# Stable polynomial

A polynomial is said to be stable if either:

The first condition provides stability for (or continuous-time) linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz polynomial and with the second property a Schur polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is determined by applying any one of several stability criteria.

## Properties

• To test if a given polynomial P (of degree d) is Schur stable, it suffices to apply this theorem to the transformed polynomial
$Q(z)=(z-1)^d P\left({{z+1}\over{z-1}}\right)$

obtained after the Möbius transformation $z \mapsto {{z+1}\over{z-1}}$ which maps the left half-plane to the open unit disc: P is Schur stable if and only if Q is Hurwitz stable and $P(1)\neq 0$. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the Jury test or the Bistritz test.

• Necessary condition: a Hurwitz stable polynomial (with real coefficients) has coefficients of the same sign (either all positive or all negative).
• Sufficient condition: a polynomial $f(z)=a_0+a_1 z+\cdots+a_n z^n$ with (real) coefficients such that:
$a_n>a_{n-1}>\cdots>a_0>0,$

is Schur stable.

• Product rule: Two polynomials f and g are stable (of the same type) if and only if the product fg is stable.

## Examples

• $4z^3+3z^2+2z+1$ is Schur stable because it satisfies the sufficient condition;
• $z^{10}$ is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;
• $z^2-z-2$ is not Hurwitz stable (its roots are -1,2) because it violates the necessary condition;
• $z^2+3z+2$ is Hurwitz stable (its roots are -1,-2).
• The polynomial $z^4+z^3+z^2+z+1$ (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth roots of unity
$z_k=\cos\left({{2\pi k}\over 5}\right)+i \sin\left({{2\pi k}\over 5}\right), \, k=1, \ldots, 4 \ .$
Note here that
$\cos({{2\pi}/5})={{\sqrt{5}-1}\over 4}>0.$
It is a "boundary case" for Schur stability because its roots lie on the unit circle. The example also shows that the necessary (positivity) conditions stated above for Hurwitz stability are not sufficient.