FRACTIONAL CALCULUS BASED STABILIZATION TECHNIQUE APPLIED TO SUPPRESS CHAOS IN CHAOTIC CIRCUITS

2010 ◽  
Vol 24 (24) ◽  
pp. 4861-4879
Author(s):  
MOHAMMAD SALEH TAVAZOEI ◽  
MOHAMMAD HAERI ◽  
SAEID JAFARI
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Chaotic Oscillations ◽  
Practical Applications ◽  
Stability Theorems ◽  
Single Input ◽  
Chaotic Circuits ◽  
The Stability ◽  
Linearized Model ◽  
Stabilization Technique

This paper deals with a new fractional calculus based method to stabilize fixed points of single-input 3D systems. In the proposed method, the control signal is determined by fractional order integration of a linear combination of the system linearized model states. The tuning rule for this method is based on the stability theorems in the incommensurate fractional order systems. The introduced technique can be used in suppression of chaotic oscillations. To evaluate the performance of the proposed technique in practical applications, it has been experimentally applied to control chaos in two chaotic circuits.

scholarly journals Stability of Fractional Order Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Margarita Rivero ◽  
Sergei V. Rogosin ◽  
José A. Tenreiro Machado ◽  
Juan J. Trujillo
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
State Of The Art ◽  
Point Of View ◽  
Fractional Order Systems ◽  
Short Period ◽  
Systems And Control ◽  
Different Types ◽  
The Stability ◽  
And Control

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

Design and Application of Fractional Order PIλDµ Controller in Grid-Connected Inverter System

2017 ◽  
Author(s):  
Zhifeng Pan ◽  
Xiaohong Wang ◽  
Thi Thu Giang Hoang ◽  
Ying Luo ◽  
Yangquan Chen ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Frequency Domain Analysis ◽  
Discretization Method ◽  
Vector Method ◽  
Practical Applications ◽  
Three Phase ◽  
Grid Connected Inverter ◽  
Fractional Order Controller ◽  
Two Parameters

In comparison with the traditional controller PID, the fractional order controller PIλDμ is added with two parameters (μ and λ), that increases the flexibility of the controller. The larger adjustable space makes it more favorable to the control of the nonlinear system like the three-phase grid-connected inverter. However, since the fractional calculus operator is an irrational function on the complex plane, it can’t be implemented directly in simulation or in practical applications. In this paper, the advantages of the fractional order controller PIλDμ will be analyzed, then the fractional calculus operator is fitted by the frequency domain analysis method. Based on the vector method, the fractional order controller PIλDμ of the grid-connected inverter is designed. Simultaneously, the optimal controller parameters are searched with the ITAE and IAE as the performance index. And finally, the results are compared with those of the traditional controller PID. In consideration of the defects of fractional algorithm and single discretization method, a hybrid discretization method is proposed in order to ensure that the discretized controller can keep the same time-domain response and frequency characteristics as the designed controller. The experimental results show that the proposed method has the dynamic and static characteristics better than the traditional controller PID, which proves that the application of fractional order controller in three-phase grid-connected inverter is effective and feasible.

scholarly journals Razumikhin Stability Theorem for Fractional Systems with Delay

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
D. Baleanu ◽  
S. J. Sadati ◽  
R. Ghaderi ◽  
A. Ranjbar ◽  
T. Abdeljawad (Maraaba) ◽  
...  
Keyword(s):  
Time Delay ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Stability Theorem ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Science And Engineering ◽  
Fractional Systems ◽  
Razumikhin Theorem ◽  
The Stability

Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.

scholarly journals Stability and Stabilizing of Fractional Complex Lorenz Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Stability Theory ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Fractional Order Systems ◽  
Caputo Derivatives ◽  
Stability And Stabilization ◽  
The Stability

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

Non-adhesive lotus and other hydrophobic materials

2008 ◽  
Vol 366 (1870) ◽  
pp. 1539-1556 ◽  
Author(s):  
David Quéré ◽  
Mathilde Reyssat
Keyword(s):  
Solid Surface ◽  
Room Temperature ◽  
Water Repellency ◽  
Short Review ◽  
Water Repellent ◽  
Practical Applications ◽  
Hydrophobic Materials ◽  
Volatile Liquid ◽  
Air Cushion ◽  
The Stability

Superhydrophobic materials recently attracted a lot of attention, owing to the potential practical applications of such surfaces—they literally repel water, which hardly sticks to them, bounces off after an impact and slips on them. In this short review, we describe how water repellency arises from the presence of hydrophobic microstructures at the solid surface. A drop deposited on such a substrate can float above the textures, mimicking at room temperature what happens on very hot plates; then, a vapour layer comes between the solid and the volatile liquid, as described long ago by Leidenfrost. We present several examples of superhydrophobic materials (either natural or synthetic), and stress more particularly the stability of the air cushion—the liquid could also penetrate the textures, inducing a very different wetting state, much more sticky, due to the possibility of pinning on the numerous defects. This description allows us to discuss (in quite a preliminary way) the optimal design to be given to a solid surface to make it robustly water repellent.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

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