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|volume = 51
|pages = 281–297
}}</ref>
|author = P. Baranyi, Y. Yam and P. Várlaki
|title = Tensor Product model transformation in polytopic model-based control
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|pages = 240
|isbn = 978-1-43-981816-9 (IN PRINT)
}}</ref> for qLPV control theories. It transforms a function (which can be given via [[Closed-form expression|closed formulas]] or [[neural network]]s, [[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
|author = D. Tikk, P.Baranyi, R. J. Patton
|title = Approximation Properties of TP Model Forms and its
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|year = 2007
|pages = 221–331
}}</ref>
A free [[MATLAB]] implementation of the TP model transformation can be downloaded at [http://tptool.sztaki.hu/] or 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]]
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 theories]].
The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions"
|author = P. Baranyi and L. Szeidl and P. Várlaki and Y. Yam
|title = Definition of the HOSVD-based canonical form of polytopic dynamic models
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|address = Budapest, Hungary
|month = July 3–5
}}</ref>
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 theories]].
==Definitions==
;Finite element TP function: A given function <math>\mathbf{f}({\mathbf{x}})</math>, where <math>\mathbf{x}\in R^N</math>, is a TP function if it has the structure:
:: <math>\mathbf{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
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namely it generates core tensor <math>\mathcal{S} </math> and weighting functions of <math> \mathbf{w}_n(x_n) </math> for all <math> n=1 \ldots N </math>. Its free [[MATLAB]] implementation is downloadable at [http://tptool.sztaki.hu/] or at [[MATLAB]] Central [http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool].
If the given model does not have TP structure (i.e. it is not in the class of TP functions), then the TP model transformation determines its approximation
:: <math>\mathbf{F}(\mathbf{x}) \approx \mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(x_n),</math>
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==Properties of the TP model transformation==
* It is a non-heuristic and tractable numerical method firstly proposed in control theory.<ref name=
* It transforms the given function into finite element TP structure. If this structure does not exist, then the transformation gives an approximation under the constrain on the number of elements.
* 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 that can be carried out in a routine fashion.
* It generates the HOSVD-based canonical form of TP functions
|author = L. Szeidl and P. Várlaki
|title = HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems
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