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.