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:

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.

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.