Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

Introduction

In recent decades, many studies on analysis and synthesis problems relating to the multiple model (also called multimodels) approach have been undertaken. This has been motivated by the desire to establish the design problems in numerical terms. These have become possible as a result of the convex polytopic representation of the multiple model approach and the development of effective numerical resolution tools based on convex optimization software.

Automation relies on the concept of a model representing the internal dynamic behavior of a system. For example, a system is modeled through a mathematical relationship of input/output behaviors, or through an equation relating to its change of state. The dilemma is then to choose between the reliability of the model using the real process and the adequacy of this model expressed in mathematical form. Faced with this problem, automation engineers are often led, based on physical considerations, to consider certain classes of systems according to structural restrictions (such as linearity and convexity), leading to model approximations. Thus, linear representation (LTI models) has been widely used. The nonlinear model is thus represented by a single linear model, obtained from first approximation and close to an operating point. The disadvantage of such an approach is its local aspect, the linear model only being a local description of system behavior.

In recent years, a general approach based on multiple LTI models (linear or affine) around various function points has been proposed. This so-called multiple model (also called multimodel or Takangi-Sugeno fuzzy model) approach is a convex polytopic representation, which can be obtained either directly from a nonlinear mathematical model, through mathematical transformation or through linearization around various operating points. Many studies relating to the stability of this class of nonlinear systems have been published in recent years. Initially, these works were inspired by linear system control techniques, leading to the use of studies following quadratic and nonquadratic Lyapunov approach. Numerous approaches have been developed to study the stability of different system categories, such as uncertain, nonlinear, bilinear, varying parameter and delayed systems.

This book deals with the stability analysis and synthesis of control laws and observers for multiple models. Our approach is essentially based on Lyapunov’s second method and LMI formulation. Uncertain multiple models with unknown inputs are also studied. Quadratic and nonquadratic Lyapunov functions are considered. Interest in the quadratic method stems from the fact that it is easy to use the synthesis search parameters, which can be set out as a convex optimization problem in LMI form. However, the quadratic method has turned out to be very conservative, in that this approach ignores all of the data contained in the activation functions. These constraints become still more conservative if we add performance constraints of the closed-loop system.

In order to increase confidence in the quadratic method, multiple model stability studies are carried out by considering nonquadratic Lyapunov functions.

This book consists of five chapters dealing exclusively with continuous-time multimodels. It is set out as follows:

– Chapter 1 provides an introduction to multiple model representation and the tools used. Various methods used to obtain a multiple model, LMI tools and various control laws are presented.

– Chapter 2 presents different stability conditions. Using quadratic and nonquadratic Lyapunov functions, suitable conditions for stability are proposed for multiple models with uncertain parameters.

– Chapter 3 is integrally dedicated to observer synthesis. Unmeasurable decision variables and the separation principle are tackled. State estimation in the presence of unknown and uncertain inputs is covered. Various techniques are examined and LMI synthesis conditions are proposed. Some illustrative examples are included.

– Chapters 4 and 5 deal with stabilization through full state and output feedback controllers. The conditions obtained are expressed in the bilinear form (BMI), then LMI design conditions are proposed. Closed-loop multiple model performance is ensured through the placement of poles in LMI regions. The synthesis of robust control laws is also tackled by considering two types of parametric uncertainty (structured and interval uncertainties).

Various examples are proposed in order to illustrate the theoretical developments.

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 transformation when an analytical model is available.

These models use state–space representation. Thus, studies dealing with the stability analysis of multiple models as well as synthesis of controllers and observers adopt state–space representation in order to extend to nonlinear systems, results widely used in the linear domain.

Continuous-time and discrete-time multiple models, are generally of the form:

[1.1]

[1.2]

where is the state variables vector, the input vector and the decision variables vector. represent activation functions such that

1.2.1. Construction of multiple models by identification

Black box models are identified from inputs and outputs data around various operating points. Independently of the type of chosen model, this identification requires an optimal structure to be found, as well as an estimation of parameters and a validation of the final model [GAS 01, JOH 00, JOH 03, MUR 97, TAK 85]. In our case, the model is nonlinear relative to the parameters. Some iterative nonlinear optimization techniques are used according to the available data a priori. Identification methods for the unknown parameters are generally based on the minimization of a functional of the difference between the estimated output of the multiple model ym(t) and the measured output of the system y(t). The criterion commonly used is the minimization of the quadratic error:

[1.3]

where H is the observation time and θ is the parameters vector