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, orall 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.
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 ( z + 1 z − 1 ) obtained after the Möbius transformation z ↦ z + 1 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 ) ≠ 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 + ⋯ + a n z n with (real) coefficients such that: a n > a n − 1 > ⋯ > 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. 4 z 3 + 3 z 2 + 2 z + 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 + 3 z + 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 unityNote here thatIt 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.
