- Research Article
- Open Access

# Annular Bounds for Polynomial Zeros and Schur Stability of Difference Equations

- Ke Li
^{1}and - Jin Liang
^{2}Email author

**2011**:782057

https://doi.org/10.1155/2011/782057

© K. Li and J. Liang. 2011

**Received:**7 October 2010**Accepted:**30 October 2010**Published:**21 November 2010

## Abstract

We investigate the monic complex-coefficient polynomial of degree in the complex variable and obtain a new annular bound for the zeros of , which is sharper than the previous results and has clear advantages in judging the Schur stability of difference equations. In addition, examples are given to illustrate the theoretical result.

## Keywords

- Difference Equation
- Characteristic Equation
- Positive Root
- Polynomial Equation
- Characteristic Polynomial

## 1. Introduction

It is well known that many discrete-time systems in engineering are described in terms of a difference equation, and the characteristic equation for the difference equation plays a key role in the study of the behaviors of the solutions, especially the stability of the solutions, to the discrete-time systems. Since the characteristic equations for difference equations are closely related to some complex polynomials, the estimates of the bound for the moduli of various complex polynomial zeros have been investigated by many researchers (cf. e.g., [1–8] and references therein). In the study on this issue, one of meaningful research ideas is to indicate such a common property of a lot of polynomials by a few very special polynomials. Using this idea, a good annular bound by estimating the largest nonnegative zeros of four specific polynomials is given in [8] recently. As a continuation of this work and our paper [4], in this paper we investigate further the location of the zeros of complex-coefficient polynomials on the basis of such a research idea and establish a new annular bound theorem (Theorem 3.1), which improves the previous corresponding result and has clear advantages in judging the Schur stability of difference equations. Examples are given to illustrate the advantages of the new result.

## 2. Preliminaries

Without losing the generality, we assume that , or, equivalently, .

Basic notations are as follows.

: ,

: the modulus of a complex number ,

: the set of all zeros of ,

: with ,

: the smallest positive integer such that in ,

: the largest positive integer such that in ,

: the smallest positive integer such that in ,

: the largest positive integer such that in ,

: the integer part of a real number .

for any positive integers , , and sequence . This notation is logical and useful in the note.

Remark 2.1.

By Descartes' rule of signs, it is easy to see that for each , the polynomial , , has a unique positive zero.

We denote by , , , and the unique positive zero of , , , and , respectively.

## 3. Main Result

The following result is established in [8].

Theorem A (see [8]).

, and .

Theorem 3.1.

Let , , and , . Then

- (i)
- (ii)
- (iii)
the annular bound of original polynomial can be further improved by iterative procedure.

Proof.

which imply that (3.1) is hold. So (i) is proved.

where and .

we get by combining(3.13) and (3.14).

we have .

This means that (iii) is true.

- (a)
When , it follows from (3.14) and (3.15) that for every , , that is, , that is, is -stable. Similarly, we can draw the same conclusion when , and when or .(b) By the similar arguments in the proof of (iii) of Theorem 3.1, the results in (a) can be improved. This also provides an iterative algorithm to test the r-stability and Schur stability of polynomials. (c) The question "What happens to Theorem 3.1 when , , and " is worth considering further.

Example 3.3.

The following examples show the advantages of Theorems 3.1 over Theorem A in analyzing the Schur stability of difference equations (discrete-time systems).

Example 3.4.

So Theorem A cannot guarantee the stability of such a system.

Example 3.5.

which cannot determine the instability of this system.

Example 3.6.

Consequently, such a difference equation (discrete-time system) is Schur stable.

## Declarations

### Acknowledgments

This work was supported partially by the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.