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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
|title = TP model transformation as a way to LMI based controller design
|journal = IEEE
|date=April 2004
|volume = 51
|number = 2
|pages = 387–400
|doi=10.1109/tie.2003.822037
|s2cid = 7957799
}}</ref><ref name="springer.com">{{cite book |doi=10.1007/978-3-319-19605-3|title=TP-Model Transformation-Based-Control Design Frameworks|year=2016|last1=Baranyi|first1=Péter|isbn=978-3-319-19604-6}}</ref><ref name="P. Baranyi 2014, pp. 934-948">{{cite journal |doi=10.1109/TFUZZ.2013.2278982|title=The Generalized TP Model Transformation for T–S Fuzzy Model Manipulation and Generalized Stability Verification|journal=IEEE Transactions on Fuzzy Systems|volume=22|issue=4|pages=934–948|year=2014|last1=Baranyi|first1=Peter|doi-access=free}}</ref><ref name=compind>{{cite journal
|
|author2 = D. Tikk
|author3 = Y. Yam
|author4 = R. J. Patton
|title = From Differential Equations to PDC Controller Design via Numerical Transformation
|journal = Computers in Industry
|year = 2003
|volume = 51
|issue = 3
|pages = 281–297
|doi=10.1016/s0166-3615(03)00058-7
}}</ref><ref name=ykc00>{{cite
|author1=P. Baranyi
|
|
|year = 2013
|pages = 240
|isbn = 978-1-43-981816-9
}}</ref> as key concept for
|
|author2 = P. Baranyi
|author3 = R. J. Patton
|title = Approximation Properties of TP Model Forms and its Consequences to TPDC Design Framework
|journal = Asian Journal of Control
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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 [
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
The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions",<ref name=canon1>{{cite
|
|author2 = L. Szeidl
|author3 = P. Várlaki
|author4 = Y. Yam
|title = Definition of the HOSVD-based canonical form of polytopic dynamic models
|
|date=July 3–5, 2006
|pages = 660–665
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}}</ref> on which further information can be found [[HOSVD based canonical form of TP functions and qLPV models|here]]. It has been proved that the TP model transformation is capable of numerically reconstructing this [[HOSVD]] based canonical form.<ref name=canon3 /> 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.<ref
==Definitions==
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that is, using compact tensor notation (using the [[tensor product]] operation <math>\otimes</math> of <ref name=Lath00>{{cite journal
|
|author2 = Bart De Moor
|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 = 1253–1278
|doi=10.1137/s0895479896305696
|citeseerx =10.1.1.3.4043
}}</ref> ):
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Here <math>\mathcal{Y}=\mathcal{F}({\mathbf{x}})</math> is a tensor as <math>\mathcal{Y}\in \mathcal{R}^{L_1\times L_2\times \ldots L_O}</math>, thus the size of the core tensor is <math>\mathcal{S}\in \mathcal{R}^{I_1\times I_2\times \ldots \times I_N \times L_1\times L_2\times ... \times L_O}</math>. The product operator <math> \boxtimes </math> has the same role as <math> \otimes </math>, but expresses the fact that the tensor product is applied on the <math> L_1\times L_2\times ... \times L_O</math> sized tensor elements of the core tensor <math>\mathcal{S}</math>. Vector <math>\mathbf{x} </math> is an element of the closed hypercube <math>\Omega=[a_1,b_1]\times[a_2,b_2]\times ... \times[a_N,b_N]\subset R^N</math>.
;Finite element convex TP function or model: A TP function or model is convex if the
:: <math> \forall n : \sum_{i_n=1}^{I_n} w_{n,i_n}(x_n) = 1 </math> and <math>w_{n,i_n}(x_n) \in [0,1] .</math>
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:: <math>\mathcal{F}(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(x_n) </math>,
namely it generates the core tensor <math>\mathcal{S} </math> and the weighting functions <math> \mathbf{w}_n(x_n) </math> for all <math> n=1 \ldots N </math>. Its free [[MATLAB]] implementation is downloadable at [https://web.archive.org/web/20120229061018/http://tptool.sztaki.hu/] or at [[MATLAB]] Central [http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool].
If the given <math>\mathcal{F}(\mathbf{x})</math> does not have TP structure (i.e. it is not in the class of TP models), then the TP model transformation determines its approximation:<ref name=ykc01 />
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* It generates the HOSVD-based canonical form of TP functions,<ref name=canon1 /> which is a unique representation. It was proven by Szeidl <ref name=canon3>{{cite journal
|author1=L. Szeidl |author2=P. Várlaki
|
|journal = Journal of Advanced Computational Intelligence and Intelligent Informatics
|year = 2009
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|number = 1
|pages = 52–60
|doi=10.20965/jaciii.2009.p0052
}}</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:▼
|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
:* 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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==References==
<references/>
Baranyi, P. (2018). Extension of the Multi-TP Model Transformation to Functions with Different Numbers of Variables. Complexity, 2018.
==External links==
* [https://drive.google.com/drive/folders/1f4yZsIVv2_QLJg9o898ehzqa7j3EQ68j?usp=sharing TPtoolBoxMATLAB]
[[Category:Control theory]]
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