Control Notes

The Lambda tuning rules, sometimes also called Internal Model Control (IMC)* tuning, offer a robust alternative to tuning rules aiming for speed, like Ziegler-Nichols, Cohen-Coon, etc. Although the Lambda and IMC rules are derived differently, both produce the same rules for a PI controller on a self-regulating process.

While the Ziegler-Nichols and Cohen-Coon tuning rules aim for quarter-amplitude damping, the Lambda tuning rules aim for a first-order lag plus dead time response to a set point change. The Lambda tuning rules offer the following advantages:

• The process variable will not overshoot its set point after a disturbance or set point change.
• The Lambda tuning rules are much less sensitive to any errors made when determining the process dead time through step tests. This problem is common with lag-dominant processes, because it is easy to under- or over-estimate the relatively short process dead time. Ziegler-Nichols and Cohen-Coon tuning rules can give really bad results when the dead time is measured incorrectly.
• The tuning is very robust, meaning that the control loop will remain stable even if the process characteristics change dramatically from the ones used for tuning.
• A Lambda-tuned control loop absorbs a disturbance better, and passes less of it on to the rest of the process. This is a very attractive characteristic for using Lambda tuning in highly interactive processes. Control loops on paper-making machines are commonly tuned using the Lambda tuning rules to prevent the entire machine from oscillating due to process interactions and feedback control.
• The user can specify the desired response time (actually the closed loop time constant) for the control loop. This provides one tuning factor that can be used to speed up and slow down the loop response.

Unfortunately, the Lambda tuning rules have a drawback too. They set the controller’s integral time equal to the process time constant. If a process has a very long time constant, the controller will consequently have a very long integral time.  Long integral times make recovery from disturbances very slow.

It is up to you, the controls practitioner, to decide if the benefits of Lambda tuning outweigh the one drawback. This decision must take into account the purpose of the loop in the process, the control performance objective, the typical size of process disturbances, and the impact of deviations from set point.

Below are the Lambda tuning rules for a PI controller. Although Lambda / IMC tuning rules have also been derived for PID controllers, there is little point in using derivative control in a Lambda-tuned controller. Derivative control should be used if a fast loop response is required, and should therefore be used in conjunction with a fast tuning rule (like Cohen Coon). Lambda tuning is not appropriate for obtaining a fast loop response.  If speed is the objective, use another tuning rule.

To apply the Lambda tuning rules for a self-regulating process, follow the steps below. Also, please read the paragraph in red text following the tuning equations.

1. Do a step-test and determine the process characteristics

a) Place the controller in manual and wait for the process to settle out.

b) Make a step change in the controller output (CO) of a few percent and wait for the process variable (PV) to settle out. The size of this step should be large enough that the PV moves well clear of the process noise/disturbance level. A total movement of five times the noise/disturbances on the process variable should be sufficient.

c) Calculate the process characteristics as follows:

Process Gain (gp)

Convert the total change in PV to a percentage of the measurement span.

gp = change in PV [in %] / change in CO [in %]

Dead Time (td)

Note: Make this measurement in the same time-units your controller’s integral mode uses. E.g. if your controller’s integral time is in minutes, use minutes for this measurement.

Find the maximum slope of the PV response curve. This will be at the point of inflection. Draw a line tangential through the PV response curve at this point. Extend this line to intersect with the original level of the PV before the step in CO. Take note of the time value at this intersection.

td = time difference between the change in CO and the intersection of the tangential line and the original PV level

Time Constant (tau)

Calculate the value of the PV at 63% of its total change. On the PV reaction curve, find the time value at which the PV reaches this level

tau = time difference between intersection at the end of dead time, and the PV reaching 63% of its total change

Note: Make this measurement in the same time-units your controller’s integral mode uses. E.g. if your controller’s integral time is in minutes, use minutes for this measurement.

Step Test for Lambda Tuning

d) Repeat steps b) and c) two more times to obtain good average values for the process characteristics. If you get vastly different numbers every time, do even more step tests until you have a few step tests that produced similar values. Use the average of those values.

2. Pick a desired closed loop time constant (taucl) for the control loop

A large value for taucl will result in a slow control loop, and a small taucl value will result in a faster control loop. Generally, the value for taucl should be set between one and three times the value of tau.

Use taucl = 3 x tau to obtain a very stable control loop. If you set taucl to be shorter than tau, the advantages of Lambda tuning listed above soon disappear.

3. Calculate PID controller settings using the equations below

Controller Gain (Kc)

Kc = tau/(gp x (taucl + td))

Integral Time (Ti)

Ti = tau

Derivative Time (Td)

Td = zero.

Important Notes!

• The tuning equations above are designed to work on controllers with interactive or noninteractive algorithms, but not controllers with parallel (independent gains) algorithms.
• The rules calculate controller gain (Kc) and not proportional band (PB). PB = 100/Kc.
• The rules assume the controller’s integral setting is integral time Ti (in minutes or seconds), and not integral gain Ki (repeats per minute or repeats per second). Ki = 1/Ti.

Read this posting for more details.

If your controller is different from the above, simple parameter conversions will allow you to use the Lambda rules.

Stay tuned!

Jacques Smuts – Author of the book Process Control for Practitioners

*Rivera, D.E., M. Morari, and S. Skogestad, Internal Model Control 4. PID Controller Design, Industrial Engineering and Chemical Process Design and Development, 25, p. 252, 1986