Read e-book online Advances in Modelling and Control of Non-integer-Order PDF

By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

ISBN-10: 3319098993

ISBN-13: 9783319098999

ISBN-10: 3319099000

ISBN-13: 9783319099002

This quantity provides chosen facets of non-integer, or fractional order structures, whose research, synthesis and purposes have more and more develop into a true problem for varied study groups, starting from technology to engineering. The spectrum of functions of the fractional order calculus has exceedingly extended, actually it might be tough 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 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 components to the mathematical conception of fractional-order platforms. within the 3rd a part of this quantity, a number of latest ends up in approximation, modeling and simulations of fractional-order platforms is given. The fourth half offers new strategies to a few difficulties in controllability and keep an eye on 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, specifically for discrete-time platforms. the ultimate, 6th a part of this quantity offers 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 tactics 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 functions that was once equipped through the dep. of electric, keep an eye on and computing device Engineering, Opole college of expertise, 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

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M , A(ζ) := y ∈ C 1 (Ω [yj ] ∈ C(Ω ∂Ωn = ζ, ai Dti and ∇D [y] = (∇D [y1 ], . . , ∇D [ym ]), ∇D = n i=1 α (·) i ei · C , {ei : i = 1, . . , 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).

22 E. Girejko, D. Mozyrska, and M. Wyrwas In order to formulate results on viability we need the following definitions. Let us denote by I an open interval in R. Definition 8. Let K ⊂ Rn be nonempty and f : I × Rn → Rn . The subset K is fractionally viable with respect to f if for any x0 ∈ Rn there exists [t0 , T ] ⊆ R, such that (6) has at least one solution such that x : [t0 , T ] → K, for t ∈ [t0 , T ], where t0 > 0. We say then that x is a viable solution of (6) in K. The idea of viability of fractional differential equations can be expressed by using the concept of tangent cone.

M. Differentiating with respect to ε and taking ε = 0, we get m 0= j=1 Ωn ∂F (y(t), ∇D [y](t), t)T js [ps ](t) ∂yj n ∂F + α (·) α (·) i=1 i ∂C [yj ] ai Dti i (y(t), ∇D [y](t), t)C ai Dti T js [ps ] (t) dt. (5) Applying Corollary 1 we obtain n i=1 ∂F Ωn α (·) αi (·) ∂C [yj ] ai Dti n = i=1 Ωn i (y(t), ∇D [y](t), t)C ai Dti αi (·) ti Dbi ∂F (y(τ ), ∇D [y](τ ), τ ) n 1−αi (·) T js [ps ](t) · ti Ibi + T js [ps ] (t)dt α (·) i ∂C [yj ] ai Dti (t)T js [ps ](t)dt ∂F (y(τ ), ∇D [y](τ ), τ ) αi (·) ∂C [yj ] ai Dti i=1 ∂Ω n (t) · ν i d(∂Ωn ).

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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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