Abstract:
A sensor for detecting the position and speed of a part mobile along at least one measurement direction includes an estimator ( 38 ) adapted to provide estimation of position and speed based on a displacement model of the target during a period of observation T obs , the model relating the position of the target at a time t included in the period of observation T obs , at least to the position and the speed to be estimated.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a sensor and a method for measuring position and speed. 
     DESCRIPTION OF THE RELATED ART 
     There exist sensors of position and speed of a movable part comprising:
         at least one excitation inductor suitable for inducing a magnetic excitation field as a function of an excitation current or voltage of this inductor,   at least one target made of conducting or magnetic material suitable for modifying as a function of its position the magnetic excitation field, this target being secured to the movable part,   at least one first transducer suitable for transforming the magnetic field modified by the target into an electrical measurement signal, and   a first estimator able to estimate, on completion of an observation period T obs , the position and the speed of the target on the basis of N samples of the electrical measurement signal and of N samples of the excitation current and/or voltage, these samples being taken during the observation period T obs  and N being an integer greater than two.       

     In existing sensors, the magnetic excitation field is an alternating signal of frequency f 0  and the estimator is a synchronous demodulator adjusted to the frequency f 0  so as to extract the amplitude of the component of frequency f 0  from the measured electrical signal. This extracted amplitude is representative of the position of the target. 
     Synchronous demodulators make it possible to eliminate the additive noise present in the measured electrical signal. By additive noise is meant here the noise related to various spurious phenomena which gets added to or superimposed on the theoretical electrical signal which ought to be obtained in the absence of noise. The elimination of the additive noise by the synchronous demodulator is all the more efficacious the longer the observation period. 
     Additionally, the position of the target must be substantially constant throughout the observation period. In the converse case, the displacements of the target during the observation period are averaged by the sensor over the observation period, in such a way that the accuracy in the measurement of the position decreases. 
     So, the greater the speed of displacement of the target, the shorter the chosen observation period must be so as to consider that over this observation period, the position of the target is constant. However, shortening the observation period is detrimental to the elimination of additive noise. 
     Thus, existing sensors are rather inaccurate for measuring the position of a fast-moving target. 
     SUMMARY OF THE INVENTION 
     The invention is aimed at remedying this drawback by proposing a more accurate sensor when the target is moving fast. 
     The object of the invention is therefore a sensor of position and/or speed of a movable part in which the estimator is able to establish the estimate of the position and/or of the speed as a function of a model of displacement of the target during the observation period T obs , this model linking the position of the target at an instant t included in the observation period T obs  to at least the position and the speed to be estimated. 
     By virtue of the use of a displacement model, the estimator of the above sensor takes account of the fact that at least the speed of displacement of the target is not zero during the observation period to establish the estimate of the position and speed of the target. It is therefore no longer necessary to choose an observation period short enough for the speed to be almost zero over this period. Thus, the above sensor can use a longer observation period than that of the existing sensors, so as to obtain a more accurate measurement without however being impeded by the fact that the target is moving during the observation period. 
     The embodiments of this sensor can comprise one or more of the following characteristics:
         the estimator is able to use a sliding observation period shifted temporally by at most ((N−1)/N)*T obs  with respect to the previous observation period used to estimate the position and the speed of the target;   the sensor comprises an excitation unit suitable for generating the excitation current and/or voltage in such a way that the spectral energy density of the magnetic excitation field is spread over several frequencies included in a frequency band whose width is at least 2/(N*T obs ) this frequency band containing at least 80% of the energy of the magnetic excitation field;   the width of the frequency band is at most equal to 2/T obs ;   the excitation unit is able to generate a random or pseudo-random sequence of excitation current and/or voltage, in such a way that the N samples of excitation current and/or voltage form a random or pseudo-random series of values, and the estimator is able to estimate the position and/or the speed by projecting a vector {right arrow over (D)} formed of N samples of the electrical measurement signal onto at least one vector of a pseudo-inverse matrix of which a term of the form (M T M) −1  is precalculated for several estimates, where M is a matrix, “exponent T” is the matrix transposition function and “exponent −1” is the inverse function;   at least one second transducer suitable for transforming solely the modifications of the magnetic excitation field that are independent of the displacement of the target along the measurement direction into an electrical reference signal, a second estimator suitable for estimating the value of the multiplicative factor on the basis of the electrical reference signal, and a compensator suitable for compensating for the amplitude variations in the electrical measurement signal that are caused by the variations in the multiplicative factor as a function of the estimated value of this multiplicative factor;   the compensator comprises a regulator suitable for modifying the magnetic excitation field as a function of the deviation between a reference setpoint and the estimated value of the multiplicative factor;   the first estimator is able to automatically increase the length of the observation period when the estimate of the speed of the target decreases;   the target exhibits a break in conductivity that is not collinear with the measurement direction between two materials of different conductivities.       

     These embodiments of the sensor furthermore exhibit the following advantages:
         using a sliding observation period makes it possible to obtain a faster sensor, capable of estimating the position and the speed at time intervals of less than ((N−1)/N).T obs ;   using a magnetic excitation field whose spectral energy intensity is spread makes it possible to improve the immunity to noise of the measurement;   limiting to 2/T obs  the width of the frequency band in which the energy spectrum of the magnetic excitation field is spread makes it possible to avoid wasting energy unnecessarily and therefore to reduce the consumption of the sensor;   using a random or pseudo-random sequence of excitation current and/or voltage makes it possible to precalculate a part of the pseudo-inverse matrix, this subsequently accelerating the execution of the calculations for estimating the position and speed;   the use of a compensator makes it possible to increase the accuracy of the measurement;   the use of a regulator makes it possible to fulfill the functions of a compensator and increases the linearity of the sensor;   modifying the length of the observation period as a function of the estimate of the speed makes it possible to increase the sensitivity of the sensor for low speeds; and   the use of a target made of materials of different conductivities makes it possible to limit the sensitivity of the sensor to temperature variations.       

     The subject of the invention is also a method for measuring the position and/or speed of a movable part with the aid of the above sensor, this method comprising, on completion of the observation period T obs , a step of estimating the position and/or the speed of the target on the basis of N samples of the electrical measurement signal and of N samples of the excitation current and/or voltage, these samples being taken during the observation period T obs  and N being an integer greater than two, this position and/or speed estimate being dependent on a model of displacement of the target during the observation period, this model linking the position of the target at an instant t included in the observation period to at least the position and the speed to be estimated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will be better understood on reading the description which follows, given solely by way of example while referring to the drawings in which: 
         FIG. 1  is a schematic illustration of the architecture of a sensor of position and speed of a movable part; 
         FIG. 2  is an end-on view of a target, of the inductor and of the transducers of the sensor of  FIG. 1 ; 
         FIG. 3  is a flowchart of a method for measuring the position of a movable part with the aid of the sensor of  FIG. 1 ; and 
         FIG. 4  is a schematic illustration of another embodiment of a target for measuring a position and an angular speed. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       FIG. 1  represents a sensor  2  of the position and of the speed of a movable part  4 . 
     Here, by way of illustration, the part  4  moves in translation in a vertical direction represented by the arrow X. 
     The sensor  2  comprises:
         an inductor  10  suitable for inducing a periodic or alternating magnetic excitation field;   a target  12  made of conducting materials that is suitable for modifying as a function of its position within the magnetic excitation field;   a transducer  14  suitable for transforming the magnetic field modified by the target  12  into an electrical measurement signal;   a reference transducer  16  suitable for transforming solely the modifications of the magnetic excitation field that are independent of the displacement of the target  12  in the direction X into an electrical reference signal; and   a signal excitation and processing circuit  18 , linked to the inductor  10  and to the transducers  14  and  16 .       

     The part  4  and the target  12  are fixed to one another, so as to move in an identical manner in the direction X. 
     The inductor  10  and the transducers  14  and  16  are fixed to a plane support  20  arranged opposite a plane face  22  of the target  12 . The support  20  is mechanically independent of the target  12 , in such a way that this target can move freely opposite the support  20  in the direction X. Preferably, the support  20  is made of material that is transparent to electromagnetic fields. 
     The target  12 , the inductor  10  and the transducers  14  and  16  will be described in greater detail with regard to  FIG. 2 . 
     The circuit  18  comprises:
         a controllable excitation unit  24  suitable for generating an alternating current I exc  and a voltage U exc  for exciting the inductor  10 ;   a compensator  26  suitable for compensating for the variations in the electrical measurement signal that are caused by non-additive defects as a function of a reference setpoint and of an estimate of the amplitude of these non-additive defects;   a unit  28  for adjusting the reference setpoint;   an estimator  30  suitable for estimating the value (Â 0 ) of the non-additive defects on the basis of the electrical signal generated by the transducer  16 ; and   an analog-digital converter  32  linked between the transducer  16  and inputs of the estimator  30  to transform the electrical reference signal into a digital reference signal.       

     The non-additive defects give rise here to a variation in the value of a multiplicative factor  A  of the position of the target in the electrical measurement signal. 
     The excitation unit  24  is suitable for generating the current I exc  and the voltage U exc  in such a way that the spectral energy densities of the current I exc  and the voltage U exc  are spread continuously over a frequency band [f min ; f max ]. This frequency band [f min ; f max ] contains at least 80%, and preferably 90%, of the energy of the current I exc  and of the voltage U exc . In this way, the spectral energy density of the magnetic excitation field is also spread continuously over the same frequency band, in such a way that at least 80%, and preferably 90%, of the energy of the magnetic excitation field lies in this frequency band [f min ; f max ]. 
     The width of the frequency band is chosen equal to 2/T obs , where T obs  is a predefined observation period. The frequency f min  is nonzero and preferably greater than 10 kHz. The middle frequency f mid  of the band [f min ; f max ] is equal to the inverse of the response time of the sensor. Thus, the position of the band [f min ; f max ] will be chosen as a function of the response time desired or possible for this sensor. 
     The frequency f mid  is given by the following relation:
 
 f   mid =( f   min   +f   max )/2  (1)
 
     The compensator  26  of non-additive defects is, for example, here a regulator suitable for controlling the unit  24  as a function of the deviation between the reference setpoint delivered by the unit  28  and of an estimate Â 0  of the value of the multiplicative factor  A  delivered by the estimator  30 . 
     For this purpose, the compensator  26  has an input connected to the unit  28 , another input connected to an output of the estimator  30  and a control output connected to an input of the unit  24 . 
     The estimator  30  has an input connected to an output of the converter  32  to receive the digital reference signal and an input connected to the outputs of the unit  24  to receive the voltage U exc . 
     The circuit  18  also comprises:
         an analog-digital converter  36  linked to the transducer  14  to convert the electrical measurement signal into a digital measurement signal; and   an estimator  38  suitable for calculating the estimate {circumflex over (X)} 0  of the position and the estimate {circumflex over (V)} 0  of the speed of the target  12  on the basis of the digital measurement signal and the voltage U exc .       

     The estimator  38  therefore has an input linked to an output of the converter  36 , an input linked to the outputs of the unit  24 , and two outputs for delivering the estimates {circumflex over (X)} 0  and {circumflex over (V)} 0  to circuits outside the sensor  2 . 
     Here, by way of illustration, the estimates {circumflex over (X)} 0  and {circumflex over (V)} 0  are delivered to the input of a unit  40  for additional processings that is suitable for calculating with greater accuracy a new estimate {circumflex over (V)}′ 0  of the speed of the target and an estimate Â′ 0  of the acceleration of the target. The unit  40  also delivers as output the estimate {circumflex over (X)} 0 . 
     The details of the various functions of the units of the circuit  18  will be apparent on reading the description offered with regard to  FIG. 3 . 
     The circuit  18  is also associated with a memory  42  intended to store the samples forming the digital measurement and reference signals as well as the whole set of data required for the calculations to be executed by the estimators  30  and  38 . 
       FIG. 2  represents the face  22  of the target  12  arranged opposite a corresponding surface of the support  20 .  FIG. 2  also represents the arrangement of the inductor  10  and of the transducers  14  and  16  opposite this face  22 . 
     In  FIG. 2 , the elements already described with regard to  FIG. 1  bear the same numerical references. 
     The face  22  is parallel to the vertical direction X of displacement of the target  12  and also parallel to the surface of the support  20 . 
     This face  22  is divided into two sections S def  and S mes  arranged one alongside the other and extending vertically. A vertical limit  50  separates these two sections. 
     The section S def  is made by juxtaposing a material  44  of conductivity C 1  and a material  46  of electrical conductivity C 2  (represented by shading in  FIG. 2 ). 
     The materials  44  and  46  are arranged one alongside the other so as to form a break  48  in conductivity extending parallel to the direction X. For examples the materials  44  and  46  form two bands of constant width extending vertically parallel to the direction X. 
     The materials  44  and  46  are chosen in such a way that the conductivities C 1  and C 2  are very different from one another. Preferably, the ratio of the conductivity C 2  to the conductivity C 1  is greater than or equal to 1000. For example, here, the material  44  is an electrical insulant whose conductivity C 1  is less than 10 −10  S/m while the material  46  is an electrical conductor whose conductivity C 2  is greater than 10 6  S/m, such as copper. 
     The section S mes  is also made from the same two materials  44  and  46 . However, in the section S mes  the material  46  is laid out so as to form a horizontal band of constant width extending in a direction Y perpendicular to the direction X and parallel to the surface of the support  20 . 
     The material  44  is arranged in the section S mes  so as to form two horizontal bands extending in the direction Y and juxtaposed respectively above and below the horizontal band formed with the aid of the material  46 . 
     This layout of the materials  44  and  46  in the surface S mes  makes it possible to create two breaks  52 ,  54  in conductivity that are not collinear with the direction X. Here, these breaks in conductivity  52 ,  54  are parallel to the direction Y. 
     The inductor  10  is here formed of a coil with one or more turns whose winding axis is perpendicular to the face  22 . The section of the inductor  10  opposite the face  22  is extensive enough to make it possible to induce a magnetic field of substantially uniform excitation in the target  12 . 
     The transducers  14  and  16  are arranged inside the windings of the inductor  10 . 
     The transducer  14  is arranged at least opposite one of the breaks in horizontal conductivity  52  or  54 , so as to be sensitive to the displacements of these breaks in conductivity in the direction X. 
     The transducer  16  is conversely arranged opposite the break  48 , so as to be insensitive to the displacements of the target  12  in the direction X. 
     The transducers  14  and  16  are each formed by coils mounted in a differential manner. 
     More precisely, the transducer  14  is formed of two coils  56  and  58  linked in series but wound in opposite directions, in such a way that if the same magnetic field crosses the coils  56  and  58 , the electrical signal generated by the transducer  14  is zero. Such a transducer makes it possible to obtain a linear operating zone around the zero magnetic field. 
     Here, the coil  56  is mounted opposite the break in conductivity  52 , while the coil  58  is mounted opposite the break in conductivity  54 , so as to increase the sensitivity of the transducer  14  to the displacements of the target  10  in the direction X. 
     In a similar manner, the transducer  16  is formed of two coils  60  and  62  mounted in a differential manner. 
     The coil  60  is arranged so as to be solely opposite the material  44  and the coil  62  is placed so as to be opposite the break  48 , whatever the displacements of the target  12  in the direction X. 
     The terminals for linking the inductor  10  and the transducers  14  and  16  to the circuit  18  are represented by small circles in  FIG. 2 . 
     The relations and notations which are used subsequently in this description will now be introduced. 
     The electromotive force e mes (t) developed by the transducer  14  at an instant t is given by the following relation: 
     
       
         
           
             
               
                 
                   
                     
                       e 
                       mes 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       A 
                       ⁢ 
                       
                         
                           ⅆ 
                           
                             
                               I 
                               exc 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           ⅆ 
                           t 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           X 
                           ) 
                         
                       
                     
                     + 
                     
                       noise 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where:
         A is the multiplicative factor whose amplitude varies as a function of non-additive defects;       

     
       
         
           
             
               ⅆ 
               
                 
                   I 
                   exc 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
             
             
               ⅆ 
               t 
             
           
         
       
         
         
           
              is the first derivative with respect to time of the current I exc (t); 
             f(X) is a transduction function giving the image of the flux across the transducer  14  as a function of the position X of the target  12 ; and 
             “noise(t)” is the additive noise which is superimposed on the signal theoretically obtained in the absence of additive noise. 
           
         
       
    
     The value of the factor A depends, for example, on the geometry of the target  12 , the conductivities C 1  and C 2 , and the distance separating the transducer  14  from the surface of the target  12  (“Lift Off”). 
     The electromotive force e def (t) developed by the transducer  16  at the instant t is given by the following relation: 
     
       
         
           
             
               
                 
                   
                     
                       e 
                       def 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       A 
                       ⁢ 
                       
                         
                           ⅆ 
                           
                             
                               I 
                               exc 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           ⅆ 
                           t 
                         
                       
                     
                     + 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         noise 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where the various terms of this relation have already been defined with regard to relation (2). 
     It will be rioted that the electromotive force e def (t) does not depend on the position X of the target  12  since the break in conductivity  48  opposite the transducer  16  is parallel to the direction X. 
     The transduction function f(X) modulates the amplitude of the electrical measurement signal as a function of the position X, this function f(X) can be modeled experimentally in a static state. For example, the target  12  is brought to a position x 1  and maintained at this position while the amplitude (i.e. the peak value) of the electromotive force e mes (t) is measured. Thereafter, the target is displaced to a position x 2  and the previous operations are repeated. 
     By way of illustration, subsequently in this description, it is assumed that the function f(X) is defined by the following relation:
 
 f ( x )=α X   (4)
 
     where α is an experimentally measured constant coefficient. 
     The displacement of the target  12  during the observation period T obs  is modeled with the aid of a displacement model. This displacement model is defined by the following general relation: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       g 
                       ⁡ 
                       
                         [ 
                         
                           
                             X 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           , 
                           
                             
                               
                                 ∂ 
                                 X 
                               
                               
                                 ∂ 
                                 t 
                               
                             
                             ⁢ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           , 
                           
                             … 
                             ⁢ 
                             
                               
                                 
                                   ∂ 
                                   i 
                                 
                                 ⁢ 
                                 X 
                               
                               
                                 ∂ 
                                 
                                   t 
                                   i 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           , 
                           t 
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     t 
                     ∈ 
                     
                       [ 
                       
                         
                           - 
                           
                             T 
                             obs 
                           
                         
                         , 
                         0 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where:
         X(t) is the position of the target along the direction X at the instant t belonging to the observation period [−T obs ;0];   g is the displacement model;       

     
       
         
           
             
               X 
               ⁡ 
               
                 ( 
                 0 
                 ) 
               
             
             , 
             
               
                 
                   ∂ 
                   X 
                 
                 
                   ∂ 
                   t 
                 
               
               ⁢ 
               
                 ( 
                 0 
                 ) 
               
             
             , 
             
               … 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     ∂ 
                     i 
                   
                   ⁢ 
                   X 
                 
                 
                   ∂ 
                   
                     t 
                     i 
                   
                 
               
               ⁢ 
               
                 ( 
                 0 
                 ) 
               
             
           
         
       
         
         
           
              correspond respectively to the position, the speed, the second derivative, . . . , the i th  derivative of the position at the instant t=0, that is to say at the end of the observation period. This position and this or these derivatives are those to be estimated. 
           
         
       
    
     The model of displacement of the target during the observation period T obs  can, for example, be obtained by polynomial expansion. In that case, the model will be of the following form: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         X 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             i 
                             = 
                             N 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 ∂ 
                                 i 
                               
                               ⁢ 
                               X 
                             
                             
                               ∂ 
                               
                                 t 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             0 
                             ) 
                           
                           ⁢ 
                           
                             
                               t 
                               i 
                             
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     t 
                     ∈ 
                     
                       [ 
                       
                         
                           - 
                           
                             T 
                             obs 
                           
                         
                         , 
                         0 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Subsequently in the description, it is assumed that the target  12  can move at high speed during the observation period but that the acceleration during this same observation period is negligible. Under these conditions, the displacement model adopted is the following:
 
 X ( t )= X (0)+ V (0) t   (7)
 
     where X(0) and V(0) are respectively the position and speed of the target at the instant t=0. 
     It will be pointed out that this displacement model conveys the knowledge possessed by the designer of the sensor  2  regarding the displacements of the target  12  during the period T obs . 
     The sampling frequency of the analog-digital converters  32  and  36  is denoted f ech . This frequency f ech  is greater than 5/T obs  and preferably greater than 100/T obs  or 1000/T obs . The number N of samples taken during the period T obs  is therefore greater than 5 and preferably greater than 100 or 1000. N must be greater than 2 at the minimum. 
     The sampling instants are denoted t i , where to corresponds to the current instant (t=0), that is to say to the end of the observation period T obs , while t N-1  corresponds to the start of the observation period, that is to say to the instant −T obs . 
     {right arrow over (D)} is a vector of the N successive samples of the electrical measurement signal. {right arrow over (D)} is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     D 
                     -&gt; 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             D 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             D 
                             
                               N 
                               - 
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             D 
                             1 
                           
                         
                       
                       
                         
                           
                             D 
                             0 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where D i  represents the value of the electrical measurement signal sampled at the instant t i . 
     {right arrow over (E)} is a vector of the N successive samples of the time derivative of the current I exc . This vector is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     E 
                     -&gt; 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             E 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             E 
                             
                               N 
                               - 
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             E 
                             1 
                           
                         
                       
                       
                         
                           
                             E 
                             0 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where E i  is the sample of the time derivative of the current I exc  at the sampling instant t i . 
     {right arrow over (B)} is a vector of the N successive noise samples, defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     B 
                     -&gt; 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             B 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             B 
                             
                               N 
                               - 
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             B 
                             1 
                           
                         
                       
                       
                         
                           
                             B 
                             0 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where B i  represents the amplitude of the additive noise at the sampling instant t i . Unlike the vectors {right arrow over (D)} and {right arrow over (E)}, this vector {right arrow over (B)} is random. 
     By using relations (2), (4) and (7), the following matrix relation linking the vectors {right arrow over (D)} and {right arrow over (B)} can be written: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           D 
                           -&gt; 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             M 
                             · 
                             
                               P 
                               -&gt; 
                             
                           
                           + 
                           
                             B 
                             -&gt; 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       α 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             T 
                                             obs 
                                           
                                         
                                         · 
                                         α 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       α 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         
                                           N 
                                           - 
                                           2 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   N 
                                                   - 
                                                   2 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                 T 
                                                 obs 
                                               
                                             
                                             
                                               N 
                                               - 
                                               1 
                                             
                                           
                                         
                                         · 
                                         α 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         
                                           N 
                                           - 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                   
                                     
                                         
                                     
                                   
                                 
                                 
                                   
                                     
                                         
                                     
                                   
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       α 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             - 
                                             
                                               T 
                                               obs 
                                             
                                           
                                           
                                             N 
                                             - 
                                             1 
                                           
                                         
                                         · 
                                         α 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         1 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       α 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         AE 
                                         0 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       X 
                                       ⁡ 
                                       
                                         ( 
                                         0 
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       V 
                                       ⁡ 
                                       
                                         ( 
                                         0 
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             B 
                             -&gt; 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where {right arrow over (P)} is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     P 
                     -&gt; 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             X 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             V 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The matrix M is defined in relation ( 11 ). 
     It is possible to determine the estimates {circumflex over (X)} 0  and {circumflex over (V)} 0  respectively of the position and speed of the target  12  in the least squares sense by using the so-called “pseudo-inverse” procedure. This procedure is for example described in the following bibliographic reference: 
     R. M. Pringle, A. A. Rayner, “Generalized Inverse Matrices”, London, Griffin, 1971. 
     The estimates {circumflex over (X)} 0  and {circumflex over (V)} 0  are obtained with the aid of the following relation:
 
 {circumflex over (P)} =( M   T   M ) −1   M   T   D=QD   (13)
 
     where:
         {circumflex over (P)} is the estimate vector,   Q is the pseudo-inverse matrix,     T  denotes the transposition function,     −1  denotes the matrix inverse function.       

     The estimate vector {circumflex over (P)} is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     P 
                     ^ 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               X 
                               ^ 
                             
                             0 
                           
                         
                       
                       
                         
                           
                             
                               V 
                               ^ 
                             
                             0 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The pseudo-inverse matrix Q is defined by the following relation:
 
 Q =( M   T   M ) −1   M   T   =[{right arrow over (Q)}   1   ;{right arrow over (Q)}   2 ]  (15)
 
     where {right arrow over (Q)} 1  and {right arrow over (Q)} 2  are orthogonal vectors corresponding respectively to the first and to the second column of the matrix Q. 
     A model h(t) for the evolution over time of the value of the multiplicative factor A is also defined with the aid of the following relation:
 
 A=h ( t )  (16)
 
     It will be assumed here that the variations in the value of the multiplicative factor  A  exhibit a negligible acceleration. This is represented by the following model:
 
 h ( t )= A (0)+ VA (0) t   (17)
 
     where:
         A(0) is the value of the multiplicative factor at the sampling instant t 0 , and   VA(0) is the rate of evolution of the amplitude of the multiplicative factor at the sampling instant t 0 .       

     {right arrow over (D)} d  is a vector of the N successive samples of the electrical reference signal defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       -&gt; 
                     
                     ⁢ 
                     d 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             D 
                             
                               dN 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             D 
                             
                               dN 
                               - 
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             D 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             D 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             D 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where D di  represents the value of the electrical reference signal at the instant t i . 
     {right arrow over (B)} d  is a vector of additive noise samples present in the electrical reference signal and is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     
                       B 
                       -&gt; 
                     
                     d 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             B 
                             
                               dN 
                               - 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             B 
                             
                               dN 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             B 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             B 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             B 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     where B di  is the amplitude of the additive noise at the sampling instant t i . Unlike the vector {right arrow over (D)} d , the vector {right arrow over (B)} d  is random. 
     In a similar manner to what was described with regard to relation (11) it is possible with the aid of relations (3) and (17) to establish the following matrix relation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             D 
                             d 
                           
                           -&gt; 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               M 
                               d 
                             
                             · 
                             
                               
                                 P 
                                 -&gt; 
                               
                               d 
                             
                           
                           + 
                           
                             
                               B 
                               d 
                             
                             -&gt; 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       E 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           T 
                                           obs 
                                         
                                       
                                       ⁢ 
                                       
                                         E 
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       E 
                                       
                                         N 
                                         - 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           
                                             
                                               ( 
                                               
                                                 N 
                                                 - 
                                                 2 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             
                                               T 
                                               obs 
                                             
                                           
                                           
                                             N 
                                             - 
                                             1 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         E 
                                         
                                           N 
                                           - 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                         
                                     
                                   
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                   
                                     
                                         
                                     
                                   
                                 
                                 
                                   
                                     
                                       E 
                                       1 
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           
                                             T 
                                             obs 
                                           
                                           
                                             N 
                                             - 
                                             1 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         E 
                                         1 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       E 
                                       0 
                                     
                                   
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       A 
                                       ⁡ 
                                       
                                         ( 
                                         0 
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       VA 
                                       ⁡ 
                                       
                                         ( 
                                         0 
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           + 
                           
                             
                               B 
                               -&gt; 
                             
                             d 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where P d  is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     P 
                     d 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             VA 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     As previously, it is possible to determine the estimates Â 0  and {circumflex over (V)}A 0  respectively of the value and of the rate of variation of the multiplicative factor  A  minimizing the deviation between the model represented by relation (20) and the samples of the electrical reference signal by using the pseudo-inverse procedure. 
     According to this procedure, these estimates are given by the following relation:
 
 {circumflex over (P)}   d =( M   d   T   M   d ) −1   M   d   T   D=QD   (22)
 
     where:
         the matrix M d  is defined in relation (20), and   the matrix Q d  is the pseudo-inverse matrix.       

     The vector {circumflex over (P)} d  is defined by the following relation: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       ^ 
                     
                     d 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             0 
                           
                         
                       
                       
                         
                           
                             
                               V 
                               ^ 
                             
                             ⁢ 
                             
                               A 
                               0 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The pseudo-inverse matrix Q d  is defined by the following relation:
 
 Q   d =( M   d   T   M   d ) −1   M   d   =[{right arrow over (Q)}   d1   ;{right arrow over (Q)}   d2 ]  (24)
 
     where:
         {right arrow over (Q)} d1  and {right arrow over (Q)} d2  correspond to the vectors defined respectively by the first and second columns of the matrix Q d .       

     The various parameters, that are known in advance, of the above relations are recorded in the memory  42 . For example, the memory  42  contains the value of the following parameters: α, N, T obs  and  A . A is known since the latter is equal to the reference setpoint of the unit  28 . 
     The memory  42  also contains matrices K and K d  defined by the following relations:
 
K=(M T M) −1   (25)
 
K d =(M d   T M d ) −1   (26)
 
     The matrices K and K d  are independent of the values of the vector {right arrow over (E)} when the latter is formed of samples forming a random or pseudo-random series. 
     The value of the period T obs  chosen typically lies between 1 s and 100 μs, and, preferably, it lies between 0.3 ms and 500 μs for a target moving at a speed greater than 1 m/s and preferably at a speed greater than 100 m/s. Here, the value of the period T obs  is dependent on the estimate {circumflex over (V)} 0 . More precisely, if the estimate {circumflex over (V)} 0  increases, the period T obs  is automatically shortened and when the estimate {circumflex over (V)} 0  decreases, the period T obs  is automatically lengthened. This task is, for example, carried out by the estimator  38 . 
     The operation of the sensor  2  will now be described with the aid of  FIG. 3 . 
     Continually, during a step  70 , the compensator  26  compares the reference setpoint delivered by the unit  28  with the estimate Â 0  delivered by the estimator  30 . As a function of the deviation between this setpoint and this estimate, the compensator  26  controls the excitation unit  24 , so as to maintain the amplitude of the multiplicative factor  A  equal to the reference setpoint. 
     At each current sampling instant to, the circuit  18  undertakes a phase  72  of processing the electrical signals delivered by the transducers  14  and  16 . 
     At the start of the phase  72 , during a step  74 , the transducers  14  and  16  transform the magnetic excitation field modified by the target  12  into an electromotive force e mes (t) and e def (t), respectively. During step  74 , these electromotive forces are sampled at the instant t i  by the converters  32  and  36 , so as to obtain the values e mes (t 0 ) and e def (t 0 ). 
     During a step  76 , these samples e mes (t 0 ) and e def (t 0 ) are recorded in the memory  42  as values D 0  and D d0 . 
     During a step  78 , the estimator  38  then constructs on the basis of the samples recorded in the memory the vector {right arrow over (D)} and the estimator  30  constructs the vector {right arrow over (D)} d . 
     During a step  80 , the circuit  18  reads off the value E 0 . Accordingly, the circuit  18  reads off the value of the voltage U exc  at the instant t 0  and stores it as value of E 0 . Specifically, the voltage U exc  is proportional to the derivative of the current I exc  with respect to time and to the value of the inductance of the inductor  10 . 
     Thereafter, during a step  82 , the estimator  38  constructs the vector {right arrow over (E)} and records it in the memory  42 . 
     During a step  84 , the estimator  38  calculates the new matrix M on the basis of the vector {right arrow over (E)}, of the value of the coefficient α, of the value A, and of the observation period T obs  recorded in the memory  42 . 
     Thereafter, during a step  86 , the estimator  38  calculates the pseudo-inverse matrix Q on the basis of relation (15). More precisely, during this step  86 , the estimator  38  multiplies the matrix K prerecorded in the memory  42  by the matrix M T . Thus, on completion of step  86 , the vectors {right arrow over (Q)} 1  and {right arrow over (Q)} 2  are known. 
     On the basis of the matrix Q, during a step  88 , the estimator  38  estimates the position and the speed of the target. More precisely, during an operation  90 , the estimator  38  projects the vector {right arrow over (D)} onto the vector {right arrow over (Q)} 1  to obtain the estimate {circumflex over (X)} 0 . During an operation  92 , the estimator  38  also projects the vector {right arrow over (D)} onto the vector {right arrow over (Q)} 2  to obtain the estimate {circumflex over (V)} 0 . 
     The estimates {circumflex over (X)} 0  and {circumflex over (V)} 0  are delivered by the sensor  2  to the additional-processing unit  40 . 
     During a step  94 , the unit  40  hones the estimates delivered by the sensor  2 . More precisely, during an operation  96 , the unit  40  verifies whether the estimate {circumflex over (V)} 0  is not less than a predetermined threshold S 1 . In the affirmative, the unit  40  calculates a more accurate estimate {circumflex over (V)} 0  of the speed of the target on the basis of the estimates {circumflex over (X)} 0 , {circumflex over (X)} 1 , . . . , {circumflex over (X)} m  where the index m is an integer strictly greater than N. {circumflex over (X)} 1  represents the value of the estimate of the position of the target delivered by the sensor  2  at the sampling instant t i . 
     In the converse case, that is to say if the estimate {circumflex over (V)} 0  is greater than the threshold S 1 , said estimate is not modified, so that {circumflex over (V)}′ 0  is equal to {circumflex over (V)} 0 . The operation  96  makes it possible to improve the accuracy of the speed estimate if the value estimated of this speed by the sensor  2  is low. For example, the threshold S 1  is equal to 0.01 m/s. 
     During an operation  98 , the unit  40  also calculates an estimate Â′ 0  of the acceleration of the target  12  on the basis of the  m  previous estimates {circumflex over (X)} 1  of the position and/or of the speed {circumflex over (V)} 1 . 
     Thereafter, during a step  100 , the unit  40  delivers the estimates {circumflex over (X)} 0 , {circumflex over (V)}′ 0  and Â′ 0 . 
     In parallel with steps  84  to  100 , the estimator  30  estimates the value of the multiplicative factor A at the instant t 0 . 
     More precisely, during a step  104 , the estimator  30  calculates the matrix M d  on the basis of the vector {right arrow over (E)} stored. Thereafter, during a step  106 , the estimator  30  calculates the pseudo-inverse matrix Q d  as defined in relation (24). For this purpose, during step  106 , the estimator  30  multiplies the stored matrix K d  by the transpose of the matrix M d . 
     Thereafter, during a step  108 , the estimator  30  estimates the value Â 0  and the rate {circumflex over (V)}A 0  of variation of the multiplicative factor A. More precisely, during an operation  110 , the estimator  30  projects the vector {right arrow over (D)} d  onto the vector {right arrow over (Q)} d1  to obtain the estimate Â 0 . Likewise, during an operation  112 , the estimator  30  projects the vector {circumflex over (D)} d  onto the vector {circumflex over (Q)} d2  to obtain the estimate {circumflex over (V)}A 0 . 
     Thereafter, during a step  114 , the estimator  30  dispatches the estimate Â 0  to the compensator  26  which uses this estimate during step  70  to keep the value of the factor  A  constant. 
     Phase  72  is repeated at each sampling instant, in such a way that a new estimate of the position, of the speed and of the acceleration of the target  12  is constructed at each new sampling instant. 
       FIG. 4  represents a target  120  adapted to estimate the position and the angular speed of a part rotating about an axis  122 . 
     The target  120  here has the form of a disk divided into zones formed from materials of various conductivities and arranged one with respect to the other so as to form two sections S mes  and S def . 
     S mes  is here an annulus whose left half is formed for example by the material  46  while the right half is formed with the aid of the material  44 . The juxtaposition of these two materials  44  and  46  creates two breaks in conductivity  126  and  127  which extend radially. 
     The surface S def  is placed at the center of the surface S mes . This surface S def  is formed by a circular central pad made from the material  44 . This central pad is surrounded by a complete annulus made with the aid of the material  46 . Such a configuration of the surface S def  creates a break  130  in conductivity that is circular and centered about the axis  122 . 
       FIG. 4  also represents an inductor  134  suitable for creating a magnetic field of substantially uniform excitation and two differential transducers  136  and  138 . 
     The transducer  136  comprises, as was described with regard to  FIG. 2 , two coils mounted in series and wound in opposite directions to one another, in such a way that when these two coils are crossed by the same magnetic field, the electrical signal at the terminals of the transducer  136  is zero. 
     The transducer  136  is arranged opposite the surface S mes , so as to deliver an electrical measurement signal as a function of the rotational displacement of the breaks in conductivity  126  and  127 . 
     Like the transducer  136 , the transducer  138  is formed of two coils linked in series and wound in opposite directions. However, the transducer  136  is placed opposite the surface S def , so as to deliver an electrical reference signal independent of the angular position of the disk  120 . 
     The inductor  134  plays the role of the inductor  10  of  FIG. 1  and the transducers  136  and  138  play respectively the role of the inductors  14  and  16  of  FIG. 1 . The operation of a position and angular speed sensor using the target  120  will therefore not be described here in greater detail. 
     Numerous other embodiments are possible. 
     For example, it is not necessary for the transducers  14  and  16  or  136  and  138  to be formed of two coils mounted in a differential manner. 
     The material C 1  described as insulating can, for example, be air, thereby simplifying the fabrication of the target  12  or  120 . 
     If the non-additive disturbances play a rather unimportant role in the accuracy of the measurement, the sensor  2  can be simplified by dispensing with the transducer  16  as well as the converter  32 , the estimator  30 , the compensator  26  and the adjustment unit  28 . 
     The compensation of the non-additive defects has been described here as being carried out with the aid of a compensator  26  making it possible to keep the value of the multiplicative factor  A  constant. As a variant, such a compensation can also be carried out by dividing the electromotive force e mes (t) by the amplitude of the electromotive force e def (t). 
     In the latter case, a voltage divider is introduced between on the one hand, the converter  36  and the estimator  30  and on the other hand, the estimator  38 . This voltage divider performs the division of the electromotive force e mes (t) by the amplitude of the electromotive force e def (t) obtained as output from the estimator  30 . The compensator  26  and the adjustment unit can then be dispensed with. 
     It is also possible to compensate for the non-additive defects by simply multiplying the electromotive force e mes (t) by a finite expansion of the amplitude of the inverse of e def (t). A finite expansion such as this can take the following form:
 
(1−ε)/A
 
     The transduction function f(X) has been described as being solely proportional to X. However, other forms of relations are possible. For example, the transduction function can have the following form:
 
 f ( x )=α X+β 
 
where α and β are known coefficients.
 
     Here the coefficients of the function f(X) have been described as constant. However, should these coefficients not be constant, it is possible to provide a unit for adjusting the value of these coefficients as a function, for example, of the estimate {circumflex over (X)} 0 . 
     The inductor and the transducer  14  have been described as being formed with the aid of distinct windings. As a variant, the same winding is used at one and the same time in the guise of inductor suitable for creating the magnetic excitation field and in the guise of transducer for measuring the magnetic excitation field modified by the target. 
     Here, the electrical breaks in conductivity have been described as being formed by juxtaposing materials of different electrical conductivities. As a variant, these electrical breaks in conductivities can be obtained by forming ribs or scores on the surface of a single conducting material. Also as a variant, these breaks in conductivity can, if desirable, be buried inside the conducting material. 
     If the acceleration of the target is not negligible during the period T obs , it is possible to add a term to the displacement model described here representing the acceleration of the target during the observation period. Other terms representing higher-order derivatives of the position with respect to time can also be added to the displacement model if their contribution is not negligible for the estimate of the position of the target. In the latter cases, the estimator will then deliver an estimate for the position and the speed of the target as well as an estimate for these higher-order derivatives of the position. 
     Here, the displacement model has been described as being obtained with the aid of a polynomial decomposition. However, if the displacement of the target comprises vibratory modes, a displacement model can be obtained with the aid of Fourier series. 
     Here, given that the winding axis of the transducer  16  is perpendicular to the face  22  of the target, this transducer  16  is sensitive solely to the non-additive defect in this direction perpendicular to the face  22 . As a variant, one or more other additional transducers having winding axes that are not collinear with that of the transducer  16  are provided so as to measure the non-additive defect amplitude in non-collinear directions. 
     Likewise, several excitation inductors may be provided. 
     It is also possible to use several transducers such as the transducer  14 , for example, to improve the angular resolution. 
     The sections S mes  and S def  have been described as being adjacent to one another. As a variant, these surfaces are non-adjacent and, for example, borne by targets spaced mutually apart spatially. 
     It is not necessary that the excitation inductor be arranged on the same support as that used to support the transducers  14  and  16 . For example, the excitation inductor can be placed on the other side of the target with respect to the side where the transducers  14  and  16  are situated. 
     It is also possible to control the unit  24  in such a way as not to spread the spectrum of the magnetic excitation field. For example, in that case, the excitation current I exc  is a pure sinusoid at a frequency f 0 . In this situation, the estimator  30  is replaced with a synchronous demodulator able to filter the electrical signal received so as to extract therefrom the amplitude of the component of frequency f 0 . 
     If the vector {right arrow over (E)} then repeats at regular intervals T e , it is possible to estimate the position every interval T e  only. In this way, since at the end of the interval T e , the vector {right arrow over (E)} is identical to that at the start of this interval, it is not necessary to recalculate the pseudo-inverse matrix, thereby making it possible to accelerate the calculations. 
     The target  12  has been described here as being fixed to the movable part. As a variant, the target  12  is integral with the movable part and forms only one single block with this part. 
     The spectral spreading of the magnetic excitation field has been described as being continuous. As a variant, it can be discrete. 
     It is also possible to calculate the values E i  instead of reading them off at the output of the excitation unit  24 . It is possible to calculate the values E i  when the evolution over time of the magnetic excitation field is known in advance. This may, for example, be the case when the magnetic excitation field is periodic. 
     As a variant, the target can be made of magnetic material. 
     The sensor described above can be adapted to deliver solely an estimate of the position of the target or solely an estimate of the speed of the target. In this variant, the vector {right arrow over (D)} is then projected solely either onto the vector {right arrow over (Q)} 1  or onto the vector {right arrow over (Q)} 1  depending on the estimate that one seeks to obtain.