This paper presents a new and a more general formulation of the min and max operations for gradual intervals, initially proposed for the special case of triangular fuzzy intervals. Gradual intervals generalize the conventional interval representation by increasing its specificity, thereby making it possible to represent the imprecision and the uncertainty via the notion of gradualness. In this context, a new interpretation of fuzzy intervals through the notion of gradual intervals is developed. The originality of the proposed methodology lies in the extension of inter-interval relations to the gradual case according to midpoint-radius representation.

Recently, gradual numbers have been introduced as a means of extending standard interval computation methods to fuzzy and gradual intervals. However, it is well known that the practical use of standard interval arithmetic operators, just like their fuzzy extension, gives results that are more imprecise than necessary and, in some cases, even counterintuitive. In this paper, we combine the concepts of gradual numbers and Kaucher arithmetic on extended intervals to define extended gradual interval arithmetic, where subtraction and division operators are, respectively, the inverse operators of the addition and the multiplication. They are applied to the inversion of a linear regressive model and to a control problem that is based on the inversion of a linear model.

The work in this paper deals with the learning of gradual rules in the framework of data classification. Gradual rules are well suited to express constraints between numerical quantities. They are here used to constrain the shape of classes to be modeled. More precisely, it is proposed to represent convex polygon-shaped classes by means of "If-Then" classification gradual rules. The latter, learnt from training data, constitute elementary classifiers able to solve oneclass problem with two attributes. General classification problems are thus addressed by combining partial decisions of elementary classifiers. The approach is illustrated with the classification of radar images.

This paper presents a fuzzy interval linear regression method to combine climate temperature models. The study is carried out using air temperature data recorded yearly during the 20th century in the La Plata Basin. The objective of the study is to provide realistic predictions of the air temperature in the 21st century, taking into account five climate models to envelope the predicted data. The input to the fuzzy interval model is the central value for each climate model. The output observed data if the central value, the lower and upper limits, representing 90% of the dataset within a region. The output to the fuzzy interval regression model represents the uncertainty in a trapezoid shaped membership function, in which the core interval envelop all the observed central data values and the support interval envelopes 90% of observed data. The fuzzy regression parameters may be trapezoid shaped or crisp values and are computed such that the global uncertainty is minimized. A standard linear regression model is also be used for comparison and validation. The method has shown to be useful to handle the uncertainty management in climate model better than the linear regression despite its wider uncertainty range in all cases.

In this paper, 2 data fusion strategies from SAR images are investigated through application to measurement of displacement field due to the Kashmir earthquake (Mw=7.6, 2005). Firstly, the 3D displacement field at the Earth's surface is retrieved by a linear inversion, using the measurements from sub-pixel image correlation and differential interferometry. In addition to the generalized least square method, a fuzzy approach is applied to represent the measurement uncertainty. Secondly, the geometry of the fault is optimized by a non linear inversion, using the same measurements. The inter-comparisons between strategies and approaches are performed in order to highlight the advantages and disadvantages of each strategy and approach.

Fuzzy regression using possibilistic concepts allows the identification of models from uncertain data sets. However, some limitations still exist about the possible evolution of the output spread with respect to inputs. We present here a modified form of fuzzy linear model whose output can have any kind of output spread tendency. The formulation of the linear program used to identify the model introduces a modified criterion that assesses the model fuzziness independently of the collected data. These concepts are used in a global identification process in charge of building a piecewise model able to represent every kind of output evolution.

This paper aims at expressing MIN and MAX operations when triangular fuzzy intervals are taken as inputs, that is when Zadeh's 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.

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.

This paper presents an educational library to be used with the MATLAB/Simulink package from the Mathworks company for the design of fuzzy controllers. This free library, named FlouLib, has its foundations in a clear representation of the fuzzy interfaces (fuzzification, inference, defuzzification). First, Mamdani-type conventional fuzzy logic controllers are presented, including when rulebases are chained. Then, from the modal equivalence principle, advanced Takagi-Sugeno fuzzy controllers are introduced especially those relying on input-output linearization techniques. Examples, all available in FlouLib demos, are used to show the potentialities of FlouLib for control engineering curricula.

This part aims at designing a fuzzy inverse controller for fuzzy linear interval systems. The controller is designed from the fuzzy interval ranges of the system parameters. Indeed, for a given degree of confidence on the fuzzy system parameters, the control objective can be viewed as maintaining the system output within a tolerance envelope, around the exact trajectory, specified by a degree of preference on the fuzzy trajectory. The validity of the proposed method is illustrated by simulation examples.

This two part paper proposes a methodology of designing and analyzing an inverse controller for stable fuzzy interval linear systems according to fuzzy interval arithmetic. The first part aims at revisiting the solving of first order fuzzy linear equations. The general solution is expressed in a compact form which is directly used for deriving the condition of the solution existence. The suggested method is inspired from the a-cut principle where the fuzzy interval arithmetic is defined in terms of well-established interval arithmetic operations on closed intervals of real numbers. The so-built solution is useful for the inverse controller synthesis dealt with in part II of this paper. The validity of the proposed method is illustrated by a simulation example.

The aim of this paper is to examine the feedback linearization technique for unknown non affine-in-control systems. As the nonlinear plant is assumed to be unknown, an adaptive Takagi-Sugeno fuzzy system is used to represent it. Based on the error between the nonlinear plant and the adapted fuzzy model, the parameters of the latter are updated according to Lyapunov synthesis. Then, at each sampling time, the linearizing controller is obtained by inversion of the fuzzy model. The closed-loop control structure stability is guaranteed using Lyapunov analysis. An illustrative simulation example is given to demonstrate the feasibility of the proposed method.

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.

This paper is concerned with the use of fuzzy inputs in fuzzy logic controllers. A precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs. Two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed. The proposed concepts are applied to control a first order process with a PI controller. This simple process is chosen to clearly illustrate the behavior of the closed-loop system using fuzzy inputs for fuzzy reference and fuzzy measurement. Finally, a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system. Although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers, experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed-loop fuzzy control.

Examines the control problem of discrete-time feedback linearizable nonlinear systems for which there are no available analytic models. Based on the ability of fuzzy systems to approximate any nonlinear mapping, the unknown nonlinear system is represented by a Takagi-Sugeno fuzzy model which is identified using input-output data. Then, a fuzzy feedback linearizing controller, based on fuzzy model inversion, is used to linearize the system. The key concept is the combination of this fuzzy feedback linearizing controller with a simple linear controller resulting in a robust feedback control structure. The equivalence between the feedback linearization technique and the internal model control methodology is then established. Finally, simulation results are given to demonstrate the feasibility of the proposed method.

Deals with H/sub /spl infin// control for discrete-time nonlinear systems using an input-output linearization concept. Based on the ability of fuzzy systems to approximate any nonlinear mapping, the unknown nonlinear system is represented by a Takagi-Sugeno fuzzy model. The control problem is then tackled using a feedback linearization of the fuzzy nonlinear system. However, this feedback linearization can not guarantee robustness in the presence of uncertainties and disturbances. In order to solve this problem, an additive H/sub /spl infin// control component is added to the linearizing controller to attenuate uncertainty and disturbance influence on the control performances. In the paper, the H/sub /spl infin// control component is designed in terms of a modified Riccati inequality. Finally, an illustrative example is included to demonstrate the feasibility of the proposed method.

Presents a method for fuzzy model identification whose main characteristic is the incremental creation and automatic distribution of the fuzzy sets in the domain of each input variable. The output rule parameters are optimized by a least square error minimization. The fuzzy sets are determined in such a way that the resulting fuzzy rule base remains reasonably readable by domain experts. The methodology is applied to numerical examples found in the literature.

Establishing relationships between blood pressure variations and body acceleration and heart rate measurements is a key issue to determine the drug quantity to inject to cure patients suffering from hypertension, i.e. high blood pressure. For this purpose, a fuzzy linguistic model is developed from experimental data with a method which leads to a small number of linguistic terms and rules, easily understandable by doctors. The software instrument developed has predictive performances quite similar to those obtained with statistical models. In addition, it provides a unified formalism that will probably simplify the introduction of other features in the drug quantity decision system.

The problem of learning control is addressed for a class of nonlinear systems for which there is no available analytic model. Based on the ability of fuzzy systems to approximate any nonlinear function, a global architecture is proposed where the unknown input-output linearizing controller is replaced by a fuzzy system whose rule conclusions are learned on line. A direct learning algorithm is implemented in which the parameter evolution is guided simultaneously by the tracking error and a prediction error of the fuzzy system output. During the learning phase, additive components, such as bounding control and sliding mode control are introduced to guarantee system's stability and parameter convergence. Stability of the closed loop system is proved according to Lyapunov theory. Simulation results for the inverted pendulum are included to demonstrate the method applicability.

A reconstruction methodology is proposed for dealing with size reduction in fuzzy rulebases. Elimination of redundant fuzzy rules is always at the root of size reduction. As a matter of fact, it is thus important to be aware of the different kinds of redundancy that may appear in fuzzy rulebases. Two types of redundancy are distinguished, i.e. interpolation redundancy and overlap redundancy. On the contrary to usual methods that are only able to deal with overlap redundancy, the presented method is efficient for both redundancy types. Furthermore, the proposed reconstruction principle induces the obtention of readable linguistic rules. The method performance is illustrated by means of two examples.

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.>

This paper introduces sensors employing a fuzzy numeric to symbolic interface. The fundamental design considerations for this kind of fuzzy symbolic sensor, or fuzzy sensor, are formally presented. Then, the use of these components for fuzzy control is discussed and illustrated.