Discrete PI Controller from Continous PI Controller

Hello!

I'm trying to derive the discrete PI controller equation of this block from the following scheme, but I can't get the same result.

On continous domain, I obtained the following relationship.

Considering Backward Euler,

Then replacing,

This is different from the above-mentioned block. It's very probable that I'm wrong, because I'm very new on discrete domain topics, but if someone can help me, I will appreciate it so much.

Thanks!

Answer 1

[Nithin]

Hi @Fabián,

I am assuming that the blocks depicted in squares are Gain blocks in Simulink.

I don't exactly understand what the '1/s' block does in your block diagram. If it is similar to a gain block in Simulink then you are misunderstanding the Backward Euler method. In the s-domain, the derivative 's' can be approximated by 'z-1/z' when converting to the (z)-domain using the Backward Euler method. This approximation implies that the differential operator 's' is replaced by 'z-1/z', effectively transforming a continuous-time system into a discrete-time system.

In this context, '1/z' or 'z^(-1)' in the Backward Euler method corresponds to a unit delay block in Simulink. Therefore, you need to add a unit delay block from the result of the sum of the control signal and integral gain coefficient back to the sum, as shown in the figure below (variable names might be different from what you have used), to achieve the same result as a discrete PI Controller block.

Refer the following documentation to understand about the ‘Discrete PI Controller’:

https://www.mathworks.com/help/sps/ref/discretepicontroller.html

I hope this resolves your query.

[Sam Chak]

@Nithin, the '1/s' block implies integration in continuous-time domain. The OP @Fabián probably wants to apply the backward Euler numerical integration technique to a continuous-time Proportional–Integral (PI) Controller..

[Nithin]

Hey @Sam Chak,

@Fabián did mention that he is trying to model a Discrete PI Controller. So, I feel that '1/s' block here doesn't refer to the intergrater block. Moreover, the equation provided has a '1/s' element in it which is why I assumed it to be so.