Dynamical distributed controller for the synchronization problem of integer and fractional order partial differential equation systems

2021 ◽  
pp. 014233122110320
Author(s):  
Juan Pablo Flores-Flores ◽  
Rafael Martínez-Guerra
Keyword(s):  
Fractional Order ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Multiple Systems ◽  
Infinite Dimensional ◽  
Integer Order ◽  
Synchronization Problem ◽  
Distributed Controller ◽  
A Chain ◽  
The Stability

In this work we present a methodology to design dynamical distributed controllers for the synchronization problem of systems governed by partial differential equations of integer and fractional order. We consider fractional systems whose space derivatives are of integer order and time derivatives are of fractional commensurate order. The methodology is based on finding canonical forms by means of a change of variable, such that a distributed controller can be designed in a natural way in the form of a chain of integrators. To study the stability of the integer order closed loop system, we propose to use the spectral and semi-group theory for infinite dimensional Hilbert spaces. On the other hand, the stability criterion previously established by Matignon is used for the stability analysis of the fractional order case. Additionally, we tackle the synchronization problem of multiple systems, which is reduced to a problem of generalized multi-synchronization. To illustrate the effectiveness of the proposed methodology, examples and numerical results are given.

scholarly journals Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yi-Fei Pu ◽  
Ji-Liu Zhou ◽  
Patrick Siarry ◽  
Ni Zhang ◽  
Yi-Guang Liu
Keyword(s):  
Differential Equation ◽  
Image Processing ◽  
Partial Differential Equation ◽  
Fractional Order ◽  
Differential Calculus ◽  
Order Partial Differential Equation ◽  
Texture Image ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Integer Order

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

scholarly journals CentrifugationSynchronization Control of Fractional Order Chaotic Systems Based on Memristor and Its Hardware Implementation

2021 ◽  
pp. 289-297
Author(s):  
Zhaohan zhang, Huiling Jin
Keyword(s):  
Fractional Order ◽  
Hardware Implementation ◽  
Chaotic Systems ◽  
Control Method ◽  
Sliding Mode ◽  
Sliding Mode Controller ◽  
Controlled System ◽  
Integer Order ◽  
The Stability ◽  
Dynamic Phenomena

This paper studies the synchronization control of fractional order chaotic systems based on memristor and its hardware implementation. This paper takes the complex dynamic phenomena of memristor turbidity system as the research background. Starting with the integer order memristor system, the fractional order form is derived based on the integer order turbid system, and its dynamics is deeply studied. At the same time, the turbidity phenomenon is applied to the watermark encryption algorithm, which effectively improves the confidentiality of the algorithm. Finally, in order to suppress the occurrence of turbidity, a fractional order sliding mode controller is proposed. In this paper, the sliding mode controller under the function switching control method is established, and the conditions for the parameters of the sliding mode controller are derived. Finally, the experimental results analyze the stability of the controlled system under different parameters, and give the corresponding time-domain waveform to verify the correctness of the theoretical analysis.

scholarly journals A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1322 ◽  
Author(s):  
Wojciech Przemysław Hunek ◽  
Łukasz Wach
Keyword(s):  
Fractional Order ◽  
Mimo Systems ◽  
Inverse Model ◽  
Integer Order ◽  
Transmission Zeros ◽  
Control Paradigm ◽  
Model Control ◽  
The Stability ◽  
Research Challenge ◽  
Stable Control

The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given H- and σ -inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology.

ANTICIPATING SYNCHRONIZATION OF INTEGER ORDER AND FRACTIONAL ORDER HYPER-CHAOTIC CHEN SYSTEM

2012 ◽  
Vol 26 (32) ◽  
pp. 1250211 ◽  
Author(s):  
PENGZHEN DONG ◽  
GANG SHANG ◽  
JIE LIU
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Chen System ◽  
Integer Order ◽  
Research Fields ◽  
Long Time ◽  
Anticipation Time ◽  
Error System ◽  
The Stability

Such a problem, how to resolve the problem of long-term unpredictability of chaotic systems, has puzzled researchers in nonlinear research fields for a long time during the last decades. Recently, Voss et al. had proposed a new scheme to research the anticipating synchronization of integral-order nonlinear systems for arbitrary initial values and anticipation time. Can this anticipating synchronization be achieved with hyper-chaotic systems? In this paper, we discussed the application of anticipating synchronization in hyper-chaotic systems. Setting integer order and commensurate fractional order hyper-chaotic Chen systems as our research objects, we carry out the research on anticipating synchronization of above two systems based on analyzing the stability of the error system with the Krasovskill–Lyapunov stability theory. Simulation experiments show anticipating synchronization can be achieved in both integer order and fractional order hyper-chaotic Chen system for arbitrary initial value and arbitrary anticipation time.

scholarly journals A foundational approach to the Lie theory for fractional order partial differential equations

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Rosario Antonio Leo ◽  
Gabriele Sicuro ◽  
Piergiulio Tempesta
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Number ◽  
Fractional Order ◽  
Jet Space ◽  
Lie Theory ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Independent Variables ◽  
Fractional Differential

AbstractWe provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

D-Stability Based LMI Criteria of Stability and Stabilization for Fractional Order Systems

2015 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Feasible Solution ◽  
Order System ◽  
Linear Matrix ◽  
Fractional Order Systems ◽  
Matrix Inequalities ◽  
Integer Order ◽  
Stability And Stabilization ◽  
The Stability ◽  
Conditions Of Stability

This paper considers the stability and stabilization of fractional order systems (FOS) with the fractional order α: 0 < α < 1 case. The equivalence between stability of fractional order systems and D–stability of a matrix A in specific region is proven. The criteria of stability and stabilization of fractional order system are presented. The conditions are expressed in terms of linear matrix inequalities (LMIs) which can be easily calculated with standard feasible solution problem in MATLAB LMI toolbox. When α = 1, the results reduce to the conditions of stability and stabilization of integer order systems. Numerical examples are given to verify the effectiveness of the criteria. With the approach proposed in this paper, we can analyze and design fractional order systems in the same way as what we do to the integer order system state-space models.

scholarly journals Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System

Electronics ◽  
2020 ◽  
Vol 9 (6) ◽  
pp. 880 ◽  
Author(s):  
José Luis Echenausía-Monroy ◽  
Guillermo Huerta-Cuellar ◽  
Rider Jaimes-Reátegui ◽  
Juan Hugo García-López ◽  
Vicente Aboites ◽  
...  
Keyword(s):  
Fractional Order ◽  
Fractional Integration ◽  
Basin Of Attraction ◽  
Parameter Variation ◽  
Order System ◽  
Bifurcation Parameter ◽  
Stable States ◽  
Integer Order ◽  
The Stability ◽  
Integration Order

In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

Sign in / Sign up

Export Citation Format

Share Document