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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 Kalman decomposition ： ウィキペディア英語版 Kalman decomposition In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces. == Notation == The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is : $\backslash dot(t)\; =\; Ax(t)\; +\; Bu(t)$ : $\backslash ,\; y(t)\; =\; Cx(t)\; +\; Du(t)$ where : $\backslash ,\; x$ is the "state vector", : $\backslash ,\; y$ is the "output vector", : $\backslash ,\; u$ is the "input (or control) vector", : $\backslash ,\; A$ is the "state matrix", : $\backslash ,\; B$ is the "input matrix", : $\backslash ,\; C$ is the "output matrix", : $\backslash ,\; D$ is the "feedthrough (or feedforward) matrix". Similarly, a discrete-time linear control system can be described as : $\backslash ,\; x(k+1)\; =\; Ax(k)\; +\; Bu(k)$ : $\backslash ,\; y(k)\; =\; Cx(k)\; +\; Du(k)$ with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices $\backslash ,\; (A,\; B,\; C,\; D)$. Let the order of the system be $\backslash ,\; n$. Then, the Kalman decomposition is defined as a transformation of the tuple $\backslash ,\; (A,\; B,\; C,\; D)$ to $\backslash ,\; (\backslash hat,\; \backslash hat,\; \backslash hat,\; \backslash hat)$ as follows: : $\backslash ,\; \}AT$ : $\backslash ,\; \}B$ : $\backslash ,\; \}\; =\; D$ $\backslash ,\; T$ is an $\backslash ,\; n\; \backslash times\; n$ invertible matrix defined as : where * $\backslash ,\; T\_$ is chosen so that the columns of $\backslash ,\; \backslash begin\; T\_\backslash end$ are a basis for the reachable subspace. * $\backslash ,\; T\_\; T\_\}\backslash end$ are a basis for the unobservable subspace. * $\backslash ,\; T\_$ is chosen so that is invertible. By construction, the matrix $\backslash ,\; T$ is invertible. It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then $\backslash ,\; T\; =\; T\_$, making the other matrices zero dimension. 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「Kalman decomposition」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.