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 -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp