Stability and Multiscroll Attractors of Control Systems via the Abscissa

Edgar-Cristian Díaz-González,Carlos Arturo Loredo-Villalobos,Baltazar Aguirre-Hernández,Eric Campos-Cantón,Jorge Antonio López-Rentería

doi:10.1155/2017/6743734

Abstract

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss–Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.

Highlights

Consider the parametric dynamical system ẋ = f (x, μ), (1)where x ∈ Rn is the state vector, μ ∈ Rm is a parameter vector, and f is an enough smooth vector field
The importance of studying Hurwitz polynomials is due to its usefulness in the stability analysis of linear systems: if the characteristic polynomial of a linearized system is Hurwitz it is asymptotically stable
Maxwell [1] posed the problem in the following way: How can one find the necessary and sufficient conditions to decide whether a polynomial has all its roots with negative real part? A solution was given by Hurwitz [2] and it is known as the Routh–Hurwitz criterion

Summary

Introduction

Where x ∈ Rn is the state vector, μ ∈ Rm is a parameter vector, and f is an enough smooth vector field. The importance of studying Hurwitz polynomials is due to its usefulness in the stability analysis of linear systems: if the characteristic polynomial of a linearized system is Hurwitz (roots with negative real part) it is asymptotically stable. This has motivated researchers working on applications seeking such polynomials. In this paper we use the abscissa of stability of Hurwitz polynomials to study the stability of systems in order to generate multiscroll attractors. Therein the relationship is the following inequality σp󸀠 < σp which is used to obtain a lower bound for the abscissa of stability of a polynomial or an interval family of Hurwitz polynomials.

Preliminaries

Abscissa of Hurwitz Polynomials

A Lower Bound of the Abscissa of Stability of a Polynomial

The Abscissa to Generate Instability and Multiscrolls Attractors

Conclusion