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# allmargin gives incorrect answers for some discrete time systems

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Carlos Felipe Rengifo
on 15 Aug 2024

Commented: Paul
on 9 Sep 2024 at 13:17

Hi,

Consider the following code:

s = tf("s");

gs = 1/(s*(s+1));

gz = c2d(gs, 4, "zoh");

allmargin(gz)

gclz = zpk(feedback(gz,1,-1));

abs(gclz.P{1})

In R2024a, the stable flag given by allmargin is equal to one, indicating that the closed loop system is stable. On the other hand, the last sentence of the above code shows that the closed-loop system has a pole outside the unit circle, and therefore it is unstable. I do not have this issue in R2022b. Why the difference?

Sincerely

Carlos

### Accepted Answer

Paul
on 15 Aug 2024

Hi Carlos,

As far as I can tell, what's happening is as follows:

For a discrete time system, the default Focuse is Focus = [0 pi/Ts], that is the upper bound is the Nyquist frequency.

Then, allmargin uses an undocumented call to damp to compute the "natural frequency" of the closed loop poles. In your example, we have

s = tf("s");

gs = 1/(s*(s+1));

gz = c2d(gs, 4, "zoh");

clp = pole(feedback(gz,1))

wn = damp(clp,4) % Ts = 4;

I'm not exactly sure yet what wn is supposed to represent in the context of this problem.

Nevertheless, both elements of wn are greater than the upper bound of Focus, which is

pi/4

Hence, allmargin excludes both closed loop poles from the unit circle test, and since there are no poles left it returns Stable = 1, as you've observed.

I'd have to do some more digging to get a better idea of what they mean with that wn computation, or anyone can as damp is an ordinary m-file.

Also, at present I don't see a way to use the explicit Focus name/value pair to override the default. As best I can tell, if Focus(2) on input is greater than pi/Ts, then it's set to pi/Ts internally.

I'd say that this situation looks very suspicious, but I'd need to get a better understanding of what's happening with damp before going so far as to say that there is a bug, though it certainly looks like that's the case.

##### 1 Comment

Paul
on 16 Aug 2024

Edited: Paul
on 16 Aug 2024

The bottom of the doc page for damp show how it computes the natural frequency of the poles of a discrete-time sytem:

s = log(z)/T

wn = abs(s)

where z is the discrete time pole. Clearly, the idea is use the inverse of z = exp(s*T) and then get the natural frequency of the "equivalent" continuous time pole.

We can use the symbolic toolbox to see exactly what's happening:

syms Ts positive

z = sym(-1.2); % one of the closed loop poles in question

s = log(z)/Ts

We see that the complex part of s is pi/Ts, independent of the actual value of z (for z real and less than 0). Hence, the abs(s) will ALWAYS be greater than pi/Ts, and the allmargin closed-loop stability test will not include any real, negative poles.

Of course, the above is also true for poles that are inside the unit circle, so those won't be included in the stability test either, though I guess that's not important.

I think MathWorks screwed this up and it's a SERIOUS bug that you should report. If you do so, please post back as a comment to your Question with a summary of their response.

### More Answers (1)

Carlos Felipe Rengifo
on 16 Aug 2024

##### 2 Comments

Paul
on 19 Aug 2024

What was MathWorks response to your bug report, if you don't mind me asking?

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