Two Iterative Methods for Solving Linear Interval Systems

Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where à is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solves Ãx̃=b̃, where à is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution x̃ for linear interval systems, where Ãx̃=b̃; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.

An edge determination algorithm for exact computation of the frequency response of linear interval systems

A novel algorithm, called the edge determination algorithm, for exact computation of the frequency response of a linear interval system is proposed. The algorithm formulates candidate curves for the frequency response boundaries as cubic Bezier curves. The edge determination algorithm operates on the cubic Bezier control points of these curves to obtain those, or their parts, that are on the frequency response boundaries. It presents the frequency response boundaries as an array whose entries are the cubic Bezier control points of the curves on the boundaries. Examples for two different cases are presented to illustrate the mechanics and validity of the algorithm.

SIMPLE LMI-BASED SYNCHRONIZATION OF FRACTIONAL-ORDER CHAOTIC SYSTEMS

This paper presents a simple scheme for synchronization of fractional-order chaotic systems. The scheme utilizes a recently developed LMI (Linear matrix inequality) stabilization theorem for fractional-order linear interval systems to design a linear controller. In contrast to existing schemes in the literature, the present scheme is straightforward and does not require that nonlinear parts of synchronization error dynamics are cancelled by the controller. The fractional-order Rössler, Lorenz, and hyperchaotic Chen systems are used as demonstrative examples. Numerical results illustrate the effectiveness of the present scheme.