Tensor product model transformation
In mathematics, the tensor product model transformation was proposed by Baranyi and Yam
as key concept for higher order singular value decomposition of functions. It transforms a function 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.
A free MATLAB implementation of the TP model transformation can be downloaded at or an old version of the toolbox is aviable at MATLAB Central . 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 theory.
The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", 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. 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 :
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 : 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:
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. 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, then the TP model transformation determines its approximation:
where trade-off is offered by the TP model transformation between complexity and the approximation accuracy. The TP model can be generated according to various constrains. Typical TP models generated by the TP model transformation are:
- HOSVD canonical form of TP functions or TP model,
- Various kinds of TP type polytopic form or convex TP model forms.
Properties of the TP model transformation
- It is a non-heuristic and tractable numerical method firstly proposed in control theory.
- 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.
- It can be executed uniformly, 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, which is a unique representation. It was proven by Szeidl 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:
- The above point can be extended to TP models. Since the core tensor is dimensional, but the weighting functions are determined only for dimensions, namely the core tensor is constructed from dimensional elements, therefore the resulting TP form is not unique.
- The core step of the TP model transformation was extended to generate different types of convex TP functions or TP models, in order to focus on the systematic modification of the convex hull instead of developing new LMI equations for feasible controller design. 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 via discarding the higher-order singular values, in the same manner as the tensor HOSVD is used for complexity reduction.