Mohamed El-Koujok
email: mohamed.elkoujok@ens2m.fr

Noureddine Rafael Gouriveau
email: rgourive@ens2m.fr

Zerhouni Femto-St
email: zerhouni@ens2m.fr

A Neuro-Fuzzy Self Built System For Prognostics: a Way To Ensure Good Prediction Accuracy by Balancing Complexity and Generalization

In maintenance field, prognostics is recognized as a key feature as the prediction of the remaining useful life of a system allows avoiding inopportune maintenance spending. However, it can be a non trivial task to develop and implement effective prognostics models including the inherent uncertainty of prognostics. Moreover, there is no systematic way to construct a prognostics tool since the user can make some assumptions: choice of a structure, initialization of parameters… This last problem is addressed in the paper: how to build a prognostics system with no human intervention, neither a priori knowledge? The proposition is based on the use of a neuro-fuzzy predictor whose architecture is partially determined thanks to a statistical approach based on the Akaike information criterion. It consists in using a cost function in the learning phase in order to automatically generate an accurate prediction system that reaches a compromise between complexity and generalization capability. The proposition is illustrated and discussed.

I. INTRODUCTION

The growth of reliability, availability or safety of a system is a determining factor in regard with the effectiveness of industrial maintenance. Thus, the implementation of provisional strategies is a good way to improve the availability of processes, to ensure the smallest variations of products qualities or the direct costs falling [START_REF] Leger | Integration of maintenance in the enterprise: towards an enterprise modeling based framework compliant with proactive maintenance strategy[END_REF]. Furthermore, durable development involves the integration of economical strategies beside social and environmental ones for the optimization of processes. This major provocation of triple performance outlined a new and interesting research area: concepts like corrective or preventive maintenance were progressively completed by predictive and proactive maintenance [START_REF] Muller | Formalisation of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF] and prognostics is now recognized as a key feature in maintenance strategies as the estimation of the provisional reliability of equipment as well as its remaining useful life allows avoiding inopportune spending.

A central problem can be pointed out: the accuracy of a prognostics system is related to its ability to predict the degradation of equipment: starting from a "current situation", a prognostics tool must be able to forecast the "future possible situations". From the research point of view, many developments exist to support these prognostics or forecasting activities [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF][START_REF] De Gooijer | 25 years of time series forecasting[END_REF][START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF][START_REF] Vachtsevanos | Intelligent Fault Diagnostic and Prognosis for Engineering Systems[END_REF]. However, choosing an efficient technique depends on classical constraints that limit the applicability of the tools: available data-knowledge-experiences, complexity and dynamic of systems, available monitoring devices, implementation requirements (precision, computation time…).

Following that and according to literature, neuro-fuzzy (NF) systems appear to be very promising prognostic tools: NFs learn from examples and attempt to capture the subtle relationships among data. NFs are computationally effective techniques and are thereby well suited for practical problems, where it is easier to gather data than to formalize the behavior of the system being studied. Actual developments confirm the interest of using NFs in forecasting applications [START_REF] Wang | Prognosis of machine health condition using neuro-fuzzy systems[END_REF][START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF]. (Note that the purpose of the authors is not to present NFs systems as the single tools for prognostics, but as adequate to face with the implementation restriction and requirements.)

Nevertheless and in spite of the capabilities of NFs systems, building a NFs model for prediction is not a trivial task: various fuzzy structures can be used, the nature and quantity of inputs, as well as the form of the membership's functions have to be chosen, different learning algorithms exist, a random initialization (or expert made) must be done… Moreover, these choices are critical as they directly have an impact on the accuracy of the predictions of the system. This is the problem addressed in this paper: how to build a prognostics system with no human intervention, neither a priori knowledge?

The proposition is based on the use of an evolving NFs system that starts from scratch, i.e., for which any assumption on its structure is necessary. A statistical approach based on the Akaike criterion (AIC) is used in order to automatically generate an accurate prediction system that reaches a compromise between complexity and generalization capability.

The paper is organized in four main parts. First, the concept of prognostics is clarified in order to provide the lector with the limit of this work. As the efficiency of a prognostics system is highly dependent on its ability to perform good predictions, the relationship between prognostics and prediction is also explained and the use of first order Takagi-Sugeno neuro-fuzzy systems in prognostics applications is justified. Following that, the next section is dedicated to the proposition of a self built NFs system for prognostics. For that purpose, the ways of building such models are discussed as well as the learning aspects that influence the prediction performances and an evolving neuro fuzzy system is pointed out. The method to balance complexity and generalization capability is then presented and the whole is finally illustrated in the last part: various experiments on industrial predictions benchmarks are used in order to discuss the proposition and to compare it with the classical Auto Regressive eXogenous approach (ARX).

II. TOWARDS NF-SYSTEMS FOR PROGNOSTICS

The concept of prognostics Prognostics is traditionally related to fracture mechanics and fatigue. It started to be brought up by the modal analysis community as an intersting field [START_REF] Farrar | A Coupled Approach to Developing Damage Prognosis Solutions[END_REF]. In this "meaning", prognostics is called the prediction of a system's lifetime. Prognostics can also be defined as a probability measure: the chance that a machine operates without a fault or failure up to some future time [START_REF] Lin | Recursive filters for a partially observable system subject to random failure[END_REF]. This "probabilistic prognostics value" is all the more an interesting indication as the fault or failure can have catastrophic consequences (e.g. nuclear power plant) and maintenance manager need to know if inspection intervals are appropriate. Some authors introduce prognostics as a process that allows the a priori reliability modeling and thereby enables to estimate the remaining time to underpass a limit fixed by the practitioner or by past experiences [START_REF] El-Koujok | Error estimation of a neuro-fuzzy predictor for prognostic purpose[END_REF][START_REF] Chinnam | A neurofuzzy approach for estimating mean residual life in condtitionbased maintenace systems[END_REF].

Finally, although there are some divergences in literature, prognostics can be defined as proposed by the International Organization for Standardization: "prognostics is the estimation of time to failure and risk for one or more existing and future failure modes" [START_REF]ISO, 13381-1. Condition monitoring and diagnostics of machines, prognostics -Part1: General guidelines[END_REF]. In this acceptation, prognostics is also called the "prediction of a system's lifetime" as it is a process whose objective is to predict the remaining useful life (RUL) before a failure occurs given the current machine condition and past operation profile [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF].

Prognostics and prediction

All definitions proposed here before assimilate prognostics to a "prediction process": a future situation must be caught. It obviously supposes that the current situation can be grasped (practically, it's the synthesis of a detection process and of measured data of the system). Following that and according to previous works, two salient aspects of prognostics appear [START_REF] Dragomir | Framework for a distributed and hybrid prognostics system[END_REF]:

1. prognostics is mostly assimilated to a prediction process (a future situation must be caught), 2. prognostics is based on the failure notion, which implies that it is associated with a limit of acceptability (the predicted situation must be assessed with regard to a referential). Thus, prognostics could be split into 2 sub-activities: a first one to predict the evolution of a situation at a given time, and a second one to assess this predicted situation with regards to an evaluation referential. Let's resume (figure 1):

-identification: a situation is captured by the detection process and additional current measures, -prediction: the situation is forecasted in time, -assessment: a situation is evaluated by the use of performance criteria, -prognostics: a predicted situation is assessed.

In addition, a central problem appears: the accuracy of a prognostics system is related to its ability to approximate and predict the degradation of an equipment; the prediction phase is a critical one. Developments of this paper emphasizes on this aspect of prognostics. In pratice, the prediction step must be supported by operational tools which choice is important. Fig. 1 Prognostics as a prediction and assessment process [START_REF] Dragomir | Framework for a distributed and hybrid prognostics system[END_REF] First order TS systems for prediction / forecasting The aim of this part is not to dress an exhaustive overview of prediction approaches but to explain the orientations of works that have been taken. For more details on prognostics approaches, see [START_REF] Vachtsevanos | Intelligent Fault Diagnostic and Prognosis for Engineering Systems[END_REF][START_REF] Lebold | Open standards for condition-based maintenance and prognostics systems[END_REF].

Real systems are complex and their behaviour is often non linear, non stationary. These considerations make harder a modelling step, even impossible. Yet, a prediction tool must deal with it. Moreover, monitoring systems have evolved and it is now quite easy to online gather data. According to all this, data-driven approaches have been increasingly applied to prediction problems in general and to machine prognostics in particular, and works emphasize on the interest of using hybrid systems for prediction purpose. More precisely, first order Takagi-Sugeno (TS) fuzzy models have shown improved performances over conventional approaches [START_REF] Wang | Prognosis of machine health condition using neuro-fuzzy systems[END_REF][START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF], and they appear to be adequate to perform the degradation modeling step of prognostics [START_REF] El-Koujok | Error estimation of a neuro-fuzzy predictor for prognostic purpose[END_REF].

First order Takagi-Sugeno systems: principle

A first order TS model can be seen as a multi-model structure consisting of linear models that are not necessarily independent [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF]. It is based on the fuzzy decomposition of the input space. For each part of the state space, a fuzzy rule can be constructed to make a linear approximation of the input. The global output is a combination of the whole rules.

A TS model is composed of 5 layers. Consider figure 2. This model has two inputs variables. Two membership functions (antecedent fuzzy sets) are assigned to each one of them. The TS model is finally composed of two fuzzy rules. (That can be generalized to the case of n inputs and N rules). The rules perform a linear approximation of inputs:

1 n i 1 i n i i i0 i1 1 in n R : if x is A and ... and x is A THEN y =a +a .x +...+a .x (1)
where R i is the i th fuzzy rule, N is the number of rules,

X t = [x 1 , x 2 , …,
x n ] T is the input vector, A i j denotes the antecedent fuzzy sets, j= [1,n], y i is the output of the i th linear subsystem, and a iq are its parameters, q= [1,n].

In layer 1, let assume Gaussian antecedent fuzzy sets to define the regions of fuzzy rules in which the local linear submodels are valid:

i i * i2 j j j ȝ =exp -4 x-x (ı ) § • ¨© ¹ (2) Π Π Ν Ν Σ y x 1 x 2 R 1 R 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A L1 L2 L3 L4 L5 Π Π Ν Ν Σ y x 1 x 2 R 1 R 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A L1 L2 L3 L4 L5 Π Π Ν Ν Σ y x 1 x 2 R 1 R 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A Π Π Π Π Ν Ν Ν Ν Σ y x 1 x 2 R 1 R 2 R 1 R 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
Fig. 2 A First-order TS model where σ i j is the spread of the membership function, and x i* is the focal point (center) of the i th rule antecedent. The firing level (τ i ) and the normalized firing level (λ i ) of each rule are obtained as follows (layers 2 and 3):

IJ =ȝ (x )×...×ȝ (x ) i i1 1 in n , N i i j j=1 Ȝ = IJ IJ ¦ (3)
The model output is the weighted averaging of individual rules' contributions (layers 4 and 5):

T N N i i i e i i=1 i=1 y= Ȝ y = Ȝ x ʌ ¦ ¦ (4)
where π i = [a i0 , a i1 , …, a in ] is the vector parameter of the i th sub-model, and x e = [1 X T ] T is the expanded data vector.

A TS model has two types of parameters. The non-linear parameters are those of the membership functions (the center and the spread deviation for a Gaussian membership). These kinds of parameter are referred to as premise or antecedent parameters. The second types of parameters are the linear ones that form the consequent part of each rule (a iq in eq. 1).

III. A SELF BUILT NF SYSTEM FOR PREDICTION

Fitting a neuro-fuzzy systems: critical steps

Assuming that a TS model can approximate an input-output function, in practice, this kind of model must be tuned to fit to the studied problem (prediction in our case). This implies at least four complementary tasks to be performed.

1. Choice of input signals. This step aims at determining which variables are relevant to model the desired input-output relation, i.e., to identify the exogenous variables of the phenomenon. It can be expert made or result from processing techniques (feature extraction and selection). This problem is not addressed in this paper, but for subsequent explanations, let note X = [x 1 , x 2 , …, x n ] the vector of exogenous variables that enables to estimate y d , the endogenous variable.

2. Inputs definition. Time series predictions are usually made by considering some past values of each variable. In order to make a prediction, one has to select the set of regressors for modeling (ϕ s ), within the global set of potential regressors (ϕ) of both the exogenous and endogenous variables.

Formally, and assuming h the horizon of prediction, θ the vector parameter of the model, at any time t, prediction for time t+h can be expressed as in equation 5.

3. Design of the structure. In a few worlds, one has to choose the number and type of membership functions for each input (layer 1) and the number of rules which depends on the numbers of connections in between layers 2 and 3. The design of a TS model is thereby quite flexible. Nevertheless, choices made at this step directly influence the accuracy performance of the model as well as its complexity (as more membership functions and rules are chosen, as more parameters have to be tuned). Note that thanks to its generalization capabilities, Gaussian membership's functions are usually used.

4. Fitting the parameters. The parameters of a TS model must be tuned to fit to the studied problem. This is the aim of the learning procedure and different approaches can be used. In all cases, the consequent parameters of the system (the linear ones) are tuned by using a least squares approach.

d d y 1 1 x 1 s s n n x n y t 1 y t q x t 1 x t p y t h f x t 1 x t p ϕ θ ϕ ϕ - - °- - °+ = ∈ = ® °- - °! ! " ! ( ) ( ) ( ) ( ) ˆ( ) ( , ), ( ) ( ) (5)

Reducing human influence thanks to evolving algorithms

Whereas the first of these four steps is not studied in this paper, the second one is discussed in section IV. Followings paragraphs present the ways of identifying TS model and enables to discuss the above points 2 and 3.

The simplest method to construct a TS fuzzy system is the "mosaic scheme": the user defines the architecture of the model and the antecedent's parameters values [START_REF] Espinosa | Fuzzy Logic, Identification and Predictive Control, Advances in Industrial Control[END_REF]. This approach must be leaved aside to reduce the influence of practitioner. Gradient descent algorithms have been adapted to the TS structure in order to calculate the antecedent parameters by the standard back-propagation procedure (see for example the ANFIS system proposed by [START_REF] Jang | Neuro-fuzzy modeling and control[END_REF]). In the same way, genetic algorithms can be used to compute the fitting of antecedent parameters. These approaches require the user to choose the number of membership's functions, and to initialize various algorithms parameters. To face with it, clustering approaches (and some genetic algorithms) have been adapted to neurofuzzy systems, the basic idea being to use a learning data set to automatically generate the adequate structure of the TS model (number of membership functions, and of rules). However, in practical applications, the learning process is effective only if sufficient data are available and, when trained, such a TS model is fixed. Thereby, if the behavior of input and/or output changes significantly with regards to the learning phase (like in a degradation process), predictions can suffer from the lack of representative learning data. In order to continuously integrate the dynamic of signals, evolving algorithms have finally been developed [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF][START_REF] Kasabov | Denfis: Dynamic evolving neural-fuzzy inference system and its application for timeseries prediction[END_REF]. These algorithms are based on clustering methods and therefore, do not require the user to define the structure of the TS model. In opposition to all previous approaches, they do not need a complete learning data set to start the identification process of the TS model (start from scratch): they have a flexible structure that evolves with the data gathered from the system. Data are collected continuously which enables to form new rules or to modify an existing one.

This kind of self constructing evolving predictors are thereby very interesting for prognostics applications where it is very difficult, even impossible, to formalize the behavior of the system. A particular evolving TS model is that one proposed by [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF][START_REF] Angelov | Evolving fuzzy systems from data streams in real-time[END_REF]: the evolving extended TS system (exTS).

Theoretical exTS backgrounds

The learning procedure of exTS is composed of two phases: [START_REF] Leger | Integration of maintenance in the enterprise: towards an enterprise modeling based framework compliant with proactive maintenance strategy[END_REF] an unsupervised data clustering technique is used to adjust the antecedent parameters, (2) the supervised recursive least squares learning method updates consequent parameters. Both algorithms can not be fully detailed in this paper.

The exTS clustering phase processes on the global inputoutput data space: z = [x T ; y T ] T , z ∈ R n+m , where n+m defines the dimensionality of the input/output data space. Each one of the sub-model of exTS operates in a sub-area of z. This clustering algorithm is based on the calculus of a potential which is the capability of a data to form a cluster (antecedent of a rule). The procedure starts from scratch and, as more data are available, the model evolves by replacement or upgrade of rules. This enables the adjustment of the antecedent parameters.

The exTS model is used for online prediction: at step k, equation 4 can be expressed as follows:

T T N N k 1 i i i e i k k i 1 i 1 y y x λ λ π ψ θ + = = = = = ¦ ¦ ˆ (6) T T T T T k 1 e 2 e n n k ȥ =[Ȝ x , Ȝ x ,...,Ȝ x
] is a vector of the inputs, weighted by normalized firing (λ) of the rules, and

T T T T k 1 2 N k ș =[ʌ , ʌ ,..

.,ʌ ] ˆˆˆâ

re parameters of the sub-models. An estimation of the parameters based on k data samples is obtained by applyong the following RLS procedure:

T k k 1 k k k 1 k k 1 C y k 23 θ θ ψ ψ θ - + - = + - = ˆˆ( ) ; , ,... (7)
T T k k 1 k 1 k k k 1 k k 1 k C C C C 1 C ψ ψ ψ ψ - - - - = - + (8)
with initial conditions

T T T T 1 1 2 N k 1 0 C I θ π π π = = = Ω [ , ,..., ]
,

,
where Ω is a large positive number, ΩΙ a R(n+1)×R(n+1) covariance matrix.

Complexity of an exTS system

The architecture of an exTS neuro-fuzzy system is singular in that the number of rules is equal to the number of membership's functions by input. Thereby and assuming that exTS is a firstorder TS model, the number of linear parameters can be simply determined by the relation nb l = N×(n+1), where, N is the number of rules and n, the number of inputs. As being Gaussian, each membership has 2 parameters to be tuned and the total amount of non-linear parameters is expressed by nb nl = 2×n×N. The total number of parameters of an exTS can be interpreted as an indicator of the complexity of the model:

exTS l nl nb nb nb N 3 n 1 = + = × + ( ) (8)

IV. BALANCING COMPLEXITY AND GENERALIZATION CAPABILITY

Problem statement

According to previous work [START_REF] El-Koujok | Development of a prognostic tool to perform reliability analysis[END_REF][START_REF] Angelov | Evolving computational intelligence systems[END_REF], exTS is an effective forecasting tool: it can perform accurate predictions while minimizing the user's assumptions. Nevertheless, whereas the number of rules is automatically computed in the clustering phase of the exTS learning procedure, the number of inputs must be set by the practitioner (which is the right set of regressors? -see the second critical step defined in previous page). According to equation 8, this implies a more or less complex fuzzy structure and the prediction performance follows from it. Thereby, the purpose of this section is to propose a way to automatically generate an accurate exTS prediction system that reaches a compromise between complexity and generalization capability. The approach is based on the Akaike Information Criterion (AIC).

Akaike Information Criterion

Akaike Information Criterion (AIC) has been introduced in order to provide a mathematical formulation of the principle of parsimony in the field of model construction [START_REF] Akaike | A new look at the statistical model identification[END_REF]. This criterion enables to judge from the quality of fit of an estimator and can be used with prediction models. It is defined as follows:

( )

AIC S log L 2 p = × + × (9)
where, S is the number of data samples used when fitting the model, p, the number of independently adjusted parameters within the model, L a likelihood function of the model, i.e., a loss function that traduces the accuracy of past estimations:

S 2 d i 1 1 L y y S = = - ¦ ( ) (10)
According to equations 9 and 10, AIC is composed of two parts. The first one (S.log(L)) enables to weight the importance of the accuracy of the estimator (/ predictor), whereas the second one (2×p) aims at weighting the complexity of the model: -AIC is as low, as the accuracy is good (logarithmically), -AIC is as high, as the number of parameters is high (linearly). Various identification / prediction models can be ranked according to their AIC, the more suitable been that of the lowest ones (suitable in the sense of "compromise in between accuracy and complexity"). Consider figure 3 for an illustration.

Note that the AIC criterion can be implemented without the aid of subjective judgment.

Balancing complexity and accuracy of an exTS predictor

As stated in equation 5, in order to build a prediction model, one has to define the set of regressors of both the exogenous and endogenous variables.

Let note O = {p x1 , p x2 , …, q y } the order of the model containing the number of past values used for prediction for each input. Depending on this choice, many forecasting systems can be constructed. In addition, and thanks to equation 8, the AIC of all these potential models can be easily computed during the learning phase. Following that, one is able to determine which model presents the lowest AIC and is thereby providing a compromise between complexity and accuracy. This procedure is depicted in figure 4.

V. EXPERIMENTS AND DISCUSSION

Object

The aim of previous sections was to point out a suitable prediction technique for prognostics purpose, for which any assumption or intervention of the practitioner is necessary, i.e., to propose a way to go through self built systems. That can be reached thanks to two aspects that are illustrated in this section:

-the evolving exTS model is an accurate prediction tool that outperforms classical approaches, -the procedure of input selections derived from the AIC criterion enables to automatically generate a fuzzy architecture that balance accuracy and complexity. The Auto Regressive eXogenous approach (ARX) has been used for comparisons: this model is well known and extensively employed in modeling and prediction of times series. Given a set of inputs x i , an ARX prediction can be expressed as:

n i i i=1 A(w). y(t)= B w x t ¦ ˆ( ). ( ) (11)
where,

q 1 y 1 q y p x i 1 i i i 1 p x i A w 1 w w B w w w w x t x t τ α α β β τ - - - - - = + + + = + + = - " " ( ) ( ) . ( ) ( )
As for the exTS model, let note O = {p x1 , p x2 , …, q y } the order of an ARX model, with total amount of parameters:

ARX y x x 1 n nb q p p = + + + " (12)
Two real experimental data sets have been used for tests. In both cases, the aim of predictions is to approximate a physical phenomenon by learning data gathered from the system. That can be assimilated to the prediction step of the prognostics process.

First data sets: industrial dryer

The first data set is issued from an industrial dryer. It has been contributed by Jan Maciejowski from Cambridge University [24]. Three exogenous variables (x 1 , x 2 , x 3 ) are linked to one endogenous variable (y):

x 1 : fuel flow rate, x 2 : hot gas exhaust fan speed, x 3 : rate of flow of raw material, y: dry bulb temperature. This data set has been used to evaluate the ability of an exTS system to predict a non-linear function. This has been made with comparison with the ARX model. For both models, the procedure of input selection based on AIC criterion has been applied in order to identify the a priori more suitable structure, i.e., that one that balances complexity and accuracy (let name them the exTS comp and ARX comp models). According to equations 8 and 12, the complexity of the models can be easily calculated. The prediction performance was assessed by using the root mean square error criterion (RMSE) which is the most popular prediction error measure and the coefficient of determination (R2) which is a measure of how well future outcomes are likely to be predicted by the model:

The data set contains 876 samples. The learning phase was stopped after 500 data samples, and the remaining samples served to test the models at one step ahead. Results are shown in table 1 and figure 5. According to accuracy performance indicators and as shown in figure 5, there is no doubt that the ARX model is not able to catch the behaviour of the dry bulb temperature (R2=-4,4846 !), whereas the exTS model presents good results, even if partially defined thanks to the AIC criterion approach.

ARX comp exTS comp O = {p x1 , p x2 , p x3 , q y } { 9 , 1 , 6 , 6 } { 1 ,
Considering our final applicative objective (prognostics of failures), to be capable to carry out predictions on such a signal is of good omen: real systems are complex and have generally a nonstationary and non-linear behaviour, what makes difficult a modelling phase. Working with this data series constitutes a first step to the specification of a prognostics system able to take into account the dynamic of real systems without prior knowledge and human intervention.

Second data sets: air temperature in a mechanical system

The second data set is issued from an hair dryer. It has been contributed by W. Favoreel from the Kuleuven Unversity [25].

The air temperature of the dryer is linked to voltage of the heating device. This data have inside one exogenous variable (x 1 ) and one endogenous variable (y):

x 1 : voltage of the heating device, y: air temperature. This data set has been used to evaluate the ability of the input selection procedure based on AIC to reach a compromise between generalization and complexity. Similarly to the first data set, the procedure has been applied to both models ARX and exTS and leaded to two a priori more suitable structures ARX comp and exTS comp . In order to extract more solid conclusions from the tests and expand the comparison to a same model (and not from one to another), the most accurate models have also been searched by scanning all the set of possible inputs (thanks to the set of potential regressors ϕ). Let name them the ARX accur and exTS accur models.

The data set contains 1000 samples. The learning phase was stopped after 500 data samples, and the remaining samples served to test the models at one step ahead. Results are shown in table 2 Results enable to notice that, whatever the model of prediction is, the accuracy grows as the complexity increases: the more the RMSE is near from zero (or R2 near from 1), the more parameters the model has. This seems to mean that accuracy and complexity are actually correlated and that there is no way to maximize the satisfaction on both criteria.

Following that, the interest of using the automatic input selection based on AIC can be examined. Whereas the most accurate prediction model for both systems (ARX accur and exTS accur ) has much more parameters than this one obtained by the input selection procedure (ARX comp and exTS comp ), the accuracy of prediction are quite the same. As for an example, consider figure 7 that shows the probability density function (pdf) of the error of prediction for exTS comp and exTS accur models. Both pdf curves are much closed together. In other worlds, the proposed approach really enables to balance complexity and generalization capability and directly choose an adequate prediction structure.

Remind that it can be very difficult to define a prediction structure for prognostics purpose. The proposed approach, combined with evolving capability of the exTS system provides practitioners with a simple methodology to directly go through a suitable prediction system (self built).

Conclusions

The purpose of the work reported in this paper is to point out an accurate prediction technique to perform the approximation and prediction of the degradation of an equipment. Indeed, in one hand, choosing an efficient technique for prognostics depends on constraints that limit the applicability of the tools, and in the other hand, there is no systematic way to construct a prognostics tool since the user can make some assumptions: choice of a structure, initialization of parameters… This is the problem addressed in the paper: how to build a prognostics system with no human intervention, neither a priori knowledge?

According to real implementation restrictions, neurofuzzy systems appear to be well suited for practical problems where it is easier to gather data (online) than to formalize the behavior of the system being studied. More precisely, the paper points out the accuracy of the exTS model in prediction. The exTS model has a high level of adaptation to the environment and to the changing data. It is thereby an efficient tool for complex modeling and prediction. Moreover, any assumption on the structure of exTS is necessary, which is an interesting characteristic for practical problems in industry.

Nevertheless, practitioners have to choose the inputs of a NF predictor. In order to face with it, an approach based on the use of the AIC criterion is proposed. It consists in building a cost function that takes into account simultaneously the accuracy of predictions and the complexity of the model. Various inputs can easily be tested in a computational procedure in order to automatically generate an accurate prediction system that reaches a compromise between complexity and generalization capability.

Finally, the whole paper aims at proposing a self-built system for prognostics. Developments are at present extended in order to test the proposition on real degradation data. In addition, other criterions are studied in order to directly integrate the underlying idea of the procedure into the learning algorithms.

Fig. 3

 3 Fig. 3 Evolution of AIC

Fig. 4

 4 Fig. 4 Procedure to build a "suitable" exTS

Fig. 5

 5 Fig. 5 Industrial dryer -predictions at one step ahead

Fig. 7

 7 Fig. 6 Air temperature -input selection results

Table 1 .

 1 Industrial dryer -simulation results

	1 , 2 , 3 }

Table 2 .

 2 and figure6. Air temperature -simulation results

		ARX comp	exTS comp
	O = {p x1 , q y }	{ 18 , 2 }	{ 1 , 3 }
	nb parameters p	20	117
	AIC	-6,5405	-1775
	RMSE test	0,1066	0,0962
	R2 test	0,9820	0,9854
		ARX accur	exTS accur
	O = {p x1 , q y }	{ 85 , 1 }	{ 4, 3 }
	nb parameters p	86	242
	AIC	-6,3487	-1116
	RMSE test	0,0819	0,0402
	R2 test	0,9894	0,9974


Thibaut Divoux

David Tamarii

Catherine Barentin

Sébastien Manneville

Transient Shear Banding in a Simple Yield Stress Fluid

Keywords: numbers: 83.60.La, 83.50.Ax, 83.50.Rp

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Yield stress fluids (YSF) encompass a large amount of everyday-life and industrial complex fluids ranging from hair gels, cosmetic creams, or toothpastes to many food products or even concrete and drilling muds. They respond elastically below a certain stress threshold σ c , known as the yield stress, whereas they flow as liquids if stressed above σ c . The behavior of YSF still raises many fundamental and practical issues, in particular in the vincinity of this solid-to-fluid transition, often referred to as the unjamming transition [START_REF] Weeks | chapter in Statistical Physics of Complex Fluids[END_REF]. Recently it was proposed to categorize YSF into two groups: (i) those with aging and memory effects that present thixotropic properties (clay suspensions, colloidal gels, adhesive emulsions) and (ii) those for which such effects can be neglected, which are called simple YSF (foams, carbopol "gels", nonadhesive emulsions) [START_REF] Ragouilliaux | [END_REF]3]. Since they present glassylike features, thixotropic materials have received a lot more of attention in the past decade [4][5][6]. Their yielding properties result from a competition between aging and shear-induced rejuvenation which leads to a shear banding phenomenon involving a critical shear rate γc [7][8][9][10]. Indeed, for an imposed shear rate γ > γc , the whole fluid flows, while for γ < γc , one observes the coexistence of a solid region with a vanishing local shear rate γloc = 0 and a fluidized region sheared at γloc = γc . Besides, various models in terms of hidden local degrees of freedom such as the fluidity [11] or the structural "λ parameter" [4,7] account for the corresponding steady-state velocity profiles. On the other hand, simple YSF show a continuous yielding transition in the sense that the local shear rate can take any nonzero value. In that case, the flow may appear as localized only due to stress gradients that result from the strong curvature of wide-gap cylindrical cells: in small-gap geometries, steady-state flow profiles are homogeneous above yielding [10,[12][13][14][15]. Nonetheless, lately, subtle effects have been reported that somewhat blur the above distinction. First, non-thixotropic nonadhesive emulsions, for which magnetic resonance velocime-try coupled to wide-gap rheology shows simple yielding behavior [12], were found to present shear banding when confined into microchannels [16]. Second, thixotropic Laponite suspensions, that exhibit shear banding in a rough geometry, were shown to slowly evolve towards homogeneous flow in the presence of smooth boundaries [17]. These findings evidence the influence of confinement and boundary conditions on the yielding scenario and call for a systematic investigation of the spatiotemporal dynamics of YSF.

In this Letter we revisit the case of a simple YSF, namely a carbopol "gel". Indeed, in spite of a huge body of rheological work, no thorough temporally and spatially resolved study has ever been conducted on simple YSF. In particular, it is generally assumed that the flow remains homogeneous during its evolution towards steady state. Here shear-rate controlled experiments show that (i) the transient regime from solidlike to liquidlike behavior involves a shear-banded flow; (ii) the duration τ f of this transient regime decreases as a power law of the imposed shear rate: τ f ∝ γ-α ; and (iii) the flow always turns out to be homogeneous after this transient shear banding. This fluidization scenario is very robust and the exponent α does not depend on the gap width or on the boundary conditions. Our results not only confirm the idea that one cannot observe steady-state shear banding without thixotropy but also demonstrate that a simple YSF can present shear banding in a transient regime that can last surprisingly as long as 10 5 s.

Experimental set-up and protocol -Our working system is made of carbopol ETD 2050 dispersed in water at 1% wt. This system is referred to as a carbopol "gel" in the literature although its microstructure shows a honeycomb-like assembly of swollen polymer regions with typical size 5-20 µm [18,19]. Carbopol "gels" are known to be non-aging, non-thixotropic simple YSF [13,20]. Here our samples are seeded with micronsized glass spheres [21] in order to use ultrasonic speckle ve- locimetry (USV) [22] simultaneously to standard rheological measurements with an Anton Paar MCR301 rheometer. The addition of such acoustic contrast agents only slightly stiffens the system (by at most 25%) and rheological measurements performed without glass spheres show the exact same trends as in Figs. 1 and2.

The carbopol gel is poured into the gap of a rough Couette cell (rotating inner cylinder radius 23.9 mm, gap width e = 1.1 mm and height 28 mm). The surface roughness of 60 µm obtained by gluing sandpaper to both walls was chosen to be of the order of the microstructure size in order to avoid wall slip. To ensure that the strain accumulated during loading into the cell has no influence, the sample is systematically presheared for 1 min at +1000 s -1 and for 1 min at -1000 s -1 . We then check that a reproducible initial state is reached by measuring the viscoelastic moduli G ′ = 100 ± 10 Pa and G ′′ = 12 ± 2 Pa for 2 min. Finally the sample is left at rest for 1 min before starting the experiment.

Constitutive behavior -Figure 1 shows that the Herschel-Bulkley model accounts perfectly for the flow curve of our carbopol system in good agreement with previous works [13,[18][19][20]. In particular, the absence of a stress plateau in the flow curve at low shear rates in a rough geometry is usually taken as the signature of simple yielding behavior [3,15]. However we observe a slight hysteresis between decreasing and increasing shear-rate sweeps for γ 1 s -1 (see inset of Fig. 1). Such an effect is also noticeable but not discussed in previous reports [ 13,20]. We checked that the size of the hysteresis cycle decreases as the waiting time is increased from 100 s to 500 s per point. We shall see below that it may take a surprisingly long time to reach steady state at low shear rates. Therefore we will attribute the slight hysteresis in the flow curve to a critical slowing down of the system dynamics close to yielding and not to any memory effect.

Start-up experiments -The above sweep test shows that a better method to probe yielding is to impose a constant shear rate γ over a long duration (> 500 s) and to record the shear stress σ(t). Meanwhile we measure profiles at about 15 mm from the cell bottom through USV [22]. Such a combined technique gives access to both global and local dynamics of the fluidization process. As shown in Fig. 2 for γ = 0.7 s -1 , the shear stress follows a two-step relaxation which can be analyzed in light of the velocity profiles: (i) for t 2000 s, a shear band is present at the rotor side and slowly expands across the gap [Fig. 2(a)-(c)] and (ii) for t 2000 s, the velocity profiles becomes linear within a few 100 s [Fig. 2(d)] and the flow remains homogeneous and stationary up to experimental uncertainty for the rest of the experiment [Fig. 2(e)]. The "fast" fluidization at t ≃ 2000 s is better seen in Fig. 3(a) where the width δ of the shear band is plotted against time. Comparing Fig. 2 and Fig. 3(a), we interpret the plummet of σ(t) around t = 2000 s as the rheological signature of the end of shear-banding.

The velocity profiles of Fig. 2(a),(b) clearly point to the presence of strong wall slip at the rotor even though rough surfaces were used. The complete data analysis [START_REF]). The effective shear rate is defined as γeff = [v(0)-v(e)]/e. The width of the shear band is obtained as the abscissa of the intersection between[END_REF] presented in Fig. 3 The above experiments and analysis were repeated for shear rates ranging from 0.01 to 20 s -1 and for various gap widths e = 1.1-3 mm and wall roughnesses ("smooth" raw Plexiglas with a roughness of about 15 nm or 60 µm rough sandpaper). The scenario described in Figs. 2 and3 was found to be very robust. Note however that, depending on the applied shear rate, the bump in σ(t) does not always show as clearly as in Fig. 2: for γ 2 s -1 , it gets buried into the initial decay while for γ 0.2 s -1 the stress jump becomes very small. Therefore local velocity measurements are essential to assess the nature of the stress relaxation towards steady state.

In all cases a shear band is observed transiently over a "fluidization time" τ f defined as the time when homogeneous flow is reached as indicated in Fig. 3(a). The width of the fluidized zone keeps a similar evolution as γ is varied, although the initial value of δ seems to depend on γ [Fig. 4(a)]. As a major result of this Letter, we observe that τ f decreases as a power law for increasing shear rates [Fig. 4(b)]. For γ 1 s -1 , τ f is larger than

FIG. 4: (a) Width δ of the shear band normalized by the gap width e vs time for different applied shear rates γ = 3.0, 2.0, 1.5, 1.0, 0.7 (gray), 0.2, 0.08 s -1 from left to right. Both walls have a roughness of 60 µm. The time interval between two successive velocity profiles was set to be at least 1 s and inversely proportional to γ in order to focus on the long-time relaxation. For each curve, the fluidization time τ f is extracted as indicated in Fig. 3 100 s and fluidization times as long as 10 5 s were measured for γ 0.1 s -1 . Such ultraslow dynamics at low shear rates undoubtedly account for the small hysteresis in the flow curve of Fig. 1 as well as for the stress relaxations reported in Fig. 2 and in [13]. The absence of any clear dependence on the gap width as well as the fact that flow curves and stress relaxations very close to those of Figs. 1 and2 were measured in both cone-and-plate and plate-plate geometries (data not shown) imply that the fluidization process does not significantly depend on geometry-induced stress and strain heterogeneities.

Discussion -As already implied by recent works [13,20], we confirm through local velocity measurements in small-gap Couette cells that steady-state velocity profiles are homogeneous for a simple YSF. More importantly our study brings about a number of findings that contradict the simple picture of a fast, continuous transi-tion from solidlike to liquidlike behavior under a constant shear rate. A central result is the scaling law τ f ∝ γ-α for the fluidization time. From the experimental point of view, it means that it is crucial to check whether the flow has indeed reached a steady-state, especially at low shear rates, before drawing conclusions from standard rheological data such as the flow curve or the stress response. Note that the stress in Fig. 2 has not fully relaxed after 10 4 s even though the velocity profiles measured roughly at middle height of the Couette cell are perfectly stationary. This suggests that fluidization may not occur in a fully homogeneous way along the vorticity direction due to end effects at the cell bottom or at the free surface.

The above fluidization is reminiscent of the slow fragmentation and erosion process triggered by slippery walls in thixotropic Laponite suspensions [17], for which the characteristic time was also shown to scale as a power law with a similar exponent but with a critical shear rate γ * below which fluidization is never observed: τ f ∝ ( γγ * ) -2.1 [26]. Together with the recent simulation results on sheared attractive particles showing transient shear banding [27] and earlier observations of ultraslow transients in concentrated emulsions [28], the present findings suggest that a generic yielding mechanism involving fluidization times with a critical-like behavior may be at play and go beyond the distinction between simple and thixotropic YSF based on the steady-state.

Providing a model for such a mechanism, in particular for the exponent α, appears as a major challenge for future work. Although α is very robust for a given batch as shown in Fig. 4, it should be mentioned that values of α ≃ 3 were obtained for a different sample preparation, hinting that α is a material parameter that may vary from one simple YSF to the other. Up to now theoretical approaches have focused on steady states [7,11,[START_REF] Bocquet | [END_REF] or on the viscosity bifurcation under imposed shear stress [9,25] so that a comparison of our experiments with existing models is not yet available. Still the most striking feature of the transient regime is the presence of shear banded flows with strong wall slip in spite of rough boundary conditions that are usually assumed to prevent slip phenomena. This observation suggests a yielding mechanism where the initially solidlike material suddenly fails at the inner wall independently of boundary conditions and leaves a strongly sheared lubrication layer at the rotor which then fluidizes the solidlike sample. Moreover the sudden acceleration of δ(t) around τ f [Fig. 4(a)] and the absence of gap dependence rule out propagative or diffusive processes but rather points to an avalanche-like behavior although no thixotropy is present. Such ingredients should be included in future tentative models for the observed dynamics.

FIG. 1 :

 1 FIG.1:The flow curve, shear stress σ vs the shear rate γ, obtained by decreasing γ from 100 to 0.01 s -1 (•) and then increasing γ from 0.01 to 100 s -1 (•) with a waiting time of 100 s per point. Note the slight hysteresis for γ 1 s -1 which is the signature of a significant slowing down of the system dynamics near the yielding transition (see text). The gray line is the best fit of the decreasing part by the Herschel-Bulkley model: σ = σc + A γn , with σc = 26.9 Pa, n = 0.55 and A = 9.8 Pa.s -n . Inset: zoom over the low shear rate regime. Both walls have a roughness of 60 µm.

FIG. 2 :

 2 FIG. 2: The shear-stress σ vs time t for a shear rate γ = 0.7 s -1 applied at t = 0. Insets: velocity profiles v(r), where r is the distance to the rotor, at different times [symbol, time (s)]: ( , 27); (△, 1013); (⋄, 1730); (▽, 2250); (•, 6778); (•, 9413). The arrow indicates the rotor velocity v0. Both walls have a roughness of 60 µm. The two-step relaxation of σ(t) is strongly correlated to the evolution of v(r) (see text).

FIG. 3 :

 3 FIG. 3: Analysis of the velocity profiles recorded every 18 s during the experiment at γ = 0.7 s -1 shown in Fig. 2. (a) Width δ of the fluidized shear band normalized by the gap width e vs time. The shear band invades the whole gap over a characteristic time τ f . (b) Evolution of the effective shear rate γeff normalized by the imposed shear rate γ vs time. (c) Local shear rates γloc close to the walls vs time. (d) Slip velocities vs normalized by the rotor velocity v0 vs time. In(c) and (d), gray (black resp.) symbols correspond to the rotor (stator resp.). The errorbars indicate the uncertainty due to our spatial resolution of 40 µm and to the procedure used to fit the velocity profiles[START_REF]). The effective shear rate is defined as γeff = [v(0)-v(e)]/e. The width of the shear band is obtained as the abscissa of the intersection between[END_REF].

  (b)-(d) shows that smaller but noticeable slip is also present at the stator and that slip velocities as well as the local shear rate close to the rotor decrease concomittantly to the growth of the fluidized band [Fig. 3(c),(d)]. Wall slip becomes completely negligible only when homogeneous flow is reached [Fig. 3(b)].

  FIG.4:(a) Width δ of the shear band normalized by the gap width e vs time for different applied shear rates γ = 3.0, 2.0, 1.5, 1.0, 0.7 (gray), 0.2, 0.08 s -1 from left to right. Both walls have a roughness of 60 µm. The time interval between two successive velocity profiles was set to be at least 1 s and inversely proportional to γ in order to focus on the long-time relaxation. For each curve, the fluidization time τ f is extracted as indicated in Fig.3(a). (b) Fluidization time τ f vs the applied shear rate γ for various gap widths and boundary conditions (BC). The gray line is a power-law fit τ f = β/ γα with α = 2.3 ± 0.1 and β = 272 ± 11 s 1-α .

We thank the Anton Paar company for lending us an MCR301 rheometer, Y. Forterre for providing the carbopol and E. Bertin, L. Bocquet, A. Colin, P. Chaudhuri and N. Taberlet for fruitful discussions.