Implementing differentiation and integration in a digital control system?

Question

In a computer control system (microcontroller-based), the sampling time of the inputs to the system determine whether to treat the system as a discrete system or continuous. From what I have learnt, when \$t_s \rightarrow 0\$, you can treat the discrete control system as a continuous time system.

My question is, how do I implement what an op-amp integrator and differentiator would do on a microcontroller.

Differentiation - Can I simply calculate the gradient between adjacent sampled values, or is it more complicated?

Integration - Similarly, can I add up all the previous values which have been sampled?

Answer 1

Integration is easy especially if each sample is the same time apart, just keep adding each new sample to the integration value. Note you may need to deal with integral wind-up where your integral gets massive. This is especially true if you are doing a PID controller.

For the differential taking the difference between sample n and sample n-1 is the very first step. You may need to consider noise though, if you have a lot or your system is to sensitive you may wish to filter it.

Answer 2

It sounds like what you're trying to do is implementing standard numerical calculations. No need to try and mimic the analog version of it, just apply the standard formulae for both operations. For differentiation $$\frac{f(x+t_0)-f(x)}{t_0}$$ In other words, the difference between the sample divided by the sampling time. For the integration you need to multiply the incoming sample with the sampling time, and add it to the last result of you're integrator, like so: $$y[n] = y[n-1] + t_0*x[n-1]$$ There are more methods of integration and differentiation, but these are pretty standard ones. The lower the sampling time is and the quicker your microcontroller is, the more you approximate continuous-time scenario.

Answer 3

My question is, how do I implement what an op-amp integrator and differentiator would do on a microcontroller.

An op-amp integrator has a CR time constant and the digital equivalent is this: -

Because the op-amp integrator also inverts, the "sign change" should be built into the digital circuit if you want to be exact.

A true differentiator (mathematically) will amplify noise dramatically in a simple digital differentiator so I'd consider using an integrator (as described above) and subtract its output from the input to obtain "differentiation". For very slow moving inputs, the integrator output will equal the input hence the subtracted output will be zero. For very fast input signals, the integrator output will be very small and the differentiated output will be almost the same magnitude as the input.