I'm struggling to find information on the use of the lambda norm; i.e. if I'm not mistaken
||x||\lambda= (\int0T e-\lambda t ||x(t)||2 dt)1/2,
where \lambda>0 and ||.|| is the 2 norm.
What advantage does it have over the Lp norm topology and why is it preferred over Lpnorms in certain contexts? For example, in [1], I have stumbled upon the following phrase which confuses me: "However, it is well known that the lambda norm leads generally to low convergence rates."
I find this similar to exponential forgetting in identification/estimation schemes, except in this case we're interested in the signal norm; the way it is used in [1] is to prove uniform convergence of a function sequence {f0,f1,f2,...} with bounded support ([0,T]); i.e. limk\to\infty ||fk-f*||\lambda = 0.
Why wouldn't we consider, say L2 convergence? Is it because it is easier to work with, or does it have some other physical meaning (like the energy interpretation of L2) that makes it useful?
Thanks!
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Ref.
[1] A. Tayebi - Adaptive iterative learning control for robot manipulators, Automatica, 2014.
[2] W.-J. Cao and J.-X. Xu - On Functional Approximation of the Equivalent Control Using Learning Variable Structure Control, IEEE Trans. on Automatic Control, 2002.
I can guess, that such norm can be useful for optimisation of transient response of the system. I suppose, that in certain cases you would want to for example ignore oscillations arround the steady state and be interested only in the time of reaching that state.
However, I think that it also could be the case of "I don't know how to use it but I can compute it" :) Such norm, especially for linear finite dimensional systems is relatively easy to compute. You can use the theorem of "shift in the complex domain" for transfer function in order to remove lambda and then use Parseval theorem (or James-Nichols-Philps theorem, which is derived from Parseval's) and expres the norm using the coefficients of the transfer function (and of course lambda).
There is a theorem in Laplace transform. Let "L" denote laplace transform and L(g(t))= G(s). then L(e(-\lambda t)g(t))=G(s+\lambda).
I have one more idea but again with infinite time. if x(t) is not in Lp e-'\lambda tx(t) might be. It can be useful especially for L2 because its a hilbert space. So you can potentially use some kind of approximation technique available in Hilbert space on e-'\lambda tx(t) and then multiply the result by e\lambda t. We have attempted such approach for non integer order system approximation however with very mixed results.
value of lambda chosen will also vary based on the system and situation. So there will not single method for finding it but it will vary based on situation
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