By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

ISBN-10: 3319098993

ISBN-13: 9783319098999

ISBN-10: 3319099000

ISBN-13: 9783319099002

This quantity provides chosen points of non-integer, or fractional order structures, whose research, synthesis and functions have more and more develop into a true problem for varied learn groups, starting from technological know-how to engineering. The spectrum of purposes of the fractional order calculus has highly accelerated, in truth it might be tough to discover a science/engineering-related topic region 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 amount is subdivided into six components, reflecting specific features 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 parts to the mathematical thought of fractional-order structures. within the 3rd a part of this quantity, a number of latest leads to approximation, modeling and simulations of fractional-order platforms is given. The fourth half offers new strategies to a few difficulties in controllability and keep watch over of non-integer order platforms, specifically fractional PID-like keep watch over. The 5th half analyzes the steadiness of non-integer order structures and a few new effects are provided during this vital admire, particularly for discrete-time structures. the ultimate, 6th a part of this quantity offers a spectrum of purposes of the noninteger order calculus, starting from bi-fractional filtering, specifically of electromyographic signs, in the course of the thermal diffusion and advection diffusion methods 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 was once geared up via the dep. of electric, regulate and machine Engineering, Opole collage of expertise, Opole, Poland.

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**Additional 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**

4. (14) Observe that functional (12) is invariant under a gauge transformation: α (·) j y˜j (t) = yj (t) + C f (t), aj Dtj j = 1, . . , 4, where f : Ω → R is an arbitrary function of class C 2 in all of its argument. Therefore, by Theorem (2), we conclude that 4 αj (·) tj Dbj j=1 Ejf (F ) = 0, Noether’s Second Theorem for VOFVP 45 where Ejf (F ) are Euler–Lagrange expressions (14). Equations Ej (F ) = 0 do not uniquely determine the vector function y. Therefore, as it is in the case of the classical electromagnetic ﬁeld, to avoid this lack of uniqueness, the additional condition should be imposed on y = (y1 , y2 , y3 , y4 ).

Then lim t¯h = t and h→0 (−α) lim (t¯h − a)h h→0 = (t − a)−α . Viable Solutions to Fractional Diﬀerence and Diﬀerential Equations 21 Proposition 5. Let α ∈ (0, 1], h > 0, f being continuous with integrable f on + 1. Then some [a, T ] with T > 0, and t¯h := a + (1 − α)h + nh, where n = t−a h C α aD f α a Δh,∗ f (t) = lim h→0 (t¯h ) . Proposition 6. Let α ∈ (0, 1], h > 0, f being continuous with integrable f on + 1. Then some [a, T ] with T > 0, and t¯h := a + (1 − α)h + nh, where n = t−a h RL α a D f ¯ (t) = lim (a Δα h f ) (th ) .

4. (14) Observe that functional (12) is invariant under a gauge transformation: α (·) j y˜j (t) = yj (t) + C f (t), aj Dtj j = 1, . . , 4, where f : Ω → R is an arbitrary function of class C 2 in all of its argument. Therefore, by Theorem (2), we conclude that 4 αj (·) tj Dbj j=1 Ejf (F ) = 0, Noether’s Second Theorem for VOFVP 45 where Ejf (F ) are Euler–Lagrange expressions (14). Equations Ej (F ) = 0 do not uniquely determine the vector function y. Therefore, as it is in the case of the classical electromagnetic ﬁeld, to avoid this lack of uniqueness, the additional condition should be imposed on y = (y1 , y2 , y3 , y4 ).

### 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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