By Tadeusz Kaczorek (auth.), Wojciech Mitkowski, Janusz Kacprzyk, Jerzy Baranowski (eds.)
This quantity provides numerous elements of non-integer order platforms, often referred to as fractional structures, that have lately attracted an expanding cognizance within the clinical neighborhood of platforms technology, utilized arithmetic, keep watch over concept. Non-integer platforms became appropriate for plenty of fields of technology and know-how exemplified by means of the modeling of sign transmission, electrical noise, dielectric polarization, warmth move, electrochemical reactions, thermal methods, acoustics, and so forth. The content material is split into six elements, each of which considers one of many presently appropriate difficulties. within the first half the belief challenge is mentioned, with a distinct specialize in optimistic structures. the second one half considers balance of sure sessions of non-integer order structures with and with no delays. The 3rd half is targeted on such vital elements as controllability, observability and optimization specifically in discrete time. The fourth half is concentrated on allotted platforms the place non-integer calculus results in new and fascinating effects. the following half considers difficulties of options and approximations of non-integer order equations and platforms. the ultimate and so much huge half is dedicated to functions. difficulties from mechatronics, biomedical engineering, robotics and others are all analyzed and solved with instruments from fractional structures. This quantity got here to fruition due to excessive point of talks and fascinating discussions at RRNR 2013 - fifth convention on Non-integer Order Calculus and its purposes that came about at AGH college of technology and know-how in Kraków, Poland, which used to be prepared via the college of electric Engineering, Automatics, desktop technology and Biomedical Engineering.
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Additional info for Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland
1000, at which we see that the trajectory is tending to the equilibrium point (0, 0). One can choose the function V (x1 , x2 ) = x21 + x22 that is positive deﬁnite and decrescent. 3)) . Therefore by Theorem 2 the trivial solution of the considered system is asymptotically stable. Example 2. 05) = 1. Then the point (0, 0) is the equilibrium point of the considered system. The values of x1 and x2 for n = 1, . . , n are displayed in Figures 2(a) and 2(b), respectively. Figure 2(c) shows the phase portrait of (x1 , x2 ) for n = 0, .
Aα = A + αI n and c j (α ) = (−1) j j + 1 (5) Definition 1. The fractional system (1) is called positive if and only if xk ∈ ℜ n+ and yk ∈ ℜ +p , k ∈ Z + for any initial conditions x0 ∈ ℜ n+ and all input sequences uk ∈ ℜ m + , k ∈ Z+ . Lemma 1.  If 0 < α < 1 then c j (α ) > 0 for j = 1,2,... Theorem 1.  The fractional (1) is positive for 0 < α < 1 if and only if Aα = A + αI n ∈ ℜ n+× n , B ∈ ℜ n+× m , C ∈ ℜ +p × n , D ∈ ℜ +p × m . (6) Remark 1. Note that the equation (4) describe a linear discrete-time system with increasing number of delays in state.
Theory & Appl. , LNEE 257, pp. 53–64. 1007/978-3-319-00933-9_5 © Springer International Publishing Switzerland 2013 54 2 A. Ruszewski Problem Formulation Consider the fractional discrete-time linear scalar systems with two delays, described by the homogeneous equation Δα xi +1 = a0 xi + a1 xi −1 + a2 xi − 2 , α ∈ (0, 1), i ∈ Z + (1) with the initial condition x−l (l = 0,1, 2) , where xi ∈ ℜ is the state vector, while a0 , a1 and a2 are real coefficients. In this paper the following definition of the fractional difference [13, 14] will be used Δα xi = i k α (−1) k x i−k , (2) k =0 where α ∈ ℜ is the order of the fractional difference and α α!
Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland by Tadeusz Kaczorek (auth.), Wojciech Mitkowski, Janusz Kacprzyk, Jerzy Baranowski (eds.)