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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 Tensor product model transformation ： ウィキペディア英語版 Tensor product model transformation In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam for quasi-LPV (qLPV) control theory. It transforms a function (which can be given via closed formulas or neural networks, fuzzy logic, etc.) into TP function form if such a transformation is possible. If an exact transformation is not possible, then the method determines a TP function that is an approximation of the given function. Hence, the TP model transformation can provide a trade-off between approximation accuracy and complexity. A free MATLAB implementation of the TP model transformation can be downloaded at () or at MATLAB Central (). A key underpinning of the transformation is the higher-order singular value decomposition.〔 Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridging between identification and polytopic systems theories. The TP model transformation is uniquely effective in manipulating the convex hull of polytopic forms, and, as a result has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness in modern LMI based control theory. Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality. Further details on the control theoretical aspects of the TP model transformation can be found here: TP model transformation in control theory. The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", on which further information can be found here. It has been proved that the TP model transformation is capable of numerically reconstructing this HOSVD based canonical form.〔 Thus, the TP model transformation can be viewed as a numerical method to compute the HOSVD of functions, which provides exact results if the given function has a TP function structure and approximative results otherwise. The TP model transformation has recently been extended in order to derive various types of convex TP functions and to manipulate them. This feature has led to new optimization approaches in qLPV system analysis and design, as described here: TP model transformation in control theory. ==Definitions== ;Finite element TP function: A given function $f(\backslash in\; R^N$, is a TP function if it has the structure: :: $f(\backslash mathbf)=\backslash sum\_^\; \backslash sum\_^\; \backslash ldots\; \backslash sum\_^\; \backslash prod\_^N\; w\_(x\_n)\; s\_,$ that is, using compact tensor notation (using the tensor product operation $\backslash otimes$ of ): :: $f(\backslash mathbf)=\backslash mathcal\backslash mathop\_^N\backslash mathbf\_n(x\_n),$ where core tensor $\backslash mathcal\backslash in\; \backslash mathcal^$ is constructed from $s\_$, and row vector $\backslash mathbf\_n(x\_n),\; (n=1\; \backslash ldots\; N)$ contains continuous univariate weighting functions $w\_(x\_n),(i\_n=1\; \backslash ldots\; I\_n)$. The function $w\_(x\_n)$ is the $i\_n$-th weighting function defined on the $n$-th dimension, and $x\_n$ is the $n$-the element of vector $\backslash mathbf$. Finite element means that $I\_n$ is bounded for all $n$. For qLPV modelling and control applications a higher structure of TP functions are referred to as TP model. ;Finite element TP model (TP model in short): This is a higher structure of TP function: :: $\backslash mathcal(\backslash mathbf)=\backslash mathcal\backslash boxtimes\_^N\backslash mathbf\_n(x\_n).$ Here $\backslash mathcal=\backslash mathcal(\backslash in\; \backslash mathcal^$, thus the size of the core tensor is $\backslash mathcal\backslash in\; \backslash mathcal^$. The product operator $\backslash boxtimes$ has the same role as $\backslash otimes$, but expresses the fact that the tensor product is applied on the $L\_1\backslash times\; L\_2\backslash times\; ...\; \backslash times\; L\_O$ sized tensor elements of the core tensor $\backslash mathcal$. Vector $\backslash mathbf$ is an element of the closed hypercube $\backslash Omega=()\backslash times()\backslash times\; ...\; \backslash times()\backslash subset\; R^N$. ;Finite element convex TP function or model: A TP function or model is convex if the wighting functions hold: :: $\backslash forall\; n\; :\; \backslash sum\_^\; w\_(x\_n)\; =\; 1$ and $w\_(x\_n)\; \backslash in\; ()\; .$ This means that $f(\backslash mathbf)$ is inside the convex hull defined by the core tensor for all $\backslash mathbf\; \backslash in\; \backslash Omega$. ;TP model transformation: Assume a given TP model $\backslash mathcal\; =\; \backslash mathcal(\backslash mathbf)$, where $\backslash mathbf\backslash in\; \backslash Omega\; \backslash subset\; R^N$, whose TP structure maybe unknown (e.g. it is given by neural networks). The TP model transformation determines its TP structure as :: $\backslash mathcal(\backslash mathbf)=\backslash mathcal\backslash boxtimes\_^N\backslash mathbf\_n(x\_n)$, namely it generates the core tensor $\backslash mathcal$ and the weighting functions $\backslash mathbf\_n(x\_n)$ for all $n=1\; \backslash ldots\; N$. Its free MATLAB implementation is downloadable at () or at MATLAB Central (). If the given $\backslash mathcal(\backslash mathbf)$ does not have TP structure (i.e. it is not in the class of TP models), then the TP model transformation determines its approximation:〔 :: $\backslash mathcal(\backslash mathbf)\; \backslash approx\; \backslash mathcal\backslash boxtimes\_^N\backslash mathbf\_n(x\_n),$ where trade-off is offered by the TP model transformation between complexity (number of components in the core tensor or the number of weighting functions) and the approximation accuracy. The TP model can be generated according to various constrains. Typical TP models generated by the TP model transformation are: * HOSVD canonical form of TP functions or TP model (qLPV models), * Various kinds of TP type polytopic form or convex TP model forms (this advantage is used in qLPV system analysis and design). 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「Tensor product model transformation」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.