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Suppose we have a controller whose TF is $$ G(s) = \frac{1}{(s^2+1)} $$ This yields two marginally stable poles at \$ s = \pm j \$

Now let's discretize the controller through z transform, after which we obtain an expression that looks similar to

$$ \frac{z-1}{(z-e^{-jT})(z-e^{jT})} $$

So our poles in the Z domain are \$ z = e^{\pm jT} \$

Let's vary our T, for \$T = \frac{\pi}{2}\$, we have z = -/+ j, exactly as our continuous time controller.

But for \$T = \pi\$, we see that both of the poles of z are at - 1.

What is the effect of this aliasing from continuous time controller to discrete time controller? i.e. what happens to the output this for this case

What are used to prevent aliasing?

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  • \$\begingroup\$ Why are you starting with a marginally stable controller? \$\endgroup\$ – copper.hat Jan 30 '15 at 6:29
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A discrete time controller inherently depends on sample time T, therefore, also the stability margin. There is a condition which is related to Shannon's theorom. That is the so called strip condition. Your maximal possible (allowed) sampletime is calculated by $$T \leq \frac{\pi}{\omega_0} = T_{\max},$$ which is in your case $$T \leq \pi,$$ since $$\omega_0 = 1.$$ If you go with T beyond this limit you have the effect of aliasing. This is due to periodicity, then, the z-transform does not map uniquely the strip onto the complex z-plane, hence, information get lost. A possible effect that might happen to your output after violating the strip condition is that oscilations, so called hidden oscilations, arrise. The controller can not intervene if sample time is too high.

I recommend the book from Katsuhiko Ogata "Discrete-time control systems", and there chapter 3 respective aliasing and further artefacts. Regarding strips, see chapter 4 "Mapping between the s plane and the z plane".

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  • \$\begingroup\$ Feel free to elaborate and provide reference. This sounds extremely interesting! \$\endgroup\$ – Rodrigo Amarante Jan 30 '15 at 8:51
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It is also often the case that you can't add anti-aliasing filters in the feedback path to mitigate aliasing in real time controllers as this will add to the loop delay. In synchronously sampled real time controllers, the sampling of the feedback is performed at the same period (or multiple of) the controller update rate, for example, with some dead-time after switching (for switching converters) to ensure that the feedback has stabilized.

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If you don't attenuate sensor signal components higher than half your sampling rate they can potentially be aliased as lower frequency and acted on by your control system. Imagine 1 kHz noise being aliased as 10 Hz in a discrete time motor position controller, leading to a real 10 Hz oscillation on the motor shaft.

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