By Blas M. Vinagre, YangQuan Chen (Eds.)
Read or Download 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 PDF
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Extra info 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
Let q be the least common multiple of the q n; then νn= n/q, where n and q are positive integer numbers. ν, with ν=1/q (a differential equation with ν=1/2 is said semi-differential ). The coefficients and orders do not coincide necessarily with the previous ones, since some of the coefficients can be zero (1 ). b) the νn are irrational numbers but multiples of the a 0<ν<1. 6) we can perform the inversion quite easily, by following the steps: a) Transform H(s) into H(z), by substitution of s v for z (2 ).
A given periodic function, with period T, can be considered as a sum of delayed versions of a given basic wavelet. 14) that shows that the differintegrated of a periodic function can be obtained by convolving the wavelet with the differintegrated of the comb signal. To obtain the corresponding Fourier series we must find the differintegration jω 0 t of the exponential e for all t∈R. The FT of this function is 2πδ(ω-ω0). δ(ω-ω0). 15) Using the FT in eq. (jn T )αδ(ω - n ) T T T -∞ 41st IEEE CONFERENCE ON DECISION AND CONTROL 44 TUTORIAL WORKSHOP #2.
The binomial coefficients α are given by: k α = (-1)k. (-α)k (1 ) k k! 11) Although in the deductions of the previous results we used the LT, its validity is wider, allowing their use with other functions, as it is the case of the exponential defined for all the time. 5). 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).