A New Asymptotic Stability Criterion for Linear Discrete-Time Systems

In this brief, a new criterion to determine whether the roots of a polynomial
$p(z^{-1})$
with real coefficients are strictly inside the unit circle is presented. It is based on representing the polynomial in a form corresponding to the characteristic polynomial of a closed-loop systems in which all-pass transfer functions are included in both direct chain and feedback loop. The novel and simple criterion for assessing the asymptotic stability of linear discrete-time systems is proved applying the small-gain theorem in the discrete-time domain.