# 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.

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