Understanding Pole Speed in Digital Control System Design

[NeuralNet]

When designing a State Observer for a control system the observer poles ($eig(A-LC)$) should typically be about 10 times faster than the controller poles ($eig(A-BK)$).

But when designing a digital control system what does it mean for the poles to be faster? For the analog case it simply means that to get more negative on the real axis, but since the s-plane is mapped to the z-plane, how does one determine how a pole is faster?

 

[milesyoung]

Vertical lines in the s-plane (real part = constant) maps to circles of constant radius in the z-plane with the origin as their center.

In the z-plane, as you move from within the unit circle to outside of it, you move from the stable region to the unstable, i.e. the time constant of the pole(s) increase in magnitude as you move away from the origin.

 

[NeuralNet]

Okay, so the mapping from the s-plane to z-plane is as follows:
$$z=e^{sT}$$

And from s-plane to z-plane:
$$s=\frac{1}{T}ln(z)$$

Where $T$ is the sampling period.

So if the observer poles are supposed to be "10 time faster" than the controller poles can I do the following, given a digital control system:
1. Determine the controller poles.
2. Map them to the s-plane.
4. Take the real part of the poles and multiply by 10. These will be the continuous Observer Poles.
5. Map the continuous observer poles back to the z-plane. These are the discrete Observer Poles.

Would that work?

 

[milesyoung]

1. Determine the observer poles.

The "controller poles" yeah?

4. Take the real part of the poles and multiply by 10. These will be the continuous Observer Poles.

You're going to affect the damping ratio of the poles as well if you do that. You can move them out along a loci of constant damping ratio (constant angle) until you get the time constant you want.

But yes, you're free to map back and forth as you please.

 

[NeuralNet]

The "controller poles" yeah?

Yes, I meant "controller poles" (which I have edited).


You have answered my question. Thank you very much.