Sylvie Galichet

Fuzzy controllers: synthesis and equivalences

It has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy. In particular, any given linear control can be achieved with a fuzzy controller for a given accuracy. The aim of this paper is to show how to automatically build this fuzzy controller. The proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller. The main idea is to equate the output of the fuzzy controller with the output of the PI controller at some particular input values, called modal values. The rule base and the distribution of the membership functions can thus be deduced. The analytic expression of the output of the generated fuzzy controller is then established. For Sugeno-type fuzzy controllers, precise equivalence is directly obtained. For Mamdani-type fuzzy controllers, the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values. The usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed. >

Nonlinear internal model control: application of inverse model based fuzzy control

This paper investigates the possible applications of dynamical fuzzy systems to control nonlinear plants with asymptotically stable zero dynamics using a fuzzy nonlinear internal model control strategy. The developed strategy consists in including a dynamical Takagi-Sugeno fuzzy model of the plant within the control structure. In this way, the controller design simply results in a fuzzy model inversion. In this framework, the originality of the presented work lies in the use of a dynamical fuzzy model and its inversion. In order to be able to implement the control structure, two crucial points have to be addressed in the considered fuzzy context, on the one hand the model representation and identification, on the other, the model inversion. As the fuzzy system can be viewed as a collection of elementary subsystems, its inversion is approached here in a local way, i.e., on the elementary subsystems capable to provide an inverse solution. In this case, the inversion of the global fuzzy system is thus tackled by inversion of some of its components. By doing so, exact inversion is obtained and offset-free performances are ensured. In order to guarantee a desired regulation behavior and robustness of stability of the control system, the fuzzy controller is connected in series with a robustness filter. The potential of the proposed method is demonstrated with simulation examples.

Structure identification and parameter optimization for non-linear fuzzy modeling

This work presents a method for non-linear fuzzy model identification. The main characteristic of the method is the automatic determination of the number and position of the fuzzy sets in the domain of each variable. The resultant fuzzy rule base allows model interpretation by domain experts. The main contribution of this work is a formulation that allows the optimization of output parameters by a least-squares error (LSE) minimization. A numerical solution of the LSE problem is developed based on the singular value decomposition of the regressor matrix. The whole methodology is applied to some numerical examples found in the literature.

Application of fuzzy logic control for continuous casting mold level control

This paper deals with the problem of molten metal level control in continuous casting. Under normal circumstances, proportional integral derivative (PID) control performs quite well, but abnormal conditions (in particular nozzle clogging/unclogging) require manual intervention. Indeed, when the flow of matter into the mold increases suddenly, the PID controller is not always able to prevent large level variations that can even lead to mold overflow. So, a fuzzy controller has been designed using the expert knowledge of the operators for controlling the process during disturbed phases. The paper discusses both the design of the fuzzy logic controller and its integration with the PID in a global control architecture. Results from simulation and successful online implementation are presented.

Inverse controller design for fuzzy interval systems

This paper aims at designing and analyzing an inverse controller for stable inversible (minimum phase) fuzzy interval linear and/or multilinear systems. The controller is designed from the fuzzy interval ranges of the system parameters using an /spl alpha/-cut methodology. Indeed, for a given /spl alpha/-cut of the fuzzy system parameters representing an uncertainty level, the control objective can be viewed as maintaining the system output within a tolerance envelope, around the exact trajectory, specified by the degree of preference /spl alpha/ on the fuzzy trajectory. The stability is ensured in the way that the controller restricts the system output divergence within the tolerance envelope. The validity of the proposed method is illustrated by simulation examples.

Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents

Abstract This paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions. As pointed out in recent works, such fuzzy systems can be decomposed into collections of multi-linear subsystems. In this paper, an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem. Based on this analytical methodology, an algorithm is proposed for computing inverse solutions. As the inversion is handled analytically, the exactness of the obtained solutions is guaranteed. Furthermore, according to the decomposability of the studied fuzzy systems, all inverse solutions are found. Finally, whatever the fuzzy system under consideration, there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations (no solution, one unique solution, multiple solutions, an infinity of solutions). The proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved. In other words, with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis, decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure.

A revisited approach to linear fuzzy regression using trapezoidal fuzzy intervals

Conventional Fuzzy regression using possibilistic concepts allows the identification of models from uncertain data sets. However, some limitations still exist. This paper deals with a revisited approach for possibilistic fuzzy regression methods. Indeed, a new modified fuzzy linear model form is introduced where the identified model output can envelop all the observed data and ensure a total inclusion property. Moreover, this model output can have any kind of spread tendency. In this framework, the identification problem is reformulated according to a new criterion that assesses the model fuzziness independently from the collected data distribution. The potential of the proposed method with regard to the conventional approach is illustrated by simulation examples.

MIN and MAX Operators for Fuzzy Intervals and Their Potential Use in Aggregation Operators

This paper aims at expressing MIN and MAX operations when triangular fuzzy intervals are taken as inputs, that is when Zadehs extension principle is considered. First presented approach consists in representing fuzzy intervals by means of their profiles. In this context, a computation algorithm can be easily derived for implementing the MIN and MAX operations. Another methodology based on interval relations is then proposed for determining a general analytical expression of the MIN and MAX operations. The potential use of these expressions in the framework of uncertain aggregation operators is illustrated with the two-additive Choquet integral.

Fuzzy Feedback Linearizing Controller and Its Equivalence With the Fuzzy Nonlinear Internal Model Control Structure

Fuzzy Feedback Linearizing Controller and Its Equivalence With the Fuzzy Nonlinear Internal Model Control Structure This paper examines the inverse control problem of nonlinear systems with stable dynamics using a fuzzy modeling approach. Indeed, based on the ability of fuzzy systems to approximate any nonlinear mapping, the nonlinear system is represented by a Takagi-Sugeno (TS) fuzzy system, which is then inverted for designing a fuzzy controller. As an application of the proposed inverse control methodology, two popular control structures, namely, feedback linearization and Nonlinear Internal Model Control (NIMC) are investigated. Moreover, the paper points out that, under some conditions, both of the control structures are equivalent and naturally implement a Smith predictor in the presence of time delays.

Performing Approximate Reasoning with Words

When defining a term set as a (mite family of fuzzy sets on a universe of discourse, two description levels are introduced: the level of the referential and the level corresponding to the symbolic term set. Depending on the level which is privileged, two different views of reasoning with fuzzy set labels can be thought of: Zadeh’s view of approximate reasoning which takes place at the level of the universe of discourse, and another view where fuzzy sets are manipulated in a more symbolic way at the term set level, which would corresponds to the idea of computing with words recently advocated by Zadeh also. The two views are contrasted in this paper, and their differences are laid bare.