Stable polynomial

Last updated October 26, 2024

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.
Hadamard product: The Hadamard (coefficient-wise) product of two Hurwitz stable polynomials is again Hurwitz stable.^[1]

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 and 2) because it violates the necessary condition;
$z^{2}+3z+2$ is Hurwitz stable (its roots are −1 and −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.

Stable matrices

Just as stable polynomials are crucial for assessing the stability of systems described by polynomials, stability matrices play a vital role in evaluating the stability of systems represented by matrices.

Hurwitz matrix

A square matrix A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part.

Schur matrix

Schur matrices is an analogue of the Hurwitz matrices for discrete-time systems. A matrix A is a Schur (stable) matrix if its eigenvalues are located in the open unit disk in the complex plane.

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In signal processing and control theory, the Bistritz criterion is a simple method to determine whether a discrete, linear, time-invariant (LTI) system is stable proposed by Yuval Bistritz. Stability of a discrete LTI system requires that its characteristic polynomial

In control system theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations.

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References

↑ Garloff, Jürgen; Wagner, David G. (1996). "Hadamard Products of Stable Polynomials Are Stable". Journal of Mathematical Analysis and Applications. 202 (3): 797–809. doi: 10.1006/jmaa.1996.0348 .

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[1] Garloff, Jürgen; Wagner, David G. (1996). "Hadamard Products of Stable Polynomials Are Stable". Journal of Mathematical Analysis and Applications. 202 (3): 797–809. doi: 10.1006/jmaa.1996.0348 .

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