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Podlubný, Igor; Petráš, Ivo; Vinagre, Blas M.; O’Leary, Paul; Dorčák, Ľ.

doi:10.1023/a:1016556604320

Cited by 501 publications

(46 citation statements)

References 11 publications

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“…The fractional calculus that allows us to consider integration and differentiation of any order, not necessarily integer, has been the object of extensive study for analyzing not only anomalous diffusion on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials; see [1][2][3] and references therein), but also fractional phenomena in optimal control (see, e.g., [4][5][6]). As indicated in [2,5,7] and the related references given there, the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields.…”

Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

“…As indicated in [2,5,7] and the related references given there, the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional derivatives in time.…”

Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

“…It is significant to study this class of problems, because, in this way, one is more realistic to describe the memory and hereditary properties of various materials and processes (cf. [4,5,12,13]). …”

Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

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Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

Wang

Chen

2011

Adv Differ Equ

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In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20

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Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

Section: K(t − S)g(s U(s) U(κ 2 (S)))ds T ∈ [0 T] U(0) + H(u)mentioning

confidence: 99%

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Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

Wang

Chen

2011

Adv Differ Equ

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“…Implementation of such fractional-order controllers can be achieved using analog or digital means, but still poses a greater challenge to the engineer compared to having integer-order controllers. Continuoustime fractional-order controllers can be implemented by using approximations such as the continuous fractional approach, and Oustaloup recursive approach and its modifications [35,36]. Discrete-time implementation, on the other hand, can be achieved through the use of FIR and IIR filters, and other discretization methods such as the Tustin method and step or impulse response invariants [37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Posicast control of a class of fractional-order processes

et al. 2013

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Abstract:The number of studies on the control of fractional-order processesprocesses having dynamics described by differential equations of arbitrary orderhas been increasing in the past two decades and it is now ubiquitous. Various methods have emerged and have been proven to effectively control such processes usually resulting in fractional-order controllers similar to their conventional integer-order counterparts, which include, but are not limited to fractional PID and fractional lead-lag controllers. However, such methods require a lot of computational effort and fractional-order controllers could be challenging when it comes to their synthesis and implementation. In this paper, we propose a simple yet effective delay-based controller with the use of the Posicast control methodology in controlling the overshoot of a fractional-order process of the class : {P( ) = 1/( α + )} having orders 1 < α < 2. Such controllers have proven to be easy to implement because they only require delays and summers. In this paper, the Posicast control methodology introduced in the past few years is modified to minimize the overshoot of the processes step response to a level that is acceptable in control engineering and automation practices. Furthermore, proof of the existence of overshoot for such class of processes, as well as the determination of the peak-time of the open-loop response of a fractional-order process of the class is presented. Validation through numerical simulations for a class of fractional-order processes are presented in this paper.

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“…S1 and Fig. S1), FOCs offer increased flexibility in modeling and designing various electrical devices and systems, such as filters, 12,13 oscillators, 14,15 neural circuits, 2 transmission lines, 16 supercapacitors and batteries, [17][18][19][20][21] impedance matching networks, 22 phase-locked loops, 23 and proportional-integral-derivative controllers, 24 and open the door to several unconventional properties that cannot be obtained using traditional circuit elements. For example, unlike a conventional capacitor, an FOC supports a capacitive memory, which enables temporal history of signals to be "stored" by electrical circuits, and therefore, it can be used to more accurately mimic/model electrical pulses communicated by neurons 2 and memory regeneration phenomena observed in dielectrics.…”

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confidence: 99%