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.

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 Sliding Mode Control and Modified Generalized Projective Synchronization of a New Fractional-Order Chaotic System

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Junbiao Guan ◽  
Kaihua Wang
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Secure Communication ◽  
Sliding Mode ◽  
Projective Synchronization ◽  
Control Technique ◽  
Order System ◽  
Mode Control ◽  
The Stability

A new fractional-order chaotic system is addressed in this paper. By applying the continuous frequency distribution theory, the indirect Lyapunov stability of this system is investigated based on sliding mode control technique. The adaptive laws are designed to guarantee the stability of the system with the uncertainty and external disturbance. Moreover, the modified generalized projection synchronization (MGPS) of the fractional-order chaotic systems is discussed based on the stability theory of fractional-order system, which may provide potential applications in secure communication. Finally, some numerical simulations are presented to show the effectiveness of the theoretical results.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

scholarly journals Relation between stability and resilience determines the performance of early warning signals under different environmental drivers

2015 ◽  
Vol 112 (32) ◽  
pp. 10056-10061 ◽  
Author(s):  
Lei Dai ◽  
Kirill S. Korolev ◽  
Jeff Gore
Keyword(s):  
Early Warning ◽  
Alternative Stable States ◽  
Basin Of Attraction ◽  
Environmental Drivers ◽  
Ecological Communities ◽  
Stable States ◽  
Warning Signals ◽  
Early Warning Signals ◽  
The Stability ◽  
Stability And Resilience

Shifting patterns of temporal fluctuations have been found to signal critical transitions in a variety of systems, from ecological communities to human physiology. However, failure of these early warning signals in some systems calls for a better understanding of their limitations. In particular, little is known about the generality of early warning signals in different deteriorating environments. In this study, we characterized how multiple environmental drivers influence the dynamics of laboratory yeast populations, which was previously shown to display alternative stable states [Dai et al., Science, 2012]. We observed that both the coefficient of variation and autocorrelation increased before population collapse in two slowly deteriorating environments, one with a rising death rate and the other one with decreasing nutrient availability. We compared the performance of early warning signals across multiple environments as “indicators for loss of resilience.” We find that the varying performance is determined by how a system responds to changes in a specific driver, which can be captured by a relation between stability (recovery rate) and resilience (size of the basin of attraction). Furthermore, we demonstrate that the positive correlation between stability and resilience, as the essential assumption of indicators based on critical slowing down, can break down in this system when multiple environmental drivers are changed simultaneously. Our results suggest that the stability–resilience relation needs to be better understood for the application of early warning signals in different scenarios.

Stability and resonance analysis of a general non-commensurate elementary fractional-order system

2020 ◽  
Vol 23 (1) ◽  
pp. 183-210 ◽  
Author(s):  
Shuo Zhang ◽  
Lu Liu ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Order Parameters ◽  
Fractional Order System ◽  
Order System ◽  
Stability Conditions ◽  
Order Restriction ◽  
Resonance Analysis ◽  
General Kind ◽  
The Stability

AbstractThe elementary fractional-order models are the extension of first and second order models which have been widely used in various engineering fields. Some important properties of commensurate or a few particular kinds of non-commensurate elementary fractional-order transfer functions have already been discussed in the existing studies. However, most of them are only available for one particular kind elementary fractional-order system. In this paper, the stability and resonance analysis of a general kind non-commensurate elementary fractional-order system is presented. The commensurate-order restriction is fully released. Firstly, based on Nyquist’s Theorem, the stability conditions are explored in details under different conditions, namely different combinations of pseudo-damping (ζ) factor values and order parameters. Then, resonance conditions are established in terms of frequency behaviors. At last, an example is given to show the stable and resonant regions of the studied systems.

Model Predictive Control of General Fractional Order Systems Using a General Purpose Optimal Control Problem Solver: RIOTS

2019 ◽  
Author(s):  
Sina Dehghan ◽  
Tiebiao Zhao ◽  
YangQuan Chen ◽  
Taymaz Homayouni
Keyword(s):  
Optimal Control ◽  
Model Predictive Control ◽  
Fractional Order ◽  
Predictive Control ◽  
Order System ◽  
Problem Solver ◽  
Fractional Order Systems ◽  
Application Process ◽  
Order Model ◽  
Integer Order

Abstract RIOTS is a Matlab toolbox capable of solving a very general form of integer order optimal control problems. In this paper, we present an approach for implementing Model Predictive Control (MPC) to control a general form of fractional order systems using RIOTS toolbox. This approach is based on time-response-invariant approximation of fractional order system with an integer order model to be used as the internal model in MPC. The implementation of this approach is demonstrated to control a coupled MIMO commensurate fractional order model. Moreover, the performance and its application process is compared to examples reported in the literature.

scholarly journals Complex Behavior Analysis of a Fractional-Order Land Dynamical Model with Holling-II Type Land Reclamation Rate on Time Delay

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Li Wu ◽  
Zhouhong Li ◽  
Yuan Zhang ◽  
Binggeng Xie
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Land Reclamation ◽  
Variable Order ◽  
Transformation Rate ◽  
Complex Behavior ◽  
Bifurcation Parameter ◽  
Type Transformation ◽  
The Stability ◽  
Variable Order Fractional Derivative

In this paper, a fractional-order land model with Holling-II type transformation rate and time delay is investigated. First of all, the variable-order fractional derivative is defined in the Caputo type. Second, by applying time delay as the bifurcation parameter, some criteria to determine the stability and Hopf bifurcation of the model are presented. It turns out that the time delay can drive the model to be oscillatory, even when its steady state is stable. Finally, one numerical example is proposed to justify the validity of theoretical analysis. These results may provide insights to the development of a reasonable strategy to control land-use change.

The Inverted Decoupling Based Fractional Order Two-Input-Two-Output IMC Controller

2017 ◽  
Author(s):  
Dazi Li ◽  
Xingyu He
Keyword(s):  
Frequency Domain ◽  
Fractional Order ◽  
Disturbance Rejection ◽  
Internal Model ◽  
Design Method ◽  
Internal Model Control ◽  
Order System ◽  
Single Input Single Output ◽  
Integer Order ◽  
Model Control

Many processes in the industry can be modeled as fractional order, research on the fractional order become more and more popular. Usually, controllers such as fractional order PID (FOPID) or fractional active disturbance rejection control (FADRC) are used to control single-input-single-output (SISO) fractional order system. However, when it comes to fractional order two-input-two-output (TITO) processes, few research focus on this. In this paper, a new design method for fractional order control based on multivariable non-internal model control with inverted decoupling is proposed to handle non-integer order two-input-two-output system. The controller proposed in this paper just has two parameters to tune compared with the five parameters of the FOPID controller, and the controller structure can be achieved by internal model control (IMC) method which means it is easy to implement. The parameters tuning method used in this paper is based on frequency domain strategy. Compared with integer order situation, fractional order method is more complex, because the calculation of the frequency domain characteristics is difficult. The controller proposed in this paper is robust to process gain variations, what’s more, it provides ideal performance for both set point-tracking and disturbance rejection. Numerical results are given to show the performance of the proposed controller.

LAG SYNCHRONIZATION OF THE FRACTIONAL-ORDER SYSTEM VIA NONLINEAR OBSERVER

2011 ◽  
Vol 25 (29) ◽  
pp. 3951-3964 ◽  
Author(s):  
HAO ZHU ◽  
ZHONGSHI HE ◽  
SHANGBO ZHOU
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Stability Theorem ◽  
Nonlinear Observer ◽  
Order System ◽  
Fractional Order Systems ◽  
Lag Synchronization ◽  
Systematic Method ◽  
Fractional System ◽  
The Stability

In this paper, based on the idea of nonlinear observer, lag synchronization of chaotic fractional system with commensurate and incommensurate order is studied by the stability theorem of linear fractional-order systems. The theoretical analysis of fractional-order systems in this paper is a systematic method. This technique is applied to achieve the lag synchronization of fractional-order Rössler's system, verified by numerical simulation.

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.

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