By Tadeusz Kaczorek (auth.), Wojciech Mitkowski, Janusz Kacprzyk, Jerzy Baranowski (eds.)

This quantity offers a number of elements of non-integer order structures, often referred to as fractional platforms, that have lately attracted an expanding cognizance within the medical neighborhood of structures technological know-how, utilized arithmetic, keep watch over conception. Non-integer platforms became proper for plenty of fields of technological know-how and know-how exemplified by means of the modeling of sign transmission, electrical noise, dielectric polarization, warmth move, electrochemical reactions, thermal tactics, acoustics, and so forth. The content material is split into six components, each of which considers one of many at present 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 periods of non-integer order structures with and with no delays. The 3rd half is concentrated on such vital elements as controllability, observability and optimization particularly in discrete time. The fourth half is concentrated on disbursed structures the place non-integer calculus results in new and engaging effects. the following half considers difficulties of suggestions and approximations of non-integer order equations and structures. the ultimate and so much broad half is dedicated to purposes. difficulties from mechatronics, biomedical engineering, robotics and others are all analyzed and solved with instruments from fractional platforms. This quantity got here to fruition because of excessive point of talks and fascinating discussions at RRNR 2013 - fifth convention on Non-integer Order Calculus and its purposes that happened at AGH collage of technological know-how and know-how in Kraków, Poland, which used to be equipped via the school of electric Engineering, Automatics, machine technological know-how 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**

**Example text**

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. [17] If 0 < α < 1 then c j (α ) > 0 for j = 1,2,... Theorem 1. [17] 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.

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. 05 + n)) . Therefore by Theorem 1 the trivial solution of the considered system is stable. Stability of Fractional Diﬀerence Systems with Two Orders 5 51 Conclusions Using the Lyapunov direct method we studied the stability of the Caputo nonlinear fractional diﬀerence systems with two orders. We stated the suﬃcient conditions for uniform stability, uniformly asymptotic stability, and globally uniformly asymptotic stability for such systems.

Kybernetes: The International Journal of Systems & Cybernetics 38(7/8), 1059–1078 (2009) 11. : Realization problem for fractional continuous-time systems. Archives of Control Sciences 18(1), 43–58 (2008) 12. : Realization problem for positive multivariable discrete-time linear systems with delays in the state vector and inputs. Int. J. Appl. Math. Comp. Sci. 16(2), 101–106 (2006) 13. : Realization problem for positive discrete-time systems with delay. System Science 30(4), 117–130 (2004) 14. : Realization problem for positive fractional discrete-time linear systems.