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The tensor product (TP) model transformation defines and numerically reconstructs the Higher-Order Singular Value Decomposition (HOSVD) of functions. It plays the same role with respect to functions as HOSVD does for tensors (and SVD for matrices). The need for certain advantageous features, such as rank/complexity reduction, trade-offs between complexity and accuracy, and a manipulation power representative of the TP form, has motivated novel concepts in TS fuzzy model based modelling and control. The latest extensions of the TP model transformation, called the multi- and generalised TP model transformations, are applicable to a set functions where the dimensionality of the outputs of the functions may differ, but there is a strict limitation on the dimensionality of their inputs, which must be the same. The paper proposes an extended version that is applicable to a set of functions where both the input and output dimensionalities of the functions may differ. This makes it possible to transform complete multicomponent systems to TS fuzzy models along with the above-mentioned advantages.

The appearance of the Singular Value Decomposition (SVD) was one of the largest breakthroughs in matrix algebra [

The above-mentioned extensions and variations of the TP model transformation were primarily applied to fuzzy model complexity reduction [

It is executable on models given by equations or soft computing based representations, such as fuzzy rules or neural networks or other black-box models. The only requirement is that the model must provide an output for each input (at least on a discrete scale, see Section

It will find the minimal complexity, namely, the minimal number of rules of the TS fuzzy model. If further complexity reduction is required, it provides one of the best trade-offs between the number of rules and approximation error.

It works like a principle component analysis, in that it determines the order of the components/fuzzy rules according to their importance.

It is capable of deriving the antecedent fuzzy sets according to various constraints. For instance, it can be used to define different convex hulls, a capability which has recently been shown to play an important role in control theory.

It is capable of transforming the given model to predefined antecedent fuzzy sets (pseudo-TP model transformation)

It is capable of transforming a set of models simultaneously, while common antecedent fuzzy sets are derived for all models.

Based on the above, various theories and applications have emerged using the TP model transformation. Further computational improvements were proposed in [

One of the key advantages of the TP model transformation is that is capable of finding the minimal complexity of all components of the system and guarantees the same antecedent system for all components. This is a very typical requirement in design or stability verification methodologies, that is, the model, controller, and observer need to have the same antecedent system, hence, convex representation. Therefore, the simultaneous manipulation of the components with the multi-TP model transformation or the generalised TP model transformation (that combines all variants of the TP model transformation) yields further possibilities for control performance optimisation [

Despite the above advantages, a crucial limitation of the generalised TP model transformation is that it can only be applied to a set of systems which have the same number of inputs. For instance, consider four different systems given with different representations, as shown in Figure

Multi TP model transformation.

A further generalisation proposed in this paper can be applied to systems like in the example given in Figure

Proposed extension of the TP model transformation.

Recenly proposed SOS-type (Sum-of-sqares) TS fuzzy LPV models are also widely applied in fuzzy control theories [

The following notations are used in the paper:

Scalar:

Vector:

Matrix:

Tensor:

Set:

Index

Index

Interval:

Space:

In the case of spaces:

In the case of vectors:

In the case of tensors:

Grid:

Pair

Discretised function

Types of the weighting functions are as follows:

SN: sum normalised

NN: nonnegativeness

NO: normalised

CNO: close to normalised

RNO: relaxed normalised

INO: inverse normalised

IRNO: inverse relaxed normalised.

For further details, refer to [

Assume that a set of functions is given as

The goal of the TP model transformation is to transform

under the following constraints given on the weighting functions.

Weighting function systems

Weighting function systems

Weighting function systems

The types (i.e., SN, NN, NO, CNO, RNL, INO, and IRNO) of the weighting function systems

Thus, (

Discretisation of all

Discretise the predefined weighting functions over the dimensions of

This step is executed in the same way as in the case of the original TP model transformation; see [

Execute the following steps in each dimension

Lay out tensors

If

Execute SVD on

As a matter of fact, if nonzero singular values are discarded then it is only an approximation. Let

If

and, according to the conditions, execute SN, NN, NO, CNO, and complexity trade-off by discarding singular values in the same way as in the original TP model transformation:

Again, if nonzero singular values are discarded then it is only an approximation. Let

Finally,

where

where

This step is the same as in the multi-TP model transformation [

Then we achieved the goal. We have the TP model form of all functions with the given constraints:

The convex hull manipulation and the complexity trade-off are done in the second step. Therefore the approximation accuracy is controlled here by the discarded nonzero singular values. However, the discarded nonzero singular values lead to approximation error. If the given weighting function system is not sufficient (i.e., the number of the weighting functions is less than the rank of that dimension) then we arrive at an approximation only. The use of the pseudoinverse guarantees, however, that it will be the best approximation.

Consider a multicomponent system with input vector

In order to have a systematic notation, we denote the input vector of System

System

The input vector of System

This system is given by formulas such as

The input vector of System

This is given by a fuzzy logic model. Assume that two rules are given (

IF

Further assume that the membership functions are in Ruspini partition:

The input vector of System

This is a black-box model that can provide

The goal of the example is to transform all the four systems to TS fuzzy representations (or TP model if the resulting weighting functions cannot be represented as antecedent fuzzy states), with the following conditions:

All systems must have the same antecedent function system on the input interval of

The same antecedent function system of variable

where “

The only requirement for the weighting function system of the input

It is worth emphasizing again that the previous methods for TP model representation cannot be applied in the present case, since the elements of the input vectors are different.

Let us define grid

Thus the number of points on the discretisation grid is

Let us discretise the systems over the rectangular grid defined by vectors

where the first three dimensions are assigned to the input variables and the last dimension is assigned to the output vector. The discretisation of System

where the first two dimensions are assigned to the input variables

The discretisation of System

where the first two dimensions are assigned to the input variables

Let us discretise the predefined weighting function as well:

Lay out tensors

Create

Execute SVD on

The result of this step to be used later is

Let

Lay out tensors

Then execute HOSVD on each

The result of this step is

Let

Then having the discretised tensors and weighting functions of all systems we can numerically reconstruct the weighting functions [

Thus, we have achieved our goal:

The proposed TP model transformation can be executed on a set of models where the dimensionality of the inputs may differ. The proposed TP model transformation has all the advantages of the previous ones, including easy convex hull manipulation, complexity trade-offs, pseudo TP model transformation, and automatic and numerical execution.

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the FIEK Program (Center for Cooperation between Higher Education and the Industries at the Széchenyi István University, GINOP-2.3.4-15-2016-00003).