By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

This quantity offers chosen facets of non-integer, or fractional order structures, whose research, synthesis and purposes have more and more turn into a true problem for varied examine groups, starting from technological know-how to engineering. The spectrum of functions of the fractional order calculus has exceedingly multiplied, in reality it might be not easy to discover a science/engineering-related topic region 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 amount is subdivided into six elements, reflecting specific features of the fractional order calculus. the 1st half encompasses a unmarried invited paper on a brand new formula of fractional-order descriptor observers for fractional-order descriptor continous LTI platforms. the second one half offers new parts to the mathematical conception of fractional-order structures. within the 3rd a part of this quantity, a number of latest ends up in approximation, modeling and simulations of fractional-order structures is given. The fourth half offers new recommendations to a couple difficulties in controllability and keep watch over of non-integer order platforms, particularly fractional PID-like keep an eye on. The 5th half analyzes the soundness of non-integer order structures and a few new effects are provided during this vital appreciate, 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, specifically of electromyographic indications, throughout the thermal diffusion and advection diffusion approaches 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 functions that used to be equipped via the dept of electric, keep watch over and computing device Engineering, Opole college of know-how, Opole, Poland.

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**Extra info for Advances in Modelling and Control of Non-integer-Order Systems: 6th Conference on Non-integer Order Calculus and Its Applications, 2014 Opole, Poland**

**Sample text**

We assume that functions arbitrary, independent C 1 functions deﬁned on Ω js C βjis (·) 1 ¯ [ps ] are of the C class on Ωn . To formulate the second Noether ci and ai Dti theorem we make use of the formal adjoint operator T˜ js of T js : n f (t)T js [ps ](t) dt = Ωn αi (·) js [ci ti Dbi ps (t) cjs (t)f (t) + · f ](t) dt + [·] i=1 Ωn ps (t)T˜ js [f ](t) dt + [·] = (4) Ωn ¯ n ; R), ti I1−αi (·) [cjs ·f ] ∈ C 1 (Ω ¯n ; R), and where [·] represents boundfor f ∈ C 1 (Ω i bi ary terms; this can be obtained by the integration by parts (see Corollary 1).

In [2, 15] generalizations of the fractional Hamilton principle are proposed and appropriated Euler–Lagrange equations were proved. The works [13, 14] are devoted to problems of the calculus of variations with functionals given by multi-dimensional deﬁnite integrals involving partial derivatives of variable fractional order. An analog of the ﬁrst Noether theorem for variable order fractional variational problem is proved in [16]. , [9, 11]. These identities Noether called “dependencies”. For example, the Bianchi identities, in the general theory of relativity, are examples of such “dependencies”.

Tn ) be such that t ∈ Ωn , where Ωn = (a1 , b1 ) × · · · × (an , bn ) is a subset of Rn . Moreover, let us deﬁne the following sets: Δi := (ti , τ ) ∈ R2 : ai ≤ τ < ti ≤ bi , i = 1 . . , n. ¯i ; R , i = 1, . . , n, In the following we assume that αi : Δ¯i → [0, 1], αi ∈ C 1 Δ t ∈ Ωn and f : Ωn → R. Definition 4. The left Riemann–Liouville partial integral of variable fractional order αi (·, ·) with respect to the ith variable ti , is given by ti αi (·,·) [f ](t) ai Iti := 1 (ti − τ )αi (ti ,τ )−1 f (t1 , .