A Generalized Markov Stability Criterion for Linear Systems

It is said to be Hurwitz stable if the real part of all zeros of the polynomial (1) are negative; It is said to be Schur stable if all zeros of the polynomial (1) lie within unit circle. By matrix theory, Gantmacher (1964), algebraic criteria on Hurwitz stability for continuous linear systems can be directly derived from calculation of a Cauchy index of a rational fraction ) (x R , namely ) (x R I +∞ ∞ − , in the real axis. For the discrete-time systems, Nour-Eldin (1971) has shown that a linear system is Schur stable if and only if the Cauchy index of a rational fraction in the interval ) 1 , 1 ( + − equals the order of the denominator. Inspired by Nour-Eldin's work, some reduced criteria on Schur stability using combination of Markov parameters with some linear relationships of the system coefficients have been presented by Anderson et al (1976, 1990).