(3-2.8)

PB

100 100 R

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

PB Ú

(3-2.9)

PB

m(t ) = m + KC e (t ) + KC t IR Ú e (t ) dt (3-2.10)

The transfer function for the classical PI controller is

M (s) tI s + 1

1ˆ

Ê

GC (s) = = KC 1 + = KC (3-2.11)

Ë tI s ¯

E(s) tI s

To summarize, proportional“integral controllers have two tuning parameters: the

gain or proportional band, and the reset time or reset rate. The advantage is that the

integration removes the offset. About 85% of all controllers in use are of this type.

The disadvantage of the PI controller is related to the stability of the control loop.

Remembering that the ultimate gain, KCU, is considered the limit of stability

(maximum value of KC before the system goes unstable), theory predicts, and prac-

tice con¬rms, that for a PI controller the KCU is less than for a proportional

controller. That is,

> KCU

KCU P PI

The addition of integration adds some amount of instability to the system; this

is presented in more detail in Chapter 5. Therefore, to counteract this effect, the

controller must be tuned somewhat less aggressively (smaller KC). The formulas we

use to tune controllers will take care of this.

48 FEEDBACK CONTROLLERS

3-2.3 Proportional“Integral“Derivative Controller

Sometimes another mode of control is added to the PI controller. This new mode

of control is the derivative action, also called the rate action, or pre-act. Its purpose

is to anticipate where the process is heading by looking at the time rate of change

of the error, its derivative. The describing equation is

de (t )

KC

m(t ) = m + KC e (t ) + Ú e(t ) dt + K tD (3-2.12)

C

tI dt

where tD = derivative (or rate) time. The time unit used is generally minutes;

however, some manufacturers use seconds.

The proportional“integral“derivative (PID) controller has three terms, KC or PB,

R

tI or t I , and tD, which must be adjusted (tuned) to obtain satisfactory control. The

derivative action gives the controller the capability to anticipate where the process

is heading, that is, to look ahead by calculating the derivative of the error. The

amount of anticipation is decided by the value of the tuning parameter, tD.

Let us consider the heat exchanger shown in Fig. 3-1.1 and use it to clarify what

is meant by “anticipation.” Assume that the inlet process temperature decreases by

some amount and the outlet temperature starts to decrease correspondingly,

as shown in Fig. 3-2.6. At time ta the amount of the error is positive and small.

Consequently, the amount of control correction provided by the proportional and

integral modes is small. However, the derivative of this error, that is, the slope of

the error curve, is large and positive, making the control correction provided by the

derivative mode large. By looking at the derivative of the error, the controller knows

that the controlled variable is heading away from the set point rather fast, and con-

sequently, it uses this fact to help in controlling. At time tb the error is still positive

and larger than before. The amount of control correction provided by the propor-

tional and integral modes is also larger than before and still adding to the output

of the controller to open the steam valve further. However, the derivative of the

error at this time is negative, signifying that the error is decreasing; the controlled

variable has started to come back to the set point. Using this fact, the derivative

mode starts to subtract from the other two modes since it recognizes that the error

is decreasing. This controller results in reduced overshoot and oscillations around

the set point.

PID controllers are recommended for use in slow processes (processes with long

time constants), such as temperature loops, which are usually free of noise. Fast

processes (processes with short time constants) are easily susceptible to process

noise. Typical of these fast processes are ¬‚ow loops and liquid pressure loops. For

these processes with noise, the use of derivative action will amplify the noise and

therefore should not be used for these processes.

The transfer function of a PID controller is given by

M (s) 1

Ê ˆ

GC (s) = = KC 1 + + t Ds (3-2.13)

Ë ¯

E(s) tI s

To summarize, PID controllers have three tuning parameters: the gain or pro-

portional band, the reset time or reset rate, and the rate time. PID controllers should

49

TYPES OF FEEDBACK CONTROLLERS

Figure 3-2.6 Response of heat exchanger temperature to a disturbance.

50 FEEDBACK CONTROLLERS

not be use in processes with noise. An advantage of the derivative mode is that it

provides anticipation. Another advantage is related to the stability of the system.

Theory predicts, and practice con¬rms, that the ultimate gain with a PID controller

is larger than that of a PI controller. That is,

> KCU

KCU PID PI

The derivative terms add some amount of stability to the system; this is presented

in more detail in Chapter 5. Therefore, the controller can be tuned more aggres-

sively now. The formulas we™ll use to tune controllers will take care of this.

3-2.4 Proportional“Derivative Controller

The proportional“derivative (PD) controller is used in processes where a

proportional controller can be used, where steady-state offset is acceptable but

some amount of anticipation is desired, and no noise is present. The describing

equation is

de (t )

m(t ) = m + KC e (t ) + KC t D (3-2.14)

dt

and the transfer function is

M (s)

GC (s) = = KC (1 + t D s) (3-2.15)

E(s)

Based on our previous presentation on the effect of each tuning parameter on the

stability of systems, the reader can complete the following:

KCU ? KCU

PD P

3-3 RESET WINDUP

The problem of reset windup is an important and realistic one in process control.

It may occur whenever a controller contains integration. The heat exchanger control

loop shown in Fig. 3-1.1 is again used at this time to explain the reset windup

problem.

Suppose that the process inlet temperature drops by an unusually large amount;