# PID Controllable

If I have a control problem, how can I know if it's solvable by PID techniques or it needs other modern approaches?

I think this basically boils down to whether the cost function I want to minimize can be minimized by PID operations or not; is that right?

• I think in most cases the question to ask is whether PID control is necessary rather than whether it is adequate. I just read the wikipedia article on PID control and it seems to include the information you need to figure out how it works. If you're asking for commercial or large scale applications and you don't understand the systems involved, you should probably have an engineer making that decision for each individual project. Main takeaways at any rate is that PID systems do not require error to produce control. Can you explain a bit more what you are asking/don't understand?
– K H
Dec 15, 2018 at 0:28
• I'm looking to mathematics of the problem formulation. What properties would make it uncontrollable with PID? Dec 15, 2018 at 0:38
• If you have a control problem, then you must define the input and feedback with some measured output error in time and amplitude. Then by choice of feedback order to reducing the effects of higher order terms towards 1st order with feedback you can optimize the output. Eg use accelermeter feedback to control acceleration, position sensor to control position. Dec 15, 2018 at 0:59

Cost function is what Optimal Control is concerned about and is not applicable in PID Control.

PID control has this disadvantage that it has no knowledge of the plant or process system function. It operates entirely on the error signal, e(t)= set_point - measured_point and tries to minimize that. It's by no means the optimal solution all the time, and if you don't know your desired state beforehand, you can't use the PID.(Because you can't form the error signal e(t))

Begin by looking into Wikipedia on

• Controllability

• PID Controller

to have a better insight.

You need to define "solvable". There are certainly ways to frame a question to "modern" control design methods that will cause a PID controller to pop out -- or at least to cause a controller with transfer function $$H_c(s)=\frac{a_2 s^2 + a_1 s + a_0}{s(s+\omega_d)}$$ I'm pretty sure you just say "I have a plant with an uncontrolled integrator (i.e., a modeled offset) and I need a reduced-order controller of no more than order 2". Then turn the crank, and -- presto! -- you have a "PID" controller.

Whether the PID controller you get from that design effort will be adequate to your needs, or worse than a more complicated "modern" controller, or better, depends on the problem at hand and your skill with the math.