The proportional band is a band around the setpoint of the PXZ where

the output is between 0% and 100%. The percentage of output is proportional

to the amount of error between the setpoint variable (SV) and

the process variable (PV). Outside of the proportional band the output

is either 0% or 100%

The proportional band on the PXZ is equidistant from the main setpoint

as illustrated below.

An example of proportioning would be a vehicle approaching a stop

sign at an intersection. If the driver were traveling at 50mph and only

applied his brakes once at the intersection, his car would skid through

the intersection before coming to a full stop. This illustrates how On/Off

control acts. If, however, the driver started slowing down some

distance before the stop sign and continued slowing down at some

rate, he could conceivably come to a full stop at the stop sign. This

illustrates how proportional control acts. The distance where the speed

of the car goes from 50 to 0 MPH illustrates the proportional band. As

you can see, as the car travels closer to the stop sign, the speed is

reduced accordingly. In other words, as the error or distance between

the car and the stop sign becomes smaller, the output or speed of the

car is proportionally diminished. Figuring out when the vehicle should

start slowing down depends on many variables such as speed, weight,

tire tread, and braking power of the car, road conditions, and weather

much like figuring out the proportional band of a control process with its

many variables.

The width of the proportional band depends on the dynamics of the

system. The first question to ask is, how strong must my output be to

eliminate the error between the setpoint variable and process variable?

The larger the proportional band (low gain), the less reactive

the process. A proportional band too large, however, can lead to

process wandering or sluggishness. The smaller the proportional

band (high gain), the more reactive the output becomes. A proportion-

al band too small, however, can lead to over-responsiveness leading to

process oscillation.

Proportional band to large

A proportional band which is correct in width approaches main setpoint

as fast as possible while minimizing overshoot. If a faster approach to

setpoint is desired and process overshoot is not a problem, a smaller

or narrower proportional band may be used. This would establish an

over-damped system or one where the output would change greatly,

proportional to the error. If process overshoot cannot be tolerated and

the approach to setpoint does not have to be quick, a larger or wider

proportional band may be used. This would establish an under-damped

system or one where the output would change little, proportional to the

error.

To Calculate Proportional Band:

Pr o p or t i o n  a (as a percentage) Proportional Band

                                                              _____________X

                                                               Input Range

                                                             _____________X

              3%                            =           100C

                                                 Proportional Band

                                                (as a percentage)

Proportional Band Range=               _________________X 1000

                                                          100%

30°C = 3% x 1000°C

          100%

With the proportional band alone, the process tends to reach equilibrium

at some point away from the main setpoint. This offset is due to

the difference between the output needed to maintain setpoint and

the output of the proportional band at setpoint. In the case of the PXZ

controller where the proportional band is equidistant from the main

setpoint, the output is around 50%. If anything more or less than 50%

output is required to maintain setpoint, an offset error will occur. Integral action eliminates this offset.      See the diagrams below.

Integral action eliminates offset by adding to or subtracting from the

output of the proportional action alone. This increase or decrease in

output corrects for offset error within the proportional band in establishing

steady-state performance at setpoint. It is not intended to correct

for process disturbances. See the following diagram.

Integral Time is the speed at which the controller corrects for offset. A

short integral time means the controller corrects for offset quickly. If

the integral time is too short, the controller would react before the

effects of previous output shifts, due to dead time or lag, could be

sensed causing oscillation. A long Integral time means the control

corrects for offset over a long time. If the integral time is too long, the

offset will remain for some time causing slow responding or sluggish

control. See the diagram below.

In the case of a process upset, proportional only or proportional-integral

action cannot react fast enough in returning a process back to

setpoint without overshoot. The derivative action corrects for disturbances

providing sudden shifts in output which oppose the divergence

of the process from setpoint. See the diagram below.

_______________________________

The derivative action changes the rate of reset or integration proportional

to the rate of change and lag time of the system. By calculating

the rate of change of the process and multiplying it by the lag time

which is the time it takes the controller to sense an output change,

the controller can anticipate where the process should be and

change the output accordingly. This anticipatory action speeds up and

slows down the effect of proportional only and proportional-integral

actions to return a process to setpoint as quickly as possible with

minimum overshoot. See the diagram below.

Derivative time is the amount of anticipatory action needed to return a

process back to setpoint. A short derivative time means little derivative

action. If the derivative time is too short, the controller would not

react quickly to process disturbances. A long derivative time means

more derivative action. If the derivative time is too large, the controller

would react too dramatically to process disturbances creating

rapid process oscillation. A process which is very dynamic such as

pressure and flow applications is more efficiently controlled if the

derivative action is turned off because of the oscillation problem

which would result.

Tuning the PXZ, as with any PID loop, requires tuning each parameter

separately and in sequence. To achieve good PID control manually,

you can use the trial and error method explained below.

Set Integral Time = 0 (off)

Set Derivative Time = 0 (off)

Start with a large Proportional Band value which gives very sluggish

control with noticeable offset and tighten by decreasing the value in

half. Analyze the process variable. If the control is still sluggish, tighten

by decreasing the value in half again. Continue with the same procedure

until the process starts to oscillate at a constant rate. Widen

the Proportional Band by 50%, or multiply the setting 1.5 times. From a

cold start, test and verify that the Proportional Band allows maximum

rise to setpoint while maintaining minimum overshoot and offset. If not

completely satisfied, fine-tune the value, up or down, as needed and

test until correct. The Proportional Band is now tuned.

Start with a large Integral Time value which gives very sluggish

response to process offset and tighten by decreasing the value in half.

Analyze the process variable. If the response to process offset is still

sluggish, tighten by decreasing the value in half again. Continue with

the same procedure until the process starts to oscillate at a constant

rate. Increase the Integral Time value by 50%, or multiply the setting

1.5 times. From a cold start, test and verify that the Integral Time

allows maximum elimination of offset with minimum overshoot. If not

completely satisfied, fine-tune the value, up or down, as needed and

test until correct. The Integral Time is now tuned.

Do not add Derivative Time if the system is too dynamic. Start with a

small Derivative Time value which gives sluggish response to process

upsets and double the value. Analyze the process variable. If the

response to process upsets is still sluggish, double the value again.

Continue with the same procedure until the process starts to oscillate

at a quick constant rate. Decrease the Derivative Time value by 25%.

From a cold start, test and verify that the Derivative Time value allows

maximum response to process disturbances with minimum overshoot. If

not completely satisfied, fine-tune the value, up or down, as needed and

test until correct. Note that the Derivative Time value is usually

somewhere around 25% of the Integral Time value.

Another tuning method is the closed-loop cycling or Zeigler-Nichols

method. According to J.G. Zeigler and N.B. Nichols, optimal tuning is

achieved when the controller responds to a difference between setpoint

and the process variable with a 1/4 wave decay ratio. That is to

say that the amplitude of each successive overshoot is reduced by 3/4

until stabilizing at setpoint. The procedure is explained below.

1. Integral Time=0

Derivative Time=0

2. Decrease the Proportional Band to the point where a constant rate

of oscillation is obtained. This is the response frequency of the

system. The frequency is different for each process.

3. Measure the Time Constant which is the time to complete one

cycle of the response frequency. The Time Constant will be defined

as “T” when calculating Integral and Derivative Times.

4.Widen the Proportional Band until only slightly unstable. This is

the

Proportional Band’s Ultimate Sensitivity. The Proportional Band’s

Ultimate Sensitivity width will be defined as “P” when calculating the

actual Proportional Band.

5.Use the following coefficients in determining the correct PID settings

for your particular application.