By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

ISBN-10: 3319098993

ISBN-13: 9783319098999

ISBN-10: 3319099000

ISBN-13: 9783319099002

This quantity offers chosen points 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'd be demanding to discover a science/engineering-related topic quarter the place the fractional calculus had no longer been integrated. 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 facets of the fractional order calculus. the 1st half features 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 offers new components to the mathematical concept of fractional-order platforms. within the 3rd a part of this quantity, a number of latest leads to approximation, modeling and simulations of fractional-order structures is given. The fourth half provides new options to a couple difficulties in controllability and keep an eye on of non-integer order structures, specifically fractional PID-like regulate. The 5th half analyzes the soundness of non-integer order platforms and a few new effects are provided during this very important admire, specifically for discrete-time platforms. 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, during 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 energetic viewers of the RRNR'2014, the sixth convention on Non-integer Order Calculus and Its purposes that used to be geared up through the dep. of electric, regulate and laptop Engineering, Opole college of know-how, Opole, Poland.

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**Extra 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**

**Sample text**

Corollary 2. Let 2 > α > 32 and p ∈ C[a, b] be an arbitrary positive function. In the L2 (a, b) - space an inﬁnite, countable and orthonormal basis of continuously diﬀerentiable eigenfunctions of operator T: {φk ; k ∈ N} corresponding to Regular Sturm-Liouville Problem with Riemann-Liouville Derivatives simple, positive eigenvalues λ11 > following series are convergent: 1 λ2 > ... > ∞ k=1 k ∈ N exists. ; 31 f, φk = ∀f ∈ L2 (a, b), f (x)φk (x)dx, (31) a 1 <∞ (λk )2 (32) and relation |φk (x)| < M+ |λk | (33) holds for certain M+ ∈ R and any k ∈ N, x ∈ [a, b].

N} is ai Dti the standard basis in Rn . We assume that F is a Lagrangian of class C 1 : ¯n −→ R F : Rm × Rmn × Ω (x1 , x2 , t) −→ F (x1 , x2 , t), and 1−αi (·) ti Ibi ∂F (y(τ ),∇D [y](τ ),τ ) α (·) i ∂C a Dt i i [yj ] ¯n , i = (t) is continuously diﬀerentiable on Ω 1, . . , n, j = 1, . . , m. Following theorem states that if a function minimizes (or maximizes) functional (1), then it necessarily must satisfy (2). Noether’s Second Theorem for VOFVP 41 Theorem 1 (Cf. [14]). If y˜ ∈ A(ζ) minimizes (or maximizes) the functional (1) then y˜ satisfies the following equations n ∂F αi (·) ∂F (y(τ ), ∇D [y](τ ), τ ) (y(t), ∇D [y](t), t) + (t) = 0, ti Dbi αi (·) ∂yj ∂C [yj ] ai Dti i=1 (2) for t ∈ Ωn , j = 1, .

New York (1961) 24. : Fractional integrals and derivatives. Gordon and Breach, Yverdon (1993) 25. : Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013) 26. : Sturm-Liouville Theory. Mathematical Surveys and Monographs, vol. 121. American Mathematical Society (2005) Noether’s Second Theorem for Variable Order Fractional Variational Problems Agnieszka B. Malinowska1 and Tatiana Odzijewicz2 1 2 Faculty of Computer Science Bialystok University of Technology 15-351 Bialystok, Poland Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro 3810-193 Aveiro, Portugal Abstract.

### Advances in Modelling and Control of Non-integer-Order Systems: 6th Conference on Non-integer Order Calculus and Its Applications, 2014 Opole, Poland by Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

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