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:
- all its roots lie in the open left half-plane, or
- all its roots lie in the open unit disk.
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 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 is Schur stable, it suffices to apply this theorem to the transformed polynomial
- Necessary condition: a Hurwitz stable polynomial has coefficients of the same sign.
- Sufficient condition: a polynomial with coefficients such that
- Product rule: Two polynomials f and g are stable if and only if the product fg is stable.
- Hadamard product: The Hadamard product of two Hurwitz stable polynomials is again Hurwitz stable.
Examples
- is Schur stable because it satisfies the sufficient condition;
- is Schur stable but it does not satisfy the sufficient condition;
- is not Hurwitz stable because it violates the necessary condition;
- is Hurwitz stable.
- The polynomial is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth roots of unity
Stable matrices