in a ramp fashion as expressed by the

equation and shown in the ¬gure. Note that when t = tI the controller output

becomes

m(t = t I ) = 50% + KC + KC

Thus, in an amount of time equal to tI, the integral mode repeats the action taken

by the proportional mode. The smaller the value of tI, the faster the controller

integrates to repeat the proportional action. Realize that the smaller the value of

tI, the larger the term in front of the integral, KC/tI, and consequently, the faster the

integral term moves the controller output.

46 FEEDBACK CONTROLLERS

Figure 3-2.5 Response of level under P and PI control.

To explain why the PI controller removes the offset, consider the level control

system used previously to explain the offset required by a P controller. Figure 3-2.5

shows the response of the level under P and PI controllers to a change in inlet ¬‚ow

from 150 gpm to 170 gpm. The response with a P controller shows the offset, while

the response with a PI controller shows that the level returns to the set point, with

no offset. Under PI control, as long as the error is present, the controller keeps

changing its output (integrating the error). Once the error disappears, goes to zero,

the controller does not change its output anymore (it integrates a function with a

value of zero). As shown in the ¬gure, at time = 3 min, the error disappears. The

signal to the valve must still be 60%, requiring the valve to deliver 170 gpm. Let us

look at the PI equation at the moment the steady state is reached:

KC

m(t ) = 50% + KC (0) + (0) dt

tI Ú

= 50% + 0 + 10% = 60%

The equation shows that even with a “zero” error, the integral term is not zero but,

rather, 10%, which provides the required output of 60%. The fact that the error is

zero does not mean that the value of the integral term is zero. It means that the

integral term remains constant at the last integrated value. Integration means area

under the curve, and even though the level is the same at t = 0 and at t = 3 min, the

value of the integral is different (different areas under the curve) at these two times.

The value of the integral term times KC/tI is equal to 10%. Once the level returns

to the set point, the error disappears and the integral term remains constant.

Integration is the mode that removes the offset.

47

TYPES OF FEEDBACK CONTROLLERS

Some manufacturers do not use the reset time for their tuning parameter. They

R

use the reciprocal of reset time, which we shall refer to as reset rate, t I ; that is,

1

t IR = (3-2.7)

tI

The unit of t IR is therefore 1/time or simply (time)-1. Note that when tI is used and

faster integration is desired, a smaller value must be used in the controller. However,

when t IR is used, a larger value must be used. Therefore, before tuning the reset

term, the user must know whether the controller uses reset time (time) or reset rate

(time)-1; tI and t I are the reciprocal of one another, and consequently, their effects

R

are opposite.

As we learned in Section 3.2.1, two terms are used for the proportional mode

(KC and PB), and now we have just learned that there are two terms for the inte-

gral mode (tI and t IR). This can be confusing, and therefore it is important to keep

the differences in mind when tuning a controller. Equation (3-2.6), together with

the following equations, show the four possible combinations of tuning parameters;

we refer to Eq. (3-2.6) as the classical controller equation.

100 100

m(t ) = m + e (t ) + t I e (t ) dt