Annular Bounds for Polynomial Zeros and Schur Stability of Difference Equations
© K. Li and J. Liang. 2011
Received: 7 October 2010
Accepted: 30 October 2010
Published: 21 November 2010
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.
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  recently. As a continuation of this work and our paper , 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.
Basic notations are as follows.
3. Main Result
The following result is established in .
Theorem A (see ).
which imply that (3.1) is hold. So (i) is proved.
This means that (iii) is true.
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.
The following examples show the advantages of Theorems 3.1 over Theorem A in analyzing the Schur stability of difference equations (discrete-time systems).
So Theorem A cannot guarantee the stability of such a system.
which cannot determine the instability of this system.
Consequently, such a difference equation (discrete-time system) is Schur stable.
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).
- Boese FG, Luther WJ: A note on a classical bound for the moduli of all zeros of a polynomial. IEEE Transactions on Automatic Control 1989,34(9):998-1001. 10.1109/9.35817MathSciNetView ArticleMATHGoogle Scholar
- Cargo GT, Shisha O: Zeros of polynomials and fractional order differences of their coefficients. Journal of Mathematical Analysis and Applications 1963, 7: 176-182. 10.1016/0022-247X(63)90046-5MathSciNetView ArticleMATHGoogle Scholar
- Gardner RB, Govil NK: On the location of the zeros of a polynomial. Journal of Approximation Theory 1994,78(2):286-292. 10.1006/jath.1994.1078MathSciNetView ArticleMATHGoogle Scholar
- Liang J, Li K: A new stability criterion for polynomials with the quasi-critical condition. International Journal of Nonlinear Sciences and Numerical Simulation 2008,9(3):289-292. 10.1515/IJNSNS.2008.9.3.289MathSciNetView ArticleGoogle Scholar
- Marden M: Geometry of Polynomials, Mathematical Surveys, No. 3. 2nd edition. American Mathematical Society, Providence, RI, USA; 1966:xiii+243.Google Scholar
- Milovanovic GV, Milovanovic DS, Rassias TM: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientic, Singapore; 1994.MATHGoogle Scholar
- Rahman QI, Schmeisser G: Analytic Theory of Polynomials. Oxford University Press, Oxford, UK; 2005:xiv+742.MATHGoogle Scholar
- Sun Y-J: New result for the annular bounds of complex-coefficient polynomial zeros. IEEE Transactions on Automatic Control 2004,49(5):813-814. 10.1109/TAC.2004.828323View ArticleMathSciNetGoogle Scholar
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.