PID tuning - modified Skogestad

As mentioned in the previous post, in some cases, particularly for servo positioning systems, it's a good idea to dive a bit deeper into the theories of tuning. One method I have found useful is the Skogestad method.

All the details can be found the linked paper. This is a modified form that I have found useful for hydro-turbines in combination with the trial and error method. The method consists of:

• A step in input, plotting the output and the input.
• Finding characteristic values from that plot.
• Using a single parameter called tau_c as tuning parameter.

The reason that this method is sometimes useful for hydro turbines, is that these systems can be very dynamic and oscillating. It can therefore be very difficult to identify the 15% and 30% overshoot from the trial and error method with any accuracy at all (although not entirely impossible). The Skogestad method (modified) will give usable values in these cases, at least for Ti.

The modifications to the method are:

• Disregard everything about time delay. Although one can often find something looking like a time delay, and it can be 2-3 seconds long, this is NOT a time delay in the normal sense, and should not be treated as one. Treating it as a time delay will in most cases produce unworkable PID parameters (instability or way too much overshoot).
• Use a tau_c of approximately 5 as a starting point (subject to tuning, see further down)
  

The disadvantage of the method, is it only works for PI. It does not work for PID unless approximating the system as a second order system (which a hydro power plant definitely is not, and a rather impossible thing to do in the field anyway). A good point of the method, is that Ti always comes out with a good value. It's really only tau_c (Kp in practice) that would eventually be adjusted to fit performance specs. A tau_c of 5 is found by me as a good starting point on idealized systems. It could however be way off on real systems.

Well behaved system

Looking first at a "well behaved" system. In hydro-power plant circumstances, this will mean a very simple system, an idealized plant.

The system is set in open loop, and a step in "kappa" (guide vanes) of 10% (0.1 pu) is done. This is seen as the green line, where we also can see the speed of the servo. The red line is the turbine speed in "per units", starting at 1.0. Here we can clearly see the speed going in the opposite direction during the first 2 seconds. This is fairly typical, but it's not really a time delay. The pink line shows the 63% rise in speed and the corresponding time for that rise (approximately 5.8 seconds).

This plot is all the information we need to tune according to Skogestad, which is rather neat. We find:

• Time constant tau_1 = 5.8 seconds
• Steady state gain k = delta(n)/delta(kappa) = 0.063/0.1 = 0.63

According to the rules of Skogestad, pretending this is some kind of first/second order system, disregarding time delay and setting tau_c = 5, we get:

• Kp = (1/k)(tau_1/tau_c) = (1/0.63)(5.8/5) = 1.84
• Ti = tau_1 = 5.8 s

That's it, super simple and super fast. Tuning this plant using the trial and error approach we get Kp = 1.9 and Ti = 6.3. From a practical point of view, this is more or less identical. Both will give fairly equal performance, and both well withing specs. It's not coincidental that a well behaved system has a generator inertia constant of approximately Ta = 6, and this corresponds well to the "optimal" Ti in this case.  

A realistic system

This is a real system. It's certainly on the "fuzzy" and oscillating side, but far from unusual. Doing the same as for the idealized system, we get the plot:

We find:

• Time constant tau_1 = 9.5 seconds
• Steady state gain k = delta(n)/delta(kappa) = 0.022/0.1 = 0.22

Then with tau_c = 5 as before:

• Kp = (1/k)(tau_1/tau_c) = (1/0.22)(9.5/5) = 8.63
• Ti = tau_1 = 9.5 s

This time Ti seems reasonable, and is in line with what the trial and error method gives. Kp on the other hand seems way off, and it is. It will produce a system behaving like this:

The system will oscillate forever, the PID is unusable. Note also the slow servo speed, needed for safety reasons on this plant to prevent water hammer. Since Ti is OK, we can simply use step 5 from the trial and error approach to get a working Kp. Doing this, I find that Kp = 1.4 is as "good as it gets", and still be within performance specifications. A step response in set-point of the closed system, certainly does not look good, but the physics of the system does not allow for anything better.

In the last plot below, I have introduced anti-windup. It doesn't improve the closed loop response, but makes a smoother servo run (less wear and tear) and will have a much better performance in terms of handling disturbances because Ti is reduced considerably. This is done by trial and error.

For real systems, some trial and error usually has to be done no matter what, but Skogestad method is very helpful in obtaining Ti for systems with difficult behavior. An open loop step plot will also reveal useful information of the system. In the next post, a combined approach is shown.

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.