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| In mathematics, the '''[[tensor product]] (TP) model transformation''' was proposed by Baranyi and Yam
| | If you present photography effectively, it helps you look much more properly at the globe around you. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. Should you have just about any issues with regards to where as well as how to make use of [http://dinky.in/?WordpressDropboxBackup459000 wordpress dropbox backup], you'll be able to email us in our own web site. You've got invested a great cope of time developing and producing up the topic substance. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. If you are looking for Hire Wordpress Developer then just get in touch with him. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>Website security has become a major concern among individuals all over the world. Automated deal feed integration option to populate your blog with relevant deals. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. If this is not possible you still have the choice of the default theme that is Word - Press 3. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| <ref name=Baranyi04>{{cite journal
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| |author = P. Baranyi
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| |title = TP model transformation as a way to LMI based controller design
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| |journal = IEEE Transaction on Industrial Electronics
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| |year = 2004
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| |volume = 51
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| |number = 2
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| |month = April
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| |pages = 387–400
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| }}</ref>
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| <ref name=compind>{{cite journal
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| |author = P. Baranyi and D. Tikk and Y. Yam and R. J. Patton
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| |title = From Differential Equations to PDC Controller Design via Numerical Transformation
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| |journal = Computers in Industry, Elsevier Science
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| |year = 2003
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| |volume = 51
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| |pages = 281–297
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| }}</ref><ref name=ykc00>{{cite journal
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| |author = P. Baranyi, Y. Yam and P. Várlaki
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| |title = Tensor Product model transformation in polytopic model-based control
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| |journal = Taylor&Francis, Boca Raton FL
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| |year = 2013
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| |pages = 240
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| |isbn = 978-1-43-981816-9 (IN PRINT)
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| }}</ref> for quasi-LPV (qLPV) control theory. 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.<ref name=ykc01>{{cite journal
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| |author = D. Tikk, P.Baranyi, R. J. Patton
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| |title = Approximation Properties of TP Model Forms and its
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| Consequences to TPDC Design Framework
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| |journal = Asian Journal of Control
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| |volume = 9
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| |number = 3
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| |year = 2007
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| |pages = 221–331
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| }}</ref>
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| 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]].<ref name=Lath00 />
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| 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]].
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| The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions",<ref name=canon1>{{cite book
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| |author = P. Baranyi and L. Szeidl and P. Várlaki and Y. Yam
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| |title = Definition of the HOSVD-based canonical form of polytopic dynamic models
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| |booktitle = 3rd International Conference on Mechatronics (ICM 2006)
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| |year = 2006
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| |pages = 660–665
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| |place = Budapest, Hungary
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| |month = July 3–5
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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.
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| 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 theory]].
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| ==Definitions==
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| ;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:
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| :: <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>
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| that is, using compact tensor notation (using the [[tensor product]] operation <math>\otimes</math> of <ref name=Lath00>{{cite journal
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| |author = Lieven De Lathauwer and Bart De Moor and Joos Vandewalle
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| |title = A Multilinear Singular Value Decomposition
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| |journal = Journal on Matrix Analysis and Applications
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| |year = 2000
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| |volume = 21
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| |number = 4
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| |pages = 1253–1278
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| }}</ref> ):
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| :: <math>f(\mathbf{x})=\mathcal{S}\mathop{\otimes}_{n=1}^N\mathbf{w}_n(x_n),</math>
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| where core tensor <math>\mathcal{S}\in \mathcal{R}^{I_1\times I_2\times \ldots \times I_N}</math> is constructed from <math>s_{i_1 i_2 \ldots i_N}</math>, and row vector <math>\mathbf{w}_n(x_n), (n=1 \ldots N)</math> contains continuous univariate weighting functions <math>w_{n,i_n}(x_n),(i_n=1 \ldots I_n)</math>. The function <math>w_{n,i_n}(x_n)</math> is the <math>i_n</math>-th weighting function defined on the <math>n</math>-th dimension, and <math>x_n</math> is the <math>n</math>-the element of vector <math>\mathbf{x}</math>. Finite element means that <math> I_n </math> is bounded for all <math> n </math>. For qLPV modelling and control applications a higher structure of TP functions are referred to as TP model.
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| ;Finite element TP model (TP model in short): This is a higher structure of TP function:
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| :: <math>\mathcal{F}(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(x_n).</math>
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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>.
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| ;Finite element convex TP function or model: A TP function or model is convex if the wighting functions hold:
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| :: <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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| 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>.
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| ;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>,
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| 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 [http://tptool.sztaki.hu/] or at [[MATLAB]] Central [http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool].
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| 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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| :: <math>\mathcal{F}(\mathbf{x}) \approx \mathcal{S}\boxtimes_{n=1}^N\mathbf{w}_n(x_n),</math>
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| where trade-off is offered by the TP model transformation between complexity (number of components in the core tensor or the number of weighting functions) and the approximation accuracy. The TP model can be generated according to various constrains. Typical TP models generated by the TP model transformation are:
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| * HOSVD canonical form of TP functions or TP model (qLPV models),
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| * Various kinds of TP type polytopic form or convex TP model forms (this advantage is used in qLPV system analysis and design).
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| ==Properties of the TP model transformation==
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| * It is a non-heuristic and tractable numerical method firstly proposed in control theory.<ref name=Baranyi04 /><ref name=compind />
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| * It transforms the given function into finite element TP structure. If this structure does not exist, then the transformation gives an approximation under a constraint on the number of elements.
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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.
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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
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| |author = L. Szeidl and P. Várlaki
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| |title = HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems
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| |journal = Journal of Advanced Computational Intelligence and Intelligent Informatics
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| |year = 2009
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| |volume = 13
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| |number = 1
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| |pages = 52–60
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| }}</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:
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| :* the number of weighting functions are minimized per dimensions (hence the size of the core tensor);
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| :* the weighting functions are one variable functions of the parameter vector in an orthonormed system for each parameter (singular functions);
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| :* the sub tensors of the core tensor are also in orthogonal positions;
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| :* the core tensor and the weighting functions are ordered according to the higher-order singular values of the parameter vector;
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| :* it has a unique form (except for some special cases such as there are equal singular values);
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| :* introduces and defines the rank of the TP function by the dimensions of the parameter vector;
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| * The above point can be extended to TP models (qLPV models to determine the [[HOSVD]] based canonical from of qLPV model to order the main component of the qLPV model). Since the core tensor is <math > N+O </math> dimensional, but the weighting functions are determined only for dimensions <math > n=1 \ldots N </math>, namely the core tensor is constructed from <math > O </math> dimensional elements, therefore the resulting TP form is not unique.
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| *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.
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| * 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==
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| <references/>
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| ==External links==
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| * [http://tptool.sztaki.hu/ TP Tool – home page]
| |
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| [[Category:Control theory]]
| |
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