Complexity

Volume 2018 (2018), Article ID 8546976, 9 pages

https://doi.org/10.1155/2018/8546976

## Extension of the Multi-TP Model Transformation to Functions with Different Numbers of Variables

Széchenyi István University, Győr, Hungary

Correspondence should be addressed to Péter Baranyi; moc.liamg@iynarab.retep.forp

Received 18 September 2017; Accepted 29 January 2018; Published 19 March 2018

Academic Editor: Kevin Wong

Copyright © 2018 Péter Baranyi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The appearance of the Singular Value Decomposition (SVD) was one of the largest breakthroughs in matrix algebra [1]. Its applicability was extended to tensors in the form of the Higher-Order SVD [2] around 2000. Recently, a further extension of the SVD and HOSVD concept, known as the tensor product (TP) Model Transformation, was proposed for functions in control theory [3]. A comprehensive overview is given in [4]. Various extensions of the TP model transformation such as the bilinear-, pseudo-, multi-, and generalised TP model transformation, as well as the concept of HOSVD canonical form of TS fuzzy or TP models, were proposed in [4–7], with a special focus on TS fuzzy models in [8]. The approximation power of the TP model transformation applied to TS fuzzy models is investigated in [9].

The above-mentioned extensions and variations of the TP model transformation were primarily applied to fuzzy model complexity reduction [10, 11] and in the widely used TS fuzzy model based PDC (Parallel Distributed Compensation) control theories [12–14]. But also, in general, it has been applied to polytopic model, TP/TS fuzzy model, and LMI (Linear Matrix Inequality [15]) based control theories. The most important features of the TP model transformation are guaranteed by the key transformation step whereby a numerically reconstructed HOSVD structure is determined. Key features of the transformation are as follows:(i)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 4, Step 1).(ii)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.(iii)It works like a principle component analysis, in that it determines the order of the components/fuzzy rules according to their importance.(iv)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.(v)It is capable of transforming the given model to predefined antecedent fuzzy sets (pseudo-TP model transformation)(vi)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 [16, 17]. It has been proved in [5, 18–20] that LMI based control design theories are very sensitive for convex hulls defined by consequents (vertices) of TS fuzzy models. Thus, the convex hull manipulation capability of the TP model transformation is an important and necessary step in LMI based control design. Very effective convex hull manipulation methods were incorporated into the TP model transformation in [21–23]. Further useful control approaches and applications were published in the field of control theory [24–41]. Many powerful approaches are published on the field of sliding mode control in [29, 42, 43]. In physiological control the usability of TP model transformation has been demonstrated as well [44–49]. Various further theories and applications are studied in [50–87].

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 [18–20].

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 1. S1 is a fuzzy logic model; S2 is neural network; S3 is given by an equation; and S4 is a black-box model. All of these models have the same inputs but may have different sized output tensors. The multi-TP model transformation is capable of simultaneously transforming all systems to TP or TS fuzzy model form, such that the same antecedent sets are defined on the inputs. The generalised TP model transformation can also transform to predefined antecedent fuzzy sets.