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
Q ( z ) = ( z − 1 ) d P ( z + 1 z − 1 ) {\displaystyle Q(z)=(z-1)^{d}P\left({{z+1} \over {z-1}}\right)}
obtained after the Möbius transformation z ↦ z + 1 z − 1 {\displaystyle 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 ) ≠ 0 {\displaystyle 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 + ⋯ + a n z n {\displaystyle f(z)=a_{0}+a_{1}z+\cdots +a_{n}z^{n}} with (real) coefficients such that
a n > a n − 1 > ⋯ > a 0 > 0 , {\displaystyle 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.
Examples
4 z 3 + 3 z 2 + 2 z + 1 {\displaystyle 4z^{3}+3z^{2}+2z+1} is Schur stable because it satisfies the sufficient condition;
z 10 {\displaystyle z^{10}} is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;
z 2 − z − 2 {\displaystyle z^{2}-z-2} is not Hurwitz stable (its roots are −1 and 2) because it violates the necessary condition;
z 2 + 3 z + 2 {\displaystyle z^{2}+3z+2} is Hurwitz stable (its roots are −1 and −2). The polynomial z 4 + z 3 + z 2 + z + 1 {\displaystyle 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 ( 2 π k 5 ) + i sin ( 2 π k 5 ) , k = 1 , … , 4 . {\displaystyle z_{k}=\cos \left({{2\pi k} \over 5}\right)+i\sin \left({{2\pi k} \over 5}\right),\,k=1,\ldots ,4\,.}
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