The Time-Varying Fractional Order Difference Equations

2003 ◽  
Author(s):  
Piotr W. Ostalczyk
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
Difference Equations ◽  
Control Strategy ◽  
Closed Loop ◽  
Time Varying ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Numerical Example ◽  
Fractional Orders

In this paper we explore the linear difference equations with fractional orders being a function of time. A description of closed-loop dynamical systems described by such equations is proposed. In the numerical example a simple control strategy based on time-varying fractional orders is presented.

scholarly journals Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1751
Author(s):  
Oana Brandibur ◽  
Eva Kaslik ◽  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Discrete Version ◽  
Model Parameters ◽  
Step Size ◽  
Order Difference ◽  
Stability And Instability ◽  
Fractional Orders

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

Simulating and analyzing artificial nonstationary earthquake ground motions

1982 ◽  
Vol 72 (2) ◽  
pp. 615-636
Author(s):  
Robert F. Nau ◽  
Robert M. Oliver ◽  
Karl S. Pister
Keyword(s):  
Difference Equations ◽  
Kalman Filters ◽  
Ground Motions ◽  
Structural Response ◽  
Model Parameters ◽  
Time Varying ◽  
Nonparametric Method ◽  
Linear Difference Equations ◽  
Earthquake Ground Motions ◽  
Earthquake Accelerograms

Abstract This paper describes models used to simulate earthquake accelerograms and analyses of these artificial accelerogram records for use in structural response studies. The artificial accelerogram records are generated by a class of linear linear difference equations which have been previously identified as suitable for describing ground motions. The major contributions of the paper are the use of Kalman filters for estimating time-varying model parameters, and the development of an effective nonparametric method for estimating the variance envelopes of the accelerogram records.

scholarly journals Analysis of Atangana–Baleanu fractional-order SEAIR epidemic model with optimal control

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Chernet Tuge Deressa ◽  
Gemechis File Duressa
Keyword(s):  
Numerical Simulation ◽  
Optimal Control ◽  
Fractional Order ◽  
Control Strategy ◽  
Epidemic Model ◽  
Numerical Scheme ◽  
Order Derivative ◽  
Control Analysis ◽  
Fractional Order Derivative ◽  
Fractional Orders

AbstractWe consider a SEAIR epidemic model with Atangana–Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic.

scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

LPV-Based-Method Aiming at a Win-Win Situation for Analytical Control Theories and Intelligent Optimization Methodologies

2009 ◽  
Author(s):  
Kazuhiko Hiramoto
Keyword(s):  
Control Strategy ◽  
Closed Loop ◽  
Control Method ◽  
Control Object ◽  
Gain Scheduling ◽  
Design Parameters ◽  
Analytical Control ◽  
Time Varying ◽  
Intelligent Optimization ◽  
Gain Scheduling Control

A new collaborative control strategy between time varying design parameters in LPV plants and the feedback controllers is proposed in the present paper. As the feedback control law the gain scheduling control scheme is adopted to guarantee the closed-loop L2 gain performance against the variation of the time varying parameter in the control object. The gain-scheduling controller can be obtained in an analytical manner by solving LMIs. For the closed-loop system with the LPV plant and the gain scheduling controller Genetic algorithm (GA), known as a so-called intelligent optimization method, is applied to optimize the closed-loop response. The proposed control system has a complementary structure between the LMI-based analytical control strategy and the flexible intelligent control method that does not impair their advantages each other. In this sense a win-win situation for the LMI-based gain scheduling control and the GA-based intelligent optimization is realized in the proposed approach. A simple simulation example is presented to show the effectiveness of the proposed method.

Robust Control of a Laboratory Hydraulic Canal by Using a Fractional PI Controller

2007 ◽  
Author(s):  
Luis Sa´nchez Rodri´guez ◽  
Vicente Feliu Batlle ◽  
Rau´l Rivas Pe´rez ◽  
Miguel A´ngel Rui´z Torija
Keyword(s):  
Fractional Order ◽  
Control Strategy ◽  
Supervisory Control ◽  
Pi Controller ◽  
Time Varying ◽  
Fractional Order Control ◽  
Supervision System ◽  
Dynamical Parameters ◽  
Fractional Order Pi Controller ◽  
Fractional Controller

A fractional order control strategy has been implemented in an automated laboratory hydraulic canal characterized to present time-varying dynamical parameters. A method for tuning a fractional order PI controller that makes the hydraulic canal control system robust to process parameters variations has been proposed too. Experiments have been carried out in our laboratory hydraulic canal, and the fractional controller has been implemented in an industrial SCADA (Supervisory Control And Data Acquisition) control and supervision system. The comparison between the response of our fractional PI controller and the standard PI controller proved the effectiveness of the proposed fractional order control strategy in terms of performance and robustness.

THREE SCHEMES TO SYNCHRONIZE CHAOTIC FRACTIONAL-ORDER RUCKLIDGE SYSTEMS

2007 ◽  
Vol 21 (12) ◽  
pp. 2033-2044 ◽  
Author(s):  
YANBIN ZHANG ◽  
TIANSHOU ZHOU
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
System Parameter ◽  
Chaotic Synchronization ◽  
Linear Feedback ◽  
Linear Feedback Control ◽  
Synchronization Problem ◽  
Alternate Choice ◽  
Parameter Values ◽  
Fractional Orders

The synchronization problem of chaotic fractional-order Rucklidge systems is studied both theoretically and numerically. Three different synchronization schemes based on the Pecora–Carroll principle, the linear feedback control and the bidirectional coupling are proposed to realize chaotic synchronization. It is shown that such schemes can achieve the same aim for the same set of system parameter values (including fractional orders). This provides an alternate choice for applications of fractional-order dynamical systems in engineering fields.

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