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- Springer
- Universidade Estadual Paulista
- American Institute of Mathematical Sciences (AIMS)
- Elsevier
- Springer Verlag
- Taylor & Francis
- 19th International Conference on Methods and Models in Automation and Robotics
- Sociedade Brasileira de Física
- Associação Brasileira de Ciências Mecânicas
- MDPI - Multidisciplinary Digital Publishing Institute
- SAGE Publications
- Instituto Politécnico do Porto
- Society of Photo-Optical Instrumentation Engineers (SPIE) - Optical Engineering
- Revista Colombiana de Matemáticas
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## ‣ Statements on chaos control designs, including a fractional order dynamical system, applied to a MEMS comb-drive actuator

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica
Formato: 1837-1857

Português

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In this work, we deal with a micro electromechanical system (MEMS), represented by a micro-accelerometer. Through numerical simulations, it was found that for certain parameters, the system has a chaotic behavior. The chaotic behaviors in a fractional order are also studied numerically, by historical time and phase portraits, and the results are validated by the existence of positive maximal Lyapunov exponent. Three control strategies are used for controlling the trajectory of the system: State Dependent Riccati Equation (SDRE) Control, Optimal Linear Feedback Control, and Fuzzy Sliding Mode Control. The controls proved effective in controlling the trajectory of the system studied and robust in the presence of parametric errors.

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## ‣ Fractional order and dynamic simulation of a system involving an elastic wide plate

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Conferência ou Objeto de Conferência
Formato: 1973-1976

Português

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Numerous researchers have studied about nonlinear dynamics in several areas of science and engineering. However, in most cases, these concepts have been explored mainly from the standpoint of analytical and computational methods involving integer order calculus (IOC). In this paper we have examined the dynamic behavior of an elastic wide plate induced by two electromagnets of a point of view of the fractional order calculus (FOC). The primary focus of this study is on to help gain a better understanding of nonlinear dynamic in fractional order systems. © 2011 American Institute of Physics.

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## ‣ The fractional and nonlinear magneto-flexible rod

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Conferência ou Objeto de Conferência

Português

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#External excitation#Fourier spectra#Fractional-order calculus#Integer order#Magnetic force#Mechanical systems#Nonlinear dynamic behaviors#Pseudo-phase#Time response#Calculations#Vibration analysis

This paper deals with a system involving a flexible rod subjected to magnetic forces that can bend it while simultaneously subjected to external excitations produces complex and nonlinear dynamic behavior, which may present different types of solutions for its different movement-related responses. This fact motivated us to analyze such a mechanical system based on modeling and numerical simulation involving both, integer order calculus (IOC) and fractional order calculus (FOC) approaches. The time responses, pseudo phase portraits and Fourier spectra have been presented. The results obtained can be used as a source for conduct experiments in order to obtain more realistic and more accurate results about fractional-order models when compared to the integer-order models. © Published under licence by IOP Publishing Ltd.

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## ‣ Fractional order optimal control problems with free terminal time

Fonte: American Institute of Mathematical Sciences (AIMS)
Publicador: American Institute of Mathematical Sciences (AIMS)

Tipo: Artigo de Revista Científica

Português

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#Fractional calculus#Fractional optimal control#Free-time problem#Numerical approximations#Optimization and control

We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal- time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numer- ical methods to solve the problems are presented, and some computational simulations are discussed in detail.

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## ‣ An expansion formula with higher-order derivatives for fractional operators of variable order

Fonte: Hindawi
Publicador: Hindawi

Tipo: Artigo de Revista Científica

Português

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#Analytical error#Marchaud fractional order#Mathematical analysis#Mathematical parameters#Riemann-Liouville fractional order

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

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## ‣ Control of a novel chaotic fractional order system using a state feedback technique

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Português

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#Chaos#Control of chaos#Fractional order systems#Stability tests#Chaotic fractional-order systems#Control methodology#Feedback techniques#Fractional-order chaotic systems#Fractional-order systems#Unstable equilibrium points#Algebra

We consider a new fractional order chaotic system displaying an interesting behavior. A necessary condition for the system to remain chaotic is derived. It is found that chaos exists in the system with order less than three. Using the Routh-Hurwitz and the Matignon stability criteria, we analyze the novel chaotic fractional order system and propose a control methodology that is better than the nonlinear counterparts available in the literature, in the sense of simplicity of implementation and analysis. A scalar control input that excites only one of the states is proposed, and sufficient conditions for the controller gain to stabilize the unstable equilibrium points derived. Numerical simulations confirm the theoretical analysis. © 2013 Elsevier Ltd. All rights reserved.

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## ‣ A discrete time method to the first variation of fractional order variational functionals

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica

Português

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The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.

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## ‣ A discrete method to solve fractional optimal control problems

Fonte: Springer Verlag
Publicador: Springer Verlag

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
57.445615%

#Direct methods#Fractional calculus#Fractional optimal control#Calculations#Differentiation (calculus)#Caputo fractional derivatives#Dynamic control systems#Fractional order#Integer order#Optimal control systems

We present a method to solve fractional optimal control problems, where the dynamic control system depends on integer order and Caputo fractional derivatives. Our approach consists in approximating the initial fractional order problem with a new one that involves integer order derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example.

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## ‣ Computing Hadamard type operators of variable fractional order

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Português

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#Fractional calculus#Fractional calculus of variations#Fractional differential equations#Numerical methods#Variable fractional order#Differential equations#Differentiation (calculus)#Torque control#Fractional derivatives#Fractional derivatives and integrals#Fractional operators

We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound formula for the error is provided. We exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the Hadamard-Marchaud fractional derivative.

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## ‣ Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

Fonte: Taylor & Francis
Publicador: Taylor & Francis

Tipo: Artigo de Revista Científica

Português

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#Dynamic optimization#Fractional calculus#Fractional calculus of variations#Variable fractional order

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.

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## ‣ Fuzzy, integer and fractional order control: application on a wind turbine benchmark model

Fonte: 19th International Conference on Methods and Models in Automation and Robotics
Publicador: 19th International Conference on Methods and Models in Automation and Robotics

Tipo: Aula

Português

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This paper presents a comparison between
proportional integral control approaches for variable speed wind turbines. Integer and fractional-order controllers are designed
using linearized wind turbine model whilst fuzzy controller also takes into account system nonlinearities. These controllers
operate in the full load region and the main objective is to extract maximum power from the wind turbine while ensuring the performance and reliability required to be integrated into an electric grid. The main contribution focuses on the use of fractional-order proportional integral (FOPI) controller which
benefits from the introduction of one more tuning parameter, the integral fractional-order, taking advantage over integer order
proportional integral (PI) controller. A comparison between proposed control approaches for the variable speed wind turbines is presented using a wind turbine benchmark model in the Matlab/Simulink environment. Results show that FOPI has improved system performance when compared with classical PI and fuzzy PI controller outperforms the integer and fractional-order control due to its capability to deal with system non linearities and uncertainties.

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## ‣ Fractional order calculus: historical apologia, basic concepts and some applications

Fonte: Sociedade Brasileira de Física
Publicador: Sociedade Brasileira de Física

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2011
Português

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Fractional order calculus (FOC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. However, until recently, it has been investigated mainly from a mathematical point of view. Advances in the field of fractals have revealed its subtle relationships with fractional calculus. Nonetheless, fractional calculus is generally excluded from standard courses in mathematics, partly because many mathematicians are unfamiliar with its nature and its applications. This area has emerged as a useful tool among researchers. One of the objectives of this paper is to discuss the usefulness of fractional calculus in applied sciences and engineering. In view of the increasing interest in the development of the new paradigm, another objective is to encourage the use of this mathematical idea in various scientific areas by means of a historical apologia for the development of fractional calculus.

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## ‣ Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell

Fonte: Associação Brasileira de Ciências Mecânicas
Publicador: Associação Brasileira de Ciências Mecânicas

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2014
Português

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#Generalized thermo-visco-elasticity#Three-phase-lag model#Fractional order heat equation#Eigen value approach#Vector-matrix differential equation#Step input temperatures

This problem deals with the thermo-elastic interaction due to step input of temperature on the stress free boundaries of a homogeneous visco-elastic orthotropic spherical shell in the context of a new consideration of heat conduction with fractional order generalized thermoelasticity. Using the Laplace transformation, the fundamental equations have been expressed in the form of a vector-matrix differential equation which is then solved by eigen value approach and operator theory analysis. The inversion of the transformed solution is carried out by applying a method of Bellman et al (1966). Numerical estimates for thermophysical quantities are obtained for copper like material for weak, normal and strong conductivity and have been depicted graphically to estimate the effects of the fractional order parameter. Comparisons of the results for different theories (TEWED (GN-III), three-phase-lag model) have also been presented and the effect of viscosity is also shown. When the material is isotropic and outer radius of the hollow sphere tends to infinity, the corresponding results agree with that of existing literature.

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## ‣ Tuning and auto-tuning of fractional order controllers for industry applications

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/submittedVersion; info:eu-repo/semantics/article
Formato: application/pdf

Publicado em //2008
Português

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#Auto-tuning#PID controller#Ractional order controller#Gain variations#Robust control#Robótica e Informática Industrial

This paper deals with the design of fractional order PI²Dμ controllers, in which the orders of the integral and derivative parts, λ and μ, respectively, are fractional. The purpose is to take advantage of the introduction of these two parameters and fulfill additional specifications of design, ensuring a robust performance of the controlled system with respect to gain variations and noise. A method for tuning the PI²Dμ controller is proposed in this paper to fulfill five different design specifications. Experimental results show that the requirements are totally met for the platform to be controlled. Besides, this paper proposes an auto-tuning method for this kind of controller. Specifications of gain crossover frequency and phase margin are fulfilled, together with the iso-damping property of the time response of the system. Experimental results are given to illustrate the effectiveness of this method.

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## ‣ Fractional Order Generalized Information

Fonte: MDPI - Multidisciplinary Digital Publishing Institute
Publicador: MDPI - Multidisciplinary Digital Publishing Institute

Tipo: Artigo de Revista Científica

Publicado em //2014
Português

Relevância na Pesquisa
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This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.

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## ‣ Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena

Fonte: SAGE Publications
Publicador: SAGE Publications

Tipo: Artigo de Revista Científica

Publicado em //2007
Português

Relevância na Pesquisa
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The development of fractional-order controllers is currently one of the most promising fields of research. However, most of the work in this area addresses the case of linear systems. This paper reports on the analysis of fractional-order control of nonlinear systems. The performance of discrete fractional-order PID controllers in the presence of several nonlinearities is discussed. Some results are provided that indicate the superior robustness of such algorithms.

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## ‣ Fractional Order Generalized Information

Fonte: Instituto Politécnico do Porto
Publicador: Instituto Politécnico do Porto

Tipo: Artigo de Revista Científica

Publicado em //2014
Português

Relevância na Pesquisa
57.14984%

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.

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## ‣ Lyapunov functions for fractional order systems

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artículo de revista

Português

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Artículo de publicación ISI; A new lemma for the Caputo fractional derivatives, when 0 < a < 1, is proposed in this
paper. This result has proved to be useful in order to apply the fractional-order extension
of Lyapunov direct method, to demonstrate the stability of many fractional order systems,
which can be nonlinear and time varying.; This work has been supported by CONICYT – Chile, under the grants FB009 ‘‘Centro de Tecnología para la Minería’’ and
FONDECYT 1120453, ‘‘Improvements of Adaptive Systems Performance by using Fractional Order Observers and Particle
Swarm Optimization’’.

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## ‣ Fractional correlations based on the modified fractional order Fourier transform

Fonte: Society of Photo-Optical Instrumentation Engineers (SPIE) - Optical Engineering
Publicador: Society of Photo-Optical Instrumentation Engineers (SPIE) - Optical Engineering

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Fourier optics#Fractional order Fourier transforms#Optical correlators#Optical signal processing#Recognition techniques

Based on the modified fractional order Fourier transform
(MFROFT), we introduce a modified fractional correlation operation
(MFCO). We propose two systems to implement the MFCO experimentally.
These are analyzed theoretically to conclude that the MFCO of two
objects equals the conventional correlation between the two objects
modified by multiplying each by a quadratic phase factor that contains
the parameter that determines the fractional order. The MFCO is applied
to a rectangular slit analytically, numerically, and experimentally. We
show that the MFCO results in superior correlation peaks compared to
those of the conventional correlation operation.; RIT community members may access full-text via RIT Libraries licensed databases: http://library.rit.edu/databases/

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## ‣ Solution of Some Fractional Order Telegraph Equations

Fonte: Revista Colombiana de Matemáticas
Publicador: Revista Colombiana de Matemáticas

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2014
Português

Relevância na Pesquisa
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#Fractional order telegraph equation#Generalized differential transform method#Caputo fractional derivative#Generalized Mittag-Leffler function#Generalized Wright function

In recent years, there has been a great interest in fractional differential equations due to their frequent appearance in various fields, and their more accurate models of systems under consideration provided by fractional derivatives. In particular, fractional order telegraph equations have been considered and solved for many researchers, using different methods. In this paper we derived the solution of two homogeneous space-time fractional telegraph equations using the generalized differential transform method. The derivatives are considered in Caputo sense and the solutions are given in terms of generalized Mittag-Leffler function and the generalized Wright function. Further, various graphics are included which show the behavior of the solution obtained, and results given earlier by Momani, Odibat and Momani, Yildrim, Garg and Sharma, and Garg et al. are obtained as particular cases of ones our.

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