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- In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. 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 [1] or an old version of the toolbox is available at MATLAB Central [2]. 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 at TP model transformation in control theory. (en)
- 张量积模型轉換（tensor product model transformation）是由Baranyi和Yam提出的數學模型，是高階奇異值分解的重要概念。可以將函數（可能是解析解，或是由類神經網路或模糊逻辑所得的函數）轉換為張量積（TP）函數型式。假若找不到對應的轉換，此方式可以找到近似的張量積函數。因此張量積模型變換可以在精確度以及複雜度之間作一取捨。支撐此轉換的主要概念是（HOSVD）。 張量積模型變換除了是函數的轉換外，也是qLPV（準線性變參數控制）為基礎控制中的新概念，是識別以及多胞形（polytopic）系統理論之間的串接的重要工具。張量積模型變換在凸包多胞形式的處理上非常的有效，已有結果證明在現在以LMI（線性矩陣不等式）為基礎的控制理論中，凸包多胞形式的處理是達到最佳解以及降低保守性（conservativeness）的必要及關鍵步驟。因此，張量積模型變換在數學概念上是轉換，但在控制理論上確立了概念上的新方向，也奠定了有關最佳化的新研究方向。進一步有關張量積模型變換的理論層面說明可以參考。 張量積模型變換也激發了「張量積函數的HOSVD正則形式」（HOSVD canonical form of TP functions）的定義，進一步的資料在。已經確認張量積模型變換可以在數值形式重現基礎的正規型式。因此，可以將張量積模型變換視為是計算函數HOSVD的數值方法，若該函數存在張量積函數結構，可以找到其結構，不然，也可以找到近似解。 近來張量積模型變換已延伸到推導不同型式的凸張量積函數，並且進行對應的處理。此特點已為qLPV系統分析及設計提供了新的最佳化方式。 (zh)
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- In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. 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. (en)
- 张量积模型轉換（tensor product model transformation）是由Baranyi和Yam提出的數學模型，是高階奇異值分解的重要概念。可以將函數（可能是解析解，或是由類神經網路或模糊逻辑所得的函數）轉換為張量積（TP）函數型式。假若找不到對應的轉換，此方式可以找到近似的張量積函數。因此張量積模型變換可以在精確度以及複雜度之間作一取捨。支撐此轉換的主要概念是（HOSVD）。 張量積模型變換除了是函數的轉換外，也是qLPV（準線性變參數控制）為基礎控制中的新概念，是識別以及多胞形（polytopic）系統理論之間的串接的重要工具。張量積模型變換在凸包多胞形式的處理上非常的有效，已有結果證明在現在以LMI（線性矩陣不等式）為基礎的控制理論中，凸包多胞形式的處理是達到最佳解以及降低保守性（conservativeness）的必要及關鍵步驟。因此，張量積模型變換在數學概念上是轉換，但在控制理論上確立了概念上的新方向，也奠定了有關最佳化的新研究方向。進一步有關張量積模型變換的理論層面說明可以參考。 近來張量積模型變換已延伸到推導不同型式的凸張量積函數，並且進行對應的處理。此特點已為qLPV系統分析及設計提供了新的最佳化方式。 (zh)
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- Tensor product model transformation (en)
- 張量積模型變換 (zh)
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