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As ilustrated in controlsystemsacademy shown relation between z-domain and s-damain poles by this image. natural frequency and damping ratio of z-poles with contour for natural frequency and damping ratio given by these equations. enter image description here

However there are some special case which is real-negative pole in z-domain can not represent with one pole in s-domain also mention matlab error that state.

MatLab: Warning: The model order was increased to handle real negative poles.

from calling function d2c. Then I look into mathworks website and found this.

When you call d2c without specifying a method, the function uses ZOH by default. 
The ZOH interpolation method increases the model order for systems that have real negative poles.
This order increase occurs because the interpolation algorithm maps
real negative poles in the z domain to pairs of complex conjugate poles in the s domain.

So, my question is

  1. How do we covert real-negative poles to s-domain?
  2. How to determine matural frequency and damping ratio of z poles inside undefine region (colored wth red in first image)?
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    $\begingroup$ This is sounding like an XY question. What are you trying to actually do? Approximating systems in the $z$ domain to the Laplace domain works very well for $|z - 1| <= 0.1$ or even $0.25$, but gets increasingly absurd as the ratio between highest frequency of interest and the sampling rate get higher. $\endgroup$
    – TimWescott
    Commented Dec 1, 2023 at 22:51
  • $\begingroup$ @TimWescott As the equation and diagram are really clear and useful I just wonder what can we do about the region that not define by the equation but matlab can do it I don't understand thew methed to do so. $\endgroup$
    – M lab
    Commented Dec 2, 2023 at 5:21

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Your question, as asked, is impossible to answer without being misleading because it contains premises that are incorrect, and that utterly closes the space of possible correct and useful answers.

So I'm going to explain why those premises are wrong, while giving you your correct -- but useless -- answers.

The $z$ domain is the equivalent* of the Laplace-domain when you are dealing with a system that operates in sampled time**.

When you take a continuous-time system transfer function in the Laplace domain and use the ZOH method to convert it into the $z$ domain, the conversion is exact under three conditions:

  1. You are adding sampling at the output of the continuous-time system.
  2. You are adding a zero-order hold (ZOH) at the input of the continuous-time system.
  3. You are not adding any pure fractional-sample delay (or you are using a more sophisticated method than Matlab's c2d that takes it into account).

A point about going from Laplace to $z$ in this manner is that it is exact going from Laplace to $\mathbf z$, but you cannot reverse the process.

This irrevirsibility is not a feature of the conversion from Laplace to $z$ domains -- it is a feature of the sampler and the ZOH that you have to add to your continuous-time system model. Information is lost in the sampling process -- that process has a many-to-one mapping, and then information is lost again in the ZOH -- that process also has a many-to-one mapping. This means that the conversion from Laplace to $z$, while it can give you an exact model of the resulting system in the $z$ domain is not reversible.

This irreversibility means that Matlab's d2c function, even when using the ZOH option, can only ever be an approximation. Because of the nature of the sampling process and the ZOH spectral properties, as stated, this approximation does not hold well for $\left | z - 1 \right | > 0.1$ or so, and gets increasingly worse as that distance increases.

Specifically, all that cool and useful stuff that gets drummed into your head about the damping ratio and natural frequencies of a Laplace-domain pole pair holds less and less as $\left | z - 1 \right |$ gets large.

How do we covert real-negative poles to s-domain?

Any way we want to, while recognizing that the conversion will only ever be a good reflection of the sampled-time system for poles close to $|z - 1| = 0$.

If you are going from Laplace to sampled time, and you're using an impulse-invariant conversion or a $n^{th}$-order hold model, then a pole at $s = \alpha + j\omega$ will map to a pole at $z = e^{\left( \alpha + j\omega \right)T}$, where $T$ is the sampling rate.

If you want to have a universal expression for converting a pole anywhere on the $z$ plane to a Laplace-domain equivalent, then just take $s' = \frac{1}{T} \ln z$.

How to determine natural frequency and damping ratio of z poles inside the undefined region (colored with red in first image).

If you then want to calculate a natural frequency and damping ratio, just do it for $s'$ as calculated above. Just keep in mind that this may be intellectually satisfying, and "correct" in a very narrow way, but it will tell you little about the real system behavior.


* Well, in a sense when you are working in the $z$ domain you are working in the Laplace domain -- at least, the near-universal formulation and justification of the z domain is to derive it from the Laplace domain. You can, however, assume the $z$ domain from first principles and then derive the Laplace domain from it, if you're feeling perverse.

** Like all blanket statements, this one is only almost true -- but if the blanket is big enough to cover a person, the times where someone out there using the $z$ domain to calculate system response with pure delays but no sampling is no bigger than a pin-prick hole in the blanket.

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