Stable polynomial

<p>In the context of the <a href="/facts/Characteristic_equation_(calculus)/tBqj8HUJ">characteristic polynomial</a> of a <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> or <a href="/facts/Difference_equation/JolXC71M">difference equation</a>,  a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> is said to be stable if either:
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<ul><li>all its roots lie in the <a href="/facts/Open_set/QfTCIqMu">open</a> left <a href="/facts/Half-plane/AGknBgnw">half-plane</a>, or</li>
<li>all its roots lie in the <a href="/facts/Open_set/QfTCIqMu">open</a> <a href="/facts/Unit_disk/phrUu7yC">unit disk</a>.</li></ul>
<p>The first condition provides <a href="/facts/Stability_theory/Su5sHnxs">stability</a> for <a href="/facts/Continuous-time/RR8wpS9e">continuous-time</a> linear systems, and the second case relates to stability
of <a href="/facts/Discrete-time/RR8wpS9e">discrete-time</a> linear systems. A polynomial with the first property is called at times a <a href="/facts/Hurwitz_polynomial/t2Ys4jtI">Hurwitz-stable polynomial</a> and with the second property a Schur-stable polynomial. Stable polynomials arise in <a href="/facts/Control_theory/aIuERpn6">control theory</a> and in mathematical theory
of differential and difference equations. A linear, <a href="/facts/Time-invariant_system/NpWUk0YG">time-invariant system</a> (see <a href="/facts/LTI_system_theory/kEmj0QDC">LTI system theory</a>) is said to be <a href="/facts/BIBO_stability/RxD8h2f5">BIBO stable</a> 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 <a href="/facts/Stability_criterion/AMSTVwWT">stability criteria</a>.
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Stable polynomial open-in-new

Stable polynomial