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-stable polynomial and with the second property a Schur-stable 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

::<math> Q(z)=(z-1)^d P\left(\right)</math>

:obtained after the Möbius transformation <math>z \mapsto </math> which maps the left half-plane to the open unit disc: P is Schur stable if and only if Q is Hurwitz stable and <math> P(1)\neq 0</math>. 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 <math>f(z) = a_0+a_1 z+\cdots+a_n z^n</math> with (real) coefficients such that

::<math> a_n>a_{n-1}>\cdots>a_0 > 0,</math>

: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.

Examples

  • <math> 4z^3+3z^2+2z+1 </math> is Schur stable because it satisfies the sufficient condition;
  • <math> z^{10}</math> is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;
  • <math> z^2-z-2</math> is not Hurwitz stable (its roots are −1 and 2) because it violates the necessary condition;
  • <math> z^2+3z+2 </math> is Hurwitz stable (its roots are −1 and −2).
  • The polynomial <math> z^4+z^3+z^2+z+1 </math> (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth roots of unity

::<math> z_k=\cos\left(