# PID Gain Calculation

A component of a control loop is approximated by the following relation between input $x$, and output, $y$:

$y = 5x^2$

During normal operation, the value of input to this component ranges between $1$ and $1.5$. In stability analysis of the overall control loop, what is the gain of this component?

So, I'm not sure how to solve this problem. I don't know how to create a transfer function by performing a Laplace transform because this isn't a differential equation and there is only one component. Would really appreciate it if someone could point me in the right direction.

## 1 Answer

The input and the output of the component need to have a linear relation (such as y = 5 * x), so you have to linearize that ecuation in order to aproximate it to a linear ecuation that you could use to obtain the gain of the component.

To linearize :

If x = a;

y = (f(a) + f’(a) * (x - a))

So if you take a = 1 (one of the values you gave)

y = 5 + 10 * (x - 1) y = 10 * x - 5

So you could use a gain of 10 and add a sumator with a minus for substracting 5.

I hope this helps you.

• So, would it be best to use 10 or would you use the coefficient of x that you get with a = 1.5? – picotard Dec 18 '17 at 3:28
• It’s an aproximation, the best would be to use one coefficient for every input, but i don’t habe any idea o how to do that. It would be like an if statement in programing, but i don’t think it’s posible. But if you one to use a non-linear system... you could start in the state control world, but thats something absolutely different. – Juanea7 Dec 18 '17 at 7:00