scholarly journals On simplified forms of the fractional-order backward difference and related fractional-order linear discrete-time system description

2015 ◽  
Vol 63 (2) ◽  
pp. 423-433 ◽  
Author(s):  
P. Ostalczyk
Keyword(s):  
Fractional Order ◽  
Order System ◽  
Discrete Time System ◽  
Input Matrix ◽  
Time System ◽  
Discrete State ◽  
System Description ◽  
Numerical Examples ◽  
Discrete State Space ◽  
Space Description

Abstract In this paper three simplified forms of the fractional-order (FO) backward difference (BD) are proposed and analysed. Due to time and frequency characteristics criteria parameters of simplified forms of the FOBDs are chosen. Applications of the simplified forms of the FOBDs diminish a number of multiplications and additions needed to evaluate the FOBD. This is very important in real-time microprocessor calculations. It is proved that in a discrete state-space description of a fractional-order system one should correct the input matrix with simplified forms of the FOBD. Investigations are supported by two numerical examples

scholarly journals Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization

AIP Advances ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 045310 ◽  
Author(s):  
Adel Ouannas ◽  
Amina Aicha Khennaoui ◽  
Shaher Momani ◽  
Giuseppe Grassi ◽  
Viet-Thanh Pham
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Three Dimensional ◽  
Discrete Time System ◽  
Time System ◽  
And Control

Stability analysis of a discrete-time system with a variable-, fractional-order controller

2010 ◽  
Vol 58 (4) ◽  
pp. 613-619 ◽  
Author(s):  
P. Ostalczyk
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Discrete Time ◽  
Closed Loop ◽  
Control Strategies ◽  
Discrete Time System ◽  
Time System ◽  
Matrix Condition Number ◽  
Fractional Order Controller ◽  
Matrix Condition

Stability analysis of a discrete-time system with a variable-, fractional-order controllerVariable, fractional-order backward difference is a generalisation of commonly known difference or sum. Equations with these differences can be used to describe a variable-, fractional order digital control strategies. One should mention, that classical tools such as a state-space description and discrete transfer function cannot be used in the analysis and synthesis of such a type of systems. Equations describing a closed-loop system are proposed. They contain square matrices imitating the action of matrices in the system polynomial matrix description. This paper focuses on the stability analysis of a closed-loop SISO linear system with a controller described by the equations mentioned. A stability condition based on a transient denominator matrix condition number is proposed. Investigations are supported by two numerical examples.

Responses comparison of the two discrete-time linear fractional state-space models

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Tadeusz Kaczorek ◽  
Piotr Ostalczyk
Keyword(s):  
State Space ◽  
State Space Model ◽  
State Space Models ◽  
Discrete State ◽  
Numerical Examples ◽  
Space Model ◽  
The Difference ◽  
Time Linear ◽  
Discrete State Space ◽  
Fundamental Matrices

AbstractIn this survey we consider two fractional-order discrete state-space models of linear systems. In both cases the crucial elements are the fundamental matrices. The difference between them is analyzed. A fundamental condition for the first state-space model is given. The investigations are illustrated by the numerical examples.

A New Approximation Algorithm of Fractional Order System Models Based Optimization

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Li Meng ◽  
Dingyu Xue
Keyword(s):  
Approximation Algorithm ◽  
Fractional Order ◽  
Approximation Scheme ◽  
Original System ◽  
Order System ◽  
Fractional Order Systems ◽  
Numerical Examples ◽  
Frequency Domains ◽  
Approximate System ◽  
Phase Fitting

This paper proposes a new approximation scheme which is an extension to the well-established Charef’s approximation method for fractional order systems. The method relaxes the constraints on the locations of the pole-zero pairs of the approximate system, which provides more flexibility and space for approximate system to approach the original system as close as possible. The approximate system based on an optimization process performs not only a good magnitude fitting but also a good phase fitting. The benefits from using the proposed scheme are illustrated by numerical examples in frequency domains.

Local controllability of nonlinear discrete-time fractional order systems

2013 ◽  
Vol 61 (1) ◽  
pp. 251-256 ◽  
Author(s):  
D. Mozyrska ◽  
E. Pawłuszewicz
Keyword(s):  
Linear Approximation ◽  
Fractional Order ◽  
Discrete Time ◽  
Local Controllability ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Discrete Time System ◽  
Controllability Problem ◽  
Time System ◽  
Order Difference

Abstract The Riemann-Liouville, Caputo and Gr¨unwald-Letnikov fractional order difference operators are discussed and used to state and solve the controllability problem of a nonlinear fractional order discrete-time system. It is shown that independently of the type of fractional order difference, such a system is locally controllable in q steps if its linear approximation is globally controllable in q steps

scholarly journals An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Amina Aicha Khennaoui ◽  
A. Othman Almatroud ◽  
Adel Ouannas ◽  
M. Mossa Al-sawalha ◽  
Giuseppe Grassi ◽  
...  
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Test Method ◽  
Order System ◽  
Chaotic Attractors ◽  
Discrete Time System ◽  
Discrete Time Systems ◽  
Time Systems

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

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