Discrete Fractional Order Systems Research Articles

Allee effect and Holling type - II response in a discrete fractional order prey - predator model

A gauss type prey - predator model is considered with Allee effect and Holling type II response. Fractional order two species system is discretized. The discretized system exhibits much richer and complex dynamics than its corresponding continuous version.Bifurcation types like flip, neimark sacker and chaos exist in the discretized system.Existence of the positive fixed points is established and local stability of discrete fractional order system is discussed with the variational matrix. It is also shown that the system allows a flip bifurcation and a neimark - sacker bifurcation. Rich dynamical nature of the system is established through time plots with phase trajectory, exciting invariant circles and periodic doubling bifurcation which leads to chaos.

Fractional order unknown input filter design for fault detection of discrete fractional order linear systems

In this study, a new method is introduced to design an estimator for discrete-time linear fractional order systems, which are affected by unknown disturbances. The main goal of this study is decoupling disturbance and uncertainties from the true states for discrete fractional order systems in noisy environment. The fractional Kalman filter framework is exploited to develop a robust estimator against unknown inputs (UIs) in noisy environment. The proposed filter is exploited to detect faults in fractional order systems. Simulation results illustrate the advantages of this robust filter for state estimation and fault detection of fractional order model of ultra-capacitor (UC). The robustness of the designed filter is shown in the sense of disturbance decoupling in the presence of noise.

FPGA implementation of fractional-order discrete memristor chaotic system and its commensurate and incommensurate synchronisations

A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.

A modified Kalman filter algorithm for fractional system under Lévy noises

This paper considers the filter designing problem for discrete linear fractional order systems with non-Gaussian noises. Taking both the system noise and the measurement noise as Lévy noises instead of Gaussian white noises, through the approximating transformation for Lévy noises, a modified Kalman filter algorithm is proposed for discrete linear fractional order system. The developed results can be used to estimate the system states for discrete linear fractional order system under non-Gaussian Lévy noises with satisfactory accuracy, which are more general than some existing results. Finally, simulation results are provided to verify and illustrate the effectiveness of the obtained results.

Discrete fractional order system vibrations

A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage–current is formulated. Also a function of thermal energy FO dissipation of a FO electrical relation is discussed.

Robust Controller Design for Discrete Fractional Order System: A Disturbance Observer based Approach

This paper, investigates the one of open problems of robust controller design for discrete fractional order system. For achieving the specified goal a disturbance observer is constructed based on full state information. After that an extra compensator based on theory of disturbance observer is added to the classical state feedback. It is also shown that after applying the proposed controller closed loop system would become robust against disturbance as the estimation error gets reduced. Effectiveness of the proposed scheme is illustrated by the simulation results.

Advances in classical and analytical mechanics: A reviews of author’s results

A review, in subjective choice, of author?s scientific results in area of: classical mechanics, analytical mechanics of discrete hereditary systems, analytical mechanics of discrete fractional order system vibrations, elastodynamics, nonlinear dynamics and hybrid system dynamics is presented. Main original author?s results were presented through the mathematical methods of mechanics with examples of applications for solving problems of mechanical real system dynamics abstracted to the theoretical models of mechanical discrete or continuum systems, as well as hybrid systems. Paper, also, presents serries of methods and scientific results authored by professors Mitropolyski, Andjelic and Raskovic, as well as author?s of this paper original scientific research results obtained by methods of her professors. Vector method based on mass inertia moment vectors and corresponding deviational vector components for pole and oriented axis, defined in 1991 by K. Hedrih, is presented. Results in construction of analytical dynamics of hereditary discrete system obtained in collaboration with O. A. Gorosho are presented. Also, some selections of results author?s postgraduate students and doctorantes in area of nonlinear dynamics are presented. A list of scientific projects headed by author of this paper is presented with a list of doctoral dissertation and magister of sciences thesis which contain scientific research results obtained under the supervision by author of this paper or their fist doctoral candidates.

Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains

Abstract Two description forms of a linear fractional-order discrete system are considered. The first one is by a fractional-order difference equation, whereas the second by a fractional-order state-space equation. In relation to the two above-mentioned description forms, stability domains are evaluated. Several simulations of stable, marginally stable and unstable unit step responses of fractional-order systems due to different values of system parameters are presented.

OBSERVER FOR DISCRETE FRACTIONAL ORDER STATE-SPACE SYSTEMS

The paper is devoted to the design of an Luenberger-type observer for discrete fractional order systems. The discrete fractional order state space system is first recalled. Then its observability is discussed, and the observer equations are derived. Finally, some numerical examples are presented, and the applicability of the proposed observer is discussed. Also some numerical aspects of the observer algorithm are raised.