New Stability Tests for Discretized Fractional-Order Systems Using the Al-Alaoui and Tustin Operators

Rafał Stanisławski,Krzysztof J Latawiec,Marek Rydel

doi:10.1155/2018/2036809

Rafał Stanisławski, Krzysztof J Latawiec + Show 1 more

Open Access

https://doi.org/10.1155/2018/2036809

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Journal: Complexity	Publication Date: Nov 7, 2018
Citations: 2	License type: CC BY 4.0

Affiliation: Opole University of Technology

Abstract

This paper provides new results on a stable discretization of commensurate fractional-order continuous-time LTI systems using the Al-Alaoui and Tustin discretization methods. New, graphical, and analytical stability/instability conditions are given for discrete-time systems obtained by means of the Al-Alaoui discretization scheme. On this basis, an analytically driven stability condition for discrete-time systems using the Tustin-based approach is presented. Finally, the stability of discrete-time systems obtained by finite-length approximation of the Al-Alaoui and Tustin operators are discussed. Simulation experiments confirm the effectiveness of the introduced stability tests.

Highlights

Stable discretization of continuous-time fractional-order systems is an important issue in various areas of science and technology, including system science, signal processing, and control theory
We introduce simple, either analytically driven or purely analytical stability tests for discrete-time systems obtained by the use of the Al-Alaoui operator
These results are used to propose a stability test for systems based on the discretized Tustin operator, which can be considered as a special case of the Al-Alaoui approach

Summary

Introduction

Stable discretization of continuous-time fractional-order systems is an important issue in various areas of science and technology, including system science, signal processing, and control theory In this field, we have three main discretization operators which can generate discrete-time counterparts for continuous-time fractional-order systems, in terms of the Euler, Tustin, and Al-Alaoui methods. We introduce simple, either analytically driven or purely analytical stability tests for discrete-time systems obtained by the use of the Al-Alaoui operator These results are used to propose a stability test for systems based on the discretized Tustin operator, which can be considered as a special case of the Al-Alaoui approach.

Preliminaries

Main Results

Finite-Length Implementation of Discretization Operators

Simulation Examples

Conclusion