Laurent Foulloy

Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. This paper describes new properties of this transformation, by relating it with the well-known probability inequalities of Bienaymé-Chebychev and Camp-Meidel. The paper also provides a justification of symmetric triangular fuzzy numbers in the spirit of such inequalities. It shows that the cuts of such a triangular fuzzy number contains the “confidence intervals” of any symmetric probability distribution with the same mode and support. This result is also the basis of a fuzzy approach to the representation of uncertainty in measurement. It consists in representing measurements by a family of nested intervals with various confidence levels. From the operational point of view, the proposed representation is compatible with the recommendations of the ISO Guide for the expression of uncertainty in physical measurement.

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. >

A fuzzy approach for the expression of uncertainty in measurement

This paper deals with a fuzzy expression of uncertainty in measurement. The fuzzy approach proposed consists of representing measurements by a family of intervals of confidence stacked atop one another, that in fact define the upper bound of the probability distributions consistent with these intervals of confidence. This approach is compatible with the ISO Guide for the expression of uncertainty in measurement, and is particularly interesting because it allows both the handling of specificity and uncertainty of measurement. Moreover, fuzzy uncertainty propagation is available thanks to fuzzy arithmetic, which is a generalization of interval analysis, yielding both worst case results and best estimates at the same time. In order to simplify the propagation, a parameterized possibility distribution approximating the optimal one is proposed and compared with the probabilistic approaches.

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.

Global vision and performance indicators for an industrial improvement approach

Abstract This article organizes reflections around the notion of industrial performance. If the latter has always justified all the actions carried out in an enterprise, these actions have evolved with the evolution of the context. From a brief review of the characteristics of this context where information technology plays an expanding role, two concepts inherent to today’s performance are deduced and then analyzed: the need to consider the enterprise through a global vision on the one hand, and the importance of a continuous improvement approach of the performance of the whole or parts of the enterprise on the other hand. Then, the study will focus more particularly on the performance indicator as a fundamental tool in an improvement approach.

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.

Fuzzy symbolic sensors—From concept to applications

Abstract This paper deals with sensors which compute and report liguistic assessments of numerical acquired values. Such sensors, called symbolic sensors, are particularly adapted when working with control systems which use artificial intelligence techniques. After having reconsidered some elements of the measurement theory, this paper sets the foundations of the symbolic sensors by introducing the meaning of a lexical value and the description of a numeric measurement as two mappings that link the set of the subsets of the numerical domain with the set of the subsets of the lexical domain. It is then shown how the fuzzy subset theory provides a smart way for the treatment of symbol graduality, for measurement imprecisions and measurement validation, and for taking into account the measurement context. This approach leads to introducing a specific structure into the sensors, then called fuzzy symbolic sensors. As application, two specimens of fuzzy symbolic sensors have been successfully implemented. The first is an ultrasonic range finding sensor, which uses a procedure of management of errors and a procedure of creation of concepts by semantic relationships. The second is a colour matching sensor, which uses an interpolation method for creating the concepts by learning with a teacher.

Fuzzy modeling of measurement data acquired from physical sensors

The measurement uncertainty in physical sensors is often represented by a probabilistic approach, but such a representation is not always adapted to new intelligent systems. Therefore, a fuzzy representation, based on the possibility theory, can sometimes be preferred. We previously proposed a truncated triangular probability-possibility transformation to be applied to any unimodal and symmetric probability distribution which can be assimilated to one of the four most encountered probability laws (Gaussian, double-exponential, triangular, uniform). In this paper, we propose to build a fuzzy model of data acquired from physical sensors by applying this transformation. For this purpose, a minimum of knowledge about the probabilistic modeling of sensors is required. Three main situations are considered and for each situation, an adapted fuzzy modeling is proposed. Examples of these three situations are based on FM-chirped ultrasonic sensors.

The aggregation of complementary information via fuzzy sensors

Abstract This paper focuses on the acquisition of high-level information, i.e. information that is related to many conventional physical quantities in a non-analytical way. In these complex cases, we proposed to use fuzzy sensors which compute and report linguistic assessments of numerically acquired values. Two methods are proposed to realized the aggregation from basic measurements. The first one performs a combination of the relevant features by means of a rule-based description of the relations between them. With the second, the aggregation is realized through an interpolation mechanism that creates a fuzzy partition of the numeric multi-dimensional space of the basic features. The two methods will be applied to a linguistic description of comfort deduced from temperature and humidity measurements. Finally, these two methods will be compared in terms of their efficiency and material implementation.

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.