Stable polynomial

In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either:

all its roots lie in the open left half-plane, or
all its roots lie in the open unit disk.

The first condition provides stability for 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

The Routh–Hurwitz theorem provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in the Routh–Hurwitz and Liénard–Chipart tests.
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.
Hadamard product: The Hadamard (coefficient-wise) product of two Hurwitz stable polynomials is again Hurwitz stable.[1]

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.