Tensor product model transformation: Difference between revisions

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{{Short description|Key concept in higher-order singular value decomposition of functions}}

In mathematics, the '''[[tensor product]]''' ('''TP''') '''model transformation''' was proposed by Baranyi and Yam<ref name=Baranyi04>{{cite journal

|author = P. Baranyi

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|pages = 221–331

|doi=10.1111/j.1934-6093.2007.tb00410.x

|s2cid = 121716136

}}</ref>

A free [[MATLAB]] implementation of the TP model transformation can be downloaded at [https://drive.google.com/filedrive/dfolders/19NkG4m7Fwo6HXM4vVPh4srscbbq1In3S2ebT-eSbc/viewknwDqWaS4dLFarKITqjcBqq?usp=drive_link] or an old version of the toolbox is aviableavailable at [[MATLAB]] Central [http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool]. A key underpinning of the transformation is the [[higher-order singular value decomposition]].<ref name=Lath00 />

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<ref>A.Szollosi, and Baranyi, P. (2016). Influence of the Tensor Product model representation of qLPV models on the feasibility of Linear Matrix Inequality. Asian Journal of Control, 18(4), 1328-1342</ref><ref>A. Szöllősi and P. Baranyi: „Improved control performance of the 3‐DoF aeroelastic wing section: a TP model based 2D parametric control performance optimization.” in Asian Journal of Control, 19(2), 450-466. / 2017</ref><ref name="springer.com"/> 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]].

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|author3 = Joos Vandewalle

|title = A Multilinear Singular Value Decomposition

|journal = SIAM Journal on Matrix Analysis and Applications

|year = 2000

|volume = 21

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|pages = 52–60

|doi=10.20965/jaciii.2009.p0052

|doi-access =free

}}</ref> that the TP model transformation numerically reconstructs the [[HOSVD]] of functions. This form extracts the unique structure of a given TP function in the same sense as the HOSVD does for tensors and matrices, in a way such that:

:* the number of weighting functions are minimized per dimensions (hence the size of the core tensor);

:* the weighting functions are one variable functions of the parameter vector in an orthonormed system for each parameter (singular functions);

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==External links==

* [https://drive.google.com/drive/folders/1f4yZsIVv2_QLJg9o898ehzqa7j3EQ68j?usp=sharing TPtoolBoxMATLAB]

* [https://web.archive.org/web/20120229061018/http://tptool.sztaki.hu/ TP Tool – home page]

[[Category:Control theory]]