On the roots of a real polynomial inside the unit circle and a stability criterion for linear discrete systems

Abstract

Necessary and sufficient algebraic conditions for the roots of a real polynomial to lie inside the unit circle are given in table form. In this form, the constraints are obtained only by evaluation of second-order determinants. The connection and identity existing between the stability criterion established in this paper and that of a previously obtained criterion1 are discussed. The usefulness of the table may be found in those cases where the coefficients of the real polynomial are given in numbers. It is similar to Routh's table obtained for the continuous case. Conditions on the numbers of the roots inside, outside, or on the unit circle are also discussed, within the cases when the determinants are zero or non-zero. Also, necessary and sufficient conditions are formulated for all the roots to be inside a circle of radius σ less than unity, and also the conditions when the roots are to lie between plus and minus unity in the z plane. Various examples of discrete systems are presented which illustrate the applications of the new stability criterion as well as the other conditions formulated in this paper. In concluding the paper, various analytical stability criteria applied to linear discrete systems are enumerated and compared, with emphasis on the advantageous applications of each.