Tuning of Controllers

Controllers are tuned in an effort to match the characteristics of the control equipment to the process so that two goals are achieved:

❑ The system responds quickly to errors;
❑ The system remains stable (PV does not oscillate around the SP).

Gain

Controller tuning is performed to adjust the manner in which a control valve (or other final control element) responds to a change in error. In particular, we are interested in adjusting the gain of the controller such that a change in controller input will result in a change in controller output that will, in turn, cause sufficient change in valve position to eliminate error, but not so great a change as to cause instability or cycling.

Gain is defined simply as the change in output divided by the change in input.

Examples:

Change in Input to Controller - 10%
Change in Controller Output - 20%
Gain = 20% / 10% = 2

Change in Input to Controller - 10%
Change in Controller Output - 5%
Gain = 5% / 10% = 0.5

Gain Plot - The Picture 1 below is simply another graphical way of representing the concept of gain.

Picture 1: Graphical Representation of Gain Concept

Proportional Mode

The proportional mode is used to set the basic gain value of the controller. The setting for the proportional mode may be expressed as either:

1. Proportional Gain
2. Proportional Band

Proportional Gain

In electronic controllers, proportional action is typically expressed as proportional gain. Proportional Gain (Kc) answers the question: "What is the percentage change of the controller output relative to the percentage change in controller input?" Proportional Gain is expressed as:

Gain Kc = ∆Output[%] / ∆Input[%]

Proportional Band

Proportional Band (PB) is another way of representing the same information and answers this question: "What percentage of change of the controller input span will cause a 100% change in controller output?" Proportional Band is expressed as:

PB = ∆Input (% Span) For 100% ∆Output

Converting Between PB and Gain

A simple equation converts gain to proportional Band:

PB = 100/Gain

Also recall that:

Gain = 100%/PB


Picture 2 shows the relationship between proportional gain and proportional band.

Picture 2: Relationship of Proportional Gain and Proportional Band

Limits of proportional action

>> Responds Only to a Change in error - Proportional action responds only to a change in the magnitude of the error.
>> Does Not Return the PV to Setpoint - Proportional action will not return the PV to setpoint. It will, however, return the PV to a value that is within a defined span (PB) around the PV.

Proportional Action - Closed Loop

Loop Gain - Every loop has a critical or natural frequency. This is the frequency at which cycling may exist. This critical frequency is determined by all of the loop components. If the loop gain is too high at this frequency, the PV will cycle around the SP; i.e. the process will become unstable.

Proportional Summary

For the proportional mode, controller output is a function of a change in error. Proportional band is expressed in terms of the percentage change in error that will cause 100% change in controller output. Proportional gain is expressed as the percentage change in output divided by the percentage change in input.

PB = (∆Input, % / ∆Output, % ) x 100 = 100/Gain

∆ Controller Output = (Change in Error)(Gain)

1. Proportional Mode responds only to a change in error;
2. Proportional mode alone will not return the PV to SP.

Advantages - Simple.
Disadvantages - Error.

Settings - PB settings have the following effects:

>> Small PB (%) Minimize Offset;
>> High Gain (%) Possible cycling;
>> Large PB (%) Large Offset;
>> Low Gain Stable Loop.

Tuning - reduce PB (increase gain) until the process cycles following a disturbance, then double the PB (reduce gain by 50%).

Integral Mode

Duration of Error and Integral Mode - Another component of error is the duration of the error, i.e., how long has the error existed? The controller output from the integral or reset mode is a function of the duration of the error.

Open Loop Analysis

The purpose of integral action is to return the PV to SP. This is accomplished by repeating the action of the proportional mode as long as an error exists. With the exception of some electronic controllers, the integral or reset mode is always used with the proportional mode.

Integral, or reset action, may be expressed in terms of:

>> Repeats Per Minute - How many times the proportional action is repeated each minute;
>> Minutes Per Repeat - How many minutes are required for 1 repeat to occur.

Closed Loop Analysis

Closed Loop With Reset - Adding reset to the controller adds one more gain component to the loop. The faster the reset action, the greater the gain.

Slow Reset Example - In this example the loop is stable because the total loop gain is not too high at the loop critical frequency (Picture 3). Notice that the process variable does reach set point due to the reset action.

Picture 3: Slow Reset - Closed Loop


Fast Reset Example - In the example the rest is too fast and the PV is cycling around the SP (Picture 4).

Picture 4: Fast Reset - Closed Loop

Reset Windup

Reset windup is described as a situation where the controller output is driven from a desired output level because of a large difference between the set point and the process variable (Picture 5).

Picture 5: Reset Windup - Anti-Reset Windup


Shutdown - Reset windup is common on shut down because the process variable may go to zero but the set point has not changed, therefore this large error will drive the output to one extreme.
Startup - At start up, large process variable overshoot may occur because the reset speed prevents the output from reaching its desired value fast enough (Picture 6).

Picture 6: Reset Windup - Shutdown and Startup


Anti Reset Windup - Controllers can be modified with an anti-reset windup (ARW) device. The purpose of an anti-reset option is to allow the output to reach its desired value quicker, therefore minimizing the overshoot.

Integral (Reset) Summary

Output is a repeat of the proportional action as long as error exists. The units are in terms of repeats per minute or minutes per repeat.

Advantages - Eliminates error.
Disadvantages - Reset windup and possible overshoot.

Fast Reset (Large Repeats/Min.,Small Min./Repeat):

1.High Gain;
2.Fast Return To Setpoint;
3.Possible Cycling.

Slow Reset (Small Repeats/Min.,Large Min./Repeats):

1.Low Gain;
2.Slow Return To Setpoint;
3.Stable Loop.

Trailing and Error Tuning - Increase repeats per minute until the PV cycles following a disturbance, then slow the reset action to a value that is 1/3 of the initial setting.

Derivative Mode

Some large and/or slow process do not respond well to small changes in controller output. For example, a large liquid level process or a large thermal process (a heat exchanger) may react very slowly to a small change in controller output. To improve response, a large initial change in controller output may be applied. This action is the role of the derivative mode.

The derivative action is initiated whenever there is a change in the rate of change of the error (the slope of the PV). The magnitude of the derivative action is determined by the setting of the derivative, the mode of a PID controller and the rate of change of the PV. The derivative setting is expressed in terms of minutes. In operation, the controller first compares the current PV with the last value of the PV. If there is a change in the slope of the PV, the controller determines what its output would be at a future point in time (the future point in time is determined by the value of the derivative setting, in minutes). The derivative mode immediately increases the output by that amount (Picture 7).

Picture 7: Derivative Action is based on the rate of change in Error (Y/X)

Example

Let's start a closed loop example by looking at a temperature control system. In this example, the time scale has been lengthened to help illustrate controller actions in a slow process. Assume a proportional band setting of 50%. There is no reset at this time. The proportional gain of 2 acting on a 10% change in set point results in a change in controller output of 20%. Because temperature is a slow process the setting time after a change in error is quite long. And, in this example, the PV never becomes equal to the SP because there is no reset.

Rate Effect - To illustrate the effect of rate action, we will add there a mode with a setting of 1 minute. Notice the very large controller output at time 0. The output spike is the result of rate action. Recall that the change in output due to rate action is a function of the speed (rate) of change of error, which in a step is nearly infinite. The addition of rate alone will not cause the process variable to match the set point (Picture 8).

Picture 8: No Rate, Small Rate examples, Closed Loop



Effect of Fast Rate - Let's now increase the rate setting to 10 minutes. The controller gain is now much higher. As a result, both the IVP (controller output) and the PV are cycling. The point here is that increasing the rate setting will not cause the PV to settle at the SP (Picture 9).

Picture 9: P+D, High Rate Setting, Closed Loop Analysis


Need for Reset Action - It is now clear that reset must be added to bring process variable back to set point.

Applications - Because this component of the controller output is dependent on the speed of change of the input or error, the output will be very erratic if rate is used on fast process or one with noisy signals. The controller output, as a result of rate, will have the greatest change when the input changes rapidly.

Controller Option to Ignore Change in SP - Many controllers, especially digital types, are designed to respond to changes in the PV only, and to ignore changes in SP. This feature eliminates a major upset that would occur following a change in the setpoint.

Derivative (Rate) Summary

Rate action is a function of the speed of change of the error. The units are minutes. The action is to apply an immediate response that is equal to the proportional plus reset action that would have occurred some number of minutes I the future.

Advantages - Rapid output reduces the time that is required to return PV to SP in slow process.
Disadvantage - Dramatically amplifies noisy signals; can cause cycling in fast processes.

Settings

Large (Minutes):

1.High Gain
2.Large Output Change
3.Possible Cycling

Small (Minutes):

1.Low Gain
2.Small Output Change
3.Stable Loop

Trial and Error Tuning - Increase the rate setting until the process cycles following a disturbance, then reduce the rate setting to one-third of the initial value.