Substitute PID Controls with a Polynomial Equation/Table?

[Jay_]

TL;DR

TL;DNR: Would it be as effective to just use a polynomial function that is interpolated from an input-error-to-controller-output table instead of the effort of tuning gains of a PID controller?

So, I had a discussion with a friend of mine, neither of us are in controls but I was curious about an answer here. In a PID controller, we essentially take in an error value, do a mathematical operation on it and determine the input (controller output signal B) needed to the actuator to produce a desired output (position, temperature etc) based on the error (controller input signal A). This output from the actuator, is read by a sensor and fed back to generate the error and so on till it singles in on the setpoint value (position, temperature etc).

Now as a substitute to this whole thing, could we create a table of the desired error and controller output response, and then interpolate it have a polynomial function? So we close in on the set point more aggressively when the error is high, and less when it's lower. And we could just define this mathematically.

The only thing I could think of as to why this isn't as good is because a PID controller defines three variables (Ki, Kp and Kd), while this polynomial function would required probably 8 or 10 constants depending on the degree of the polynomial needed to make the controls good enough for our application. But are there any other reasons why this polynomial function approach (based on a table), wouldn't be preferred over tuning a PID controller?

 

[Baluncore]

A PID controller keeps track of history in the computation of the integral and the derivative. If you only have the proportional value, then the system cannot be controlled by a single input polynomial. For example, without the integral you could not eliminate a long term wandering DC error.

 

[The Fez]

Yes, what you are describing is still just proportional control. The problem lies in the fact that your goal is to have zero error, so as you approach your desired setpoint, your whole polynomial will approach zero, no mater how many terms it has. Think of it like a gas pedal. if you set your desired speed at 65, once you reach that speed, your polynomial will be equal to zero, meaning you will be giving it no gas. you will always need a non-zero error to have a non-zero input.

The other problem is that there are external disturbances in your system that will affect the output. To use the car example again, suppose you are on a long, straight, flat, deserted highway. You do some experiments and find that at 20% gas pedal, you're going 25 mph, at 40% gas pedal you're going 45 mph, etc. You can put that in a chart and interpolate, and you'll have your formula. However, the next time you're in the car, you're shuttling the entire wrestling team with all their luggage to the airport. And you're driving up a hill, in a huge headwind. Your chart will say you only need 58% gas to maintain highway speed, but you are barely topping 30 mph.

 

[DaveE]

It's a great question. But as you said "neither of us are in controls". The previous answers are great. If you want to understand the basics of control systems, you will need to study control systems a little bit. Be careful of ideas that may replace older established methods if their proponents haven't actually understood those methods (i.e. the state of the art).

PID controllers aren't just defined by Ki, Kp, and Kd. Those constants are gain terms applied to very different control laws. Ki is applied to knowledge of the error history (integrated, or cumulative, error). Kp is applied to knowledge of the current error. Kd is applied to knowledge of the trend (slope, derivative, rate of change, etc.) of the error. Your system only uses the current error, without knowledge of it's history or trend.

I'll also add that PID controllers are absolutely not "state of the art" they are just generally useful and easy to understand. Real control system engineers are working at a more general and fundamental level, which I'll call pole/zero placement in compensation networks. In the "PID" lingo, I have designed several feedback systems that I'll describe as IPDPIP, but most are just PI, which is usually good enough. This isn't as scary as it sounds if you know the fundamentals.