By Blas M. Vinagre, YangQuan Chen (Eds.)
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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
13) - suggests us to work with the step response instead of the impulse response to avoid derivatives or working with non-regular functions near the origin. 4. Stability of fractional linear time invariant continuous -time systems The study of the stability of the FLTI systems we are going to do is based on the BIBO (1 ) stability criterion that implies stability when the impulse response is absolutely integrable. The simplest FLTI system is the system with transfer function H(s) = sν with s belonging to the principal Riemann surface.
S ... 1) in order to include other kinds of exponents: rational or, generally, real (or even complex numbers). It is immediate to see that there are two forms of obtaining the extension, depending on the choice done for region of convergence for the LT: the left and right half-planes. Let us begin by considering both cases with integer powers and obtaining the corresponding inverses. With these cases, we can form the following table, where δ(t) is the Dirac impulse and u(t) is the Heaviside unit step function: table 1 To obtain a differintegration of a given function we only have to convolve it with one of the functions presented in each row.
20) For the final value, we have: x(α-1)(∞) = lim s α X(s) s →0 The proofs are very similar to the usual; so, we will not present them. 4. Differintegration of the Causal Exponential This case is exceptionally important, since it appears in the computation of the impulse and step responses of linear causal systems. We will not consider the case of integer differintegration. 10), by putting y(t) = u(t). 41st IEEE CONFERENCE ON DECISION AND CONTROL 45 TUTORIAL WORKSHOP #2. 21) We will represent this function by Eα(t,a), with E0(t,a) = eatu(t) and Eα(t,0) = δ(α-1)(t).
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 by Blas M. Vinagre, YangQuan Chen (Eds.)
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