PID tuning the simple way

Tuning that works "every" time

Throughout the years I have tried every method I could find, from text book classical analytical s-plane methods to trial and error types. It's mostly turbine governors I have been working with, both in theory and in practice. The only method I have found that works almost every time, in theory and on site, is a trial and error method. Sadly I cannot find the copy of the original text, nor can I remember where I found it. I have written down the method though, ages ago.

The method requires that you are able to set the set-point arbitrarily and can read off the process value. A graphical representation is of much help rather than just numbers, although numbers works as well. The method is also described in the manual of LVTrans. The method requires no knowledge of the system, although such knowledge certainly helps.

The first thing to do is to stabilize the system. This can usually be done by setting Kp to 1 or less, and Ti to 10 or more and Td to 0. Then set the set-point (SP) to a desired value of which you are going to tune around, the operational point OP, and let the whole system settle. It's useful to have some integral value here to get the SP and the process value (PV) equal, but not strictly necessary. When this is done, and looks OK and steady state, the tuning can start.

1. Set Ti to a really large number, or deactivate further integration if possible. In parallel form, Ki can be set to zero. Keep Td = 0. Let the system stabilize if it isn't already.
2. Do steps in SP of approximately 10% up and down from the OP and observe the PV.
3. Increase Kp bit by bit until an overshoot of the PV of about 15% is observed. That is 15% of the size of the step in SP, not 15% of the total PV.
4. With the Kp from 3, decrease Ti bit by bit until the overshoot increases to around 30%.
5. For a PI governor: Decrease Kp bit by bit until the overshoot is 5-20% or looks OK - finished.
6. For a PID: Increase Td bit by bit until the overshoot is 5-20% or looks OK - finished.

That's it. An "optimal" PID will have some overshoot, but for some applications an overshoot may not be wanted. Then simply adjust Kp or Td in step 5/6 until little or no overshoot is seen. By coincidence these steps looks very much like the graphical animation in the Wikipedia article.

Note. This method makes it possible to tune most PID controllers in most systems, but doesn't necessarily create the "best possible parameters" whatever that may be. For turbines the specs for the governor are in frequency domain (a separate analysis) and the ability to follow changes in SP and changes in frequency. Any tuning that makes the system behave according to spec is good enough. If no formal specs exists, then this method results in no better or worse parameters than any other method. It does however create good and stable results almost every time, and with little knowledge of the system.

On a second note, the tuning of systems where servos saturate can be tricky. By saturation I mean the output is restricted in amplitude and speed. Some anti-windup algorithm should then be included for best behavior of the PID controller. This should be in effect for step 4 and 5/6 if it isn't already. Most commercial PID's have anti-windup implemented.

On a third note, a pure 1st order system may not give any noticeable overshoot in step 3. If that happens, then you know it's a first order system. A typical example is PID servo positioning. For such systems you can set Kp to almost "whatever you want", and adjust Ti to achieve a response with no, or desired overshoot. If performance is specified, you simply adjust to achieve the wanted performance. In this case, it is however a good idea to dive a bit further into the theories of PID tuning, starting for instance here.

Background

Tuning a PID governor is the process of determining the proportional gain, the integral gain and the derivative gain so the process behave the way you want it to. A key element is "behave the way you want it to".

If the stability is important, then there could be industry standards specifying the stability gains. Typically there are gain margins and phase margins that are standardized. These are obtained in frequency domain analysis, either directly or through FFT analysis. An industrial standard could also specify certain time parameters, how fast the PID should be, how accurate, the overshoot and so on. If no specifications are given, then the usual way is to get the PID so "optimal" as possible while keeping it stable.

There are therefore no universal descriptions of what a proper tuned PID actually is. It will be dependent on the application, the industry and the very process it is governing. The only common denominator is stability of the system as a whole. Theory and practice are usually different when working with PIDs, especially when it comes to derivative. This is sometimes referred to as realizable forms of the PID vs mathematical/theoretical forms. 

Two popular versions of the PID exists. The standard form and the parallel form. From Wikipedia the standard form is:

and the parallel form is:

where

As long as the relations between the constants (above) are remembered and taken into account, these two are equivalent mathematically. The tuning procedure shown above uses the standard form. The parallel form could also be used as long as the relations above are taken into account. Both these forms above happen to be mathematical forms rather than realizable forms, but that is of little relevance here.