FR3133683A1 - Method for predicting the remaining life time of at least one mechanical system and associated device - Google Patents

Abstract

Method for predicting the remaining life time of at least one mechanical system and associated device One aspect of the invention relates to a method for predicting the remaining life time of at least one mechanical system from a reference database comprising a plurality of time series called target and noted series, each time series being associated to a descriptor with an integer greater than or equal to one and to a reference system similar to the mechanical system subject to the prognosis with, , and a plurality of time series called source and noted series associated with at least one system to be characterized, each series source also being associated with a descriptor with . Figure to be published with the abstract: Figure 4C

Description

Method for predicting the remaining life time of at least one mechanical system and associated device TECHNICAL FIELD OF THE INVENTION

The technical field of the invention is that of predictive maintenance.

The present invention relates to a method for prognosticating the remaining life time of at least one mechanical system and an associated data processing device, and in particular a method based on the alignments of time series and making it possible to associate with the prognosis carried out a uncertainty about said prognosis.

TECHNOLOGICAL BACKGROUND OF THE INVENTION

The first mention of a similarity-based method for RUL prognosis (for “Remaining Useful Life” in French) was implemented to adapt it to the CMAPSS dataset (Commercial Modular Aero-Propulsion System Simulation) which contains multivariate signals to study the degradation of a simulated turbojet engine. The advantages of such an approach are multiple. It allows the production of a prognosis with little available data, which is often the case in an industrial environment. In addition, as the lifespan of the product progresses, the comparison database can be enriched and as a result, without changing the method, the precision in determining the RUL will be improved. However, if the database is not sufficiently rich, the forecasts are tainted with high uncertainties: this is the major disadvantage of this method.

The literature on RUL prognosis based on similarity measurements or time series alignment uses methods solely based on data processing or hybrid methods. After the selection of operating data and the extraction of descriptors for the study, it is necessary to quantify the similarity between two time series. To do this, a large part of the state of the art uses the Euclidean distance evaluated on a sliding window to develop an RUL, a weighted sum of all the intermediate RULs obtained for each comparison between a test descriptor and a training descriptor. . In certain methods of the state of the art, a principal component analysis (PCA) makes it possible to extract a signal which is a global HI (Health Index) health indicator for all the sensor data. The signal obtained is modified by a Kalman filter then normalized. Then the PSO algorithm for particle swarm optimization (Particle Swarm Optimization) is used in conjunction with a regression according to the SVR (Support Vector Regression) method to predict the future behavior of the HI indicator. The particles created are compared to the training points to obtain a similarity score from which the weights of the fusion model are derived. The first five similarity scores are then selected as weights to obtain the final RUL. This latest work was carried out for training data up to the end of life. Still with the same principle of measuring similarity, other methods of the state of the art use deep learning to obtain the prediction of the remaining life time.

Other work has developed an approach to measuring similarity that is not based on Euclidean distance, for example using the MMD (Maximum Mean Discrepancy) measure calculated on a sliding window. This is used as a similarity measure and then the KTST (Kernel Two Sample Test) is used to define the candidates most similar to the test sequence. The RUL is calculated by adjusting the parameters of a Weibull distribution, which makes it possible to obtain the remaining life time.

In still other methods, the similarity measure used to measure the similarity between two segments of different sizes is the DTW (Dynamic Time Wrapping) measure. A rectangular matrix containing the Euclidean distance between each pair of points is created for the two segments. The path which crosses the matrix by minimizing the costs (elements of the matrix) on its passage makes it possible to obtain a set of values whose sum represents the similarity measure. Thus, it is possible to define the resemblance between two data segments. The problem of degradation rates and time jumps are also addressed by defining a fitting function for the data. The weights for the final fusion model, to obtain the RUL, are chosen by a so-called self-adaptive model. Two sets of weights are generated, each from a model of decreasing exponentials based on the similarity of a segment and the error.

State-of-the-art methods, however, have several drawbacks. In particular, and not exhaustively: these methods require a selection of descriptors in order to obtain a coherent prognosis; the measure of similarity depends on the size of the series; few methods accompany the RUL prognosis with a measure of uncertainty about this prognosis.

The invention offers a technical solution to these different problems, by allowing a set of descriptors to be taken into account, without prior selection, the selection of relevant descriptors being induced in the creation of the weighting factors. Furthermore, in the present invention, the similarity measurement between two time series is not carried out on a sliding window and does not use a simple Euclidean distance between the test points (or source series) and the training points ( or target series). As a result, this measurement is valid for any series size and the alignment procedure makes it possible to automatically deduce the interval over which to carry it out, allowing automatic temporal rescaling. Next, the similarity evaluation function of the present invention implements a new cost function developed specifically to quantify the results of series alignments. From this cost function, weights are derived to obtain a prognosis of the RUL, not from a weighted sum, but as a solution to an optimal transport problem. Finally, the present invention also allows the quantification of the uncertainty in the prognosis thus determined.

• Pour chaque descripteur , pour chaque série cible associée audit descripteur et pour chaque système à caractériser , une étape d’alignement de la série source associée au descripteur considéré à la série cible considérée de sorte à identifier, pour chaque point de la série source considérée un et un seul point de la série cible considérée, un déplacement des points sources étant associé à chaque alignement ;
• Pour chaque descripteur , pour chaque série cible associée audit descripteur et pour chaque système à caractériser , à partir de l’alignement effectué lors de l’étape précédente, une étape de détermination d’une fonction de coût quantifiant la qualité de l’alignement entre la série source associée au descripteur considéré et la série cible considérée associé au même descripteur , la fonction de coût étant déterminée à partir du déplacement des points de la série source considérée avec la série cible considérée lors de l’étape d’alignement ;
• pour chaque descripteur et pour chaque système à caractériser , une étape de sélection, à partir de la fonction de coût, de la série cible la plus proche de la série source associée au descripteur et au système considéré ;
• Pour chaque série cible sélectionnée à l’étape précédente et à partir de la fonction de coût calculée pour la série cible considérée, une étape de détermination d’une pondération associée à la série cible considérée ;
• A partir des coefficients de pondération déterminés à l’étape précédente, une étape de fusion des séries cibles sélectionnées de sorte à obtenir une estimation de la position du dernier point de la série source considérée après alignement ;
• à partir de la position du dernier point de la série source , une étape de détermination de la durée de vie résiduelle avant défaillance du système à caractériser.

For this, a first aspect of the invention relates to a method for predicting the remaining life time of at least one mechanical system from a reference database comprising a plurality of time series, called target series and noted , each target series being associated with a descriptor with with an integer greater than or equal to one and a reference system similar to the mechanical system subject to the prognosis with , and a plurality of time series called source series and noted , associated with at least one system to be characterized , each source series also being associated with a descriptor with , the process comprising:

• For each descriptor , for each target series associated with said descriptor and for each system to be characterized , a step of aligning the source series associated with the descriptor considered at the target series considered so as to identify, for each point of the source series considered one and only one point of the target series considered, a movement of the source points being associated with each alignment;
• For each descriptor , for each target series associated with said descriptor and for each system to be characterized , from the alignment carried out during the previous step, a step of determining a cost function quantifying the quality of the alignment between the source series associated with the descriptor considered and the target series considered associated with the same descriptor , the cost function being determined from the displacement of the points of the source series considered with the target series considered during the alignment step;
• for each descriptor and for each system to be characterized , a selection step, from the cost function, of the target series closest to the source series associated with the descriptor and the system considered ;
• For each target series selected in the previous step and from the cost function calculated for the target series considered, a step of determining a weighting associated with the target series considered;
• From the weighting coefficients determined in the previous step, a step of merging the target series selected so as to obtain an estimate of the position of the last point of the source series considered after alignment;
• from the position of the last point of the source series , a step of determining the residual life before failure of the system to be characterized.

In addition to the characteristics which have just been mentioned in the previous paragraph, the process according to a first aspect of the invention may present one or more complementary characteristics among the following, considered individually or in all technically possible combinations.

• Une sous-étape de projection des points de la série source et de la série cible considérées selon une direction de projection prédéterminée ;
• Pour chaque point de la série source, une sous-étape de détermination du point de la série cible le plus proche après projection ;
• Pour chaque point de la série source, une sous-étape d’advection vers le point de la série cible le plus proche après projection, l’amplitude du déplacement lors de cette sous-étape d'advection étant fixée par la distance entre le point source et le point cible considérés ;

In one embodiment, the alignment step comprises, for each descriptor , for each target series associated with said descriptor and for each system to be characterized :

• A sub-step of projecting the points of the source series and the target series considered according to a predetermined projection direction;
• For each point of the source series, a sub-step of determining the closest point of the target series after projection;
• For each point of the source series, an advection sub-step towards the closest point of the target series after projection, the amplitude of the displacement during this advection sub-step being fixed by the distance between the point source and target point considered;

These three sub-steps are repeated until each source point of the source series has joined a target point of the target series.

In one embodiment, the projection direction is an optimal projection direction associated with the descriptor considered identical to each iteration and chosen from a plurality of projection directions.

In an alternative embodiment, the projection direction is an optimal projection direction associated with the descriptor considered and chosen from a plurality of optimal projection directions associated with the descriptor considered and the alignment step comprises, after the substep d advection, when the amplitude of the displacement is less than a threshold value, a sub-step of determining a new optimal projection direction associated with the descriptor considered by the plurality of optimal projection directions associated with the descriptor considered.

In one embodiment, the step of determining the residual lifespan also includes determining an uncertainty on said residual lifespan, the uncertainty taking into account the uncertainty in determining the association between the last point of the source series and the points of the target series considered as well as the weighting associated with each selected target series.

In one embodiment, the alignment step is implemented iteratively and the cost function determined during the step of determining a cost function of the following form:

Or is the number of iterations necessary for the alignment of the source series considered with the target series considered, is the number of points in the source series considered, is the ordinate of the source point indexed by at the first iteration and is the ordinate of the source point indexed by at iteration .

In one embodiment, the weighting associated with the target series relating to the descriptor is given by the following relation:

Or is the normalized cost function associated with the series relating to the descriptor determined using the following relationship:

Or is the normalized cost function associated with the series relating to the descriptor , is the median of the cost functions of the selected target series, is the third quartile and the first quartile.

• lors d’étape de fusion des séries cibles sélectionnées, une incertitude sur la position du dernier point pour chaque série cible sélectionnée est déterminée à partir de la fonction d’assignation relative à ladite série cible, de manière à obtenir, en plus de l’estimation de la position du dernier point de la série source après alignement, une incertitude sur cette estimation ;
• lors de l’étape de détermination de la durée de vie résiduelle avant défaillance du système à caractériser, une incertitude sur l’estimation de la durée de vie résiduelle est déterminée à partir de l’incertitude sur la position du dernier point de la série source.

In one embodiment, the alignment step is an iterative step and during the alignment step, an assignment function is determined, said function representing, during the iteration , for a target series associated with the descriptor , the point of the target series closest to the last point of the source series and:

• during the step of merging the selected target series, an uncertainty on the position of the last point for each selected target series is determined from the assignment function relating to said target series, so as to obtain, in addition to the estimation of the position of the last point of the source series after alignment, an uncertainty in this estimation;
• during the step of determining the residual life before failure of the system to be characterized, an uncertainty in the estimation of the residual life is determined from the uncertainty in the position of the last point of the source series .

In one embodiment, during the determination step, the residual lifespan is determined using the Sinkhorn algorithm and the regulation factor used for this algorithm is an ideal regularization factor determined by dichotomy.

In one embodiment, each time series of the reference base is associated with one operational condition among a plurality of operational conditions, the source series being divided into a plurality of source subseries, each associated with one operational condition of the plurality of operational conditions, and the alignment step, the step of determining a cost function, the selection step, the step of determining a weighting and the merging step are applied to each sub -series in the chronological order of the sub-series, the position of the last point of the sub-series considered being considered as the position of the first point of the following sub-series, the determination of the residual life determined during the The step of determining the residual lifespan is carried out from the position of the last point of the last sub-series.

A second aspect of the invention concerns a data processing device comprising the means for implementing the method according to a first aspect of the invention.

A third aspect of the invention relates to a computer program comprising instructions which, when the program is executed by a computer, lead it to implement a method according to a first aspect of the invention.

A fourth aspect of the invention relates to a computer-readable data carrier, on which the computer program according to a third aspect of the invention is recorded.

The invention and its various applications will be better understood on reading the following description and examining the accompanying figures.

BRIEF DESCRIPTION OF THE FIGURES

The figures are presented for information purposes only and in no way limit the invention.

There shows a flowchart of a method according to the invention.

THE has show the physical quantities acquired on a system during an operating cycle.

THE And show the evolution of two descriptors as a function of the number of cycles.

THE has schematically illustrate the principle of the invention.

THE has illustrate the alignment step according to the invention.

There illustrates the movement of a point of the source series during the alignment step according to the invention.

There illustrates the alignment of a source series associated with a given descriptor with three target series associated with the same descriptor.

There illustrates the creation of a histogram based on the different associations made for the last point of the source series during the alignment step.

There illustrates the fusion of information relating to the weighting and the uncertainty on the index of the last point of the source series.

There illustrates the determination of an ideal regularization factor by dichotomy.

There illustrates the consideration of different operational conditions.

DETAILED DESCRIPTION

A first aspect of the invention illustrated in relates to a method for predicting the remaining life time (also known by the acronym RUL for “Remaining Useful Live”) in English of at least one mechanical system, for example an electromechanical actuator. As will be detailed below, the method according to the invention is based on the alignment of time series. This consists of associating the points of a series with another series from a reference base. The association is then made by minimization of a criterion between the associated points and the points to be associated. The method according to the invention can in particular be implemented using a data processing device comprising the means for implementing this method. More particularly, the data processing device comprises a calculation means, for example a processor or an ASIC card, associated with a memory, the memory being configured to contain the instructions and the data necessary for the implementation of the method according to the 'invention.

General principle of the invention

Reference database, source series and target series

The method is implemented from a reference database comprising a plurality of time series, called target series, each time series being associated with a descriptor with with an integer greater than or equal to one and a reference system with . The reference system associated with each target series is a system similar or identical to the system whose RUL we seek to predict. This reference database can be derived from accelerated tests in a given environment and/or from real operating profiles.

In addition, the method is also implemented using a plurality of time series called source series and denoted associated with each system to be characterized with , each series also being associated with a descriptor with . In the remainder of the description, the system to be characterized will be an electromechanical actuator. However, the principle of the invention applies to any mechanical system (including electromechanical systems) that can be described by a plurality of time series each relating to a descriptor.

Alignment of a set of source series with respect to each set of target and pr series onostic

Thus, for each descriptor and for each system , when aligning the source series associated with said descriptor and system audit in relation to a target series of the reference database associated with the same descriptor , the source points of the source series resulting from measurements carried out on the system to be characterized will be aligned with the target points of the target series considered.

More particularly, as already mentioned, in the following we will define a first time series any size corresponding to the source series and a plurality of second time series size each corresponding to a target series from the reference database. In addition, by way of illustration, the rest of the description will be placed in the case where And (resp. ) designates the set of values of the indices of (resp. ).

In order to constitute the time series, measurements are carried out on the system(s) to be characterized (for the source series) or on the reference system(s) for the target series (in real conditions and/or by accelerated aging). Such measures are illustrated in has which show the position measurement ( ), speed ( ) and acceleration ( ) carried out on four actuator cylinders for a given operating cycle.

From these measurements, a plurality of descriptors is defined, each descriptor being a statistical quantity resulting from the measurements which makes it possible to account for the degradation of the system considered. Two descriptors are presented to And . There represents the RMS value (or effective value) of the vibration signal (or acceleration) during the deployment of the cylinders (deploy in English) and the represents when the cylinders retract (stow in English) depending on the number of cycles. In these two figures, it clearly appears how such a descriptor makes it possible to account for wear over time.

In addition, the measurements are carried out during operating cycles (or use cycle, for example, for a system installed on an aircraft, during each flight) and a value of each descriptor is calculated for each cycle. Thus, each point of a source or target time series is associated with a cycle and a value of a descriptor. In addition, it is possible, for each descriptor, to associate a severity with each point in a time series (a method allowing this association to be carried out is for example described in application FR3098318A1). Preferably, each time series can also be associated with operational conditions corresponding to the conditions of use of the system considered.

Thus, the objective of the alignment in the method according to the invention is to find an injective function which associates with each index a point of (the source series), the index of its neighboring point in (the target series). In the chosen example where the source series has fewer points than the target series, the property is then verified. When the situation is reversed, that is to say when the source series has more points than the target series, then it is necessary to resample the target series. Function characterizes the alignment of the source series on the target series .

In the following, the following notation will be adopted. Signals captured on (an integer such that systems to be characterized (for source series) or on (an integer such that reference systems (for the target series) are extracted statistical descriptors of points for the source series and points for target series. By descriptor we mean here a mathematical quantity determined from the signals captured on the system(s) to be characterized, said mathematical quantity being chosen so as to account for a defect. Here for example, the descriptors are for example relative: to the maximum, to the minimum, to the standard deviation, to the average of the signal measured on the system to be characterized. All the signals measured on the reference systems form a noted reference database Or is the number of points in each target series, is the number of descriptors and is the number of reference systems used to constitute the reference database. Likewise, all of the signals measured on the system(s) to be characterized form a set of source series denoted Or is the number of points in each source series, is the number of descriptors and is the number of systems to characterize.

As already mentioned, it is possible to determine the state of health of a mechanical system (here an electromechanical actuator) from which the measured data (or signals) come by comparing them to those present in the reference database. The source series previously defined (on the data measured on the mechanical system to be characterized) will then be a column vector indexed by the index such as and relating to the descriptor considered, of the matrix . From where . Each target series associated with the same descriptor as for it (that is to say each time series of the reference database) will be noted .

An objective of the method according to the invention is then to align the source series on all target series of the reference database associated with the same descriptor . In other words, the source series associated with a given descriptor will be aligned with the target series associated with the same descriptor. The idea is then to order the candidates in order of resemblance to .

To achieve this alignment, a cost function (this cost function will be detailed below) is defined to associate with each assignment function (this assignment function will be detailed below) a real value. A fusion of information for each of the classified parameters is then carried out to allow the estimation of the consumed lifespan and a distribution representative of its quality. This alignment step makes it possible to extract information from the data on both the diagnosis and the prognosis of the system. Indeed, for diagnosis, the series in the database are labeled by degree of severity, once the source series is aligned with the target series for the different , the degree of severity of each point of will be assigned to the points of . Finally, for the prognosis, the abscissa of the last aligned point of the source series in the target series makes it possible to obtain the remaining life time of the actuator whose descriptors have been calculated. The RUL is then the time remaining between the last aligned point of the source series and the last point of the target series.

THE has illustrate this alignment method in an example considering only two different time series relating to the same descriptor . More particularly, the illustrates the source series to align. There illustrates a target series of the reference database on which the source series will be aligned as well as the degree of severity associated with each point of the target series. For the record, determining the degree of severity for a point in the time series is something known to those skilled in the art. An example is for example given in application FR3098318A1. The alignment is shown in where each point of the source series is associated with one and only one point of the target series (the way in which this association is made will be illustrated later). Finally, as illustrated in , a prognosis of the remaining life time (or RUL) of the system (here an actuator) is determined as a function of the number of cycles (on the abscissa) separating the last point of the source series after alignment of the last point of the target series.

Alignment, for each descriptor, of the source series in relation to the target series

The method according to the invention comprises, for each descriptor , for each target series associated with said descriptor and for each system to be characterized , a step E1 of aligning the source series associated with the descriptor considered at the target series considered so as to identify, for each point of the source series considered one and only one point of the target series considered, a movement of the source points being associated with each alignment.

• Une sous-étape de projection des points de la série source et de la série cible considérée selon une direction de projection prédéterminée ;
• Pour chaque point de la série source , une sous-étape de détermination du point de la série cible le plus proche après projection ;
• Pour chaque point de la série source , une sous-étape d’advection vers le point de la série cible le plus proche, l’amplitude du déplacement lors de cette sous-étape d’advection étant fixée par la distance entre le point de la série source et le point de la série cible considérés .

In one embodiment, alignment step E1 comprises:

• A sub-step of projecting the points of the source series and the target series considered according to a predetermined projection direction;
• For each point in the source series , a sub-step of determining the point of the target series the closest after projection;
• For each point in the source series , a sub-step of advection towards the point of the target series the closest, the amplitude of the displacement during this advection sub-step being fixed by the distance between the point of the source series and the point of the target series considered .

These three sub-steps are repeated until each source point of the source series has joined a target point of the target series. As already mentioned, during these iterations the points of the source series move so that it is possible to associate an alignment with a movement (or path) of the points of the source series during said alignment.

In one embodiment, an assignment function is used which gives the index of the point of the target series considered to be closest to the last point of the source series during the iteration sub-steps of alignment step E1. The usefulness and details of this assignment function will be detailed later in connection with the .

For the record, in the state of the art, the partial assignment (i.e. alignment) of a source series to a target series in a plurality of dimensions by a method using optimal transport techniques was proposed. This method has been applied to two- and three-dimensional image processing. In the context of the present invention, the inventors have adapted this method to apply it to time series.

The alignment step will now be detailed using the has . More particularly, on the , a source series and a target series relating to the same descriptor are represented in the form of clouds of points (in a two-dimensional space: time on the abscissa and the value of each descriptor on the ordinate), in solid lines for the points of the source series and dotted for the points of the target series . The time series are then considered as two-dimensional objects: one dimension for the time axis and one dimension for the descriptor values associated with each instant.

Additionally, a concatenation matrix can be created from vector . Each column of containing the concatenation of the vector and a vector . To each element is then associated with a temporal index . It is useful to note that, in the example of has , does not represent the absolute timestamp value of the points but a cycle number, that is to say an increment going from has (the number of points of the source). Thus, this new vector only accounts for the precedence relationship between the elements of . A target matrix is defined in a similar way with the difference that it has more points than the source matrix therefore . The creation of these matrices previously makes it possible to carry out the first step E1 of the method according to the invention using optimal transport techniques as previously described.

As mentioned previously and illustrated in , the alignment step E1 according to the invention comprises a sub-step of projecting the points of the source series and the target series considered in a predetermined projection direction. Contrary to what is done in the state of the art, during this sub-step, the source points are projected onto a projection direction that is not random but according to an optimal predetermined projection direction (this notion will be defined below ).

As illustrated in , the alignment step E1 then comprises, for each point of the source series, a sub-step of determining the closest point of the target series after projection. In order to resolve the assignment of the nearest neighbors from the points of the source series to the points of the target series, a state-of-the-art algorithm can be used, for example the algorithm developed in Boneel and Coeurjolly (2019). . Note that each point of the source series is thus associated (or assigned) to a single point of the target series, but that several points of the source series can be associated with the same point of the target series. It should also be noted that, depending on the projection direction chosen, the closest point for a point of the source series considered can vary between two iterations of the sub-steps of the alignment step E1.

As illustrated in , the alignment step E1 then comprises, for each point of the source series, an advection sub-step towards the closest point of the target series, the amplitude of the displacement during this sub-step being fixed by the distance between the source point and the target point considered. Thus, during this sub-step, the points of the source series will move towards the points of the target series assigned to them during the sub-step of determining the closest point of the target series after projection.

It is interesting to note that, in the present invention, contrary to what is done in the state of the art, the displacement of the points of the source series is no longer calculated in the projection space but directly in the final space in two dimensions as shown in . Additionally, during this movement, a displacement amount is calculated for each point in the source series and the points move a given amount on the direction created between them and their respective nearest neighbors in the target series.

An illustration of this movement during the different iterations is given in the for a point of the source series, here the last point of the source series so that towards the point of the target series having been defined as the neighbors of the point for the iteration considered (here, depending on the iteration considered, two different points of the target series are defined as neighbors of the point ). In this figure, the dotted circles represent the positions of the source point at iterations preceding the current iteration.

Function of the is the association function (already introduced) at the iteration for descriptor and the actuator . With each new iteration, the points of the source series will approach the points of the target series until they completely reach the latter. On the , the displacement vectors of the source points are represented by a gray arrow. Either a source point considered at the iteration And a target point corresponding to its nearest neighbor at the iteration . So for each point and each point , the displacement vector has as direction the straight line passing through And , is directed towards the point and, in the example serving as an illustration, has as its norm, or quantity of movement, half the Euclidean distance separating of . Thus, the momentum (represented by the norm of the gray arrow) is calculated at the by dichotomy. In the chosen example, the momentum of the point from source to iteration with Euclidean distance is given by with :

The momentum is then calculated by dividing the remaining distance between And by a postman , with in this example α=2. Either the distance traveled by a point in the source series, at each iteration a distance can be associated with a point. The sequence of distances is then created. Each of these quantities is then added to the coordinates of the points of the respective source series, which generates the movement of the points of the source series towards those of the target series.

Determining the optimal projection direction

As mentioned previously, the projection direction is not chosen randomly, but represents an optimal direction chosen from a set of projection directions made up of several directions . This optimization is carried out only during a learning phase of the algorithm. This learning phase is a multi-criteria optimization allowing the optimal projection directions to be selected. More particularly, the space of all possible projection directions is sampled for a given alignment. To this end, for each descriptor, a projection direction is selected at random then an alignment step as described previously is carried out from the projection direction thus selected. The quality of the alignment carried out using the projection direction considered is then quantified by the cost function of which will be described below. This operation is repeated for a plurality of projection directions (depending on the available computing power). Once this plurality of projection directions has been evaluated using the cost function, a plurality of optimal projection directions is selected, a projection direction being said to be optimal when the cost associated with said projection is less than a predetermined threshold.

This search could for example be carried out using the Optuna library known from the state of the art. Of course, any other resolution algorithm could be used to carry out said search.

In more detail, the procedure for determining the optimal projection direction is carried out by iterations. The equation to solve to find the set of directions is given by:

All represents the set of optimal directions determined for the descriptor and a target series relating to the same descriptor . The whole is obtained by minimizing the sum of the movement costs, on one dimension, of the different states of the source cloud at each iteration. collects these travel costs.

More specifically, for the first iteration , , for the second iteration , and for the last iteration . All on which the cost function is calculated is therefore of size . Each intermediate state of the source constituting it is the result at source cloud transport iteration on target . The function allowing this transport, and consequently the creation of the function which assigns each point of the source to the points of the target, is the function applied to the two projected clouds And . All is then determined by a multi-criteria iterative optimization procedure.

In one embodiment, the projection direction is an optimal projection direction associated with the descriptor considered identical to each iteration and chosen from a plurality of projection directions.

In one embodiment, this procedure is applied to a fixed number of optimal projection directions. The optimal projection directions then become independent of the current iteration of the alignment step E1. In one embodiment, the projection direction is an optimal projection direction associated with the descriptor and chosen from a plurality of optimal projection directions associated with the descriptor considered and the alignment step E1 comprises, after the advection sub-step, when the amplitude of the displacement is less than a threshold value, a sub-step of determining a new optimal projection direction associated with the descriptor considered by the plurality of optimal projection directions associated with the descriptor considered. In one embodiment, the displacement evaluated during each iteration is the average displacement or the median displacement of each point in the series during said iteration. However, other statistical quantities making it possible to account for the overall evolution of the displacement could be used.

In one embodiment, in order to reduce calculation costs, only two directions with And are calculated by descriptor. Thus, in this particular embodiment, only two optimal projection directions are taken into account so that the sub-step of determining a new optimal projection direction is implemented at most only once during of alignment step E1. For example, a direction optimal is used as long as the movement of the points of the source series towards the points of the target series is significant. On the other hand, as soon as this displacement passes below a predetermined threshold value a new direction of projection optimal is then chosen. This makes it possible in particular to prevent the source series from converging towards a local minimum. The number of projection directions is thus greatly reduced.

Calculation of the cost function and selection of the closest target series

Once the alignment is done for each descriptor between the source series and each target series of the reference database it is necessary to determine, for each descriptor , the degree of proximity between the target series and the source series considered. For this, the method 100 according to the invention comprises each descriptor , for each target series associated with said descriptor and for each system to be characterized , from the alignment carried out during the previous step, a step E2 of determining a cost function quantifying the quality of the alignment between the source series associated with the descriptor considered and the target series considered associated with the same descriptor , the cost function being determined from the displacement of the points of the source series considered with the target series considered during the alignment step E1.

This step E2 is preferably carried out using an indicator of internal quality developed by the inventors which allows this quantification without relying on the contribution of expert knowledge from the user or on a priori knowledge of the data involved in this quantification. For this, a matrix which contains all the intermediate states of the second dimension (i.e. the time dimension) of the source series cloud during the alignment step E1 is determined so as to obtain the following matrix:

The developed cost function applies to this matrix and produces a positive real. More particularly, the cost function is defined by:

Or is the number of iterations necessary for the alignment of the source series considered with the target series considered, is the number of points of the source series considered, is the ordinate of th point of the source at the first iteration and is the ordinate of th point of the source (of which all the points have translated) at the iteration . This cost function makes it possible to associate with a given alignment a real value representative of its quality, in other words representative of the resemblance of a source series to the target series on which the alignment is carried out, the quality being a decreasing function of the cost. So the cost function developed by the inventors makes it possible to evaluate the quality of the alignment resulting from the iterative alignment method introduced previously. This then allows the comparison of series alignments with each other.

In an exemplary embodiment used to illustrate the principles of the invention, Statistical descriptors are extracted from the raw vibration signals of each actuator to be characterized. Either the matrix defined as concatenation of descriptors to points on the columns. The matrix is then defined as and it must be aligned with the target series of the reference database . For this, it is possible to introduce a function carrying out all the sub-steps of the iterative alignment step E1 described previously. There represents a schematic diagram to explain the alignment of each descriptor of on the corresponding descriptor in the achievements database { , … } .

For each alignment the cost function is evaluated which makes it possible to obtain a cost matrix . This makes it possible to locate, for each descriptor , the most similar realization (the most similar actuator in our example) to the source series aligned among the target series contained in the reference database. It allows only the cost minima associated with each descriptor to be reported. The target series most similar to the source series for a given descriptor are therefore highlighted.

Once a cost function is associated with each alignment, it is possible to select the target series closest to the source series associated with each descriptor. For this, the method according to the invention comprises, for each descriptor and for each system to be characterized , a selection step E3, from the cost function of the target series closest to the source series associated with the descriptor and the system considered . Thus, for each descriptor, the source series relating to said descriptor is associated with the target series relating to the same descriptor for which the cost of the alignment is the lowest (in other words, the quality of the alignment is the highest). Indeed, from the matrix previously introduced, it is possible to select the best candidates, that is to say the target series closest to the source series of the set of source series.

As a reminder, one of the objectives of the method according to the invention is to determine the instant from which the system to be characterized moves to a higher degree of severity in its aging or reaches its end of life. It is from this information that the residual life before failure (or RUL) of the system could be extrapolated. Once the alignment is done and the most similar target series (and therefore the achievements) identified, it is appropriate to aggregate all this information to extract a point representing the cycle index of the system, in other words its age, at the time of the last measurement associated with the points of the source series.

Determination of weighting

To improve the results of the prognosis method, the contribution of each descriptor (that is to say of each time series of the reference database selected during the previous step E3 and associated with said descriptor) selected will be weighted according to the values present in the cost matrix . For this, the method according to the invention comprises, for each target series selected in the previous step E3 and from the cost function calculated for the target series considered, a step E4 for determining a weighting associated with the series target considered (thus the weighting is also associated with the descriptor of the target series considered).

To this end, it is first necessary to determine a vector of weightings from the minimum cost vector extract from the matrix . contains all minimum evaluations of the cost function per descriptor alignment. In one embodiment, is normalized using a normalization function, for example by normalizing by the maximum value. In another example of implementation, the normalization function is the RobustScaler function from the Python language library scikit-learn which allows you to subtract its median from a distribution and then divide it by its inter-quartile range.

In this expression, the median function used corresponds, in the usual way, to the value of which separates the matrix into two sets of equal cardinals. The inter-quartile range is defined as the extent between the first quartile and the third quartile . Once this robust scaling is applied, the normalized exponential function is applied to the vector of weights calculated previously. Thus, the weighting vector is obtained from the following equation:

The weighting vector therefore assigns a weight to each of the selected target series (each associated with a descriptor, here a descriptor of the actuators) for their resemblance to the source series, that is to say the measurements carried out on the system to be characterized.

Summary of steps taken so far

To summarize the steps E1-E4 which have just been described, the selection of the best candidates consists initially of aligning the source series with all the target series present in the database (during step E1 of alignment) then quantifying each alignment with the cost function created (during step E2 of determining the cost function). The target series (each relating to an actuator) associated with each descriptor and having the minimum cost for each descriptor are selected (during the selection step E3) and allow the creation of an optimal cost vector which, after normalization, makes it possible to obtain a weighting vector (during step E4 of determining a weighting). The latter makes it possible to weight the contribution of the information extracted from the alignment of each target series of the reference database in the information fusion model which will be described below. Thus, at this stage of the process, it is possible to align all of the descriptors of unknown source data with all of the descriptors of all of the achievements contained in the reference database. Furthermore, the developed cost function allows to quantify the quality of such alignments, whether or not there has been temporal compression (as in the case of accelerated tests) and to assign a weighting accordingly.

Determination of an index number corresponding to the last supposed cycle and, possibly, of the uncertainty associated with this determination

At the end of the previous steps, each descriptor is associated with a target series selected for its proximity to the source series and a weighting is associated with each of these target series. Determining the remaining lifespan then amounts to locating the last point of the source series within the known lifespans and behaviors of the target series in the reference database. More particularly, it is possible from the indices associated with the last aligned point of the source series for each descriptor to determine the remaining life time of the system to be characterized as a function of the target series considered.

For any source series, the alignment results make it possible to obtain a vector containing the index of the last point of the source series aligned for each descriptor . For the record, the points of the source series have been moved to those of the target series and therefore the indices of the two point clouds share the same temporal scale. The vector then serves as a basis for creating a matrix full of with a per column, which amounts to creating a matrix by the concatenation of Diracs centered in .

With variables , And already introduced, it is possible to define the matrix using the following equation:

To incorporate the weighting information determined during the previous step E4, the weighting vector is transformed into a square matrix with the weights distributed along its diagonal. To do this, it is useful to introduce the vectors of the canonical basis of . With the elements of the weight vector , the square matrix is obtained using the following relationship:

In the chosen example, the matrix is thus created with as many lines as points in the target series considered, but the proposed method can easily be extended to target series of different sizes. The dimension preserved for the matrix will be the maximum number of points in one of the target series. Furthermore, the matrix has as many columns as descriptors, it is a matrix containing the concatenation of weighted Diracs each representing the index of the last aligned point of a descriptor of the source.

To obtain a final cycle number and thus locate the system to be characterized in the probable distribution of degrees of severity and therefore estimate the residual lifespan, it is appropriate to merge the information from all the columns of so as to obtain an index number (that is to say its position on the abscissa axis) corresponding to the last supposed cycle of the system to be characterized.

For this, the method according to the invention comprises, from the weighting coefficients determined in the previous step E4, a step E5 of merging the selected target series so as to obtain an estimate of the position of the last point of the source series after alignment followed, from the position of the last point of the source series, by a step E6 of determining the residual life before failure of the system to be characterized.

However, to make a prognosis, it may also be advantageous to quantify the uncertainty associated with the determined value of the residual life of the system considered. In this embodiment, the single cycle number is no longer sufficient, which is why the Diracs (representing the exact position of the last points of the source series after alignment on each selected target series) will be replaced by probability densities . The result obtained will no longer be an integer but a density function with values in .

In order to understand the origin of the uncertainty associated with the determination of the position of the last point of the source series after alignment (and therefore of the residual lifespan), it is useful to remember that it is the fusion of all the temporal indices of the last points of each aligned descriptor coming from the same source which makes it possible to estimate the lifetime for this source from those of the candidates in the reference database. It is therefore on this quantity that a measurement of uncertainty would be relevant.

To this end, it is possible to define the random variable which represents the cycle index of the last aligned point at each iteration of the alignment step. The source series associated with a descriptor of the system to be characterized, for example , is aligned with the target series associated with the same descriptor, . An assignment function is calculated at each iteration, during the alignment step, which makes it possible to construct a sequence . The assignment represented by this function is independent of the distance from the points of the source cloud to the points of the target cloud in because it is calculated in the projected space, therefore in . The value of interest for the uncertainty measurement is therefore which represents the nearest neighbor of the last point of the source cloud in the target cloud . Thus, at the end of each alignment a vector is created . The measured uncertainty is therefore an image of the number of different values contained in the vector . These values can be put in the form of a histogram which makes it possible to estimate the confidence that one can have in the result obtained (that is to say the association between the last point of the source series and a point of the target series) as well as its stability.

At the end of the steps described above, a vector of discrete values was obtained (corresponding to the histogram mentioned previously), the latter can be considered as the sample of a population whose individuals would represent the values of the random variable .

In order to estimate the uncertainty of the overall alignment from this sample , it is necessary to determine the probability law that will follow . For this, in the chosen example, the hypothesis according to which the random variable follows a normal distribution parameterized by the mean and standard deviation of the sample Is made. Also, . However, it would be possible to use any distributions.

The probability density of the Normal distribution is a support Gaussian as presented in the following equation:

From this probability density, it is now possible to create a matrix which would be the concatenation on the columns of a distribution of the variable for each descriptor, unlike which contained weighted Diracs.

The different sub-steps which have just been described are illustrated in . At the first sub-step of the , the vector is constructed following the procedure described previously. This example shows the trajectory taken by the last point of the source cloud (in red) towards the points of the target cloud (in blue) during four iterations. To illustrate a usual case in alignment, the source point considered does not converge towards a single target point as the iterations progress. Also :

This vector is represented as a histogram in the center of the . During the second sub-step, from the sample , the average as well as the standard deviation of the sample are calculated. These results allow, during a third sub-step to parameterize the Gaussian representing the density of the probability law followed by the sample so as to obtain a matrix representing the uncertainty by a probability density of the temporal index of the last aligned point for each descriptor.

Now that the matrix representing the uncertainty by a probability density of the temporal index of the last aligned point for each descriptor has been obtained, it remains to weight each of its columns by the elements of the vector weighting determined previously. Starting from the weighting vector , it is possible to construct a matrix created by vector concatenations on the lines. The matrix is then defined as: . By defining the Hadamard product for two matrices, which allows the multiplication of the elements of two matrices terms by terms, by the symbol , the matrix is modified to create the matrix . The latter is therefore defined by the following equation (illustrated in ):

The matrix then brings together two pieces of information to measure the uncertainty of the alignment: the uncertainty linked to the variation of the value of the last aligned point of the source and the quality of the alignment provided by the weighting vector .

Thus, as has just been detailed, in one embodiment of the method according to the invention during step E5 of merging the selected target series, an uncertainty on the position of the last point for each selected target series is determined at from the assignment function relating to said target series, so as to obtain, in addition to the estimation of the position of the last point of the source series after alignment, an uncertainty on this estimation. Then, during the step of determining the residual life before failure of the system to be characterized (which will be described below), an uncertainty in the estimation of the residual life is determined from the uncertainty on the position of the last point of the source series.

One of the advantages of the method according to the invention is that it does not require an a priori selection of the relevant descriptors; the final information is obtained automatically from all of the descriptors by the conjunction of the uncertainty measurement and the alignment weighting. Also, the greater the number of descriptors, the more reliable the final prognosis result will be, with a higher calculation cost in return.

Summary of the steps previously described

In what has just been described, the source data were aligned with a database and the most similar database realizations were selected following the quantification results of each alignment. Subsequently, a set of Gaussians was obtained. Each Gaussian represents the empirical probability density associated with the index of the last aligned point of a descriptor of the unknown source on the same descriptor in the target. Therefore, it may be interesting to merge the information from these functions to obtain a distribution centered on the probable instant, and on the cycle index, corresponding to the estimated current state of the unknown source system (actuator) studied. Determining its remaining lifespan will be possible by crossing this estimate with the results of a partitioning method such as described for example in patent FR1907561 or a clustering method of time series from the reference database giving the different degrees of severity.

Determination of remaining life time

The calculations have been carried out until now with time series defined on an interval of integers. In the following will be redefined as a bounded interval of the body of the real . In the event that, as the life of the product progresses, other model series are collected to enrich the database, the calculations would then be carried out over an interval of size at least equal to .

More generally, the interval of interest can be considered unbounded. Either this interval of . According to the uncertainty measurement method, we can create a set of probability densities . Let us note the set to which they belong. The data used being sampled, the measurements have as support a finite number of points. By introducing the Dirac measure in , the previous measurements can be reformulated as a finite sum of weighted Diracs: with a vector of positive weights of size . Subsequently, the space being fixed, a discrete measure will be completely identified by its weight vector . By abuse of notation, it will be considered that . In the same way the measurement of the descriptor of another actuator in the database will be introduced. Finally will be considered as a metric space equipped with the -usual distance, otherwise called the Manhattan distance.

It is then a question of calculating a generalized centroid, a concept close to the Fréchet mean (Fréchet 1948) which will subsequently be called barycenter. The result would be a generalized barycenter of probability distributions. In a Euclidean space (here defined on the body of previous real numbers) provided with a distance , the barycenter of a set of points to which the weights are respectively associated is defined by Or is the lower bound of a set corresponding to the smallest value of a set (but not necessarily contained in the latter). Considering each as being a distribution , the result would make it possible to calculate the desired global distribution. By solving this optimization problem in Wasserstein space by replacing the usual distance by a Wasserstein distance of order squared, it is possible to determine a distribution representing this notion of barycenter.

An implementation of this resolution method was developed in the library dedicated to optimal transport by Flamary et al. 2021. The function named “ » is then used with a quadratic cost matrix and with the constraints of the optimization problem defined by all the distributions obtained previously. This function implements the resolution algorithm using the so-called Sinkhorn iterative method, well known to those skilled in the art.

Indeed, the latter is used to calculate the value of the function Or is given by the expression . When becomes too small, then the elements of can achieve machine precision and introduce numerical stability problems. In doing so, they become negligible and are stored in memory as zero elements. As the Sinkhorn algorithm progresses, this results in zero division during the final normalization of the two marginals of the calculated transport plan.

Failing to be able to find an algorithmic way to prevent this behavior, the inventors have developed a method in which an ideal regularization factor is determined for the problem by dichotomy, as illustrated by the .

The algorithm described above in pseudo-code details the procedure for selecting the parameter ε developed by the inventors. The principle behind this procedure is to find the smallest possible regularization factor, to get closer to the optimal solution, at the limit of the numerical error. Firstly, an initial value is used to calculate a first instance of the transport function. If no stability error is detected, then the regularization factor is divided by two and the procedure is started again. These steps are repeated until instability is reached. Then, the interval between the penultimate regularization value and the current value is divided by two to obtain the new ε. The procedure is repeated for a certain number of iterations determined according to the case considered.

As has just been described, it is possible, from a source series and a plurality of target series, to determine the temporal index of the last point of the source series considered. In addition, the reference database includes target series corresponding to reference systems (i.e. of the same nature and performance as the actuator to be characterized) up to the end of life. Also, the RUL is then obtained as being the difference between the known end of life time of the study system t _EOL and the estimated temporal index t _match of the last point of the source series considered after alignment.

Taking into account different operational conditions

Furthermore, to enable prognosis from data captured in heterogeneous environments, it is also important that the method can provide relevant results by considering not complete time series, but segments of time series as well as series of sizes heterogeneous. If this modeling makes sense after studying the different operational conditions in the field, a target series can be segmented into sub-series based on the disparities in the conditions of use of the monitored system.

The method according to the invention makes it possible to provide relevant results for incomplete time series, it can be used concretely for the prognosis of a system which would pass through several different operational conditions. For this, in one embodiment, each time series of the reference base is associated with an operational condition among a plurality of operational conditions, the source series being divided into a plurality of source sub-series, each associated with an operational condition of the plurality of operational conditions. In addition, the alignment step E1, the step E2 of determining a cost function, the selection step E3, the step E4 of determining a weighting and a merging step E5 are applied to each source subseries in the chronological order of the source subseries, the position of the last point of the source subseries considered being considered as the position of the first point of the following source subseries, determining the lifespan residual determined during step E6 of determining the residual lifespan being done from the position of the last point of the last source subseries.

The principle diagram is shown in . More particularly, this figure describes the previous embodiment in the case of a source series segmented into two subseries. The first source subseries obtained is compared to the target series stored in the database associated with the same operating conditions. After alignment, a first estimate of the current life cycle and therefore of the lifetime consumption is obtained.

At the end of this first phase, a new recording is carried out with new operational conditions. The corresponding source sub-series is aligned with the target series of the base associated with the same operational conditions, taking as its origin the same aging rate as that obtained in the previous phase.

This is done until the end of the system's life. During each recording, operational conditions may change.

It will be noted that, in the case of an actuator, the end of life is not necessarily the moment from which its physical integrity is compromised but represents the moment from which the actuator is no longer in operation. capable of carrying out its functions.

Also, the reference database can consist of both usual system behaviors and behaviors resulting from accelerated aging tests.

Finally, as indicated previously, this method can be coupled with the classifier presented in patent application FR1907561. Indeed, the latter made it possible to show how to determine the different degrees of severity present in the database. It is then sufficient, after alignment, to determine the corresponding class of the last point of the aligned source series from that of the corresponding target series. Thus, the current state of health of the system to be characterized is determined and the remaining life time or RUL is calculated.

Claims (13)

Method (100) for predicting the remaining life time of at least one mechanical system from a reference database comprising a plurality of time series resulting from measurements carried out on at least one mechanical reference system, called target series and noted , each target series being associated with a descriptor with with an integer greater than or equal to one and a reference system similar to the mechanical system subject to the prognosis with , and a plurality of time series resulting from measurements carried out on the at least one mechanical system subject to the prognosis, called source series and noted associated with at least one system to be characterized , each source series also being associated with a descriptor with , the process comprising:

• For each descriptor , for each target series associated with said descriptor and for each system to be characterized , a step (E1) of aligning the source series associated with the descriptor considered at the target series considered so as to identify, for each point of the source series considered one and only one point of the target series considered, a movement of the source points being associated with each alignment;
• For each descriptor , for each target series associated with said descriptor and for each system to be characterized , from the alignment carried out during the alignment step (E1), a step (E2) of determining a cost function quantifying the quality of the alignment between the source series associated with the descriptor considered and the target series considered associated with the same descriptor , the cost function being determined from the displacement of the points of the source series considered with the target series considered during said alignment step (E1);
• For each descriptor and for each system to be characterized , a step (E3) of selection, from the cost function of the target series closest to the source series associated with the descriptor and the system considered ;
• For each target series selected in the selection step (E3) and from the cost function calculated for the target series considered, a step (E4) for determining a weighting associated with the target series considered;
• From the weighting coefficients determined in the selection step (E3), a step (E5) of merging the selected target series so as to obtain an estimate of the position of the last point of the source series after alignment;
• From the position of the last point of the source series a step (E6) of determining the residual life before failure of the system to be characterized.

Method (100) according to the preceding claim in which the alignment step (E1) comprises, for each descriptor , for each target series associated with said descriptor and for each system to be characterized :

• A sub-step of projecting the points of the source series and the target series considered according to a predetermined projection direction;
• For each point of the source series, a sub-step of determining the closest point of the target series after projection;
• For each point of the source series, an advection sub-step towards the nearest point of the target series, the amplitude of the displacement during this sub-step being fixed by the distance between the source point and the target point considered;

these three sub-steps being repeated until each source point of the source series has joined a target point of the target series. Method according to the preceding claim in which, during the alignment step (E1), the projection direction is an optimal projection direction associated with the descriptor considered identical to each iteration and chosen from a plurality of projection directions. Method according to claim 2 in which the projection direction is an optimal projection direction associated with the descriptor considered and chosen from a plurality of optimal projection directions associated with the descriptor considered, and in which the alignment step (E1) comprises, after the advection sub-step, when the amplitude of the displacement is less than a threshold value, a sub-step of determining a new optimal projection direction associated with the descriptor considered by the plurality of optimal projection directions associated with the descriptor considered. Method according to the preceding claim in which the step of determining the residual lifespan also comprises determining an uncertainty on said residual lifespan, the uncertainty taking into account the uncertainty on the determination of the association between the last point of the source series and the points of the target series considered as well as the weighting associated with each selected target series. Method according to one of the preceding claims in which the alignment step is implemented iteratively and in that the cost function determined during the step of determining a cost function has the following form:

Or is the number of iterations necessary for the alignment of the source series considered with the target series considered, is the number of points of the source series considered, is the abscissa of point at the first iteration and is the abscissa of the point at iteration . Method according to one of the preceding claims in which the weighting associated with the target series relating to the descriptor is given by the following relation:

Or is the normalized cost function associated with the series relating to the descriptor determined using the following relationship:

Or is the normalized cost function associated with the series relating to the descriptor , is the median of the cost functions of the selected target series, is the third quartile and the first quartile. Method according to one of the preceding claims, in which the alignment step is an iterative step and in which, during the alignment step (E1), an assignment function is determined, said function representing, during the iteration , for a target series associated with the descriptor , the point of the target series closest to the last point of the source series and in that:

• during step (E5) of merging the selected target series, an uncertainty on the position of the last point for each selected target series is determined from the assignment function relating to said target series, so as to obtain, in addition the estimation of the position of the last point of the source series after alignment, an uncertainty in this estimation;
• during step (E6) of determining the residual life before failure of the system to be characterized, an uncertainty in the estimation of the residual life is determined from the uncertainty in the position of the last point of the source series.

Method according to one of the preceding claims in which, during step (E6) of determining the residual life before failure of the system to be characterized, the residual life is determined using a Sinkhorn and in that the regularization factor used for this algorithm is an ideal regularization factor determined by dichotomy. Method according to one of the preceding claims in which, each time series of the reference base is associated with an operational condition among a plurality of operational conditions, the source series being divided into a plurality of source sub-series, each associated with a operational condition of the plurality of operational conditions, method (100) in which the step (E1) of alignment, the step (E2) of determining a cost function, the step (E3) of selection, the the step (E4) of determining a weighting and a step (E5) of merging are applied to each source sub-series in the chronological order of the source sub-series, the position of the last point of the source sub-series considered being considered as the position of the first point of the following source sub-series, the determination of the residual life determined during step (E6) of determining the residual life being made from the position of the last point of the last source subseries. Data processing device comprising the means for implementing the method according to one of the preceding claims. Computer program comprising instructions which, when the program is executed by a computer, lead it to implement a method according to one of claims 1 to 10. Computer-readable data carrier on which the computer program according to the preceding claim is recorded.