Recursive Lagrangian Estimator of Total Least Squares

Abstract

A total least squares recursive Lagrangian 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. A k+1 estimation recursion comprises the following operations: (1) (2) (3) (4) (5) optionally (5') (6) (7) where Qa,k and Qy,k are weighting matrices; δxk, and Pk are inter-recursion variables; is an instrumental variable correlated with the variables; Qz,k, ΔAk, Kk are internal variables; α is a regularization term and λ is a forgetting parameter. Figure for abstract: Fig 2

Description

Title of the invention: Recursive Lagrangian Estimator of Total Least Squares Technical field of the invention

[0001] The present invention relates to the general field of parameter estimators in a physical system. State of the prior art

[0002] More 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.

[0003] These unknown parameters can be estimated indirectly from measurements made on the physical system, typically input measurements and output measurements.

[0004] Estimating unknown parameters is easier for linear physical systems, i.e. systems 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.

[0005] A category of linear system estimators, 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, in a non-exhaustive manner, recursive or non-recursive least squares estimators, the linear Kalman filter and recursive or non-recursive instrumental variable estimators.

[0006] OE estimators are known to produce biased estimates if the input measurements are noisy. This is why a second category of linear system estimators, called EIV estimators for "errors-in-variables," is preferred. Although they are more complex, these EIV estimators offer better performance. They are based on modeling that considers the presence of error in the measurements on both sides of the linear system equations - the inputs and the outputs. This category includes, for example, the total Kalman filter, total least squares (TLS) estimators, and total instrumental variable estimators.

[0007] Some TLS estimators rely on a singular value decomposition (SVD) approach to solve the problem. total least squares. The publication “Recursive Generalized Total Least Squares with Noise Covariance Estimation” (Rhode S. et al., 2014) illustrates for example a recursive TLS SVD estimator.

[0008] Other, less numerous, 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.

[0009] 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 Lagrange multiplier approach.

[0010] There is therefore a need to address the aforementioned drawbacks and to improve the catalog of efficient recursive TLS estimators. Statement of the invention

[0011] The invention aims to propose a recursive Lagrangian TLS estimator for estimating one or more parameters of a linear physical system.

[0012] To this end, the inventors 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 Lagrangian TLS estimator capable of benefiting from classical least squares optimization techniques, such as the forgetting factor, regularization, instrumental variables and robustness against outliers, among others.

[0013] 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 linear system, 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: (1) ^„Â+1 “ QyJ<+l + XtQa.k+]Xk <2) AAS+1= W ^k+1 ~ ^+1 ~ <4) +wxL f' optionally (5') = pM(jn + aP^ (6) = 5xk + Kk+l ( yfc+1 - Ak+lxk - Àk+lôxk ) -^-+1 ~ xk+^+i( yk+i-Ak+lxk ) where k is the recursion index; Qa>k and Qy>k are weighting matrices; ôxk, Ak and Pk are inter-recursion variables; C,, 5 is an instrumental variable correlated with the Ak variables; Qz,k, AAk, Kk are internal calculation variables; a is a regularization term and X is a positive forgetting parameter less than or equal to 1.

[0014] Such a device thus makes it possible to obtain a recursive Lagrangian TLS estimator for estimating parameters of a linear physical system in real time. In particular, the number of operations per recursion remains limited; the operational complexity is of the order of O(n2).

[0015] Such a TLS estimator can in particular be used as a “virtual” sensor of a parameter not directly measured.

[0016] Note that X=1 removes any omission; a=0 removes any regularization and À =Ài । removes any correction of noise correlated to the measurement signals.

[0017] Correspondingly, a method for estimating the parameter of a physical system considered to be linear, comprises the following steps: - obtain successive input measurements Ak and successive output measurements yk of the linear system, - estimate one or more parameters xk of the physical system recursively from the input and output measurements, in which method a k+1 estimation recursion comprises the following operations: (2) aa â . +i = -LM (3) À+i= ^+1 " ^^+1 <4) (5) has) optionally (5') p^ = Pk+}(ln + aP^ (6) 5xk+] = ôxk + Kk+l- Ak+lxk - Àk+Ï5xk) xk+i - xk+^k+ï ( yk+rAk+ ) •

[0018] 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 transposed into features of process.

[0019] In one embodiment, Qak and Qyk 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.

[0020] In a particular embodiment, Qak 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 Qak and Qy k from the input and output measurements respectively. The estimation of the unknown parameters is improved.

[0021] Of course, it is also possible to use prefixed Qak and Qyk matrices, 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.

[0022] In one embodiment, X is 1 and / or a is 0 and / or Ckf is Å^- This makes it possible to deactivate certain optimization tools taken from the literature on classical least squares.

[0023] In another embodiment, the forgetting parameter X is strictly less than 1 and / or the regularization term a is strictly positive small compared to 1 and / or the instrumental variable Ckj is equal to A^ where a is a positive integer. One or more optimization tools are thus activated.

[0024] In one embodiment, the measurement unit comprises a sub-unit for preprocessing measurements acquired from one or more sensors, to generate the successive input Ak and output yk measurements, the pre-processing sub-unit being configured to perform at least one of the following operations: - time alignment of input or output measurements by interpolation of the acquired measurements and / or by subsampling of the acquired measurements, and - a partial selection of measurements to generate the successive input Ak and output yk measurements, the selection being a function of a concomitant behavior of the physical system.

[0025] Through temporal alignment, the risk of error bias inherent in the joint use of non-concomitant measurements is reduced.

[0026] Partial selection makes it possible to exclude measurements likely to introduce errors, in particular measurements which result from conditions which do not respect the linearization hypotheses of the physical system.

[0027] These pre-processings result in an improvement in the estimation.

[0028] In one embodiment, the physical system is a vehicle and the partial selection selects only measurements concomitant with rectilinear movement. of the vehicle. Indeed, a linear modeling of the vehicle can be considered 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.

[0029] 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 motor vehicle, without a dedicated sensor.

[0030] In this case, the measuring unit can be connected to sensors to measure an engine torque of the vehicle, 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.

[0031] By way of illustration, estimating the mass of the vehicle makes it possible to determine the vertical load experienced 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.

[0032] For example, knowledge of the overall vehicle load makes it possible to better predict 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 use and wear.

[0033] 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 geodetic fields, to name a few.

[0034] At least a portion of the methods of the invention may be computer-implemented. 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 program code thereon. computer-readable for implementing the above method.

[0035] 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

[0036] Other aims, 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:

[0037] [Fig-1] illustrates a physical system considered as linear, for an implementation of the invention;

[0038] [Fig.2] illustrates, using a flowchart, steps in a method for estimating parameter of a physical system, according to embodiments;

[0039] [Fig.3] illustrates, using a flowchart, more detailed steps of a process parameter estimation of a physical system, according to embodiments;

[0040] [Fig.4] illustrates a physical system of the motor vehicle type, the com linear behavior is defined by Newton's second law;

[0041] [Fig.5] illustrates the application of the estimation method to the system of [Fig.4]; and

[0042] [Fig.6] illustrates a computer hardware architecture of a system or device estimation according to embodiments.

[0043] Detailed description of at least one embodiment

[0044] One aspect of estimation theory focuses on linear systems, which can be physical systems whose behavior is linear by nature or physical systems whose behavior is reduced, by approximation or simplification, to linear behavior.

[0045] [Fig.l] 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 has dimensions m * n.

[0046] The example of [Fig.4] described below illustrates a motor vehicle as a linear physical system. In this example, n = 3 and m = 1.

[0047] 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 estimated.

[0048] The system 100 comprises for this purpose a device 110 for estimating parameters 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.

[0049] 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'.

[0050] The processing unit 120 can perform computational processing on the estimates xk obtained in order to have specific parameter values ​​for decision-making.

[0051] By way of illustration, estimating the mass of an electric vehicle using the device 110 allows the processing unit 120 to calculate the overall load of the vehicle and to activate or authorize certain driving modes accordingly, for the purpose of saving the battery.

[0052] 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 wirelessly (for example, a Wifi network or Bluetooth link - commercial names), to the external unit 120. For example, the processing unit 120 may be implemented in a remote processing server which manages a fleet of vehicles and transmits back to them commands acting on them.

[0053] 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.

[0054] The estimation device 110 comprises one or more sensors 112, a measurement unit 114 and an estimation unit 116.

[0055] 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 [Fig.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.

[0056] 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.

[0057] All or part of the measurements Ak and yk may correspond to acquired measurements o(t) transmitted by the sensors 112. Nevertheless, some of the measurements Ak and yk are obtained by computational processing of the acquired measurements o(t), as will be apparent from the example described below.

[0058] For this purpose, the measurement unit 114 may comprise a pre-processing sub-unit 115.

[0059] 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 Ak and output yk.

[0060] According to one embodiment of the invention, the recursive scheme of the estimator comprises the following computational operations for the k+1 recursion: (2) AAt+, = - - A» ,xÀ - ) At+i ~ At+1_ <4) +wx,y' (6) = ^k + Kk+ï ( yk+ [ - Ak+ixk - Âk+lôxk ) (7) x^ = xk + Kk+l ( yk+} -Ak+1xk )

[0061] Qak and Qyk are weighting matrices. They allow to take into account heterogeneous noise distributions on the input and output data.

[0062] In one embodiment, these are input and output measurement error (or noise) covariance matrices respectively.

[0063] These matrices can be prefixed, that is to say 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 series of tests and / or using the technical specifications of the sensors 160 involved.

[0064] In an embodiment offering more precise estimations, the matrices Qak and Qyk are evolving matrices, that is to say 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 comprise 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.

[0065] The matrices Qa>k and Qy>k can be completely recalculated at each recursion or updated, thus resembling a recursive reevaluation.

[0066] The variables ôxk, Ak and Pk of the estimator 116 are inter-recursion variables, i.e. transmitted from one recursion to the other. In a Lagrangian TLS approach, ôxk is an estimate of the incremental correction of the parameters xk between two recursions, Ak is a matrix corrected by Ak taking into account the predicted residual matrix AAk (below), and Pk is a noise covariance matrix.

[0067] Qzk, AAk, Kk are, for their part, internal calculation variables.

[0068] X 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, serves to exponentially forget past data, so that greater importance is given to more recent estimates. The smaller the parameter X, the greater the forgetting.

[0069] Beyond forgetting old information, the X parameter also makes it possible to converge more quickly towards the true parameters in the case of poorly initialized parameters, as well as to smooth parameter estimates when the system is corrupted by too much noise.

[0070] For an application without forgetting old measurements during recursions, X is set to 1. For an application with forgetting, X can be set to a value strictly between 0 and 1.

[0071] The recursive scheme of the estimator can be seen as a function QCLk+l, y 2 j °ù 'cs 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.

[0072] The estimation unit 116 thus produces the estimate xk during the recursion k, intended for the processing unit 120.

[0073] The estimation device 110 can thus be seen as a virtual sensor.

[0074] The appendix below demonstrates that these recursive computational operations correspond to a Lagrangian estimator of total least squares.

[0075] [Fig. 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.

[0076] 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=l), in particular the parameters x0 and the inter-recursion variables ôx0, Po.

[0077] In one embodiment, x0 is set to the plausible values ​​of the corresponding unknown parameters. “Plausible” means any value in the range of values ​​that can be expected for the parameter concerned. By way of illustration, with reference to the example of [Fig.4], the mass of a vehicle is set at 1000 kg.

[0078] ôxq is initialized with a zero vector and Ào with a zero matrix.

[0079] The covariance matrix Po 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.

[0080] The initial step 200 also makes it possible to recover calculation parameters such as the forgetting factor X and the weighting matrices Qa, Qy if they are prefixed. These pa calculation parameters may be stored in memory of the estimation device 110.

[0081] Step 210 consists of the estimation device 110 obtaining the successive input measurements Ak and the successive output measurements yk.

[0082] 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.

[0083] These input Ak and output yk measurements 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 carrying out the computational operations described above.

[0084] This results in estimated parameters xk, which are communicated to the processing unit 120 in step 230.

[0085] Step 240 then consists of incrementing the recursion counter k, before looping back to step 210 to perform the next recursion.

[0086] The recursion frequency may be set by an operator of the estimation device 110. In the example of the vehicle of [Fig.4], a recursion period of between 10 milliseconds (ms) and 500 ms is compatible with off-the-shelf sensors and real-time operation.

[0087] [Fig.3] illustrates, using a flowchart, more detailed steps of a method for estimating the parameter of a linear physical system, according to embodiments. This method is always implemented by the estimation device 110.

[0088] Step 200 of initializing the recursive £ estimator remains unchanged.

[0089] Step 210 of obtaining input Ak and output yk measurements comprises a sub- step 212 of acquiring the measurements o(t) and a sub-step 214 of pre-processing the acquired measurements.

[0090] The acquisition 212 of the measurements o(t) consists for example of obtaining the measurements carried out by the sensors 112.

[0091] Depending on the sensors implemented, the measurements acquired may have different sampling frequencies and / or may not be synchronized in time (for example to the nearest millisecond).

[0092] This is why these acquired measurements o(t) can be preprocessed in step 214.

[0093] In one embodiment, the preprocessing comprises a time alignment of input or output measurements. This is the purpose of step 2140: ensuring that the input Ak and output yk measurements that will be provided to the estimator 116 correspond to the same time instant, in order to reduce estimation biases.

[0094] 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(ti) and o(t2) available on either side (time ti) of the time (tk) corresponding to the recursion k. Typically, the preprocessed measurement ok corresponds to o(tl) + [o(t2)+o(ti)] * [(tk-ti) / (t2-ti)].

[0095] 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 sub-sampling of the acquired measurements, for example equal to the ratio between the two aforementioned frequencies.

[0096] Interpolation and subsampling can be combined to provide preprocessed acquired measurements. In particular, subsampling advantageously precedes interpolation.

[0097] In one embodiment, the preprocessing comprises the calculation 2142 of one or more input measurements Ak and / or output yk, as a function of the acquired measurements o(t), advantageously the preprocessed measurements ok. This involves converting the acquired data into useful data of the linear system y = Ax.

[0098] This sub-step 2142 makes it possible to generate the input Ak and output yk measurements which could not be acquired directly by the sensors 112. In the example of [Fig.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.

[0099] In another embodiment, the preprocessing comprises the partial selection 2144 of measurements to generate the successive input Ak and output yk measurements. 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 physical system.

[0100] The partial selection 2144 can be carried out on the measurements o(t) acquired from the sensors 112 or on the preprocessed acquired measurements ok 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.

[0101] The sorting criteria for the selection are determined in sub-step 2146, and applied during the selection sub-step 2144.

[0102] Sub-step 2146 is for example a function of the acquired measurements o(t) (possibly pre-processed, ok).

[0103] In the example of [Fig.4], the sorting criterion is the value of the yaw rate of the vehicle. Substep 2146 calculates for example 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.

[0104] The measurements (acquired o(t), ok or input / output Ak, yk) are discarded when the vehicle is not moving rectilinearly. This results in the selection of measurements which are concomitant with a low yaw rate, typically below a threshold value, for example 0.5 radiants per second, preferably 0.1 radiants per second.

[0105] 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.

[0106] The selected input Ak and output yk measurements are then provided as input to the estimation step 220.

[0107] The optional sub-step 222 is implemented when the weighting matrices Qa>k and Qy>k are not prefixed. In this case, they are estimated from the input Ak and output yk measurements.

[0108] Typically, an input noise covariance matrix Qa>k can be updated from the input measurements Ak, 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.

[0109] When the matrices Qak and Qy k are available, they are provided, together with the input Ak and output yk measurements, as inputs to the recursive estimation algorithm 224 already described.

[0110] 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.

[0111] [Fig.4] represents a physical system of the motor vehicle type, the com behavior defined by Newton's second law (balance of forces) can be reduced to linear behavior with some simplifying assumptions (for example zero wind speed).

[0112] The forces involved include:

[0113] the inertial force Finertie whose value is (m+me)ax, with m the mass of the vehicle and me the equivalent mass of the rotating parts,

[0114] the traction force Ftract whose value is ' °where Tmot is the engine torque, y ? wheel is the transmission ratio, r|motrice is the mechanical or motor transmission efficiency (typically fixed between 0.95 and 0.99) and rroue the wheel radius. The force of traction can be negative in case of engine braking. In this case, it is a braking force,

[0115] the aerodynamic force or resistance Faero whose value is 1« / „ < 24 C , °ù Pair is the density of the ambient air, vx is the longitudinal velocity- 2 ' air ' Vvent zd vehicle longitudinal speed, vvent is the longitudinal wind speed, Af is the frontal area of ​​the vehicle and Cd is the aerodynamic coefficient of the vehicle,

[0116] the rolling resistance Fron whose value is mgC,,cos(0), with g the gravitational constant, the rolling resistance coefficient and 0 the slope angle, and

[0117] the gravitational force F^ whose value is mgsin(O).

[0118] The balance of forces leads to Finertie = F^t + Faero + Fron + Fgrav, and thus to the model following linear: , either motor mC 2PairAfCd y — rrnuc A =[«X g kH m mC rr

[0119] 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.

[0120] 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 and Cd (Af and Pctir being known).

[0121] We can therefore apply the method of [Fig.2] and [Fig.3]. [Fig.5] illustrates the application of this method to the system of [Fig.4].

[0122] The same reference numbers correspond to the same steps / sub-steps.

[0123] The on-board sensors 112 of the vehicle acquire measurements of the engine torque T mot(t), the longitudinal acceleration ax(t) and the angular speeds of the engine 0e(t) and wheel 0r(t). These measurements are transmitted to the measurement unit 114 via the CAN (Controller Area Network) bus of the vehicle.

[0124] In step 2146, the pre-processing unit 115 calculates the yaw rate rp(t) at a predefined pre-processing frequency.

[0125] At step 2140, the different measurements Tmot(t), ax(t), ^(t), ^(t) and data rp(t) are temporally aligned to the instant tk of recursion k, for example by sub-sampling and / or interpolation as described above.

[0126] The angular speeds 8ek, 8^ make it possible to calculate the transmission ratio y at step 2142.

[0127] The angular speed 8^ allows the longitudinal speed of the vehicle vx>k to be calculated.

[0128] A selection 2144 of the pre-processed measurements Tmotjk, aXjk and those calculated y, vXjk is carried out from the calculated yaw rate ipk so as to only keep the measurements and data concomitant with a rectilinear movement of the vehicle.

[0129] The transmitted measurements / data Tmot>k, ax>k, y, vx>k are processed in optional step 222 to determine the noise covariance matrices Qak and Qy k.

[0130] The six data are used by the estimator Ë at step 224 (the other input data - rroue, pmotrice, g - being fixed values) to obtain an estimate of the unknown parameters xk, here an estimate of the mass m and the resistance coefficients Cn- and Cd (Af and even being known).

[0131] As indicated above, the estimator Ë can benefit from optimization / improvement tools taken from the literature on classical least squares. The forgetting parameter X is already introduced in the scheme proposed above.

[0132] 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 Pk and lead to an overall divergence of the estimator. A regularization term a (zero to remove any regularization or positive) is thus introduced during an additional operation (5') to correct the covariance matrix Pk: Ë(xk, ÔXb Àb Pb Qyk+}, Q^+y yk+f a' ' ^+1 = Qyji+l'^ XIQa,k+lXk (2) AA<+I = ' Ak+iXk - À A ) Ak+l ~ At+1 ' AAt+l ?k+l = k+Àk+l^k ) pk+1=pMan+^yl (6) 5xk+ï = ôxk + Kk+} - Ak+{xk - Ak+{5xk) xk+i - xk+^k+ï ( yk+l~Ak+ Ak )

[0133] a is generally small compared to 1 to achieve fine regularization, for example a=0,l.

[0134] Another tool is known as instrumental variables. This correction avoids biased estimates in the presence of correlated measurement noise of a way or another 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 Ë may not be respected. Instruments Çk 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 Çk are strongly correlated with the variables Ap typically Çk is worth the matrix À of a previous recursion. For example, Çk+1=ÂA.3 or Àk_5-

[0135] The following example includes both the forgetting factor X, the regularization a and the instruments Çk. Of course, these different tools can be used separately.

[0136] The adapted recursive scheme is as follows: Qyjc+r Qaji+VA:+b ' (i)ô / ,-Q t Ak (2) AAk+ ] = - Q , .¾ Q ' .. ( y - Ak+ - Àkôxk ) (3) ^+t “ Ak+i - AAk+l <4) ^"•P^PkAk. + aPkA ^xk+i = ^xk + Kk+Ak+r ^+,5¾) *£+1 “ xk + Kk A -\+1)

[0137] 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 estimates. 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 is adjusted by the operator. As above, although all the tools presented are combined in this example, they can be used separately.

[0138] The adapted recursive scheme is as follows: ^xh QaX+V ^+1' k+Ÿ k+V (0) - Ak+Ak / n '\ / \ [ 1' l e To+d (0) q{ek^) = \ , . (1) QzJi+\ - Qy,k+Ï + XkQak^^^ <2) AAt+l = - (3) Âl+l - ^k+ï ' ^k+i <6) 6xm = 3¾ + Kt+, (yt+i - AMxt - ÀMBxt ) xk+i= xk + ^k+i(yM-AMxk)

[0139] A tool variant concerns the multiple factors of forgetting.

[0140] The forgetting factor X can be declined for each parameter of the vector x to be estimated. The multiple forgetting factors are then noted Xi5 i being the index of the parameter x; in the vector x={x;}. They make it possible to attribute individual forgetting levels to each estimated parameter.

[0141] The forgetting factor X can also be scalable, its value adapting from one recursion to another. The multiple forgetting factors are then noted Xk. They are used to react to parameters that vary over time.

[0142] These two properties can also be combined. The multiple forgetting factors are then denoted Xkji. We also denote Ak={Xkji}.

[0143] 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: k- Qyjt+Y Q-aJi+V ^+1' -^+1' (0) Ê&+1 = yÂ.+1 " Ak+lxk (0”) J - ; 1 11¾^II2 k+U ^maxâ- fl (1) ^+1 Qyk+\+ (2) AA<+! = ' ( yk+1 ' A^Xk ~ ) (3) = AÀ.+1 - AAft+I Kk+h =----(WUi---—-- / (5) Pk+ÿj ~ (.Pkj.i ” k+\-jAk+ IjPkij ) / Aki (6) 0¾] = dxk + Kk+^yk+ï-Ak+xxk-Àk+A5x^ (7) xk+ï = xk + Kk+ï[ yk+ï-Ak+xxk )

[0144] [Fig.6] illustrates a computer hardware architecture of a system 100 or an estimation device 110 according to embodiments.

[0145] 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 604 RAM memory, of the RAM or video RAM (VRAM) type, for storing the executable code of 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 LO 606 inputs / outputs 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 returning the value of unknown parameters, and / or a loudspeaker for returning audio content and / or a keyboard or any other pointing means allowing the operator to interact; and - one or more sensors 112 as described previously if they are physically integrated into the device 600.

[0146] Preferably, the communication bus 601 ensures communication and interoperability between the different elements included in the device 600 or connected thereto. The representation of the bus is not limiting and, in particular, the central unit can be used to communicate instructions to any element of the computing device 600 directly or by means of another element of the computing device.

[0147] The executable code stored in memory 603 can be received by means of the network of communication, 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.

[0148] 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). When the power is switched on, 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.

[0149] Of course, the invention is in no way limited to the embodiment variants described above, the person skilled in the art being able in particular to isolate or freely combine the above-mentioned characteristics, or to substitute equivalents for them.

[0150] APPENDIX

[0151] The linear system considered is always y=Ax.

[0152] The aforementioned publication Shen Y. et al. provides an iterative estimator of total least squares.

[0153] We consider the following notations: Wy a positive symmetric weighting matrix (for example of output measurement noise covariance), Wa a positive symmetric weighting matrix (for example of input measurement noise covariance),^ ; =W;?+(x ( 0 t JmY Concerning the iterations, we adopts the following notations: x(i+1'=x®+ôx(i+1' for the updated parameters, A® =A-AA® for the corrected input matrix and (y-Ayû) = ^A-AA^xW for the corrected output vector. The operator 0 corresponds to the Kronecker product, and Im is the identity matrix of dimension m*m (m the dimension of y).

[0154] The explanations of Shen Y. et al., and in particular formulas 14, 15a and 15b, result in the following iterative procedure:

[0155] (equations 1) HA-AA0) Wz(A-AA(0) (A-AAo)) Wz(y-Ay<û) / 7 y -1 T = HA-AA(0) W^A-AA'0) j (A-AA(O) Wz(y-A(z)x<') - AA(z)x<0 ) A^+|) = ( (A-AA(Ô) rWz(A-&dîy} ) (A-AA(0)rWz(y-AA(Ox^ )

[0156] (equations 2) AyC+0 = WjWz(y-Ax(i> - (A-AA(i))dx <M)) = W~'(xO) Wz[y-Ax(^ - ( A-AA(0)ôx(z+1) ) AA*- l+ s ■ = vec1 Aati+1 1

[0157]

[0158] To move from the iterative estimator to the recursive estimator, we separate the input measurements A and output y into two batches: (equations 3)

[0159] Ay — A^AA?] and - AA? - 21 Similarly, ^ZN = zi And 221 Xf. Xy

[0160] The components of equations 1 can be written:

[0161] (equations 4) Ayy W ZnAn “ A । W^jA j + A^ WZZA2 ÀNWZN(yN-AA{^ AA?x?) +Àlwz2(y2-A^f)

[0162] Considering the following lemma: (A+BCD) * = A1 - A *B(C *+DA *B) *DA *, the first of equations 4 allows us to obtain the following expression: (equation 5) / ~T ~ V* ~T i 1 ~ -ZV1- WHERE! ~T ~ vf

[0163] Combining the expressions of equations 4 and 5 in equations 1, we obtain: either -P& ( W'^ + À^p^+ pX - ( ù 22 + )\PtÀ2 j Wz2(y, - AAVM° )

[0164] Since , j À 'rt \ Z^k-" Z. I .4^ / y ZjZ. 1 Z. I 4+n = -p^i RAz2+ÀPÀr)' + either (equation 6) xO+1) =xp +1 > +pX ( W^h^PjAJ ) 4 (yy AA?^

[0165] Similarly, we obtain: (equation 7) = axj^ + WZ2 + y1^ - )

[0166] We can remove the iteration index 'i' in favor of a recursion index 'k' by considering an additional measure An+i, yN+i and by linking the two sets of measures

[0167]

[0168]

[0169]

[0170]

[0171] of successive iterations as follows: xp+i) = : xk and xp+1) = :xk+1 In a recursive context where data is collected at each recursion, the Kronecker product (x0 J ) can be simplified to x. Also, the noise covariance matrix Wz can be written: (equation 8) ~ ^>^4-1 + Xk^a.,k+Ak' It follows from equations 2: (equations 9) AAt+) = - - A^-À^ )•«« K» = Am - M+! By introducing the gain matrix (equation 10) ~T l , ~ ~T . &k+\ = PkAk+l\ WZJc+i + ^k+ïPkAk+l) recursive parameters and recursive parameter increments can be written: (equations 11) *k+i = xk + Kk+l(yk+fAk+ixk) andôxk+l = ôxk + Kk+i(yk+ï-Ak+lxk-Àk+lôxk) Furthermore, equation 5 allows us to rewrite the covariance matrix P: (equation 12) Pk+i = Pk-KkJik+iPk 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:

[0172] Setting Q=W', the estimator £ is thus formed from equations 8 to 12.

Claims

Claims

1. Device (110) for estimating the parameter of a physical system (100) considered to be linear, comprising: - a measurement unit (114) configured to obtain successive input measurements Ak and successive output measurements yk of the linear system, - an estimation unit (116) configured to estimate one or more parameters xk of the physical system recursively from the input and output measurements, in which an estimation recursion k+1 comprises the following operations: <2> aa^ । = - 1¾ - Aifixk ) (3) Àt+i ~ ^k+i " AAk+i ^k+î ~ k+i(^Qzj^^ optionally (5') = + f ^xk+l = ^-xk + ^+1(-^+) ' Ak+lxk ' Ak+iÔXk ) xk+1 - xk+% k+1 ( yk+ s ~Ak+ixk ) where k is the recursion index, Qa>k and Qy>k are weighting matrices; ôxk, À, and Pk are inter-recursion variables; C, , is an instrumental variable correlated to the variables Àk; Qz,k, AAk, Kk are internal calculation variables;a is a regularization term and X 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 Qak 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 Qak 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 / . is equal to 1 and / or the regularization term a is equal to 0 and / or the instrumental variable C , is equal to X , K+1 K'+ i

5. Device (110) according to one of claims 1 to 3, in which the forgetting parameter X is strictly less than 1 and / or the regularization term a is strictly positive small compared to 1 and / or the instrumental variable is equal to Ak_a 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 Ak and output yk measurements, 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 Ak and output yk measurements, 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 only selects 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, in which the measuring unit (114) is connected to sensors (112) for measuring an engine torque (Tmot) of the vehicle, a longitudinal acceleration (ax) of the vehicle as well as an angular speed (0e) of the engine and an angular speed (0r) of the wheels so as to determine a longitudinal speed (vx) of the vehicle and a transmission ratio (y).

10. Method for estimating the parameter of a physical system (100) considered as linear, comprising the following steps: - obtaining (210) successive input measurements Ak and successive output measurements yk of the linear system, - estimating (220) one or more parameters xk of the physical system recursively from the input and output measurements, method in which an estimation recursion k+1 comprises the following operations: (1) Q , =0 , , + xTQ , .¾ Ak+\xk Ak^xk) Ak+Ï ~ Ak+Ï " <4) KM = P^.Q,^, + ÀMPtCTM)1 optionally (5') Pk+l(I„ + aP^)'1 ^xk+i - sxk+KkjQ\+pAk+^ (7) ^+i=xk+Kk+i(yk+l-Ak+\xk )

11. where k is the recursion index; Qa>k and Qy>k are weighting matrices; ôxk, À and Pk are inter-recursion variables; Ç. , is an instrumental variable correlated with the variables Ak; Qz,k, AAk, Kk are internal calculation variables; a is a regularization term and X is a positive forgetting parameter less than or equal to 1. A 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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