Recursive lagrangian total least squares estimator

Abstract

The invention relates to a recursive Lagrangian total least squares estimator for estimating parameters of a physical system (100) considered to be linear, such as a motor vehicle. An estimation device (110) comprises a measurement unit (114) configured to obtain successive input measurements Ak and successive output measurements yk of the linear system, and an estimation unit (116) configured to estimate one or more parameters xk recursively from the input and output measurements. An estimation recursion k+1 comprises the following operations: (1), (2), (3), (4), (5), optionally (5'), (6), (7), where Qa,k and Qy,k are weighting matrices; δXk, Ãk and Pk are inter-recursion variables; ζk+1 is an instrumental variable correlated with the variables Ãk; Qz,k, ΔAk, Kk are internal variables; α is a regularisation term and λ is a forgetting parameter.

Description

[0001] DESCRIPTION TITLE: Recursive Lagrangian Estimator of Total Least Squares Technical field of the inventionThe present invention relates to the general field of parameter estimators in a physical system.State of the prior artMore particularly, certain parameters of a real physical system necessary for its use may not be measurable or measured, for example to avoid the costs and complexity of integrating dedicated sensors, when these exist.These unknown parameters can be estimated indirectly from measurements carried out on the physical system, typically input measurements and output measurements.The estimation of unknown parameters is easier for linear physical systems, that is to say whose behavior is described by linear equations. Nonlinear physical systems can be reduced, via simplifications, to linear systems.Also, many nonlinear physics are considered, by approximation, as linear. A category of estimators of linear systems, called OE estimators for "output error", is based on a modeling of the output error, considering the presence of error in the measurements only on one side of the equations of the linear system - the outputs. This category includes, but is not limited to, recursive or non-recursive least squares estimators, the linear Kalman filter and recursive or non-recursive instrumental variable estimators. It is known that OE estimators produce biased estimates if the input measurements are noisy. This is why a second category of estimators of linear systems, called EIV estimators for "errors-in-variables", is preferred. Although they are more complex, these EIV estimators have better performances.They are based on modeling that considers the presence of error in the measurements on both sides of the equations of the linear system - the inputs and the outputs. This category includes, for example, the total Kalman filter, total least squares (TLS) estimators, and total instrumental variable estimators. Some TLS estimators are based on a singular value decomposition (SVD) approach to solve the total least squares problem. The publication "Recursive Generalized Total Least Squares with Noise Covariance Estimation" (Rhode S. et al., 2014) illustrates, for example, a recursive TLS SVD estimator. Other, fewer TLS estimators are based on a Lagrange multiplier approach. The publication “An iterative solution of weighted total least -squares adjustment” (Shen Y. et al., 2011) describes for example an iterative Lagrangian TLS estimator.Recursive TLS estimators are preferable since they allow real-time estimations. Only recursive SVD-type TLS estimators are known to date. Due to the SVD approach, it is difficult to deploy optimization / improvement tools from the literature on classical least squares, unlike the multiplier approach. Lagrange.There is therefore a need to address the aforementioned drawbacks and to improve the catalog of efficient recursive TLS estimators.Disclosure of the inventionThe invention aims to propose a recursive TLS Lagrangian estimator for estimating one or more parameters of a linear physical system.To this end, the inventors have found that the iterative algorithm proposed by Shen Y. et al. can be cleverly converted into a recursive algorithm. This results in a new recursive TLS Lagrangian estimator capable of benefiting from classical least squares optimization techniques, such as the forgetting factor, regularization, instrumental variables and robustness against outliers, among others.In this context, the invention relates to a device for estimating the parameter of a physical system considered to be linear, comprising: a measurement unit configured to obtain successive input measurements Ak and successive output measurements yk of the system. linéaire, an estimation unit configured to estimate one or more parameters xk of the physical system recursively from the input and output measurements, wherein a k+1 estimation recursion (or recursive iteration) comprises the following operations: where k is the recursion index; Qa,k and Qy,k are weighting matrices; δxk, ^̃^^^ and Pk are inter-recursion variables; ^^^^+1 is an instrumental variable correlated to the variables ^̃^^^; Qz,k, ΔAk, Kk are internal calculation variables; α is a regularization term and λ is a positive forgetting parameter less than or equal to 1. Such a device thus makes it possible to obtain a recursive TLSLagrangian estimator to estimate in real time parameters of a linear physical system. In particular, the number of operations per recursion remains limited; the operational complexity is of the order of O(n 2). Such a TLS estimator can in particular be used as a "virtual" sensor of a parameter not directly measured. Note that λ=1 eliminates any forgetting; α=0 removes any regularization and ^^^^+1 = ^̃^^^+1 removes any correction of noise correlated to the measurement signals.Correspondingly, a method for estimating the parameter of a physical system considered as linear, comprises the following steps:- obtaining successive input measurements A k and successive output measurements y k of the linear system,- estimating one or more parameters x k of the physical system recursively from the input and output measurements, in which method a k + 1 estimation recursion comprises the following operations:(1) ^^^^,^^+1 = ^^^^,^^+1 + ^^^. ^ ^ ^ ^^ ^^,^^+1 ^^ ^^ (2) ΔA = −1^^+1 −^^^^,^^+1^^^^^^^,^^+1 (^^^^+1 − ^^^^+1^^^^ − ^̃^^^^^^^)(3) ^̃^^^+1 = ^^^^+1 − Δ^^^^+1(4) ^^ = ^^ ^^ ^^ (^^^^ ^^ −1 ^^+1 ^^ ^^+1 ^^,^^+1 + ^̃^^^+1^^^^^^^^+1) 1 (5) ^^ ^^+1 = ^^(^^^^ − ^^^^+1^̃^^^+1^^^^)optionally (5') ^^^^+1 = ^^^^+1(^^^^ + ^^^^^^+1) −1(6) ^^^^^^+1 = ^^^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^^ − ^̃^^^+1^^^^^^)(7) ^^^^+1 = ^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^).Optional features of embodiments are defined in the appended claims. Some of these features are explained below with reference to a device, while they can be translated into method characteristics.In one embodiment, Qa,k and Qy,k are input and output measurement error or noise covariance matrices, respectively. This arrangement makes it possible to take into account heterogeneous noise distributions on the input and output data. In a particular embodiment, Qa,k and Qy,k are evolving matrices, and the estimation unit comprises a matrix estimation sub-unit configured to determine for each recursive iteration the matrices Qa,k and Qy,k from the input and output measurements respectively.The estimation of the unknown parameters is improved. Of course, it is also possible to use prefixed matrices Qa,k and Qy,k, and therefore constant from one recursion to another. Such matrices are for example determined upstream of the recursive iterations, from tests and / or specifications of sensors used in the measurement unit. In one embodiment, λ is 1 and / or α is 0 and / or ^^. ^^+1is worth ^̃^^^+1. This allows to disable some optimization tools taken from the literature on classical least squares. In another embodiment, the forgetting parameter λ is strictly less than 1 and / or the regularization term α is strictly positive small compared to 1 and / or the instrumental variable ^^^^+1 is worth ^̃^^^−^^ where a is a positive integer. One or more optimization tools are thus activated.In one embodiment, the measurement unit comprises a sub-unit for pre-processing measurements acquired from one or more sensors, to generate the successive input measurements A k and output yk , the pre-processing sub-unit being configured to perform at least one of the following operations: - a temporal alignment of input or output measurements by interpolation of the acquired measurements and / or by sub-sampling of the acquired measurements, and - a partial selection of measurements to generate the successive input measurements A k and output yk , the selection being a function of a concomitant behavior of the physical system. By the temporal alignment, the risk of error bias inherent in the joint use of non-concomitant measurements is reduced. The partial selection makes it possible to exclude measurements likely to introduce errors, in particular measurements which result from conditions which do not respect linearization hypotheses of the system. physique.These preprocessings result in an improvement in the estimation. In one embodiment, the physical system is a vehicle and the partial selection only selects measurements concomitant with a rectilinear movement of the vehicle. Indeed, a linear modeling of the vehicle can be envisaged for a rectilinear movement. The acquired measurements make it possible to determine the nature of the concomitant movement, rectilinear or not. A yaw rate can be calculated at each recursion. This rate is compared to a threshold value defining the boundary between the rectilinear movement domain and the non-rectilinear movement domain. The invention finds multiple applications. In one embodiment, the physical system is a vehicle and the parameter(s) include a mass of the vehicle. Thus, a preferred application is the estimation of the mass of a vehicle, typically a land vehicle, without a dedicated sensor.In this case, the measuring unit can be connected to sensors to measure a vehicle engine torque, a longitudinal acceleration of the vehicle as well as an angular speed of the engine and an angular speed of the wheels so as to determine a longitudinal speed of the vehicle and a transmission ratio. These measurements thus obtained make it possible to estimate recursively, by use of the invention, the mass of the vehicle, a linear system resulting from the balance of forces according to Newton's second law. By way of illustration, the estimation of the mass of the vehicle makes it possible to determine the vertical load undergone by each tire. Such information in turn makes it possible to accurately estimate tire wear, tire grip and tire rolling resistance during the phases of use, in order to take appropriate measures.For example, knowledge of the overall vehicle load allows for better prediction of the battery life of electric vehicles (EVs), whose transported load can vary considerably over time. Improved routing of EVs to charging stations can then be proposed, optimizing battery usage and wear. Other automotive applications include estimating the effective rolling radius of a tire, the vehicle's center of gravity, the aerodynamic coefficient, tire grip, tire wear, tire load, tire stiffness (Kx, Dz), vehicle speed, etc. Similar applications exist in the aeronautical, aerospace, and geodesic fields, to name a few. At least part of the methods according to the invention can be implemented by computer.Accordingly, the present invention may take the form of an all-hardware embodiment, an all-software embodiment (including firmware, resident software, microcode, etc.), or an embodiment combining software and hardware aspects, all of which may be collectively referred to herein as a "circuit," "module," or "system." In addition, the present invention may take the form of a computer program product embodied in any non-transitory recording medium having computer-readable program code for performing the above method. A tangible or non-transitory medium may include a storage medium such as a hard disk drive, a magnetic tape device, or a solid-state memory device and the like.Brief description of the drawings Other objects, characteristics and advantages of the invention will appear on reading the following description, given solely by way of non-limiting example, and made with reference to the appended drawings in which: [Fig. 1] illustrates a physical system considered to be linear, for an implementation of the invention; [Fig. 2] illustrates, using a flowchart, steps of a method for estimating a parameter of a physical system, according to embodiments; [Fig. 3] illustrates, using a flowchart, more detailed steps of a method for estimating a parameter of a physical system, according to embodiments; [Fig. 4] illustrates a physical system of the motor vehicle type, the linear behavior of which is defined by Newton's second law; [Fig. 5] illustrates the application of the estimation method to the system of figure 4; and [Fig.6] illustrates a computer hardware architecture of an estimation system or device according to embodiments.Detailed description of at least one embodimentA section of estimation theory is concerned with linear systems, these being able to be physical systems whose behavior is by nature linear or physical systems whose behavior is reduced, by approximation or simplification, to a linear behavior.Figure 1 illustrates a physical system 100 considered as linear, that is to say whose behavior is in accordance with a system of linear equations y = Ax. The dimensions n and m of the vectors x and y respectively depend on the physical system considered. The matrix A is of dimension m * n.The example of Figure 4 subsequently describes a motor vehicle as a linear physical system. In this example, n = 3. et m = 1.Some parameters of the physical system are not measurable or cannot be obtained directly from other measurements. These parameters, typically those forming the vector x, can be estimés.The system 100 comprises for this purpose a device 110 for estimating the parameter of the physical system and a processing unit 120 configured to exploit the parameter(s) estimated by the device 110 for the operation of the physical system 100. According to the invention, the estimation device 110 operates recursively to estimate the unknown parameter(s) x. Subsequently, the index 'k' refers to the recursion or recursive iteration 'k'. The processing unit 120 can perform computational processing on the estimates xk obtained in order to have specific parameter values ​​for decision-making. By way of illustration, the estimation of the mass of an electric vehicle using the device 110 allows the processing unit 120 to calculate the overall load of the vehicle and consequently activate or authorize certain driving modes, for the purpose of saving energy. batterie.Although the processing unit 120 is shown embedded in the system 100, it may be external, in which case the estimates produced (and any other measurements obtained) by the device 110 may be transmitted, by wire (for example, Ethernet network) or wireless (for example, a Wifi network or Bluetooth link – trade names), to the external unit 120. For example, the processing unit 120 may be implemented in a remote processing server that manages a fleet of vehicles and transmits commands acting on them in return. Similarly, although the estimation device 110 is shown embedded in the system 100, it may be external, meaning that measurement sensors are external to the monitored system 100. The estimation device 110 comprises one or more sensors 112, a measurement unit 114 and an estimation unit 116.The sensors 112 are configured to acquire, in time t, measurements or “observations” o(t) of the physical system 100. The sensors can be embedded in the monitored physical system. In the example of figure 4, an embedded torque meter measures the engine torque of the vehicle, an accelerometer measures the longitudinal acceleration of the vehicle, and gyrometers measure the angular speeds of the engine and the wheels. Beyond this example, any type of sensor can be used. The measurement unit 114 obtains successive input measurements Ak and successive output measurements yk of the linear system for each recursion k of estimation of the unknown parameters xk . All or part of the measurements A k and yk may correspond to acquired measurements o(t) transmitted by the sensors 112. Nevertheless, some of the measurements A k and yk are obtained by computational processing of the acquired measurements o(t), as will be apparent from the example described below.To this end, the measurement unit 114 may comprise a pre-processing sub-unit 115. The measurements Ak and yk are provided as input to the estimation unit 116. This unit implements a recursive estimator ^̃^ of the Lagrangian TLS type to estimate the parameters xk recursively from the input measurements A k and output yk . According to one embodiment of the invention, the recursive scheme of the estimator comprises the following computational operations for the recursion k+1: (1) ^^^^,^^+1 = ^^^^,^^+1 + ^^^. ^ ^ ^ ^^ ^^,^^+1 ^^ ^^ (2) ΔA^^+1 = −^^^^,^^+1^^^^^^ −1^^,^^+1 (^^^^+1 − ^^^^+1^^^^ − ^̃^^^^^^^^)(3) ^̃^^^+1 = ^^^^+1 − Δ^^^^+1 (4) ^^ ^^ ^^+1 = ^^^^^̃^^^+1 (^^^^^^,^^+1 + ^̃^^^+1^^^^^̃^ ^ ^ ^ −1 ^ +1 ) 1 (5) ^^ ^^+1= ^^(^^^^ − ^^^^+1^̃^^^+1^^^^)(6) ^^^^^^+1 = ^^^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^ − ^̃^^^+1^^^^^^)(7) ^^^^+1 = ^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^)Qa ,k and Qy ,k are weight matrices. They make it possible to take into account heterogeneous noise distributions on the input and output data. In one embodiment, these are measurement error (or noise) covariance matrices at the input and output respectively. These matrices can be prefixed, i.e. fixed across the different recursions, in which case the index 'k' can be omitted. These matrices can then be determined upstream of the estimation operations by the estimator 116, typically by test series set / or using the technical specifications of the sensors 160 involved. In an embodiment offering more precise estimations, the matrices Qa,k and Qy,k are evolving matrices, i.e. they are re-evaluated or recalculated for each new recursion k.In this case, they are also provided as inputs to the recursive scheme above. The estimation unit 116 can thus include a matrix estimation sub-unit 117. This unit 117 determines for each recursive iteration the matrices Qa ,k and Qy ,k from the input Ak and output yk measurements respectively. The matrices Qa ,k and Qy ,k can be recalculated entirely at each recursion or be updated, then resembling a recursive reevaluation. The variables δxk , ^̃^^^ and Pk of the estimator 116 are inter-recursion variables, that is to say transmitted from one recursion to the other. In a Lagrangian TLS approach, δx k is an estimate of the incremental correction of the parameters xk between two recursions, ^̃^^^ is a corrected matrix of Ak taking into account the predicted residual matrix ΔAk (hereinafter), and Pk is a noise covariance matrix. Qz ,k , ΔAk, Kk are, for their part, internal calculation variables.λ is a positive forgetting parameter (or factor) less than or equal to 1. This parameter can be adjusted by an operator of the device 110. This forgetting factor, present in operations 4 and 5 of the estimator, is used to exponentially forget past data, so that more importance is given to more recent estimates. The forgetting is all the more important as the parameter λ is small. Beyond forgetting old information, the parameter λ also allows to converge more quickly towards the true parameters in case of poorly initialized parameters, as well as to smooth the parameter estimates when the system is corrupted by too much noise. For an application without forgetting old measurements over the recursions, λ is set to 1. For an application with forgetting, λ can be set to a value strictly between 0 and 1.The recursive scheme of the estimator can be seen as a function ^̃^(^^^^ , ^^^^^^, ^̃^^^, ^^^^, ^^^^,^^+1, ^^^^,^^+1, ^^^^+1, ^^) where the first four parameters of the function are the parameters to be estimated and the three inter-recursion variables, and where the last four parameters come from the input and output observation measurements. The estimation unit 116 thus produces the estimate xk during the recursion k , intended for the processing unit 120. The estimation device 110 can thus be seen as a virtual sensor. The appendix below demonstrates that these recursive computational operations correspond to a Lagrangian estimator of total least squares. Figure 2 illustrates, using a flowchart, steps of a method for estimating the parameter of a physical system considered to be linear. This method is notably implemented by the estimation device 110 described above. During an initial step 200, the recursive estimator ^̃^ is initialized.This step consists of initializing the values ​​of the variables used during the first recursion (k=1), in particular the parameters x0 and the inter-recursion variables δx0 , ^̃^0, P0. In one embodiment, x 0 is set to the plausible values ​​of the corresponding unknown parameters. By "plausible" we mean any value in the range of values ​​that can be expected for the parameter concerned. As an illustration, with reference to the example in Figure 4, the mass of a vehicle is set to 1000 kg. δx0 is initialized with a zero vector and ^̃^0 with a zero matrix. The covariance matrix P 0 is, for its part, initialized po . Id where Id is the identity matrix and po is a variance factor whose value can be set to the square of the maximum uncertainty associated with the parameters x. The initial step 200 also makes it possible to recover calculation parameters such as the forgetting factor λ and the weighting matrices Qa, Qy if they are prefixed.These calculation parameters can be stored in the memory of the estimation device 110. Step 210 consists, for the estimation device 110, in obtaining the successive input measurements A k and the successive output measurements y k . During this step, measurements o(t) are acquired by the sensors 112, transmitted to the measurement unit 114 which extracts input and output measurements from the linear model for each new recursion k. The acquired measurements o(t) can also be preprocessed to generate input and output measurements of the linear model when, for example, these input and output measurements are not directly acquired by the sensors. These input measurements A k and output yk for the next recursion are transmitted to the estimation unit 116. In step 220, the latter estimates the parameter(s) xk for the recursion k , by performing the computational operations described above.This results in estimated parameters xk, which are communicated to the processing unit 120 in step 230. Step 240 then consists of incrementing the recursion counter k, before looping back to step 210 to perform the recursion. suivante.The recursion frequency may be set by an operator of the estimation device 110. In the example of the vehicle of Figure 4, a recursion period of between 10 milliseconds (ms) and 500ms is compatible with off-the-shelf sensors and real-time operation. Figure 3 illustrates, using a flowchart, more detailed steps of a method for estimating a parameter of a linear physical system, according to embodiments. This method is still implemented by the estimation device 110. The step 200 of initializing the recursive estimator ^̃^ remains unchanged. The step 210 of obtaining input measurements A k and output y k comprises a sub-step 212 of acquiring the measurements o(t) and a sub-step 214 of pre-processing the acquired measurements. The acquisition 212 of the measurements o(t) consists for example of obtaining the measurements carried out by the sensors 112.Depending on the sensors implemented, the measurements acquired may have different sampling frequencies and / or may not be synchronized in time (e.g., to the nearest millisecond). Therefore, these acquired measurements o(t) may be preprocessed in step 214. In one embodiment, the preprocessing comprises a time alignment of input or output measurements. This is the purpose of step 2140: to ensure that the input measurements A k and output y k which will be supplied to the estimator 116 correspond to the same time instant, in order to reduce estimation biases. When the acquired measurements o(t) are not synchronous, the preprocessing can consist of an interpolation of the acquired measurements, for example a linear interpolation between the two measurements o(t 1) and o(t2) available on either side (instant t1) and (instant t2) of the instant (tk) corresponding to the recursion k.Typically, the preprocessed measurement ok corresponds to o(t1) + [o(t2)+o(t1)] * [(tk-t1) / (t2-t1)]. In the case of a sampling frequency higher than the recursion frequency, in particular in a ratio greater than 2, the preprocessing may consist of a subsampling of the acquired measurements, for example equal to the ratio between the two aforementioned frequencies. Interpolation and subsampling may be combined to provide preprocessed acquired measurements ok. In particular, subsampling advantageously precedes interpolation. In one embodiment, the preprocessing comprises the calculation2142 of one or more input measurements A k and / or output measurements yk, depending on the acquired measurements o(t), advantageously preprocessed measurements ok. This involves converting the acquired data into useful data of the linear system y = Ax. This sub-step 2142 makes it possible to generate the input measurements A and output yk which could not be acquired directly by the sensors 112.In the example of Figure 4 discussed below, step 2142 makes it possible to calculate the longitudinal speed of the vehicle from acquired measurements of the angular speed of the wheels (taking into account the radius of the wheels) and to calculate the transmission efficiency of the vehicle from acquired measurements of the angular speeds of the engine and the wheels. In another embodiment, the preprocessing comprises the partial selection 2144 of measurements to generate the successive input measurements A k and output y k . This partial selection aims to exclude irrelevant measurements, for example measurements not corresponding to the conditions or hypotheses allowing the physical system to be considered as being linear. Also, this partial selection is a function of a concomitant behavior (i.e. at the same time as the measurements concerned) of the system. physique.The partial selection 2144 can be performed on the measurements o(t) acquired from the sensors 112 or on the preprocessed acquired measurements o kissed from the sub-step 2140 or (as shown in the figure) on the input measurements Ak and output yk obtained directly from the sensors 112 (large arrow in the figure) and / or from the sub-step 2140 and / or from the calculations 2142. As a result, a subset of the measurements can finally be transmitted as input to the estimation step 220. The sorting criteria for the selection are determined in the sub-step 2146, and applied during the selection sub-step 2144. The sub-step 2146 is for example a function of the acquired measurements o(t) (possibly preprocessed, ok). In the example of Figure 4, the sorting criterion is the value of the yaw rate of the vehicle. Substep 2146 for example calculates the yaw rate for recursion k as a function of a speed difference between the rear wheels at time tk.Alternatively, it can obtain the yaw rate from a dedicated sensor 112. The measurements (acquired o(t), ok or input / output Ak, yk) are discarded when the vehicle is not moving in a straight line. This results in the selection of measurements which are concomitant with a low yaw rate, typically below a threshold value, for example 0.5 radiant per second, preferably 0.1 radiant per second e. The preprocessings mentioned above (2140, 2142, 2144) can be implemented jointly as in the example. Alternatively, only one of the preprocessings can be implemented, or two. The selected input A k and output y k measurements are then provided as input to the estimation step 220. The optional substep 222 is implemented when the weighting matrices Qa , k and Qy , k are not prefixed. In this case, they are estimated from the input A k and output y k measurements.Typically, an input noise covariance matrix Q a ,k can be updated from the input measurements A k , and thus be evaluated recursively. Similarly, an output noise covariance matrix Qy ,k can be updated from the output measurements yk , and thus be evaluated recursively. When the matrices Q a ,k and Qy ,k are available, they are provided, together with the input measurements A k and output yk , as inputs to the recursive estimation algorithm 224 already described. This results in estimated parameters xk , which are communicated to the processing unit 120 in step 230. Step 240 then consists of incrementing the recursion counter k, before looping back to step 210 to perform the next recursion.Figure 4 represents a physical system of the automobile type, whose behavior defined by Newton's second law (balance of forces) can be reduced to linear behavior with some simplifying assumptions (for example wind speed). nulle) . The forces involved include: the inertial force F inertia whose value is (m+me)ax, with m the mass of the vehicle and me the equivalent mass of the rotating parts, the traction force F d ^^ ^^^^^^ .^^.^^ ^^^^^^^^^^^^^^tr ac t have the value worth^ ^ , where Tmo t ^^^^^^^^is the engine torque, ^ is the transmission ratio, ^motor is the mechanical or motor transmission efficiency (typically fixed between 0.95 and 0.99) and r ro ue the wheel radius. The tractive force can be negative in case of engine braking. In this case, it is a braking force, the aerodynamic force or resistance F ae ro whose value is 1 2^^^^^^^^(^^^^ + ^^^^^^^^^^)2^^^^^^^^, where ^air is the ambient air density, ^x is the longitudinal speed of the vehicle, ^v en t is the longitudinal wind speed, A f is the frontal area of ​​the vehicle and C d is the aerodynamic coefficient of the vehicle, the rolling resistance F ro ll whose value is mgC rr .cos(^), with g the gravitational constant, C rr the rolling resistance coefficient and ^ the slope angle, and the force of gravity Fg rav whose value is mgsin(^). The balance of forces leads to F inertia = F tr ac t + Fae ro + F ro ll +rav , and thus to the following linear model:In this linear system, the values ​​of the vector y and the matrix A are either fixed and known values ​​(for example, g) or values ​​measurable using sensors. We seek to obtain in real time the mass m of the vehicle without using a dedicated sensor. Similarly, it is possible to obtain in real time the resistance coefficients C rr and Cd (A f and ^^^^^^^^ being known). We can therefore apply the method of Figure 2 and Figure 3. Figure 5 illustrates the application of this method to the system of Figure 4. The same reference numbers correspond to the same steps / sub-steps. The sensors 112 on board the vehicle acquire measurements of the engine torque T mot t(t), the longitudinal acceleration ax(t) and the angular speeds of the engine ^̇^^^(t) and wheel ^̇^^^(t). These measurements are transmitted to the measuring unit 114 via the vehicle's CAN (Controller Area Network) bus.In step 2146, the pre-processing unit 115 calculates the yaw rate ^(t) at a predefined pre-processing frequency. In step 2140, the different measurements T mo t(t), ax(t), ^̇^^^(t), ^̇^^^(t) and data ^(t) are temporally aligned to the recursion instant tk, for example by sub-sampling and / or interpolation as described above. The angular speeds ^̇^^^,^^, ^̇^^^,^^ make it possible to calculate the transmission ratio ^ in step 2142. The angular speed ^̇^. ^^,^^allows to calculate the longitudinal speed of the vehicle ^x ,k . A selection 2144 of the pre-processed measurements Tmo t ,k , ax ,k and those calculated ^ , ^x ,k is made from the calculated yaw rate ^k so as to keep only the measurements and data concomitant with a rectilinear movement of the vehicle. The transmitted measurements / data Tmo t ,k , ax ,k , ^ , ^x ,k are processed in the optional step 222 to determine the noise covariance matrices Qa ,k and Qy ,k . The six data are used by the estimator ^̃^ in step 224 (the other input data - r ro ue , ^mo tri ce , g - being fixed values) to obtain an estimate of the unknown parameters xk , here an estimate of the mass m and the resistance coefficients C rr and Cd (A fet ^air being known). As indicated above, the estimator ^̃^ can benefit from optimization / improvement tools taken from the literature on classical least squares.The forgetting parameter λ is already introduced in the scheme proposed above. Among these tools, regularization avoids numerical instabilities in the inversion of the matrix of operation (4). Indeed, such numerical instabilities can cause a divergence of the covariance matrix P k and lead to an overall divergence of the estimator. A regularization term α (zero to remove any regularization or positive) is thus introduced during an additional operation (5') to correct the covariance matrix P k :. α is generally small compared to 1 to achieve a fine regularization, for example α = 0.1. Another tool is known as instrumental variables. This correction avoids biased estimates in the presence of measurement noise correlated in some way with the measurement signals. Indeed, if there is a correlation between the noise and the signals, one of the additive noise assumptions taken to establish the present estimator ^̃^ may not be respected. Instruments ζ k are introduced in operation (4) and calculated for each new recursion. These instruments are chosen and replaced by appropriate input values ​​in order to eliminate the mentioned correlation and thus improve the accuracy of the estimator. The instruments ζ k are highly correlated with the variables ^̃^^^, typically ζ k is the matrix ^̃^ of a previous recursion. For example, ζ k +1=^̃^^^−3 or ^̃^^^−5.The following example includes both the forgetting factor λ, the regularization α and the instruments ζk. Of course, these different tools can be used separately. The adapted recursive scheme is as follows:. Another tool is known as Huber outlier weighting. It is used to mitigate the negative effects that the presence of outliers can have on recursive estimations. Indeed, the presence of outliers or anomalies in input or output measurements tends to degrade the performance of the estimator. A Huber weighting function is introduced at the beginning of the recursive scheme, in operations denoted (0) and (0'), and then used in operation (4) to calculate the matrix variable K. The parameter ξ is adjusted by the operator. As above, although all the tools presented are combined in this example, they can be used separately. The adapted recursive scheme is as follows: A variant of the tool concerns the multiple forgetting factors. The forgetting factor λ can be declined for each parameter of the vector x to be estimated. The multiple forgetting factors are then noted λ i , i being the index of the parameter xi in the vector x = {xi}. They allow to assign individual forgetting levels to each estimated parameter. The forgetting factor λ can also be scalable, its value adapting from one recursion to another. The multiple forgetting factors are then noted λk . They are used to react to the time-varying parameters. These two properties can also be combined. The multiple forgetting factors are then noted λ k , i . We also note Λk = {λk , i}. They are introduced at the beginning of the recursive scheme, in the operation noted (0''), then used during operations (4) and (5) where the matrices K and P are calculated coefficient by coefficient: Figure 6 illustrates a computer hardware architecture of a system 100 or an estimation device 110 according to embodiments. The device 600 comprises a communication bus 601 to which are preferably connected: - one or more central processing units 602, such as one or more CPU processors and / or one or more microprocessors; - a storage memory 603, of the ROM and / or hard disk and / or flash memory type, for storing computer programs intended to implement all or part of the operations described above; - a random access memory 604, of the RAM or even video RAM (VRAM) type, for storing the executable code of the computer programs as well as the registers adapted to record variables and parameters necessary for their execution; - a communication interface 605 connected to a network (for example CAN bus or Wifi network) in order to communicate with external equipment,for example external sensors 112 or the processing unit 120;- one or more I / O inputs / outputs 606 allowing an operator to interact with the computer programs, both in configuration and in operation. Typically, the inputs / outputs may include a screen serving as a graphical interface with the operator and restoring the value of the unknown parameters, and / or a loudspeaker for restoring audio content and / or a keyboard or any other pointing means allowing the operator to interact; and- one or more sensors 112 as described above if they are physically integrated into the device 600. Preferably, the communication bus 601 ensures communication and interoperability between the different elements included in the device 600 or connected to it. The representation of the bus is not limiting and, in particular,the central unit is usable to communicate instructions to any element of the computing device 600 directly or by means of another element of the computing device. The executable code stored in memory 603 can be received by means of the communication network, via the interface 605, in order to be stored there before execution. Alternatively, the executable code is not stored in non-volatile memory 603 but can be loaded into volatile memory 604 from a remote server via the communication network for execution directly. The central unit 602 is preferably adapted to control and direct the execution of the instructions or parts of software code of the computer program(s). Upon power-up, the program(s) which are stored in non-volatile memory 603 or on the remote server are transferred / loaded into the RAM 604, which then contains the executable code of the program(s),as well as registers for storing the variables and parameters necessary for implementing the invention. Of course, the invention is in no way limited to the variant embodiments described above, the person skilled in the art being able in particular to isolate or freely combine the aforementioned characteristics, or to substitute equivalents for them.

[0002] APPENDIX The linear system considered is always y=Ax. The aforementioned publication Shen Y. et al. provides an iterative estimator of total least squares. We consider the following notations: Wy a positive symmetric weighting matrix (for example of output measurement noise covariance), W a positive symmetric weighting matrix (for example of input measurement noise covariance), : = ^^^ − ^ 1 ⊗ ^^^^). Concerning the iterations, we adopt the following notations: x ( i+1 ) = x ( i ) + δx ( i+1 ) for the updated parameters, A ( i ) = A - ΔA ( i ) for the corrected input matrix and (^^ − Δ^^(^^)) = (^^ − Δ^^(^^)) x (^^) for the corrected output vector. The operator ⊗ corresponds to the Kronecker product, and Im is the identity matrix of dimension m*m (m the dimension of y). The explanations of Shen Y. et al., and in particular formulas 14, 15a and 15b, result in the following iterative procedure: 1) To move from the iterative estimator to the recursive estimator, we separate the input measurements A and output y into two batches: (equations 3) The components of equations 1 can be written: (equations 4)^̃^ ^^ ^^ ^^^^^^^̃^^^ = ^̃^ ^^1^^^^1^̃^1 + ^̃^ ^ 2 ^ ^^ ^^2 ^̃^2^̃^ ^^(^^) (^^) (^^) (^^ ^^ ^^^^^^(^^^^ − ΔA^^x^^) = ^̃^ ^^1^^^^1(^^1 − ΔA1x ) 1) + ^̃^ ^^(^^) (^^) 2^^^^2(^^2 − ΔA2x2) Considering the following lemma:(A+BCD) -1 = A -1 – A -1B(C -1+DA -1B) -1DA - 1 ,the first of equations 4 allows us to obtain the following expression:(equation 5)^ ^ −1 = ^^ − ^ ^^ −1 ^^ −1 (^̃^ ^^ ^^ ^^^^ ^̃^ ^^ ) 1 ^1^̃^2 (^^^^2 + ^̃^2^^1^̃^2) ^̃^2^^1where ^^1 = (^̃^1 ^^ −1 ^^ ^^1 ^̃^1) . Combining the expressions of equations 4 and 5 in equations 1, we obtain: We can remove the iteration index 'i' in favor of a recursion index 'k' by considering an additional measure AN+1 , yN+1 and batches of successive iteration measures as follows: In a recursive context where a data is collected at each recursion, the Kronecker product ( ^^ ⊗ ^^^^) can be simplified to x. Also, the noise covariance matrix WZ can be written as: (equation 8) It follows from equations 2:(equations with ^̃^^^+1 By introducing the gain matrix (equation 10) recursive parameters and recursive parameter increments can be written as: (equations 11) Furthermore, equation 5 allows us to rewrite the decovariance matrix P: (equation 12) Finally, taking the reasonable assumption of zero-mean Gaussian noise on both input and output measurements, the stochastic properties of random errors can be characterized by: By setting Q=W -1 , the estimator ^̃^ is thus formed from equations 8 à 12.

Claims

CLAIMS 1. Device (110) for estimating a parameter of a physical system (100) considered to be linear, comprising:- a measurement unit (114) configured to obtain successive input measurements A k and successive output measurements y k of the linear system,- an estimation unit (116) configured to estimate one or more parameters x k of the physical system recursively using a Lagrangian total least squares estimator, from the input and output measurements, in which a k + 1 estimation recursion of the Lagrangian total least squares estimator comprises the following operations:(1) ^^^^,^^+1 = ^^^^,^^+1 + ^^^ ^ ^ ^ ^^ ^^,^^+1 ^^ ^^ (2) ΔA^^+1 = −^^^^,^^+1^^^^^^ −1^^,^^+1 (^^^^+1 − ^^^^+1^^^^ − ^̃^^^^^^^^)(3) ^̃^^^+1 = ^^^^+1 − Δ^^^^+1 (4) ^^ ^^ ^^+1 = ^^^^^^^^+1 (^^^^^^,^^+1 + ^̃^^^+1^^^^^^^ ^ ^ ^ −1 +1) 1 (5) ^^ ^^+1= ^^(^^^^ − ^^^^+1^̃^^^+1^^^^)optionally (5') ^^^^+1 = ^^^^+1(^^^^ + ^^^^^^+1) −1(6) ^^^^^^+1 = ^^^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^ − ^̃^^^+1^^^^^^)(7) ^^^^+1 = ^^^^ + ^^^^+1(^^^^+1 − ^^^^+1^^^^)where k is the recursion index, Q a ,k and Qy ,k are weighting matrices; δxk , ^̃^^^ and Pk are inter-recursion variables; ^^^^+1 is an instrumental variable correlated with the variables ^̃^^^ ; Qz ,k , ΔAk , Kk are internal calculation variables; α is a regularization term and λ is a positive forgetting parameter less than or equal to 1.

2. Device (110) according to claim 1, wherein Qa,k and Qy,k are input and output measurement error or noise covariance matrices respectively.

3. Device (110) according to claim 1 or 2, wherein Qa,k and Qy,k are evolving matrices, and the estimation unit (116) comprises a matrix estimation sub-unit (117) configured to determine for each recursive iteration the matrices Qa,k and Qy,k from the input and output measurements respectively.4.Device (110) according to one of claims 1 to 3, in which the forgetting parameter λ is equal to 1 and / or the regularization term α is equal to 0 and / or the instrumental variable ^^^^+1 is equal to ^̃^^^+1.

5. Device (110) according to one of claims 1 to 3, in which the forgetting parameter λ is strictly less than 1 and / or the regularization term α is strictly positive and small compared to 1 and / or the instrumental variable ^^^^+1 is equal to ^̃^^^−^^ where a is a positive integer.6.Device (110) according to one of the preceding claims, in which the measurement unit (114) comprises a sub-unit (115) for pre-processing measurements acquired from one or more sensors (112), to generate the successive input measurements A k and output yk , the pre-processing sub-unit being configured to carry out at least one of the following operations: - a temporal alignment of input or output measurements by interpolation of the acquired measurements and / or by sub-sampling of the acquired measurements, and - a partial selection of measurements to generate the successive input measurements A k and output yk , the selection being a function of a concomitant behavior of the physical system.

7. Device (110) according to the preceding claim, in which the physical system (110) is a vehicle and the partial selection selects only measurements concomitant with a rectilinear movement of the vehicle. 8.Device (110) according to one of the preceding claims, wherein the physical system (100) is a vehicle and the parameter(s) include a mass of the vehicle.

9. Device (110) according to the preceding claim, wherein the measuring unit (114) is connected to sensors (112) for. measure engine torque (T mo t) of the vehicle, a longitudinal acceleration (ax) of the vehicle as well as an angular velocity ( ^̇^^^) of the engine and an angular velocity ( ^̇^^^) of the wheels so as to determine a longitudinal velocity (vx) of the vehicle and a transmission ratio (^).

10. Method for estimating a parameter of a physical system (100) considered as linear, implemented by computer and comprising the following steps: - obtaining (210), using one or more sensors, successive input measurements A k and successive output measurements yk of the linear system, - estimating (220) one or more parameters xk of the physical system recursively using a Lagrangian total least squares estimator, from the input and output measurements, method in which a k+1 estimation recursion of the Lagrangian total least squares estimator comprises the following operations: where k is the recursion index; Qa,k and Qy,k are weighting matrices; δxk, ^̃^^^ and Pk are inter-recursion variables; ^^^^+1 is an instrumental variable correlated with the variables ^̃^^^; Qz,k, ΔAk, Kk are internal calculation variables; α is a regularization term and λ is a positive forgetting parameter less than or equal to 1.

11. Computer program product incorporated in a non-transitory recording medium comprising computer-readable program code for implementing the method according to the preceding claim.

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