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.
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 unity
Note here that
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.