On 11-Nov-13 15:32, Ross Bencina wrote:
That's why I was somewhat surprised that you simply managed to
restrict the eigenvalues of the system matrix in some coordinates.
To be clear: the eigenvalues of the transition matrix only cover
time-invariant stability.
The constraint for time-varying BIBO stability is that all transition
matrices P satisfy ||TPT^-1|| < 1 where ||.|| is the spectral norm and T
is some constant non-singular change of base matrix.
Okay, for the time-varying case it seems to be the eigenvalues of
(P^T)*P in the discrete-time case, which according to Wikipedia define
the spectral norm (while in the continuous time we have the eigenvalues
of P^T+P). I'm currently a little short of time to take a precise look
at all the details. Still, in both cases we are talking about some
uniform property of eigenvalues of a symmetric matrix. In the discrete
time they need to be smaller than one to kind of make sure the next
state vector is smaller than the previous one (I think). In the
continuous time case they need to be negative to kind of make sure that
the time derivative of the state vector's length is negative.
The main reason I am suspicious is that Laroche does not even try to
cook up a change of basis matrix, or to show when it might be achieved.
It's kind of an orphan result in that paper that goes unused for showing
BIBO stability.
IIRC from briefly reading his paper, his sufficient criterion turned out
to be not applicable for the DF filters (which by themselves also didn't
seem to be time-variant BIBO-stable, IIRC), therefore he resorted to
some other approaches. That (and my own SVF investigation) led me to
consider this kind of criterion as a more or less useless one at that
time. But I may be wrong, it was a while ago and only a brief look.
Particularly suspicious is that your coordinate transformation matrix is
"built for the smallest damping", while the more problematic case seems
to occur "at the larger damping".
I'm not sure I follow you here. Smallest damping means most resonance,
where the system decays most slowly. Don't you think this would be where
the greatest problems would arise?
Because of the "shooting" effect I described earlier. I discovered it by
a numerical simulation of the system. For the low resonance the state
vector moves in an ellipse (for the "worst-case" signal I described).
The orientation and the amount of stretching of the ellipse depends on
the resonance (the lower the resonance the more the stretch). You can
suddenly switch to a lower resonance while your state vector is pointing
at such angle, that the respective position on the new ellipse is within
the "increasing radius" area. This will cause the state vector to grow.
Disclaimer: this all goes under the notice that I didn't double-check my
results or may even have forgotten some important details or simply
remember them wrong. I was posting them mostly because I thought they
might be interesting and/or usable to some extent for you (and maybe
others).
In short, using change of basis matrix T:
[1 f]
[0 1]
We have the time-varying BIBO stability constraint:
0 < f < 2, g > 0, f < k <= 2
f provides the bound on k from below.
This is also the kind of the result which I would intuitively expect at
the first thought. It's just contradicting the *unverified* results of
my earlier research, that's why I expressed my suspiciousness.
Hopefully, I'll find time to check your research in more detail.
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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