Bistritz stability criterion

In signal processing and control theory, the Bistritz criterion is a simple method to determine whether a discrete, linear, time-invariant (LTI) system is stable proposed by Yuval Bistritz. Stability of a discrete LTI system requires that its characteristic polynomial

${\displaystyle D_{n}(z)=d_{0}+d_{1}z+d_{2}z^{2}+\cdots +d_{n-1}z^{n-1}+d_{n}z^{n}}$

(obtained from its difference equation, its dynamic matrix, or appearing as the denominator of its transfer function) is a stable polynomial, where ${\displaystyle d_{n}>0}$ and ${\displaystyle D_{n}(z)}$ is said to be stable if all its roots (zeros) are inside the unit circle, viz.

${\displaystyle |z_{k}|<1,k=1,\dots ,n}$,

where ${\displaystyle D_{n}(z)=d_{n}\prod _{k=1}^{n}(z-z_{k})}$. The test determines whether ${\displaystyle D_{n}(z)}$ is stable algebraically (i.e. without numerical determination of the zeros). The method also solves the full zero location (ZL) problem. Namely, it can count the number of inside the unit-circle (IUC) zeros (${\displaystyle |z_{k}|<1}$), on the unit-circle zeros (UC) zeros (${\displaystyle |z_{k}|=1}$) and outside the unit-circle (OUC) zeros (${\displaystyle |z_{k}|>1}$) for any real or complex polynomial. The Bistritz test is the discrete equivalent of Routh criterion used to test stability of continuous LTI systems. This title was introduced soon after its presentation. It has been also recognized to be more efficient than previously available stability tests for discrete systems like the Schur–Cohn and the Jury test.

In the following, the focus is only on how to test stability of a real polynomial. However, as long as the basic recursion needed to test stability remains valid, ZL rules are also brought.