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Lambda Tuning Rules

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

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

24 Responses to “Lambda Tuning Rules”

  • Trevor:

    I have a negative relationship between my CO and my PV, in other words I raise my CO and my PV decreases. Should I use a negative Gp (and get a resultant negative Kc) or should I just use the absolute value?


  • Trevor – Very few controllers support a negative Kc. Check your controller documentation. Most likely you will have to use the absolute value of the calculated gp and configure your controller to be a direct-acting controller.

  • Tejaswinee:

    Cohen-Coon and Ziegler-Nichols can be stated as first order-+time delay process model; because we estimate same parameters like dead time, time constant and process gain. Isn’t it?

  • Tejaswinee, yes the Ziegler-Nichols, Cohen-Coon, Lambda, Minimum IAE, and many other tuning rules use the first-order + dead time process model for tuning control loops with self-regulating processes. Since the model is only an approximation of the real process, the resultant tuning settings are only approximately correct, but in most cases good enough to use without further adjustment, provided that the loop was detuned appropriately.

  • ajay:

    Hey all,

    Very nice article.

    I Have a question anybody tried to implement this method using any software.. !!!
    I am a student i am trying to implement a PID controller for controlling the flow rate of air. The problem i am facing is I want to use automatic tuning of the controller but i am not sure which method shall i use and then there are some question which can not figure out how to do it.
    so i just wanted if somebody already implemented can help me with this.
    thanks and regards

  • Ajai, I suggest you search for “relay auto tuning” on Google. Most PID controllers with integrated auto tuning use this technique in some form and it is reasonably easy to implement. The most difficult part of implementation is handling abnormal data, and very fast, very slow, noisy, and disturbed processes. But if this is a lab project you probably don’t have to worry about these factors.

  • Marco:

    I did the response test with an Oxford Instrument ITC503 temperature controller. My problem is the proportional band used by the controller is in Kelvin. How should I convert the gain?

  • Marco,
    Kc = controller gain.
    PB = proportional band.

    100%/Kc = PB[%] = PB[Kelvin] / (PV Range [Kelvin]) * 100%
    1/Kc = PB[Kelvin] / (PV Range)
    PB[Kelvin] = (PV Range) / Kc

    – Jacques

  • Marco:

    One more question. Wenn performing the Step response test, should CO changes e.g. 1%, 2%, 3%,… have the same time constant tau?

  • In theory the time constant should be the same. In practice it normally varies a bit, sometimes a lot. If the valve travels slower than the actual process responds, the apparent dead time and time constant will get longer as the step sizes increase. Also, when making really small steps such as 1% – 2%, some control valves react differently from one step top the next because of friction, stiction, and deadband. I prefer making 5% steps (sometimes larger) for tuning, but the processes don’t always allow it.

  • RobW:

    Great article. Very direct and straight forward.

    I have a question regarding intergrating processes.
    How are the lambda tuning rules applied if the process is intergrating?
    Could you outline the tuning method for this thype of process also?


  • Rob: With IMC tuning for level controllers, you would use the following settings for a PI controller:
    Kc = 1/ri x (2 taucl + td) / (taucl + td)^2
    Ti = 2 taucl + td
    Where:ri is the integration rate of the process in dPV%/(dCO% x time). Time must be in the same units as td, taucl, and your controller’s Ti.

  • Greg:

    What do you use for tau (for calculating taucl) when the process is integrating?



  • Greg – You can normally use td x 3. But be aware that some level loops have a negligibly short dead time compared to their integration rate ri. Then the IMC and modified Ziegler Nichols rules will give you a very large controller gain. If you want to use a lower gain, use a larger taucl or use the level averaging tuning rule.

  • ravi:

    To calculate “ri”, Can we used only one slope (The slope produce after making a step change in controller out put) and then what will be the procedure to calculate ri, Kc, Ti, in this case.
    An integrating process responds with a steady ramp instead of reaching a stable value, so how can we calculate “Tau” (63% of total PV change) and how can we calculate “Taucl”.

  • 1. Yes, but only if your initial slope was zero, i.e. PV1 = PV2 using the method described on this page:
    2. For integrating loops, pick a Taucl as required by the process, but make sure it is several times longer than dead time. If you don’t know what the process requires, then decide if you want to use the level as a surge tank, or you the level to remain steady and use the appropriate tuning methods.

  • umair:

    Hi, I have a question. Can we use this tuning method on cascade control like Level and Flow cascade.

  • umair:

    i use Yokogawa controller US1000 and limit of its proportional gain is 999. should i multiply my KC by 10?

  • Umair: Yes you can if you require a relatively slow response and a system that will remain stable under most adverse and interactive conditions. When needed, I have used it on cascade loops with great success.

  • No. Why do you want to do that?

  • Daniel O.:

    When performing a bump test on a liquid flow controller I get about a 20% overshoot of the PV before it settles to a steady state PV on bumps up and down. When calculating the gain of the process should i be using the maximum/minimum PV or the steady state value that occurs 5-10 seconds after the peak. Also for calculating the integral time which one would PV would you apply.

    Link to the graph below(the orange is the output change of 2% on a different scale not shown)

  • Daniel: A simple flow loop should not overshoot after a controller output step change in manual. I would question the valve positioner, it seems to be overshooting. See this article on valve problems.
    However, if this is the response you have to deal with, the safest is to use the max change in PV to calculate the process gain. For Lambda tuning’s integral time, the safest would also be to calculate the time constant from 63% of the max change in PV.

  • Clint E:

    When calculating process gain (gp) should the change in PV be representative of total range of transmitter or % change from starting point. For example if a flow transmitter is ranged 0-300 gpm and the starting PV is 100 gpm and final PV is 150 gpm – the total gpm change is 50 gpm which is a 16.67% change of total range (50/300) and 50% change from starting PV (50/100). Same question for the change in Output

  • Clint: To calculate process gain, changes in PV and CO should always be measured as a percentage of full scale.

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