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By Blas M. Vinagre, YangQuan Chen (Eds.)

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Extra resources for 41st IEEE Conference on Decision and Control, Tutorial Workshop No. 2: Fractional Calculus Applications in Automatic Control and Robotics (Las Vegas, USA, December 9, 2002): Lecture Notes

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Now, go a step ahead and consider the simple case corresponding to the fraction 41st IEEE CONFERENCE ON DECISION AND CONTROL 47 TUTORIAL WORKSHOP #2. 4) where ν is a real number. The equation s ν = a has infinite solutions that are on a circle of radius |a|1/ν . However, in the general case, we cannot assure the existence of one pole in the principal Riemann surface. This is not the case of 0<ν≤1. In this case, we may have one pole on that branch. For this reason, in the following, we shall restricting our attention to the cases in which: a) the νn are rational numbers that we will write in the form p n/q n.

Till now, we considered a differintegration of signals with LT, that exclude, for example, the sinusoids defined for all the time and other similar signals having Fourier Transform, but not Laplace Transform (or its region of convergence is degenerate). 1) but with jω instead of s. We must begin by noting that the FT of the signum function, sgn(t), is given by 2 . With jω this we can write a table similar to the one presented before. table 2 This table suggests us to introduce the following definition of differintegration of δ(t): 1 (a) = a(a+1)(a+2) ...

Some attempts were made to create a formal framework to the study of Fractional Linear Systems, but without the desired generality, coherence, and usefulness of the final results{see [2,11,21-23]}. To our knowledge, the approach we propose here to the topic is original, although it has an "already seen" character. This is because we are dealing with very well known concepts. We merely generalise them to the fractional case. With this work we intend to give a first contribution for a correct understanding of some experimental [14,22,23] results and to open a new way into modelling, simulation, and estimation in real fractional systems.