By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski
This quantity offers chosen points of non-integer, or fractional order structures, whose research, synthesis and purposes have more and more develop into a true problem for varied examine groups, starting from technology to engineering. The spectrum of functions of the fractional order calculus has exceptionally extended, actually it might be challenging to discover a science/engineering-related topic sector the place the fractional calculus had no longer been included. The content material of the fractional calculus is ranged from natural arithmetic to engineering implementations and so is the content material of this quantity. the quantity is subdivided into six components, reflecting specific elements of the fractional order calculus. the 1st half includes a unmarried invited paper on a brand new formula of fractional-order descriptor observers for fractional-order descriptor continous LTI structures. the second one half presents new parts to the mathematical idea of fractional-order platforms. within the 3rd a part of this quantity, a host of latest ends up in approximation, modeling and simulations of fractional-order structures is given. The fourth half provides new recommendations to a few difficulties in controllability and keep an eye on of non-integer order structures, particularly fractional PID-like keep an eye on. The 5th half analyzes the steadiness of non-integer order platforms and a few new effects are provided during this very important recognize, particularly for discrete-time structures. the ultimate, 6th a part of this quantity provides a spectrum of functions of the noninteger order calculus, starting from bi-fractional filtering, particularly of electromyographic signs, throughout the thermal diffusion and advection diffusion tactics to the SIEMENS platform implementation. This volume's papers have been all subjected to stimulating reviews and discussions from the lively viewers of the RRNR'2014, the sixth convention on Non-integer Order Calculus and Its purposes that used to be geared up by way of the dep. of electric, keep an eye on and computing device Engineering, Opole collage of know-how, Opole, Poland.
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Additional resources for Advances in Modelling and Control of Non-integer-Order Systems: 6th Conference on Non-integer Order Calculus and Its Applications, 2014 Opole, Poland
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Let us introduce the family of binomial functions on Z parameterized by μ > 0 and given by the values: ϕ˜μ (n) = n+μ−1 for n ∈ N0 and ϕ˜μ (n) = 0 for n < 0. n Definition 1. e. (ϕ˜α ∗ x) (n) := n n−s+α−1 x(s)q . Additionally, we deﬁne a Δ0h x (t) := x(t). s=0 n−s −α For a = 0 we will write shortly Δ−α instead of 0 Δ−α h h . Note that a Δh x : (hN)a+αh → R.
We ﬁnish with a simple illustrative example in Section 4. 2 Preliminaries Let us introduce the following triangle: Δ := (t, τ ) ∈ R2 : a ≤ τ < t ≤ b , ¯ R . and let α(t, τ ) : Δ¯ → [0, 1] be such that α ∈ C 1 Δ; Definition 1 (Left and right Riemann–Liouville integrals of variable order). Operator t α(·,·) [f ](t) a It 1 (t − τ )α(t,τ )−1 f (τ )dτ Γ (α(t, τ )) := (t > a) a is the left Riemann–Liouville integral of variable fractional order α(·, ·), while b α(·,·) [f ](t) t Ib 1 (τ − t)α(τ,t)−1 f (τ )dτ Γ (α(τ, t)) := (t < b) t is the right Riemann–Liouville integral of variable fractional order α(·, ·).