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Annular Bounds for Polynomial Zeros and Schur Stability of Difference Equations
Advances in Difference Equations volume 2011, Article number: 782057 (2010)
Abstract
We investigate the monic complexcoefficient 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.
1. Introduction
It is well known that many discretetime 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 discretetime 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 complexcoefficient 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
Throughout this paper, we let
with , , and
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 .
In order to simplify the expressions in our study, we define specially that
for any positive integers , , and sequence . This notation is logical and useful in the note.
Moreover, we write
with and ,
with ,
with and ,
with ,
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)
(3.1)

(ii)
(3.2)
where , are constants as in Theorem A;

(iii)
the annular bound of original polynomial can be further improved by iterative procedure.
Proof.
Define
where . Then it is not difficult to see that
This implies that for every we have
that is,
Hence, by (3.8), one has
which imply that
In addition, it follows from (2.4)–(2.7) that
Therefore, for each we have
which imply that (3.1) is hold. So (i) is proved.
Next we prove that (ii) holds. Actually, we have
where and .
On the other hand, since the polynomial equation has a unique positive root and
we get by combining(3.13) and (3.14).
In addition, we have
Since
we have .
In the same way, we can obtain and ; therefore,
Finally, we prove (iii). Set
with
and let be the smallest positive integer such that in . If , in analogy to (3.3) and (2.4), we can define
and , respectively. It is not difficult to see that, the unique positive root of polynomial , . Similarly, we can define , , and , respectively. Moreover, their respective positive roots , , and satisfy that
Consequently, new annular bound of , namely, with
is better than (3.1). This procedure can be applied iteratively.
can be further transformed into
respectively, and
into
until the last iteration brings no practical improvement. Obviously, when increases,
will approach the smallest and largest modulus of polynomial zero, respectively, where
denotes the unique positive root of
This means that (iii) is true.
Remark 3.2.

(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 rstability and Schur stability of polynomials. (c) The question "What happens to Theorem 3.1 when , , and " is worth considering further.
Example 3.3.
Let
where . By Theorem 3.1, we obtain
If we start the iterative procedure given in the proof of (iii) of Theorem 3.1, after five iterations, we obtain
On the other hand, by Theorem A, one only can have
The following examples show the advantages of Theorems 3.1 over Theorem A in analyzing the Schur stability of difference equations (discretetime systems).
Example 3.4.
Let the characteristic polynomial of a difference equation (discretetime system) be given by
where . Then by Theorem 3.1, we get , which implies that all zeros of lie in the open unit disk, that is, this system is Schur stable. However, by Theorem A, one has
So Theorem A cannot guarantee the stability of such a system.
Example 3.5.
Suppose the characteristic polynomial of a difference equation (discretetime system) is given by
where . Then by Theorem 3.1, we have , which implies that all zeros of are outside the open unit disk, namely, such a system is instable. By Theorem A, one has
which cannot determine the instability of this system.
Example 3.6.
Consider the following characteristic polynomial of a difference equation (discretetime system):
In this example,
Consequently, such a difference equation (discretetime system) is Schur stable.
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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).
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Keywords
 Difference Equation
 Characteristic Equation
 Positive Root
 Polynomial Equation
 Characteristic Polynomial