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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 Controllability Gramian ： ウィキペディア英語版 Controllability Gramian In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system :$\backslash dot\; =\; A\; x\; +\; B\; u$ if all eigenvalues of $A$ have negative real part, then the unique solution $W\_c$ of the Lyapunov equation :$A\; W\_c\; +\; W\_c\; A^T\; =\; -BB^T$ is positive definite if and only if the pair $(A,\; B)$ is controllable. $W\_c$ is known as the ''controllability Gramian'' and can also be expressed as :$W\_c\; =\; \backslash int\backslash limits\_0^\backslash infty\; e^\; B\; B^T\; e^\; \backslash ;\; d\backslash tau$ A related matrix used for determining controllability is :$W\_c(t\_0,\; t\_1)\; =\; \backslash int\_^\; e^\; BB^T\; e^\; \backslash ;\; d\backslash tau$ The pair $(A,B)$ is controllable if and only if the matrix $W\_c(t\_0,\; t\_1)$ is nonsingular, for any $t\_1\; >\; t\_0$.〔''(Controllability Gramian )'' Lecture notes to ''ECE 521 Modern Systems Theory'' by Professor A. Manitius, ECE Department, George Mason University.〕 A physical interpretation of the controllability Gramian is that if the input to the system is white gaussian noise, then $W\_c$ is the covariance of the state. Linear time-variant state space models of form :$\backslash dot(t)\; =\; A(t)\; x(t)\; +\; B(t)\; u(t)$, :$y(t)\; =\; C(t)\; x(t)\; +\; D(t)\; u(t)$ are controllable in an interval $()$ if and only if the Gramian matrix $W\_c(t\_0,\; t\_1)$ is nonsingular, where : $W\_c(t\_0,\; t\_1)\; =\; \backslash int\backslash limits\_^\; \backslash Phi(t\_0,\backslash tau)B(\backslash tau)B^T(\backslash tau)\backslash Phi^T(t\_0,\backslash tau)\; \backslash ;\; d\backslash tau$ == See also == * Controllability * Observability Gramian * Gramian matrix * Minimum energy control 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「Controllability Gramian」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.