Initial Conditions and Initialization of Fractional Systems

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Massinissa Tari ◽  
Nezha Maamri ◽  
Jean-Claude Trigeassou
Keyword(s):  
Initial Conditions ◽  
Fractional Order Systems ◽  
Fractional Systems ◽  
Infinite Dimensional ◽  
State Variable ◽  
Future Behavior ◽  
Fractional Integrator ◽  
True State ◽  
Fractional System ◽  
Dimensional Variable

In this paper, the initialization of fractional order systems is analyzed. The objective is to prove that the usual pseudostate variable x(t) is unable to predict the future behavior of the system, whereas the infinite dimensional variable z(ω, t) fulfills the requirements of a true state variable. Two fractional systems, a fractional integrator and a one-derivative fractional system, are analyzed with the help of elementary tests and numerical simulations. It is proved that the dynamic behaviors of these two fractional systems differ completely from that of their integer order counterparts. More specifically, initialization of these systems requires knowledge of z(ω,t0) initial condition.

Another Note on Stability of Sliding Mode Dynamics in Suppression of Fractional Chaotic Systems

2015 ◽  
Vol 743 ◽  
pp. 303-306
Author(s):  
J. Yuan ◽  
B. Shi ◽  
Yan Wang
Keyword(s):  
Stability Analysis ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Fractional Systems ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Fractional Differential ◽  
Continuous Frequency ◽  
The Stability ◽  
Sliding Mode Dynamics

This paper revisits the stability analysis of sliding mode dynamics in suppression of a classof fractional chaotic systems by a different approach. Firstly, we convert fractional differential equationsinto infinite dimensional ordinary differential equations based on the continuous frequency distributedmodel of the fractional integrator. Then we choose a Lyapunov function candidate to proposethe stability analysis. The result applies to both the commensurate fractional systems and the incommensurateones.

Approximation and Identification of Fractional Systems

2005 ◽  
Author(s):  
Amel Benchellal ◽  
Thierry Poinot ◽  
Jean-Claude Trigeassou
Keyword(s):  
Heat Transfer ◽  
Spectral Band ◽  
State Space Model ◽  
Thermal System ◽  
Fractional Systems ◽  
Space Model ◽  
Proposed Model ◽  
Fractional Integrator ◽  
Fractional System ◽  
Transfer Problems

Heat transfer problems obey to diffusion phenomenon. They can be modelled with the help of fractional systems. The simulation of these particular systems is based on a fractional integrator where the non integer behaviour acts only on a limited spectral band. Starting from frequential considerations, a more general approximation of the fractional system is proposed in this communication. It makes it possible to define a state-space model for simulation of transients, and to carry out an output-error technique in order to estimate the parameters of the model. A real application on a thermal system is presented to illustrate the advantages of the proposed model.

Recursive set-membership parameter estimation of fractional systems using orthotopic approach

2018 ◽  
Vol 40 (15) ◽  
pp. 4185-4197 ◽  
Author(s):  
Saif Eddine Hamdi ◽  
Messaoud Amairi ◽  
Mohamed Aoun
Keyword(s):  
Parameter Estimation ◽  
System Parameter ◽  
A Priori ◽  
Fractional Order Systems ◽  
Fractional Systems ◽  
Order Estimation ◽  
Set Membership ◽  
Equation Error ◽  
Bounded Error ◽  
Fractional System

In this paper, set-membership parameter estimation of linear fractional-order systems is addressed for the case of unknown-but-bounded equation error. In such bounded-error context with a-priori known noise bounds, the main goal is to characterize the set of all feasible parameters. This characterization is performed using an orthotopic strategy adapted for fractional system parameter estimation. In the case of a fractional commensurate system, an iterative algorithm is proposed to deal with commensurate-order estimation. The performances of the proposed algorithm are illustrated by a numerical example via a Monte Carlo simulation.

scholarly journals New perspectives of analog and digital simulations of fractional order systems

2017 ◽  
Vol 27 (1) ◽  
pp. 91-118 ◽  
Author(s):  
Abdelfatah Charef ◽  
Mohamed Charef ◽  
Abdelbaki Djouambi ◽  
Alina Voda
Keyword(s):  
Fractional Order ◽  
Numerical Calculations ◽  
Order Structure ◽  
Fractional Order Systems ◽  
Fractional Systems ◽  
Numerical Resolution ◽  
Fractional Integrator ◽  
Basic Ideas ◽  
Fractional Control ◽  
Digital Simulations

AbstractIn the recent decades, fractional order systems have been found to be useful in many areas of physics and engineering. Hence, their efficient and accurate analog and digital simulations and numerical calculations have become very important especially in the fields of fractional control, fractional signal processing and fractional system identification. In this article, new analog and digital simulations and numerical calculations perspectives of fractional systems are considered. The main feature of this work is the introduction of an adjustable fractional order structure of the fractional integrator to facilitate and improve the simulations of the fractional order systems as well as the numerical resolution of the linear fractional order differential equations. First, the basic ideas of the proposed adjustable fractional order structure of the fractional integrator are presented. Then, the analog and digital simulations techniques of the fractional order systems and the numerical resolution of the linear fractional order differential equation are exposed. Illustrative examples of each step of this work are presented to show the effectiveness and the efficiency of the proposed fractional order systems analog and digital simulations and implementations techniques.

Initialization of Riemann-Liouville and Caputo Fractional Derivatives

2011 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Numerical Simulations ◽  
State Vector ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Initial State ◽  
Caputo Fractional Derivatives ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Initial Conditions Problem

Riemann-Liouville and Caputo fractional derivatives are fundamentally related to fractional integration operators. Consequently, the initial conditions of fractional derivatives are the frequency distributed and infinite dimensional state vector of fractional integrators. The paper is dedicated to the estimation of these initial conditions and to the validation of the initialization problem based on this distributed state vector. Numerical simulations applied to Riemann-Liouville and Caputo derivatives demonstrate that the initial conditions problem can be solved thanks to the estimation of the initial state vector of the fractional integrator.

Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy

2013 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Initial Conditions ◽  
Dissipation Function ◽  
Distributed Energy ◽  
Initial State ◽  
Fractional Systems ◽  
Fractional Integrator ◽  
Derivative System ◽  
The Stability ◽  
Definition Of ◽  
Infinite State

This paper addresses the stability of linear commensurate order fractional systems, Dn (X) = AX 0 < n < 1, using the infinite state approach. First, the energy of a fractional integrator is defined, using the distributed energy of its initial state. Compared to the integer order case, this energy is characterized by a long memory decay, which is the characteristic feature of fractional systems. Then, it is applied to define the energy V(t) of a one derivative system. Numerical simulations exhibit the influence of initial conditions on V(t). Thanks to the definition of a dissipation function, a stability condition is derived. Finally, the general case is investigated and a weighted Lyapunov function is derived, using a positive P matrix, related to the eigenvalues of A matrix.

Stability and resonance conditions of second-order fractional systems

2016 ◽  
Vol 24 (4) ◽  
pp. 659-672 ◽  
Author(s):  
Elena Ivanova ◽  
Xavier Moreau ◽  
Rachid Malti
Keyword(s):  
Second Order ◽  
Damping Factor ◽  
Fractional Differentiation ◽  
Resonance Amplitude ◽  
Fractional Systems ◽  
Integral Number ◽  
At Resonance ◽  
Fractional System ◽  
The Stability ◽  
Normalized Gain

The interest of studying fractional systems of second order in electrical and mechanical engineering is first illustrated in this paper. Then, the stability and resonance conditions are established for such systems in terms of a pseudo-damping factor and a fractional differentiation order. It is shown that a second-order fractional system might have a resonance amplitude either greater or less than one. Moreover, three abaci are given allowing the pseudo-damping factor and the differentiation order to be determined for, respectively, a desired normalized gain at resonance, a desired phase at resonance, and a desired normalized resonant frequency. Furthermore, it is shown numerically that the system root locus presents a discontinuity when the fractional differentiation order is an integral number.

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Eric Donald Dongmo ◽  
Kayode Stephen Ojo ◽  
Paul Woafo ◽  
Abdulahi Ndzi Njah
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Lyapunov Stability Theory ◽  
The State ◽  
State Variables ◽  
State Variable ◽  
New Type ◽  
The Difference ◽  
Synchronization Scheme ◽  
Hyperchaotic Systems

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

Approximations of the optimal stopping problem in partial observation

1986 ◽  
Vol 23 (2) ◽  
pp. 341-354 ◽  
Author(s):  
G. Mazziotto
Keyword(s):  
Optimal Stopping ◽  
Measure Space ◽  
Stopping Time ◽  
Optimal Stopping Problem ◽  
Snell Envelope ◽  
Infinite Dimensional ◽  
Markov State ◽  
State Process ◽  
Partially Observed ◽  
True State

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

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