Can I implement continuous time feedback regulator in an Arduino microcontroller?

Question

Arduino is a digital mikrocontroller. But I wonder if it's possible to implement an continuous time feedback regulator in an Arduino microprocessor?

Continuous time feedback regulators such as PID:

$$ K = P(e(t)-D\frac{\mathrm{d} }{\mathrm{d} x}e(t) + I\int_{0}^{\infty} e(t) dt) $$

Or LQG regulator (this is a LQR with kalmanfilter only, not the model): $$ \dot{\hat{x}} = (A - KC)\hat{x} + Bu + Ky + Kn - KC\hat{x} $$

$$ u = r - K\hat{x} $$

Or do it need to be a digital feedback regulator? I mean....those feedback regulators works exellent by using operational amplifiers.

I know that operational amplifier works in real time. But an Arduino working in 16 Mhz speed, and that's very fast too.

Answer 1

continuous time feedback regulator in an Arduino microprocessor.

You can get at most a very accurate approximation of the continuous time system using Arduino by converting the continuous time model of your system into a discrete-time version: once you have the transfer function of you continuous time control system (filter or controller) you can operate a discretization through some well-known method such as the Bilinear transform and then obtaining the digital (dicrete-time) version to implement.

PS: do not confuse the real-time with the continuous time.

EDIT: Every time you deal with the "digital world" (memories, sample&hold processing) you need a discrete-time version of your continuous time model. You cannot directly implement it as it is (beside using an automatic code generator, but even in that case an automatic conversion continuous-to-discrete is also needed). Just to have an example, if you have a simple low-pass filter, easy to build with analog electronic components (resistor and capacitor) as: $$W(s) = \frac{1}{1+sRC}$$

to implement it into a discrete-time system (such as microcontroller) you need its discrete time version which will have comparable performance under certain conditions:

$$W(z) = {\frac {1+z^{{-1}}}{(1+2RC/T)+(1-2RC/T)z^{{-1}}}}$$

The most important thing to observe when a model of a system is converted is Nyqvist-Shannon sampling theorem.