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In quantum mechanics, the Heisenberg's uncertainty principle is an inequalities which it says that we can not measure the conjugate quantities of the particles at the same time with the precision. The fuzzy boundary condition can be a mathematical model for this type of problem. In this paper we consider the Schrödinger equation equipped with fuzzy conditions. The analytical result is derived and...
In a recent article we have introduced and used the complex fractional wavelet for QRS complex detection in ECG signal. This complex fractional wavelet is derived from the Cole-Cole distribution which is widely used in the modeling of dielectrics by a fractional power pole. This work presents an extension of QRS detection scheme using the complex Morlet wavelet function. It consists of using the real...
This paper presents generalization of the Improved Fractional Kalman Filter (ExFKF) for variable order discrete state-space systems. This generalization based on a linear discrete fractional variable order state-space system rewritten into a restricted infinite dimensional form. In order to demonstrate the effectiveness of the proposed algorithms, results of numerical experiments are presented. Proposed...
Due to adding the extra degree of freedom, the fractional order PID (FO-PID) controller can achieve better control performance than the integer order PID controller. Based on the existing tuning methods, developed for particular forms of the process, in this paper the FO-PI and FO-PD controller design method is proposed for a large class of system models, including fractional order and integer models...
The control of an isotope separation column is a difficult and challenging task, being a highly nonlinear plant with large time constants and time delays. The control of a cascade of three such columns is more difficult due to the interdependences between the columns. Previous research proved that fractional order PID controllers have been successfully implemented for a single isotope separation column...
Fractional order differential algebraic equations (FDAEs) are more complex than fractional differential equations (FDEs) on analytical and numerical analysis. In this paper, the sliding mode control theory is introduced to convert the FDAEs into FDEs firstly. Then the predictor-corrector method is used to solve FDEs. To avoid the constraint violations, the numerical results have been corrected. Furthermore,...
Parameter uncertainties and unpredictable environmental disturbances reduce control performance of real control systems. For a robust control performance, stability and disturbance rejection are two main concerns that should be addressed in practical controller design problems. This paper presents an analysis to deal with system stability and disturbance rejection control for fractional-order PI (FOPI)...
Fractional calculus is used for theoretical description of a nondestructive diagnosis method for quality assessment of electric insulation in capacitors, transformers and other devices. The new analytical approach is based on deviation of the relaxation law from the exponential (Debye) one clearly observed in experiments on the long-time relaxation. Power modes of relaxation depends on charging prehistory...
Considering that the vehicle state error derives from the different initial states and parameters between the real vehicle model and the reference model, fractional sliding model control is proposed for the active four-wheel steering vehicle. Optimal control is adopted to design the sliding mode in order to eliminate the state error caused by the different initial states. On this basis, fractional...
Dynamical properties of viscoelastic material are considered to be originated from the complex combination of elastic components and viscous components. From this idea, the time derivative of deviatoric part of the stress of elastic components have been integrated by fractional order. In this paper, several fractional derivative models for large extension are proposed based on this idea. The fractional...
The aim of the paper is to consider - in base on classical models and numerical simulations - fractional model of the type of Hegselmann and Krause (FHK) taking into account memory of states. One of our hypothesis is that fractional operators will include memory of the system to the whole process. Then we are going to examine and to do an analyze with verification of properties of this fractional...
In this paper we derive formulæ for composite numerical fractional integration and differentiation that are “polynomial accurate” in the sense that when applied to polynomials of a given degree they yield exact results. Initially, we develop a fractional equivalent to the trapezoidal rule, as well as an analytic error bound for when it is applied to arbitrary functions. Subsequently, we demonstrate...
No mixed research of hybrid and fractional-order systems into a cohesive and multifaceted whole can be found in the literature. This paper focuses on such a synergistic approach of the theories of both branches, which is believed to give additional flexibility and help the system designer. It is part II of two companion papers and focuses on fractional-order hybrid control. Specifically, two types...
In this paper we use fractional calculus to characterize diffusion in brain tissue by generalizing Fick's 2nd Law. This approach, rooted in the physics of the Continuous Time Random Walk (CTRW) Theory expresses separate measures of tissue complexity through the fractional order of the time derivative, α, and the space derivative, ß. We also calculate the entropy of the characteristic function of the...
This paper presents some results for stability analysis of fractional order polynomials using the Hermite-Biehler theorem. The possibilities of the extension of the Hermite-Biehler theorem to fractional order polynomials is investigated and it is observed that the Hermite-Biehler theorem can be an effective tool for the stability analysis of fractional order polynomials. Variable changing has been...
The procedure of constructing linearly autonomous symmetries is explicitly described and then illustrated on specific type of ordinary fractional differential equations. The results of equations classification with respect to point transformation are presented.
The Mittag-Leffler function plays a central role in fractional calculus; however its numerical evaluation still remains an expensive and challenging task. In this work we discuss the evaluation of this function for pure imaginary arguments by means of a numerical method performing the inversion of its Laplace transform on a suitably selected integral contour. By means of some numerical experiments...
A variety of fractional order models have been proposed in the literature to account for the behaviour of financial processes from different points of view. The objective of this work is to model the growth of national economies, namely, their gross domestic products (GDPs), by means of a fractional order approach. The particular case of Portugal is addressed, and results show that fractional models...
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