My prof spent 8 whole months teaching us systems modeling and PID mathematics only to tell us there's no reasonable way to set your PID values... You gotta guess and check!
There are various structured ways to set PID gains.
The mentioned use of optimal control is one, although if you go there you need to be careful about setting your costs -- LQR and other optimal schemes assume a perfect model of the plant; the higher you set Q the higher the probability that your system will ring or go unstable.
There's various robust control methods, all of which I haven't used in a Good Long While, because for the most part swept-sine measurements work nice.
The one I've used most involves taking swept-sine measurements to get the plant response in Bode plot form, then using either Bode or Nyquist plots (or both) to tune my PID (or whatever controller I'm using).
For a large class of industrial problems, swept-sine measurements won't do, either because measurements need to be undertaken on production lines that are in operation, and the operators get cranky about things like noticeable sinusoidal variations in the product (think aluminum foil or paper), or because even when operated within safe limits, large machines undergoing swept-sine measurements can be downright scary. In such cases one usually ends up using random excitation or steps (often called "bump testing" if you're in an oil refinery or a paper plant) and some sort of system identification step like ARMA.
If you do end up doing testing followed by system ID, you'll most likely get an approximate plant transfer function -- so it's wise to either use a grain of salt when doing your optimal design, or to use some robust design method or other.
Hi! Just wanted to say thanks for the stuff you post on comp.dsp. I pop in and lurk every few months, and you stick out as a super valuable contributor. I might try to actually get my news client to be able to post now that I have so much idle time at home :)
What? If you have a good model of your system then you can (relatively easily) turn it into an optimization problem and use a LQR to choose gains for you. It’s true, this method still gives you knobs you might have to tune (Q and R), but these are easily conceptualized because they slide the cost of state excursions and control effort along a Pareto boundary. That means that you can get optimal PID gains just by saying how much you penalize a state excursion vs. how much you penalize control effort (e.g. distance from your reference point v.s. fuel use, or something like that.)
It’s also certainly true that a LQR won’t help you if you have no a priori knowledge of your system, but for many of the mechanisms we need to control this isn’t a problem.
Your whole argument is based around “if you have good model...”. This where most all control theory falls on its face. Getting a good is hard. For lqr that model better be mostly linear.
Oh, you system model isn’t first order? Now you need an estimator/Kalman filter. Or more sensors. That’s just more complexity for questionable benifit, which is why lqr is beloved by academics [0]. For anything that would be adaquetly controlled with pid, stick with that. After about 2 hours fiddling with the knobs, it’ll be close enough.
This whole idea of optimality is based on bad intuition. Which states do you care about? Why? Is that more valuable than control effort? Why? Who is doing the economic analysis to determine what ultimately costs us more money? In the end, this thing are tuned just like pid: you stop when the step response looks nice. Besides all that, for a lot of systems, and in particular flexible structures with a lot of states, “penalizing state excursions” isn’t really useful intuition to begin with for almost all state space represtations. You are better off with a pid and notch filter.
[0] I decided to complete my PhD in controls so could make statements like that with at least marginal credibility.
If you have a good linear model for your system you can pull out an LQR. It does not account for how bad your linearization of the true dynamics might be.
Also from my experience you still need to tune the Q and R the same amount that you would P I and D. and I'm not convinced that it is strictly better at all. In industry at least PID absolutely dominates.
It's also important to realize that just because the LQR is derived by solving an optimization problem, that doesn't mean it gives you the best possible gains for whatever you want. You got the optimal controller to minimize a quadratic cost you made up for a linearized approximation of your system.
Iterative design (guess and check) is absolutely still the state of the art for control design, and using an LQR does not escape that.
Huh? They must have been joking. There are many cases with models that are good enough that you can calculate the gains depending on the desired response. I have done it many times.
It is definitely true for a few simple cases where the model producing simulated gain values can get pretty close, but there's an entire sub-discipline of controls theory dedicated to the trade-offs of the various tuning methodologies [1][2] because for most real-world situations, you can't predict the system response for any given set of values accurately enough to be useful. You can get the initial values set from first principles by checking your poles to avoid clear regions of instability, but once you're past that point, I believe that it's mathematically true that you can't truly find the correct gains only from first principles.
If I'm missing something in the formal background, though, I'd love to hear it. My own implementation of digital PID controllers have been based on using Ziegler-Nichols for tuning bc I was under the impression above. Any literature or methods contrary to that would definitely be welcome!
(Edited because my first line didn't make any sense)
I think in the general case, say you made a black box PID controller then attach it to some system it is supposed to control then you have to tune it, which isn't so much "guessing" but a specific refining process. If you have a model for the system under control, then sure, it's math
Depends on your system!!! If you online tune and accidentally set P a little too big it is quite easy to set off an unstable oscillation and depending on your application can get a rather expensive earth shattering kaboom...
I usually end up saying "Screw it" and adjust the value under control to remove half the error at each timestep. Surprising how often that's good enough.
Many processes like temperature controls want to settle, you just have to help them out a bit. Applying a full PID algorithm to every control problem is a bit like using TCP/IP to solve every networking problem. It will work, but you'll often do better with a domain-focused solution.
Temperature control problems are only first-order (heat->temperature), so they're much easier to control than a second order system like force->velocity->position.
