Hi Vadim,

Thanks for your feedback...

On 11/11/2013 9:52 PM, Vadim Zavalishin wrote:
[snip on the analog stuff]
For the discrete-time case the situation is more complicated, because we
can't use the continuity of the state vector function. IIRC, I also
didn't manage to build the "worst-case" signal, but there was the same
problem of the state vector becoming larger in the absence of the input


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.

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.

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?

> But, as I said, I didn't finish that
research and I could have been wrong. So just take my input FWIW.

I have made a slightly cleaner cook-up of the change of basis matrix. This seems a lot like the "uniformly negative" criteria you mentioned. It could be read that k must be uniformly positive:


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.


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