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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stable polynomial ： ウィキペディア英語版 Stable polynomial 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. The first condition provides stability for (or 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 :$Q(z)=(z-1)^d\; P\backslash left(\backslash right)$ obtained after the Möbius transformation $z\; \backslash mapsto$ which maps the left half-plane to the open unit disc: ''P'' is Schur stable if and only if ''Q'' is Hurwitz stable and $P(1)\backslash neq\; 0$. 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 $f(z)=a\_0+a\_1\; z+\backslash cdots+a\_n\; z^n$ with (real) coefficients such that: :$a\_n>a\_>\backslash cdots>a\_0>0,$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stable polynomial」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.