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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
}}</ref>▼
|s2cid = 7957799
<ref name=compind>{{cite journal▼
}}</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
|author1 = P. Baranyi
|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
|
|name-list-style=amp |title = Tensor Product model transformation in polytopic model-based control
|
|year = 2013
|pages = 240
|isbn = 978-1-43-981816-9
}}</ref> as key concept for
|
|author2 = P. Baranyi
|title = Approximation Properties of TP Model Forms and its▼
|author3 = R. J. Patton
▲|title = Approximation Properties of TP Model Forms and its Consequences to TPDC Design Framework
|journal = Asian Journal of Control
|volume = 9
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|year = 2007
|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 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]].
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
|
|
|pages = 660–665
|place = Budapest, Hungary
}}</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 name="P. Baranyi 2014, pp. 934-948"/> This feature has led to new optimization approaches in qLPV system analysis and design, as described
==Definitions==
;Finite element TP function: A given function <math>f({\mathbf{x}})</math>, where <math>\mathbf{x}\in R^N</math>, is a TP function if it has the structure:
:: <math>f(\mathbf{x})=\sum_{i_1=1}^{I_1} \sum_{i_2=1}^{I_2} \ldots \sum_{i_N=1}^{I_N} \prod_{n=1}^N w_{n,i_n}(x_n) s_{i_1,i_2,\ldots,i_N},</math>
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
|number = 4
|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>
This means that <math> f(\mathbf{x}) </math> is inside the convex hull defined by the core tensor for all <math> \mathbf{x} \in \Omega </math>.
;TP model transformation: Assume a given TP model <math>\mathcal{Y} = \mathcal{F}(\mathbf{x}) </math>, where <math>\mathbf{x}\in \Omega \subset R^N</math>, whose TP structure maybe unknown (e.g. it is given by neural networks). The TP model transformation determines its TP structure as
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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 can be executed uniformly (irrespective of whether the model is given in the form of analytical equations resulting from physical considerations, or as an outcome of soft computing based identification techniques (such as neural networks or fuzzy logic based methods, or as a result of a black-box identification), without analytical interaction, within a reasonable amount of time. Thus, the transformation replaces the analytical and in many cases complex and not obvious conversions to numerical, tractable, straightforward operations.
* 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
|
|name-list-style=amp |title = HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems
|journal = Journal of Advanced Computational Intelligence and Intelligent Informatics
|year = 2009
Line 108 ⟶ 122:
|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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:* introduces and defines the rank of the TP function by the dimensions of the parameter vector;
* The above point can be extended to TP models (qLPV models to determine the [[HOSVD]] based canonical
*The core step of the TP model transformation was extended to generate different types of convex TP functions or TP models (TP type polytopic qLPV models), in order to focus on the systematic (numerical and automatic) modification of the convex hull instead of developing new LMI equations for feasible controller design (this is the widely adopted approach). It is worth noting that both the TP model transformation and the LMI-based control design methods are numerically executable one after the other, and this makes the resolution of a wide class of problems possible in a straightforward and tractable, numerical way.
* The TP model transformation is capable of performing trade-off between complexity and accuracy of TP functions <ref name=ykc01 /> via discarding the higher-order singular values, in the same manner as the tensor HOSVD is used for complexity reduction.
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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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