By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski
This quantity provides chosen facets of non-integer, or fractional order platforms, whose research, synthesis and purposes have more and more develop into a true problem for numerous study groups, starting from technology to engineering. The spectrum of functions of the fractional order calculus has awfully improved, in truth it'd be difficult 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 quantity is subdivided into six components, reflecting specific features of the fractional order calculus. the 1st half incorporates 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 presents new parts to the mathematical idea of fractional-order structures. within the 3rd a part of this quantity, a host of latest leads to approximation, modeling and simulations of fractional-order structures is given. The fourth half offers new options to a couple difficulties in controllability and keep an eye on of non-integer order platforms, particularly fractional PID-like regulate. 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 offers 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 strategies 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 dept of electric, regulate and machine Engineering, Opole college of expertise, 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
Viable Solutions to Fractional Diﬀerence and Diﬀerential Equations 23 Remark 7. If the fractional diﬀerential equation RL α 0 D x(t) = f (t, x(t)), 0 < α < 1, t ∈ (0, T ], (14) has a viable solution in K, then for every ε > 0 there exists h > 0 such that there exists a solution y to system (a Δα h y) (nh) = f (nh, y(nh + a)) (15) that is viable in Kε , Remark 8. Let us notice that if K = Rn+ one can consider all above results as positivity ones. 5 Conclusions The problem of viability that consists in ﬁnding conditions under which there exists at least one solution to a system, which maintains in a certain set of constrains is considered.
The work was supported by Bialystok University of Technology grant G/WM/3/12. References 1. : Necessary and suﬃcient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011) 2. : Viability Theory. Birkh¨ auser, Berlin (1991) 3. : Set-Valued Analysis. , Boston (1990) 4. : Positive solutions for boundary value problems of nonlinear fractional diﬀerential equations. Journal of Mathematical Analysis and Applications 311(2), 495–505 (2005) 5.
Remark 6. Note that: a Δα h,∗ : F(hN)a → F(hN)a+(1−α)h , where α ∈ (0, 1]. The property of the composition of h-sums was proved in . Proposition 2. Let x be a real valued function defined on (hN)a , where a, h ∈ R, h > 0. For α, β > 0 the following equalities hold: −α a+βh Δh −β a Δh x (t) = −(α+β) x a Δh (t) = −β a+αh Δh −α a Δh x (t) , where t ∈ (hN)a+(α+β)h . The next proposition gives a useful identity of transforming Caputo fractional diﬀerence equations into fractional summations for the case when an order is from the interval (0, 1].