Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

Chapter 1

Multiple Model Representation

1.1. Introduction

Following the work of Zadeh [ZAD 65], there has been a high degree of success in the use of fuzzy logic in the modeling of complex/nonlinear systems and also in the synthesis of fuzzy controllers [TAK 85]. The ability of fuzzy logic to represent a wide class of systems has been demonstrated as a universal approximation. In this respect, a number of successful applications have been achieved [BUC 93, CAS 95]. Various fuzzy models can be found in literature. However, two principal models have come to light: the Mamdani and Takagi–Sugeno (T-S) [TAK 85] models. Mamdani’s fuzzy model uses fuzzy subsets for the most part, whereas the T-S type uses functions which are dependent on input variables. The most popular T-S model is the one which mostly uses a state–space or autoregressive model. This type of representation, known as multiple model representation [MUR 97], has been successfully used in all areas of automation (such as identification, control, FDI and FTC) [AKH 07a, AKH 07b, CHA 10a, CHA 09, CHA 08b, FRA 90, PAT 97].

1.2. Techniques for obtaining multiple models

Multiple models are obtained by interpolation between linear time invariant (LTI) models. Each LTI model represents an operating range which is valid around an operating point. Three methods are used to obtain a multiple model:

– by identification when input and output data is available;

– by linearization around various operating points;

– by a convex polytopic ...