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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 Cross Gramian ： ウィキペディア英語版 Cross Gramian In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is. For the stable time-invariant linear system :$\backslash dot\; =\; A\; x\; +\; B\; u\; \backslash ,$ :$y\; =\; C\; x\; \backslash ,$ the cross Gramian is defined as: :$W\_X\; :=\; \backslash int\_0^\backslash infty\; e^\; BC\; e^\; dt\; \backslash ,$ and thus also given by the solution to the Sylvester equation: :$A\; W\_X\; +\; W\_X\; A\; =\; -BC\; \backslash ,$ The triple $(A,B,C)$ is controllable and observable if and only if the matrix $W\_X$ is nonsingular, (i.e. $W\_X$ has full rank, for any $t\; >\; 0$). If the associated system $(A,B,C)$ is furthermore symmetric, such that there exists a transformation $J$ with :$AJ\; =\; JA^T\; \backslash ,$ :$B\; =\; JC^T\; \backslash ,$ then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:〔''(Cross Gramian )'' On the structure of balanced and other principal representations of SISO systems '' by K.V. Fernando and H. Nicholson; IEEE Transactions on Automatic Control; 1983〕 :$|\backslash lambda(W\_X)|\; =\; \backslash sqrt.\; \backslash ,$ Thus the direct truncation of the singular value decomposition of the cross Gramian allows model order reduction (see ()) without a balancing procedure as opposed to balanced truncation. == Note == The cross Gramian is also referred to by $W\_$. 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「Cross 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.