Is PID Control Possible for a 2nd Order System?

[Jhenrique]

PID of "2nd order"?

Exist PID control of "2nd order"? Ie., a command system of correction to error that includes a factor of correction proportional to 2nd derivative, another proportional to 1nd derivative, another proportional directly to error, another proportional to 1nd integral and another proportional to 2nd integral? Is mathematically possible to project a system so?

 

[abitslow]

Consider posting in your native language. I don't know what "2nd integral" means. I don't know what "project a system" means. (FYI, in English: 1st, 2nd, 3rd, then 4th-9th (0th), only the right-most place is considered.)
Consider a PID with (P,I,D) values of (0,1,0). If that signal is fed into another PID (0,1,0) you will have what I guess you mean by "2nd Order". 2nd Order usually refers to derivatives,(differentials), not integrals. It is ambiguous whether "2nd Order" in an integral means (∫fdx)² or ∫(∫fdx)dy (or even ∫(∫fdx)dx ).
It should be clear that any number of (P,I,D) units can be set up in a circuit (parallel and/or series) to get any "order" you wish. It is NOT at all clear to me whether most of these circuits would be effective or efficient, but that obviously depends on the exact control environment.
SO, if I interpret your question correctly: output will be proportional to a linear combination of 5 variables:
P,I, D and I→I' and D→D' that is: aP+bI+cD+eD'+fI'. I see no problem creating such a controller (using the chain rule).
I forgot to note that I am familiar with second derivative (2nd order derivative) controllers, just not second Integral (but am not a control engineer, and am far far out of school).
Consider I=b∫xdt and G=z∫IdD -- note that G is an integral with respect to the signal D (the derivative of the input). This has what I would call "mixed" order. For a "second-line" controller, its input can be the "raw" signal or some combination of that with the output of one or more "first-line" controllers, this is what I mean by a circuit. There are a HUGE number of possibilities.

 

[Jhenrique]

In the s-domain, the PID operators are (respectively) $k_P \frac{1}{s}$, $k_I$ and $k_D s$, being k the proportionality constant. A PID of 2nd order, would have the following operators in the s-domain: $k_{P2} \frac{1}{s^2}$, $k_{P1} \frac{1}{s}$, $k_I$, $k_{D1} s$ and $k_{D2} s^2$.

And an "integral of 2nd order" could be given by Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration)

Do you understand my ideia? If yes, it is possible or has some absurd?

 

[AlephZero]

If the process you want to control can be described by a differential equation of order n, using a controller of order > n can create instability, limit cycles in a nonlinear system, etc.

Most of the differential equations in physics are of order 1 or 2, so there is not much practical use for higher order controllers.

 

[Jhenrique]

If the process you want to control can be described by a differential equation of order n, using a controller of order > n can create instability, limit cycles in a nonlinear system, etc.

Most of the differential equations in physics are of order 1 or 2, so there is not much practical use for higher order controllers.

This: (http://upload.wikimedia.org/wikipedia/commons/9/91/PID_en_updated_feedback.svg) is a comum PID. I'm proposing a PID like this:

This system isn't a system of high order, or it is?

 

[AlephZero]

You can propose anything you like. I already told you why there is not much practical use for it.