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# Selected Problems of Fractional Systems Theory

10 May 2011-

TL;DR: In this paper, the realization problem for positive fractional and continuous-discrete 2D linear systems with state-feedback was formulated and the stability analysis of fractional linear systems in frequency domain was studied.

Abstract: Fractional discrete-time linear systems.- Fractional continuous-time linear systems.- Fractional positive 2D linear systems.- Pointwise completeness and pointwise degeneracy of linear systems.- Pointwise completeness and pointwise degeneracy of linear systems with state-feedbacks.- Realization Problem for positive fractional and continuous-discrete 2D linear systems.- Cone discrete-time and continuous-time linear systems.- Stability of positive fractional 1D and 2D linear systems.- Stability analysis of fractional linear systems in frequency domain.- Stabilization of positive and fractional linear systems.- Singular fractional linear systems.- Positive continuous-discrete linear systems.- Laplace transforms of continuous-time functions and z-transforms of discrete-time functions. "/b>

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TL;DR: In this paper, sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order, and the existence and uniqueness of mild solution of the system is also proved.

174 citations

### Additional excerpts

...Applying Kirchhoff’s law in closed loop (I) [28], we get u(t) = i1(t)R1 + ( i1(t) − i2(t) ) R3 + L1 d α i1(t) dtα ⇒ d α i1(t) dtα = − (R1 + R3) L1 i1(t) + R3 L1 i2(t) + u(t) L1 ....

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TL;DR: It is proved that for fractional-order nonlinear system described by Caputo's or Riemann-Liouville's definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists.

163 citations

### Cites background from "Selected Problems of Fractional Sys..."

...System (1) is called a positive system if Rn + is an invariant set (Kaczorek, 2011)....

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...System (1) is called a positive system if R + is an invariant set (Kaczorek, 2011)....

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TL;DR: This paper is an attempt to overcome the situation by reviewing the state of the art of fractional calculus and putting this topic in a systematic form, and several possible routes of future progress that emerge are tackled.

Abstract: The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

145 citations

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TL;DR: A new method of designing the adaptive backstepping controller for triangular fractional order systems with non-commensurate orders by introducing the appropriate transformations of frequency distributed model is presented.

132 citations

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TL;DR: It is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays and it is shown that the L∞-gain of a positive fractiona-order system is independent of the magnitudeof delays and fully determined by the system matrices.

Abstract: This paper addresses the stability and $L_{\infty}$ -gain analysis problem for continuous-time positive fractional-order delay systems with incommensurate orders between zero and one. A necessary and sufficient condition is firstly given to characterize the positivity of continuous-time fractional-order systems with bounded time-varying delays. Moreover, by exploiting the monotonic and asymptotic property of the constant delay system by virtue of the positivity, and comparing the trajectory of the time-varying delay system with that of the constant delay system, it is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays. In addition, it is shown that the $L_{\infty}$ -gain of a positive fractional-order system is independent of the magnitude of delays and fully determined by the system matrices. Finally, a numerical example is given to show the validity of the theoretical results.

115 citations