Stable polynomial

A polynomial is said to be stable if either:

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• all its roots lie in the left half-plane, or
• all its roots lie in the open unit disk.

The first condition defines Hurwitz (or continuous-time) stability and the second one Schur (or discrete-time) stability. Stable polynomials arise in various mathematical fields, including control theory. Indeed, a linear, time-invariant system (see LTI system theory) is said to be BIBO stable iff the denominator of its transfer function is stable. Since we consider LTI systems, the transfer function is always a rational function (i.e. a quotient between two polynomials). The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. For compactness, such stable polynomials are sometimes called Hurwitz polynomials and Schur polynomials.

Related Topics:
Hurwitz - Continuous-time - Schur - Discrete-time - Control theory - Time-invariant system - LTI system theory - BIBO stable - Iff - Denominator - Transfer function - Rational function - Quotient - Hurwitz polynomial - Schur polynomial

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Stable polynomial: Properties ►