Tensor product model transformation

In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam [1] [2][3] for quasi-LPV (qLPV) control theory. It transforms a function (which can be given via closed formulas or neural networks, 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.[4]

A free MATLAB implementation of the TP model transformation can be downloaded at or at MATLAB Central . A key underpinning of the transformation is the higher-order singular value decomposition.[5]

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.

The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions",[6] on which further information can be found here. It has been proved that the TP model transformation is capable of numerically reconstructing this HOSVD based canonical form.[7] 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. This feature has led to new optimization approaches in qLPV system analysis and design, as described here: TP model transformation in control theory.

Definitions

Finite element TP function
A given function , where , is a TP function if it has the structure:

that is, using compact tensor notation (using the tensor product operation of [5] ):

where core tensor is constructed from , and row vector contains continuous univariate weighting functions . The function is the -th weighting function defined on the -th dimension, and is the -the element of vector . Finite element means that is bounded for all . For qLPV modelling and control applications a higher structure of TP functions are referred to as TP model.

Finite element TP model (TP model in short)
This is a higher structure of TP function:

Here is a tensor as , thus the size of the core tensor is . The product operator has the same role as , but expresses the fact that the tensor product is applied on the sized tensor elements of the core tensor . Vector is an element of the closed hypercube .

Finite element convex TP function or model
A TP function or model is convex if the wighting functions hold:
and

This means that is inside the convex hull defined by the core tensor for all .

TP model transformation
Assume a given TP model , where , whose TP structure maybe unknown (e.g. it is given by neural networks). The TP model transformation determines its TP structure as
,

namely it generates the core tensor and the weighting functions for all . Its free MATLAB implementation is downloadable at or at MATLAB Central .

If the given does not have TP structure (i.e. it is not in the class of TP models), then the TP model transformation determines its approximation:[4]

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:

Properties of the TP model transformation

  • 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);
  • the sub tensors of the core tensor are also in orthogonal positions;
  • the core tensor and the weighting functions are ordered according to the higher-order singular values of the parameter vector;
  • it has a unique form (except for some special cases such as there are equal singular values);
  • introduces and defines the rank of the TP function by the dimensions of the parameter vector;

References

  1. 1 2 P. Baranyi (April 2004). "TP model transformation as a way to LMI based controller design". IEEE Transaction on Industrial Electronics. 51 (2): 387400. doi:10.1109/tie.2003.822037.
  2. 1 2 P. Baranyi and D. Tikk and Y. Yam and R. J. Patton (2003). "From Differential Equations to PDC Controller Design via Numerical Transformation". Computers in Industry, Elsevier Science. 51: 281297. doi:10.1016/s0166-3615(03)00058-7.
  3. P. Baranyi, Y. Yam & P. Várlaki (2013). "Tensor Product model transformation in polytopic model-based control". Taylor&Francis, Boca Raton FL: 240. ISBN 978-1-43-981816-9.
  4. 1 2 3 D. Tikk, P.Baranyi, R. J. Patton (2007). "Approximation Properties of TP Model Forms and its Consequences to TPDC Design Framework". Asian Journal of Control. 9 (3): 221–331. doi:10.1111/j.1934-6093.2007.tb00410.x.
  5. 1 2 Lieven De Lathauwer and Bart De Moor and Joos Vandewalle (2000). "A Multilinear Singular Value Decomposition". Journal on Matrix Analysis and Applications. 21 (4): 1253–1278. doi:10.1137/s0895479896305696.
  6. 1 2 P. Baranyi and L. Szeidl and P. Várlaki and Y. Yam (July 3–5, 2006). Definition of the HOSVD-based canonical form of polytopic dynamic models. Budapest, Hungary. pp. 660–665.
  7. 1 2 L. Szeidl & P. Várlaki (2009). "HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems". Journal of Advanced Computational Intelligence and Intelligent Informatics. 13 (1): 52–60.
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