Abstract:
A command procedure for an active magnetic bearing, the magnetic bearing comprising a series of electromagnetic actuators forming a stator, each actuator being suitable for exerting radial force on the rotor, a ferromagnetic body forming a rotor, kept free of contact between the electromagnetic actuators and suitable for being set in rotation around an axis of rotation, the rotor being suitable to undergo precession movements in particular. Sensors suitable for detecting radial displacements of the rotor and issuing position signals representative of the radial position of the rotor in relation to the actuators. Calculation of at least one actuator command signal the calculation of the command signal consisting of the application of at least one transfer function to the position signals, the transfer function containing a number of correction coefficients.

Description:
FIELD OF THE INVENTION 
     Embodiments of the present invention relate to a command procedure for a magnetic bearing, this magnetic bearing comprising a number of electromagnetic actuators forming a stator, each actuator receiving an input signal for its command and suitable for exerting radial force on the rotor; a ferromagnetic body forming a rotor, kept free of contact between the electromagnetic actuators and suitable for being set in rotation around an axis of rotation, the rotor being suitable in particular for undergoing movements of procession under the action of radial momentum induced by the radial forces exercised and applied perpendicularly to the axis of rotation; sensors for detecting radial displacement and for issuing position signals representative of the radial position of the rotor in relation to the actuators. 
     The procedure comprising the following stages: calculation, on the basis of position signals, of at least one command signal for the actuators, calculation of the command signal including the application of at least one transfer function to the position signals, the transfer function comprising a number of correction coefficients. 
     Application of one or each command signal calculated on input of an actuator for control of the radial position of the rotor. 
     Embodiments of the present invention also relate to a command device for an active magnetic bearing. The magnetic bearing comprising a number of electromagnetic actuators forming a stator, each actuator receiving an input signal for its command and being suitable for exerting radial force on the rotor; a ferromagnetic body forming a rotor, kept free of contact between the electromagnetic actuators and suitable for being set in rotation around an axis of rotation, the rotor being suitable in particular for undergoing movements of procession under the action of radial momentum induced by the radial forces exercised and applied perpendicularly to the axis of rotation; and sensors for detecting radial displacement of the rotor and for issuing position signals representative of the radial position of the rotor in relation to the actuators. 
     The command device containing means of calculation on the basis of position signals, the means of calculation being suitable for applying at least one transfer function to the position signals, the transfer function containing a number of correction coefficients, the command device being suitable for applying the or each command signal for the actuators on entry to an actuator so as to direct the radial position of the rotor. 
     For example, an active magnetic bearing allows a moveable body to be kept in a state of lift, typically an electric motor shaft, in a fixed position. It allows the shaft of the engine rotor to turn without friction or contact around an axis of rotation. In the specific domain of very high-speed motors, this type of bearing helps significantly increase the life span of the mobile mechanical parts, and thus limit maintenance operations on these parts. 
     BACKGROUND OF THE INVENTION 
     The prior art refers to a command procedure for a magnetic bearing of the type mentioned above. During such a process, an actuator directs the rotation of the rotor around its axis of rotation, this rotation corresponding to a given degree of freedom for the rotor. Excitation of the five other degrees of freedom of the rotor is an undesirable disruption that must be corrected by the means of command. To do this, the means of command, in standard form, comprise one command unit for each degree of freedom of the rotor, the command of each degree of freedom of the rotor thus being separated from the commands for the other degrees of freedom. However, for high rotor-rotation speeds, such command procedures have proved unsuitable. In fact, because of the precession movements caused by the gyroscopic effect of the rotor, coupling occurs between the degrees of freedom. For these rotation speeds, the correction of the movements of the rotor by means of these standard command procedures is therefore relatively unstable. In addition, the performance of the electric motor associated with the rotor is noticeably reduced. 
     To overcome this problem, document EP 1,771,667 B1 describes a command procedure for a magnetic bearing, within which the methods of precession of the rotor, combined with the precession movements, are taken into account by the means of command. More specifically, different correctors are modelled, each corrector taking account of the modes of precession of the rotor in the form of outside uncertainties. The command procedure therefore involves, for a given speed of rotation of the rotor, selection of the most suitable corrector. In such a procedure, correction of the precession movements of the rotor may prove selectively stable, especially in close proximity to the speeds of rotation for which the correctors have been designed. However, such a command procedure does not guarantee stability of correction throughout the full range of speeds of rotation of the rotor, especially when passing from one corrector to another. In addition, as the rotor precession modes are based on outside uncertainties, these uncertainties also constitute sources of instability for final the correction. 
     BRIEF SUMMARY OF THE INVENTION 
     Embodiments of the present invention propose a command procedure for an active magnetic bearing that ensures stability of correction of rotor precession movements regardless of speed of rotation of the rotor. 
     According to an embodiment of the present invention, there is provided a command procedure of the type mentioned above, within which at least one correction coefficient depends continually on the speed of rotation of the rotor and within which the or each command signal is suitable for commanding continuous offset of the rotor precession movements. 
     According to an embodiment of the present invention, the command procedure comprises one or more of the following characteristics, taken either in isolation or according to all technically possible combinations: 
     The said correction coefficient is a function of the moment of inertia of the rotor around the axis of rotation. 
     The said correction coefficient is a function of the moment of inertia of the rotor around an axis perpendicular to the axis of rotation. 
     During the calculation of at least one command signal for the actuators, at least the first and second command signals for the actuators are calculated simultaneously, the first command signal being suitable for injection on entry of a first actuator, the said first actuator being suitable for inducing a force on the rotor according to a first axis perpendicular to the axis of rotation, the second command signal being suitable for injection on entry of a second actuator different from the first actuator, the said second actuator being suitable for inducing a force on the rotor according to a second axis perpendicular to the first axis and to the axis of rotation. 
     The stage of calculation of at least one actuator command signal contains an intermediate calculation stage, based on the position signals, of at least a first and second command signal for inclination of the rotor in relation to the actuators, the first and second command signal for an inclination being suitable for commanding inclination of the rotor around the first and second axis respectively, the first and second actuator command signals respectively being calculated on the basis of the first and second inclination command signals respectively. 
     The calculation stage of at least one actuator command signal contains an intermediate calculation stage, based on the position signals, of at least a first and second rotor inclination movement signal in relation to the actuators, the first and second inclination movement signal being representative of the inclination of the rotor around the first and second axes respectively, each inclination command signal being calculated on the basis of the first and second inclination movement signals, the first and second inclination movement signals being processed separately for the purposes of calculating the first and second inclination command signals. 
     According to an embodiment of the present invention, there is provided a command device of the above-mentioned type, in which at least one correction coefficient depends continuously on the speed of rotation of the rotor, so that the command device is suitable for directing, via the or each command signal, the continuous offset of the rotor precession movements. 
     According to an embodiment of the present invention, the command device contains one or more of the following characteristics, taken either in isolation or according to all technically possible combinations: 
     The means of calculation are suitable for simultaneously calculating at least a first and second command signal for the actuators, the first command signal being suitable for injection on entry of a first actuator, the said actuator being suitable for inducing a force on the rotor according to a first axis perpendicular to the axis of rotation, the second command signal being suitable for injection on entry of a second actuator different from the first actuator, the second actuator being suitable for inducing a force on the rotor according to a second axis perpendicular to the first axis and the axis of rotation. 
     The means of calculation contain an inclination command unit, the inclination command unit being suitable for calculating, on the basis of the position signals, at least a first and second command signal for inclination of the rotor in relation to the actuators, the first and second inclination command signals respectively being suitable for commanding the inclination of the rotor around the first axis and the second axis respectively. 
     The means of calculation contain a signal conversion unit, the signal conversion unit being suitable for calculating a component of the first actuator command signal and a component of the second actuator command signal respectively, on the basis of the first and second inclination command signals respectively. 
     The means of calculation contain a signal conversion element, the signal conversion element being suitable for calculating at least a first and second signal of rotor inclination movement in relation to the actuators, the first and second inclination movement signals respectively being representative of the inclination of the rotor around the first and second axes respectively, and being suitable for transmission to the inclination command unit. 
     The inclination command unit is suitable for processing separately the first and second inclination movement signals for calculating the first and second command signals for an inclination. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The characteristics and advantages of embodiments of the present invention will become apparent from a reading of the following description, which is given purely as a non-limitative example and made with reference to the attached diagrams, in which: 
         FIG. 1  is a schematic representation of a set consisting of an active magnetic bearing and a bearing command device according to an embodiment of the present invention, the bearing comprising eight electromagnetic actuators and a rotor kept free of contract between the actuators; 
         FIG. 2  is a schematic representation of the command device of  FIG. 1 , comprising a command unit for inclination of the rotor in relation to the actuators; 
         FIG. 3  is a schematic representation of the inclination command unit of  FIG. 2 ; and 
         FIG. 4  is a flow chart representing the command procedure according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  represents a set  1  comprising an active magnetic bearing  10  and means  11  of commanding the bearing  10 . In the example of possible use, the active magnetic bearing  10  is radial, suitable for example for supporting a rotating shaft in a revolving electrical machine. The electrical machine presents nominal power in excess of 3 MW, for example 8 MW, together with a speed of rotation in excess of 8,000 revolutions per minutes, for example 14,000 revolutions per minute. 
     The bearing  10  contains a rotor  12 . In the example of realisation, the rotor  12  consists of the rotating shaft of the revolving electrical machine and moves in rotation around an axis Z-Z′ at a speed of rotation Ω, the axis Z-Z′ being parallel to the greatest dimension of the rotor  12 . In the next part of the description, the term “axial direction” is given to the direction defined by the axis Z-Z′, and the term “radial direction” is given to any direction defined by an axis perpendicular to the axis Z-Z′. An orthogonal reference, x-y-z, is in addition defined in relation to the axis Z-Z′, as illustrated in  FIG. 1 . 
     The rotor  12  is for example made of ferromagnetic material and is likened, in the next part of the description, to a rigid solid presenting six degrees of freedom. One of the degrees of freedom of the rotor  12  corresponds to the rotation around the axis Z-Z′. In standard form, this degree of freedom is controlled independently by an electromagnetic actuator, not represented in the figures. The rotor  12  presents a polar moment of inertia Jp and a transverse moment of inertia Jt. The polar moment of inertia Jp is defined as the moment of inertia of the rotor  12  around the axis Z-Z′. The transverse moment of inertia Jt is, meanwhile, defined as the moment of inertia of the rotor  12  around an axis perpendicular to the axis Z-Z′ belonging to the plane x-y. 
     The bearing  10  also comprises a first actuator unit  14 A, a second actuator unit  14 B, a first sensor  16 A, a second sensor  16 B, a third sensor  16 C and a fourth sensor  16 D. The actuator units  14 A,  14 B are suitable for exerting radial forces on the rotor  12  according to the directions x and y, and thus for keeping the rotor  12  in suspense. The sensors  16 A,  16 B,  16 C,  16 D are suitable for measuring the position of the rotor  12  in relation to the actuator units  14 A,  14 B at specific points according to determined directions. They are also suitable for delivering electronic position signals, these signals being representative of the radial position of the rotor  12  in relation to the actuator units  14 A,  14 B. 
     Each actuation unit  14 A,  14 B comprises at least one electromagnetic actuator. In the example of possible use, the actuation unit  14 A,  14 B respectively comprises a first electromagnetic actuator  18 A and  18 B respectively, a second electromagnetic actuator  20 A and  20 B respectively, a third electromagnetic actuator  22 A and  22 B respectively, and a fourth electromagnetic actuator  24 A and  24 B respectively. 
     As known in itself, each actuator  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B contains an excitation coil wound around a magnetic circuit with regard to the rotor  12  in order to apply radial forces to the rotor. The magnetic circuit, for example, consists of ferromagnetic material. 
     Each actuator  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B receives, at the point of entry to its coil, an input signal, more specifically an input command current. The first and second actuators  18 A,  20 A, and  18 B,  20 B respectively, receive a first input signal I X1  and a second input signal I X2  respectively. The third and fourth actuators  22 A,  24 A and  22 B,  24 B respectively receive a third input signal I Y1 , and a fourth input signal I Y2  respectively. 
     The eight actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B are fixed in relation to each other, and together form a stator. They are suitable for keeping the rotor  12  in suspense, the rotor  12  being kept free of contact between the eight actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B. 
     The first actuator  18 A,  18 B respectively is suitable for exercising radial force on the high and low regions respectively of the rotor  12 , in the direction +x. The second actuator  20 A,  20 B respectively is suitable for exercising radial force on the high and low regions respectively of the rotor  12 , in the direction −x. The third actuator  22 A,  22 B respectively is suitable for exercising radial force on the high and low regions respectively of the rotor  12 , in the direction +y. The fourth actuator  24 A,  24 B respectively is suitable for exercising radial force on the high and low regions respectively of the rotor  12 , in the direction −y. 
     The radial forces exercised by the actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B on the rotor  12  induce radial momentum on the rotor  12 . This radial momentum is applied in the radial plane radial x-y, perpendicular to the axis of rotation Z-Z′, and leads to coupling between the degrees of freedom of the rotor, especially between the degrees of freedom corresponding to rotations around axes x and y. These couplings correspond to precession movements of the rotor  12 , especially direct and indirect precession movements, known as such as commonly termed the “gyroscopic effect”. 
     The electronic position signals issued by the sensors  16 A,  16 B,  16 C,  16 D are representative of the radial position of the rotor  12  in relation to the actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B. 
     The first sensor  16 A, the second sensor  16 B respectively, is suitable for measuring the position of the upper region and the position of the lower region, respectively, of the rotor  12  along the axis x. It sends an electronic signal X 1 , X 2  respectively, representative of this position. The third sensor  16 C, the fourth sensor  16 D respectively, is suitable for measuring the position of the upper region and the position of the lower region, respectively, of the rotor  12  along the axis y. The third sensor  16 C, the fourth sensor  16 D respectively, sends an electronic signal Y 1 , Y 2  respectively, representative of this position. 
     In the method of implementation being considered, the means of command  11  are connected on one h and to each sensor  16 A,  16 B,  16 C,  16 D in order to pick up the position signals X 1 , X 2 , Y 1 , Y 2 , and on the other hand to each actuator  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B. The means of command  11  are suitable for calculating a command signal from the actuators and for applying this command signal to the input of each actuator for controlling the radial position of the rotor  12 . 
     The means of command  11  contain a first subtractor  26 A, a second subtractor  26 B, a third subtractor  26 C and a fourth subtractor  26 D. The means of command  11  also contain an actuator command device  28 , connected between the outlets from the subtractors  26 A,  26 B,  26 C,  26 D and the actuator input points. 
     Each subtractor  26 A,  26 B,  26 C and  26 D respectively receives at its non-inverting input point a reference signal X 1ref , X 2ref , Y 1ref , Y 2ref  respectively and at its inverting input point the signal X 1 , X 2 , Y 1 , Y 2  respectively. In standard form, the reference signal X 1ref , X 2ref  respectively corresponds to an instruction signal in the upper region and lower region respectively of the  12  along the axis x. In the same way, the reference signal de Y 1ref , Y 2ref  respectively corresponds to an instruction signal in the upper region and lower region respectively of the rotor  12  along the axis y. Each subtractor  26 A,  26 B,  26 C,  26 D respectively supplies, at its output point, an error signal S X1 , S X2 , S Y1 , S Y2  respectively. 
     The command device  28  receives the four error signals S X1 , S X2 , S Y1 , S Y2  at its input point. It is suitable for calculating the input command signals I X1 , I X2 , I Y1 , I Y2  and for applying these signals at the input point to the actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B, as described below. 
     As illustrated in  FIG. 2 , the command device  28  contains a first input terminal  29 A, a second input terminal  29 B, a third input terminal  29 C and a fourth input terminal  29 D. It also contains a first output terminal  30 A, a second output terminal  30 B, a third output terminal  30 C, and a fourth output terminal  30 D. The command device  28  also contains a first intermediate command signal calculation element  32 , a second intermediate command signal calculation element  34 , and a command signal calculation module  36 , connected to the outlet of the calculation elements  32 ,  34 . 
     As known in the standard form, the first calculation element  32  receives at its input point the error signals S X1 , S X2 , S Y1 , S Y2  and is suitable for generating intermediate command signals A X1 ′, A X2 ′, A Y1 ′, A Y2 ′ for the actuators. The intermediate command signals A X1 ′, A X2 ′, A Y1 ′, A Y2 ′ are suitable for commanding offset of the “negative rigidity” effect on the rotor  12 , this effect being already known and caused by the currents circulating in the actuator coils. The first calculation element contains a first amplifier  38 A, a second amplifier  38 B, a third amplifier  38 C and a fourth amplifier  38 D. 
     The amplifier  38 A,  38 B,  38 C,  38 D respectively is connected to the input terminal  29 A,  29 B,  29 C,  29 D respectively and is suitable for supplying at its output point the intermediate command signal A X1 ′, A X2 ′, A Y1 ′, A Y2 ′ respectively. 
     The first amplifier  38 A and third amplifier  38 C respectively are suitable for multiplying the signal S X1 , S Y1  respectively by a constant gain K′ 1 . The second amplifier  38 B and fourth amplifier  38 D respectively are suitable for multiplying the signal S X2 , S Y2  respectively by a constant gain K′ 2  different from the gain K′ 1 . The values of the gains K′ 1  and K′ 2  are chosen according to the standard methods of the prior art. 
     The second calculation element  34  receives at its input point the error signals S X1 , S X2 , S Y1 , S Y2  and is suitable for generating the intermediate command signals A X1 , A X2 , A Y1 , A Y2  for the actuators. The intermediate command signals A X1 , A X2 , A Y1 , A Y2  are suitable for commanding offset of the radial translational movements and the rotational movements of the rotor  12 . The second calculation element contains a first  40 A and a second  40 B adding amplifier module, a first  42 A and second  42 B regulator, and a first  44 A and second  44 B command signal separator. It also contains a signal conversion element  46 , am inclination command unit  48  and a signal conversion unit  50 , respectively connected in series. The second calculation element  34  also contains a first adding amplifier  52 A, a second adding amplifier  52 B, a third adding amplifier  54 A and a fourth adding amplifier  54 B. 
     The first adding amplifier module  40 A and the second adding amplifier  40 B respectively are connected on one hand to the two input terminals  29 A &amp;  29 B and  29 C &amp;  29 D respectively and on the other hand to the first regulator  42 A and second regulator  42 B respectively. They receive at one input point the error signal S X1 , S Y1  respectively and at their other input point the signal S X2 , S Y2  respectively, and supply at their output point a signal ST X , ST Y  respectively. The first adding amplifier module  40 A and the second adding amplifier module  40 B respectively are suitable for applying a distinct weighting coefficient to each signal present at one of their input points, and for adding together the resulting signals to supply the signal ST X , ST Y  respectively. 
     The output point of the first regulator  42 A and of the second regulator  42 B respectively is connected to the input point of the first separator  44 A and of the second separator  44 B respectively. As is already known, each regulator is for example of the PID type (Proportional Integral Derived), this type of regulator being used as the norm in the regulation of looped systems. Each regulator  42 A,  42 B presents a transfer function C 1 (p), expressed for example, with Laplace&#39;s transformation, as 
                   C   1     ⁡     (   p   )       =       K     p   ⁢           ⁢   1       +       K     i   ⁢           ⁢   1       p     +         K     d   ⁢           ⁢   1       ·   p       1   +       K     d   ⁢           ⁢   1     ′     ·   p             ,         
where K p1 , K i1 , K d1  and K d1 ′ are constant gains, as is already known. The first regulator  42 A and the second regulator  42 B are suitable for supplying at their output point a command signal AT X , AT Y  respectively for translational movement of the rotor  12  along the axis x and the axis y respectively. The first regulator  42 A and the second regulator  42 B respectively are thus suitable for independently directing the translational movements of the rotor  12  along the axis x and the axis y respectively.
 
     In a variation, each regulator  42 A,  42 B is of the PI (Proportional Integral) type. 
     One output point of the first separator  44 A and of the second separator  44 B is connected to an input point of the first adding amplifier  52 A and an input point of the third adding amplifier  54 A. The other output point of the first separator  44 A and of the second separator  44 B is connected to an input point of the second adding amplifier  52 B and an input point of the fourth adding amplifier  54 B. The first separator  44 A and second separator  44 B respectively are suitable for applying a first weighting coefficient to the signal AT X  and AT Y  and for supplying the resulting signal at the input point to the first adding amplifier  52 A and to the third adding amplifier  54 A. It is in addition suitable for applying to the signal AT X  and AT Y  respectively a second weighting coefficient, separate from the first coefficient, and for supplying the resulting signal at the input point to the third adding amplifier  52 B and to the fourth adding amplifier  54 B. 
     The signal conversion element  46  receives at its input point the error signals S X1 , S X2 , S Y1 , S Y2  and is suitable for generating a first inclination movement signal S ΦX  for the rotor  12  in relation to the actuators and a second inclination movement signal S ΦY  for the rotor  12  in relation to the actuators. The first signal S ΦX  and the second signal S ΦY  respectively are representative of the inclination of the rotor  12  around the axis x and around the axis y respectively. The element  46  contains a first subtractor module  56 A and a second subtractor module  56 B. 
     The first subtractor module  56 A, and the second subtractor module  56 B are connected on the one hand to the two input terminals  29 A &amp;  29 B and  29 C &amp;  29 D respectively and on the other hand to the input point of the inclination command unit  48 . Each subtractor module  56 A,  56 B presents two input points and one output point. The first subtractor module  56 A and the second subtractor module  56 B respectively receive at one of their input points the error signal S X1 , S Y1  respectively and at their other input point the error signal S X2 , S Y2  respectively, and supply at their output point the signal S ΦY , S ΦX  respectively. The first subtractor module  56 A and the second subtractor module  56 B are suitable for applying the same weighting coefficient to each signal present at one of its input points and for subtracting the resultant signals to supply the signal S ΦY , S ΦX  respectively. 
     The inclination command unit  48  receives the signals S ΦX , S ΦY  at its input point and is suitable for generating a first inclination command signal A ΦX  for the rotor  12  in relation to the actuators and a second inclination command signal for the rotor  12  in relation to the actuators. The first signal A ΦX  and the second signal A ΦY  respectively are suitable for commanding the inclination of the rotor  12  around axis X and around axis Y respectively. The unit  48  is also suitable for separately processing the signals S ΦX , S ΦY  in order to calculate the signals A ΦX , A ΦY . 
     As illustrated in  FIG. 3 , the inclination command unit  48  contains a first regulator  58 A, a second regulator  58 B, a third regulator  58 C and a fourth regulator  58 D. It also contains a subtractor  60  and an adding amplifier  62 . 
     The first regulator  58 A, and the second regulator  58 B respectively are connected between the output point of the first subtractor module  56 A and the positive input point of the subtractor  60  and one input point of the adding amplifier  62 . The third regulator  58 C and the fourth regulator  58 D are connected between the output point of the second subtractor module  56 B and the other input point of the adding amplifier  62  and the inverting input point of the subtractor et  60 . Each regulator  58 A,  58 B,  58 C,  58 D is also connected to a device for measuring the rotation speed Ω of the rotor  12 , not represented in the figures. In the example of possible use, each regulator  58 A,  58 B,  58 C,  58 D is PID and realised using interconnected analogue components. 
     In a variation, each regulator  58 A,  58 B,  58 C,  58 D consists of programmable logic components or of dedicated integrated circuits. 
     The first and third regulators  58 A,  58 C present a transfer function C Φ1 (p, Ω), expressed for example as follows: 
                         C   Φ1     ⁡     (     p   ,   Ω     )       =       K     p   ⁢           ⁢   Φ   ⁢           ⁢   1       ·     (     1   +     1       K     i   ⁢           ⁢   Φ1       ·   p       +           K     d   ⁢           ⁢   1   ⁢   Φ1       ⁡     (   Ω   )       ·   p       1   +         K     d   ⁢           ⁢   2   ⁢   Φ1       ⁡     (   Ω   )       ·   p           )         ,           (   1   )               
where K pΦ1  and K iΦ1  are constant gains and K d1Φ1 (Ω) and K d2Φ1 (Ω) are expressed for example as follows:
 
                       K     d   ⁢           ⁢   1   ⁢   Φ1       ⁡     (   Ω   )       =     K   ·         4   ·     J   t   2           4   ·     K     p   ⁢           ⁢   Φ1       ·     J   t       -       Ω   2     ·     J   p   2                       (   2   )                   K     d   ⁢           ⁢   2   ⁢   Φ1       ⁡     (   Ω   )       =       K   ′     ·         4   ·     J   t   2           4   ·     K     p   ⁢           ⁢   Φ1       ·     J   t       -       Ω   2     ·     J   p   2                       (   3   )               
where K and K′ being constant gains.
 
     As indicated by formulae 2 and 3, each coefficient K d1Φ1 , K d2Φ1  depends continually on the speed of rotation Ω of the rotor  12 . Each coefficient K d1Φ1 , K d2Φ1  is also a function of the polar moment of inertia Jp and of the transverse moment of inertia Jt of the rotor  12 . 
     Similarly, the second and fourth regulators  58 B,  58 D present a transfer function C Φ2 (p, Ω), different from the function C Φ1 (p, Ω), expressed for example as follows: 
                       C   Φ2     ⁡     (     p   ,   Ω     )       =       K     p   ⁢           ⁢   Φ2       ·     (     1   +     1       K     i   ⁢           ⁢   Φ2       ·   p       +           K     d   ⁢           ⁢   1   ⁢   Φ2       ⁡     (   Ω   )       ·   p       1   +         K     d   ⁢           ⁢   2   ⁢   Φ2       ⁡     (   Ω   )       ·   p           )               (   4   )               
where K pΦ2  and K iΦ2  are constant gains and K d1Φ2 (Ω) and K d2Φ2 (Ω) are expressed for example as follows:
 
                       K     d   ⁢           ⁢   1   ⁢   Φ2       ⁡     (   Ω   )       =       K   ″     ·         4   ·     J   t   2           4   ·     K     p   ⁢           ⁢   Φ2       ·     J   t       -       Ω   2     ·     J   p   2                       (   5   )                   K     d2   ⁢           ⁢   Φ2       ⁡     (   Ω   )       ⁢       K   ′′′     ·         4   ·     J   t   2           4   ·     K     p   ⁢           ⁢   Φ2       ·     J   t       -       Ω   2     ·     J   p   2                       (   6   )               
where K″ and K′″ being constant gains.
 
     The transfer function expressions C Φ1 (p, Ω) and C Φ2 (p, Ω) are obtained via several stages. During the first stage, the fundamental equations of the dynamic for the rotor  12  are written within a “fixed” Cartesian reference. This stage produces a matrix M representative of the inertia of the rotor  12  and a matrix G representative of the precession movements of the rotor  12 . The matrices M and G are non-diagonal within this reference. The matrix G depends on the speed of rotation Ω of the rotor  12 . 
     During a subsequent stage, the first change of reference occurs. More specifically, one passes from the “fixed” Cartesian reference to a “revolving” Cartesian reference to the speed of rotation Ω. This produces a new inertial matrix M′ and a new gyroscopic matrix G′. Within this “revolving” reference, the inertial matrix M′ is diagonal and the gyroscopic matrix G′ is non-diagonal. 
     During a subsequent stage, a second change of reference occurs. More specifically, one passes from the “revolving” Cartesian reference to a polar reference. There is also a movement from a real space in the mathematical sense, involving real coordinates, to a complex space in the mathematical sense, involving complex coordinates. The global matrix describing the system, which is obtained within this complex space, can then be made diagonal. In addition, this change of reference helps uncouple the cylindrical mode of the rotor  12  from the conical mode of the rotor  12 . The conical mode corresponds to the precession modes of the rotor  12 . 
     During a final stage, the real part is identified and, in the complex equations obtained, separated from the imaginary part. This produces a set of new equations. The equations are formulated in a real space and help deduce the transfer functions C Φ1 (p, Ω), C Φ2 (p, Ω) of the regulators  58 A,  58 B,  58 C,  58 D. 
     In the example of realisation in  FIGS. 1-3 , each regulator  58 A,  58 B,  58 C,  58 D presents a transfer function C Φ1 (p, Ω), C Φ2 (p, Ω) that contains no imaginary part. 
     In a variation, each regulator  58 A,  58 B,  58 C,  58 D presents a transfer function C Φ1 (p, Ω), C Φ2 (p, Ω) containing a real part and/or an imaginary part. In a particular sub-variation, the first and third regulators  58 A,  58 C each present a transfer function containing a real part only and the second and fourth regulators  58 B,  58 D each presents a transfer function containing an imaginary part only. 
     The output point of the subtractor  60  is connected to an input point on the signal conversion unit  50 . The subtractor  60  supplies, at its output point, the first inclination command signal A ΦX . The output point of the adding amplifier  62  is connected to another input point of the signal conversion unit  50 . The adding amplifier  62  supplies, at its output point, the second signal A ΦY . Each inclination command signal A ΦX , A ΦY  is thus calculated on the basis of the first S ΦX  and second S ΦY  inclination movement signal. 
     The signal conversion unit  50  receives the signals A ΦX , A ΦY  at its input point and is suitable for calculating the intermediate actuator command signals A X1 ″, A X2 ″, A Y1 ″, A Y2 ″ on the basis of the signals A ΦX , A ΦY . The intermediate command signals A X1 ″, A X2 ″, A Y1 ″, A Y2 ″ are suitable for commanding offset of the rotation movements of the rotor  12 , especially continuous offset of the precession movements of the rotor  12 . 
     As illustrated in  FIG. 2 , the signal conversion unit  50  contains a first command signal separator  64 A and a second command signal separator  64 B. 
     The input point of the first separator  64 A is connected to the output point of the subtractor  60 . One output point of the first separator  64 A is connected to an input point of the first adding amplifier  52 A, the other output point of the first separator  64 A being connected to one output point of the second adding amplifier  52 B. 
     The input point of the second separator  64 B is connected to the output point of the adding amplifier  62 . One output point of the second separator  64 B is connected to the input point of the third adding amplifier  54 A, the other output point of the second separator  64 B being connected to one input point on the fourth adding amplifier  54 B. 
     The first separator  64 A and second separator  64 B respectively are suitable for applying a first weighting coefficient to the signal A ΦX , A ΦY  respectively and supplying at the input point to the first adding amplifier  52 A and fourth adding amplifier  54 B the resultant signal A X1 ″, A Y2 ″ respectively. It is also suitable for applying a second weighting coefficient to the signal A ΦX , A ΦY  respectively, with a value opposed to that of the first coefficient, and for supplying the resultant signal A X2 ″, A Y1 ″ respectively at the input point of the second adding amplifier  52 B and third adding amplifier  54 A respectively. The output points of the adding amplifiers  52 A,  52 B,  54 A,  54 B are connected to the input point of the command signal calculation module  36 . The adding amplifiers  52 A,  52 B,  54 A,  54 B respectively supply the A X1 , A X2 , A Y1 , A Y2  respectively at their output points. 
     The signal calculation module  36  receives the signals A X1 , A X2 , A Y1 , A Y2  at its output point and is suitable for generating the input command signals I X1 , I X2 , I Y1 , I Y2  simultaneously. 
     The module  36  contains a first adding amplifier  66 A, a second adding amplifier  66 B, a third adding amplifier  66 C and a fourth adding amplifier  66 D. It also contains a first amplifier  68 A, a second amplifier  68 B, a third amplifier  68 C and a fourth amplifier  68 D. 
     One input point of the adding amplifier  66 A,  66 B,  66 C,  66 D respectively is connected to the output point of the adding amplifier  52 A,  52 B,  54 A,  54 B respectively. The other input point of the adding amplifier  66 A,  66 B,  66 C,  66 D respectively is connected to the output point of the regulator  38 A,  38 B,  38 C,  38 D respectively. The adding amplifier  66 A,  66 B,  66 C,  66 D respectively is suitable for providing an intermediate input command signal B X1 , B x2 , B Y1 , B Y2  respectively at the output point. 
     The amplifier  68 A,  68 B,  68 C,  68 D respectively is connected between the output point of the adding amplifier  66 A,  66 B,  66 C,  66 D respectively and the output terminal  30 A,  30 B,  30 C,  30 D respectively. The amplifier  68 A,  68 B,  68 C,  68 D respectively is suitable for supplying the input command signal I X1 , I X2 , I Y1 , I Y2  respectively at the output point. 
     The first amplifier  68 A and third amplifier  68 C respectively are suitable for multiplying the signal B X1 , B Y1  respectively through a constant gain K 1 . The second amplifier  68 B and fourth amplifier  68 D respectively are suitable for multiplying the signal B X2 , B Y2  respectively by a constant gain K 2 , different from the gain K 1 . The K 1  and K 2  gain values are chosen according to the standard methods of the prior art. 
     Each input command signal I X1 , I X2 , I Y1 , I Y2  is suitable for calculation on the basis of an intermediate signal B X1 , B X2 , B Y1 , B Y2 , and therefore that of an intermediate signal A X1 ″, A X2 ″, A Y1 ″, A Y2 ″. Therefore, each input command signal I X1 , I X2 , I Y1 , I Y2  is specifically suitable for commanding continuous offset of the precession movements of the rotor  12 . The command device  28  is suitable for simultaneously calculating the input command signals I X1 , I X2 , I Y1 , I Y2 . It is also suitable for directing, via each input command signal, the continuous offset of the precession movements of the rotor  12 . 
       FIG. 4  represents the stages of a procedure in one method of realising the invention, implemented by the active magnetic bearing  10 . 
     The procedure comprises an initial stage  76 , within which the position according to axes x and y of the upper region of the rotor  12  is measured by the sensors  16 A,  16 B, and the position according to axes x and y of the lower region of the rotor  12  is measured by the sensors  16 C,  16 D. The sensor  16 A,  16 B,  16 C,  16 D respectively supplies the position signal X 1 , X 2 , Y 1 , Y 2  respectively at its output point. 
     During a subsequent stage  78  the subtractor  26 A,  26 B,  26 C,  26 D respectively determines the error signal S X1 , S X2 , S Y1 , S Y2  respectively on the basis of the position signal X 1 , X 2 , Y 1 , Y 2  respectively Y 2 . 
     During a subsequent stage  80 , the signal conversion element  46  calculates the first inclination movement signal S ΦX  on the basis of error signals S X1 , S X2  and the second inclination movement signal on the basis of error signals S Y1 , S Y2 . During this same stage  80  the first adding amplifier module  40 A and the second adding amplifier module  40 B respectively, calculate the signals ST X , ST Y  respectively. 
     During a subsequent stage  82  the inclination command unit  48  separately processes the inclination movement signals S ΦX , S ΦY  and calculates each inclination command signal A ΦX , A ΦY  on the basis of signals S ΦX , S ΦY . The inclination command unit  48  therefore calculates each inclination command signal A ΦX , A ΦY  indirectly on the basis of the position signals X 1 , X 2 , Y 1 , Y 2 . As indicated by formulae (1), (2), (3), (4), (5) and (6), the unit  48  explicitly, for calculating the command signals A ΦX , A ΦY , takes account of the rotation speed of the rotor  12 . This characteristic, whatever the speed of rotation of the rotor, allows stable correction of the precession movements of the rotor. During this same stage  82  the first regulator  42 A and second regulator  42 B respectively calculate the signals AT X , AT Y  respectively. 
     During a subsequent stage  84 , the signal conversion unit  50  calculates the intermediate signals A X1 ″, A X2 ″ on the basis of the signal A ΦX . It also calculates the intermediate signals A Y1 ″, A Y2 ″ on the basis of the signal A ΦY    
     During a subsequent stage  86 , the command signal calculation module  36  simultaneously calculates the input command signals I X1 , I X2 , I Y1 , I Y2 . The input command signal I X1 , I X2 , I Y1 , I Y2  respectively is calculated specifically on the basis of the intermediate signal A X1 ″, A X2 ″, A Y1 ″, A Y2 ″ respectively. The input command signals I X1 , I X2  are therefore calculated on the basis of an inclination command signal A ΦX , and the input command signals I Y1 , I Y2  are calculated on the basis of an inclination command signal A ΦY . 
     During a subsequent stage  88 , the command device  28  applies at the input point of the actuators  18 A,  18 B,  20 A,  20 B,  22 A,  22 B,  24 A,  24 B the input command signals I X1 , I X2 , I Y1 , I Y2 . 
     By their design, the input command signals I X1 , I X2 , I Y1 , I Y2  allow the actuators to be directed to allow continuous offset of the precession movements of the rotor  12 . In addition, the signals I X1 , I X2 , I Y1 , I Y2  allow stable and uncoupled control of the direct and indirect precession movements of the rotor  12 . 
     It is thus concluded that the command procedure according to the invention ensures stability of correction of the precession movements of the rotor regardless of the rotation speed of the rotor. 
     The specialist in the field will understand that the invention is not limited to an active magnetic bearing containing eight electromagnetic actuators and four sensors, as illustrated in the present description, but applies more generally to a magnetic bearing containing at least three electromagnetic actuators and two sensors. 
     This written description uses examples to disclose the invention, including the preferred embodiments, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.