US20030140875A1

US20030140875A1 - Method for estimating the position and speed of an actuator body in an electromagnetic actuator for controlling the valve of an engine

Info

Publication number: US20030140875A1
Application number: US10/316,734
Authority: US
Inventors: Marco Panciroli; Michele Morselli; Marco Marchi
Original assignee: Magneti Marelli Powertrain SpA
Current assignee: Marelli Europe SpA
Priority date: 2001-12-14
Filing date: 2002-12-11
Publication date: 2003-07-31
Also published as: BRPI0205562B1; ITBO20010760A1; ITBO20010760A0; DE60201327T2; DE60201327D1; BR0205562A; EP1319807B1; ES2227383T3; EP1319807A1

Abstract

Method for estimating the position and the speed of an actuator body in an electromagnetic actuator for controlling a valve of an engine, according to which, starting from a known value for the position and a first moment, a value is calculated at the first moment of the magnetic flux passing through a magnetic circuit constituted by an electromagnet and by the actuator body, the value for the speed at the first moment is estimated as a function of the magnetic flux and the position at the first moment, and the value is calculated at a second moment following the first moment and separated from said first moment by an interval of time determined by adding to the value of the position at the first moment the product of the speed at the first moment for the interval of time.

Description

The present invention relates to a method for estimating the position and speed of an actuator body in an electromagnetic actuator for controlling a valve of an engine.

BACKGROUND OF THE INVENTION

As is known, experiments are currently being conducted on internal combustion engines of the type described in Italian patent application BO99A000443 filed on 4, Aug. 1999, in which the intake and exhaust valves are set in motion by electromagnetic actuators. Such electromagnetic actuators have undoubted advantages, in that they make it possible to control each valve according to a law optimised for each operating condition of the engine, whereas traditional mechanical actuators (typically camshafts) require the definition of a valve lift profile that represents an acceptable compromise for all possible operating conditions of the engine.

An electromagnetic actuator for a valve of an internal combustion engine of the type described above normally comprises at least one electromagnet capable of displacing an actuator body made of ferromagnetic material and mechanically connected to the stem of the respective valve. In order to apply a particular law of motion to the valve, a control unit drives the electromagnet with a time-variable current in order to displace the actuator body in a suitable manner.

From experimental testing it has been observed that, in order to achieve relatively high precision in controlling the valve it is necessary to have feedback control of the position of the actuator body; it is therefore necessary to have an accurate—and substantially real-time—reading of the position of said actuator body at any time. In order to achieve high performance levels from the feedback control it is furthermore preferable also to have an accurate—and substantially real-time—reading of the speed of the actuator body at any time.

In electromagnetic actuators of the type described above, the position of the actuator body is read by a laser sensor, which is, however, expensive, delicate and difficult to calibrate and is therefore unsuitable for use in mass production. Furthermore, the speed of the actuator body is estimated in a time-derivation operation on the position of said actuator body at any time. However, such an operation supplies a relatively inaccurate result in that it tends to amplify the noise present when measuring the position of the actuator body.

SUMMARY OF THE INVENTION

The aim of the present invention is to provide a method for estimating the position and speed of an actuator body in an electromagnetic actuator for controlling a valve of an engine, which does not have the drawbacks described and, in particular, is easy and economical to operate.

According to the present invention a method is provided for estimating the position and speed of an actuator body in an electromagnetic actuator for controlling a valve of an engine as claimed in

claim

1.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described with reference to the attached drawings, which illustrate a few non-exhaustive embodiments thereof, in which: [0008]
FIG. 1 is a diagrammatic view, in side elevation and in partial section, of a valve of an engine and of a corresponding electromagnetic actuator operating according to the method that is the subject-matter of the present invention; [0009]
FIG. 2 is a diagrammatic view of a control unit for the device in FIG. 1; [0010]
FIG. 3 illustrates diagrammatically a part of the control unit of FIG. 2; and [0011]
FIG. 4 illustrates a circuit diagram of a detail of FIG. 3.[0012]

DETAILED DESCRIPTION OF THE INVENTION

In FIG. 1 an electromagnetic actuator [0013] 1 (of the type described in Italian patent application BO99A000443 filed on 4, Aug. 1999) is indicated as a whole by the reference number 1, coupled to an intake or exhaust valve 2 of an internal combustion engine of a known type for displacing said valve 2 along a longitudinal axis 3 of the valve between a closed position (known and not illustrated) and a maximally open position (known and not illustrated).
The [0014] electromagnetic actuator 1 comprises a swinging arm 4 made at least partly of ferromagnetic material, which has a first end hinged to a support 5 so as to be able to oscillate about an axis 6 of rotation perpendicular to the longitudinal axis 3 of the valve 2, and a second end connected by a connector 7 to an upper end of the valve 2. The electromagnetic actuator 1 also comprises two electromagnets 8 carried in a fixed position by the support 5 so as to be arranged on opposite sides of the swinging arm 4, and a spring 9 coupled to the valve 2 and capable of holding the swinging arm 4 in an intermediate position (illustrated in FIG. 1) in which said swinging arm 4 is equidistant from the pole pieces 10 of the two electromagnets 8.
In use, the [0015] electromagnets 8 are controlled by a control unit 11 so as to exert alternately or simultaneously a force of attraction of magnetic origin on the swinging arm 4 in order to make it rotate about the axis 6 of rotation, thereby displacing the valve 2 along the respective longitudinal axis 3 and between the aforementioned maximally open and closed positions (not illustrated). In particular, the valve 2 is in the aforementioned closed position (not illustrated) when the swinging arm 4 is abutting against the upper electromagnet 8, is in the aforementioned maximally open position (not illustrated) when the swinging arm 4 is abutting against the lower electromagnet 8, and is in a partly open position when the two electromagnets 8 both have power shut off and the swinging arm 4 is in the aforementioned intermediate position (illustrated in FIG. 1) by the effect of the force exerted by the spring 9.
The [0016] control unit 11 provides feedback control, in a substantially known manner, for the position of the swinging arm 4, i.e. the position of the valve 2, on the basis of the conditioning of engine function. In particular, according to the illustration in FIG. 2, the control unit 11 comprises a reference generation block 12, a calculation block 13, a driving block 14 capable of supplying the electromagnets 8 with time-variable current, and an estimation block 15 capable of estimating—substantially in real time—the position x(t) and, where necessary, the speed v(t) of the swinging arm 4.
In use, the [0017] reference generation block 12 receives as inputs a plurality of parameters indicating the operating conditions of the engine (for example the load, the engine speed, the position of the throttle body, the angular position of the drive shaft, the temperature of the coolant) and supplies the calculation block 13 with a target value x_R(t) (i.e. a desired value) for the position of the swinging arm 4 (and therefore the valve 2).
On the basis of the target value x[0018] _R(t) for the position of the swinging arm 4 and on the basis of the estimated value x(t) of the position of the swinging arm 4 received from the estimation block 15, the calculation block 13 prepares and sends to the driving block 14 a control signal z(t) for driving the electromagnets 8. In a preferred embodiment, the calculation block 13 prepares the control signal z(t) also on the basis of an estimated value v(t) for the speed of the swinging arm 4 received from the estimation block 15.
According another embodiment, not illustrated, the [0019] reference generation block 12 supplies the calculation block 13 with either a target value x_R(t) for the position of the swinging arm 4, or a target valve x_R(t) for the speed of the swinging arm 4.
As illustrated in FIG. 3, the [0020] driving block 14 supplies power to the two electromagnets 8, each of which is composed of a respective magnetic core 16 coupled to a corresponding coil 17, for displacing the swinging arm 4 on the basis of the commands received from the calculation block 13. The estimation block 15 reads the values, as shown in detail below, either from the driving block 14, or from the two electromagnets 8, in order to calculate an estimated value x(t) for the position and an estimated value v(t) for the speed of the swinging arm 4.
The swinging [0021] arm 4 is arranged between the pole pieces 10 of the two electromagnets 8, which are carried by the support 5 in a fixed position and at a fixed distance D from one another, and therefore the estimated value x(t) of the position of the swinging arm 4 can be obtained directly with a simple operation of algebraic addition from an estimated value d(t) of the distance between a given point on the swinging arm 4 and a corresponding point on the one of the two electromagnets 8. By analogy, the estimated value v(t) for the speed of the swinging arm 4 can be obtained directly from an estimated value for the speed existing between a given point on the swinging arm 4 and a corresponding point on one of the two electromagnets 8.
In order to calculate the value x(t) the [0022] estimation block 15 calculates the two estimated values d₁(t), d₂(t) for the distance between a given point on the swinging arm 4 and a corresponding point on one of the two electromagnets 8; from the two estimated values d₁(t), d₂(t), the estimation block 15 obtains two values x₁(t), x₂(t), which generally differ from one another because of measuring errors and noise. According to a preferred embodiment, the estimation block 15 takes an average of the two values x₁(t), x₂(t), weighted if necessary on the basis of the accuracy attributed to each value x(t). By analogy, in order to calculate the value v(t) the estimation block 15 calculates the two estimated values for speed existing between a given point on the swinging arm 4 and a corresponding point on one of the two electromagnets 8; from the two estimated values for speed, the estimation block 15 obtains two values v₁(t), v₂(t), which generally differ from one another because of measuring errors and noise. According to a preferred embodiment, the estimation block 15 takes an average of the two values v₁(t), v₂(t), weighted if necessary on the basis of the accuracy attributed to each value v(t).
With particular reference to FIG. 4, which illustrates a [0023] single electromagnet 8, a description is given below of the method used by the estimation block 15 for calculating an estimated value d(t) for the distance between a given point on the swinging arm 4 and a corresponding point on the electromagnet 8, and for calculating an estimated value for the speed existing between a given point on the swinging arm 4 and a corresponding point on the electromagnet 8.
In use, when the [0024] driving block 14 applies a voltage v(t) variable over time to the terminals of the coil 17 of the electromagnet 8, a current i(t) passes through said coil 17, consequently generating a flux φ(t) over a magnetic circuit 18 coupled to the coil 17. In particular, the magnetic circuit 18 coupled to the coil 17 is composed of the core 16 of ferromagnetic material of the electromagnet 8, the swinging arm 4 made of ferromagnetic material and the air gap 19 existing between the core 16 and the swinging arm 4.
The [0025] magnetic circuit 18 has a total reluctance R defined by the sum of the reluctance of iron R_feand the reluctance of the air gap R₀; the value for the flux φ(t) circulating over the magnetic circuit 18 is connected to the value of the current i(t) circulating within the coil 17 by the following relationship (in which N is the number of turns in the coil 17):
N*i(t)=R*φ(t)
R=R _fe +R ₀
In general the value for total reluctance R depends both on the position x(t) of the swinging arm [0026] 4 (i.e. the breadth of the air gap 19, which is equal, except for a constant, to the position x(t) of the swinging arm 4), and on the assumed value for flux φ(t) . Except for negligible errors (i.e. those of a first approximation) it can be determined that the value for reluctance of iron R_fedepends solely on the assumed value for flux φ(t), while the value for reluctance of the air gap R₀depends solely on the position x(t), i.e.
R(x(t), φ(t))=R _fe(φ(t))+R₀(x(t))
N*i(t)=R(x(t), φ(t)) *φ(t)
N*i(t)=R_fe(φ(t))*φ(t)+R ₀(x(t))* φ(t)
By solving the last equation given above with regard to R[0027] ₀(x(t)), it is possible to obtain the value of the reluctance of the air gap R₀knowing the value of the current i(t), which value can easily be measured by an ammeter 20, knowing the value of N (fixed and dependent on the structural properties of the coil 17), knowing the value of the flux φ(t), and knowing the relationship between the reluctance of the iron (R_feand the flux φ (known from the structural properties of the magnetic circuit 18 and the magnetic properties of the material used, or easily determined by experimental tests).
The relationship between reluctance at the air gap R[0028] ₀and the position x can be obtained relatively simply by analysing the properties of the magnetic circuit 18 (an example of a model of the behaviour of the air gap 19 is represented by the equation given below). Once the relationship between reluctance at the air gap R₀and the position x is known, the position x can be obtained from the reluctance at the air gap R₀by applying the inverse relationship (applicable either by using the exact equation or by applying approximate numerical calculation methods). The above statements can be summarised in the following relationships (where H_fe(φ(t))=R_fe(φ((t))*φ(t)): $\begin{matrix} R_{o} (x (t)) = \frac{N \cdot i (t) - H_{fe} (ϕ (t))}{ϕ (t)} \\ R_{o} (x (t)) = K_{1} [1 - e^{- k_{2} x (t)} + k_{3} \cdot x (t)] + K_{0} \\ x (t) = R_{0}^{- 1} (R_{o} (x (t))) = R_{0}^{- 1} (\frac{N \cdot i (t) - H_{fe} (ϕ (t))}{ϕ (t)}) \end{matrix}$
The constants K[0029] ₀, K₁, K₂, K₃are constants that can be obtained in experimental tests by using a series of measurements on the magnetic circuit 18.
From the above, it is clear that if the flux φ(t) can be measured it is possible to calculate relatively easily the position x(t) of the [0030] swinging arm 4.
In a first embodiment, the flux φ(t) can be calculated by measuring the current i(t) that circulates through the [0031] coil 17 by using the ammeter 20 of a known type, measuring the voltage v(t) applied to the terminals of the coil 17 by using a voltmeter 21 of known type, and knowing the value for resistance RES of the coil 17 (a value that is easy to measure) . This method of measuring the flux φ(t) is based on the following relationships: $\begin{matrix} \frac{\partial ϕ (t)}{\partial t} = v (t) - RES \cdot i (t) \\ ϕ (T) = \int_{0}^{T} (v (t) - RES \cdot i (t)) \partial t + ϕ (0) \end{matrix}$
The conventional moment 0 is chosen so as to find out accurately the value of the flux φ(0) at said moment 0; in particular, the moment 0 is normally chosen within a period of time in which no current is flowing through the [0032] coil 17 and, therefore, the flux φ is substantially zero (the effect of any residual magnetisation is negligible), or the moment 0 is chosen according to a given position of the swinging arm 4 (typically when the swinging arm 4 is abutting against the pole pieces 10 of the electromagnet 8), in correspondence with which the value of the position x is known and therefore the value of the flux φ is known.
The method stated above for calculating the flux φ(t) is fairly accurate and fast (i.e. involving no delay); however, said method has a few problems, owing to the fact that the voltage v(t) applied to the terminals of the [0033] coil 17 is normally generated by a switching amplifier incorporated into the driving block 14 and therefore varies continuously between three values (+V_supply, 0, −V_supply) the continuous variation (with very abrupt rises and falls) of the voltage v(t) makes it very difficult to measure said voltage v(t) accurately and quickly and, consequently, to estimate the flux φ(t). In order to increase accuracy, the reading signal of the voltmeter 21 can be filtered in order to attenuate the high frequencies, but such filtering inevitably introduces a delay into the measuring process.
In another embodiment, the [0034] magnetic coil 16 is coupled to an auxiliary turn (or coil) 22, to the terminals of which another voltmeter 23 is connected; since the terminals of the turn 22 are substantially open (the internal resistance of the voltmeter 23 is so high as to be regarded as infinite without thereby introducing appreciable errors), no current flows through the turn 22 and the voltage v_aux(t) at its terminals depends solely on the time derivative of the flux φ(t), from which the flux can be deduced by means of a operation of integration (as concerns the value φ(0), see the considerations stated above): $\begin{matrix} \frac{\partial ϕ (t)}{\partial t} = v_{aus} (t) \\ ϕ (T) = \int_{0}^{T} v_{aus} (t) \partial t + ϕ (0) \end{matrix}$
From experimental tests it has been demonstrated that, in contrast to the voltage v(t) at the terminals of the [0035] coil 17, the voltage v_aux(t) is substantially direct because of the effect of magnetic inertia (particular the stray currents induced in the iron) of the magnetic circuit 18 that damp the effects of the abrupt variations in the voltage v(t) . In other words, the iron part of the magnetic circuit 18 has a low-pass filter effect that damps the abrupt variations in the voltage v(t) and makes the voltage v_aux(t) substantially direct without introducing delays in measurement.
As stated above it is clear that by using the reading of the voltage v[0036] _aux(t) of the auxiliary turn 22, calculation of the value of the flux φ(t) is more accurate and/or faster than using the reading of the voltage v(t) at the heads of the coil 17.
As well as for estimating the position x(t) of the swinging [0037] arm 4, measurement of the flux φ(t) can be used by the control unit 11 for verifying the value of the force f(t) of attraction exerted by the electromagnet 8 on the swinging arm 4, where: $\begin{matrix} f (t) = - \frac{1}{2} \cdot \frac{\partial R (x (t), ϕ (t))}{\partial x} \cdot ϕ^{2} (t) \\ f (t) = - \frac{1}{2} \cdot \frac{\partial R_{0} (x (t))}{\partial x} \cdot ϕ^{2} (t) \end{matrix}$
On the basis of the value of the position x(t) of the swinging [0038] arm 4, it is possible to calculate the value of the speed v(t) of the swinging arm 4 by using a simple time-derivative operation on the position x(t); however, the value for speed v(t) obtained with such a derivation operation has much interference, since, as is known, the derivation operation markedly amplifies high-frequency interference. To reduce the incidence of such interference it is necessary to carry out the filtering operations with low-pass type filters which, however, introduce inevitable delays in estimating the value of the speed v(t).
According to another embodiment, both the position x(t) and the speed v(t) can be calculated by using a process of calculation of the iterative type; this process is based on the equation (described above): [0039]
i(t)=R₀(x(t))*φ+H_fe(φ(t))
deriving said equation with respect to time and applying the laws of partial derivation gives the equation: [0040] $\begin{matrix} \frac{\partial i (t)}{\partial t} = \frac{\partial R_{0} (x (t))}{\partial x} \cdot \frac{\partial x (t)}{\partial t} \cdot ϕ (t) + R_{0} (x (t)) \cdot \frac{\partial ϕ (t)}{\partial t} + \frac{\partial H_{fe} (ϕ (t))}{\partial ϕ} \cdot \frac{\partial ϕ (t)}{t} \\ \frac{\partial x (t)}{\partial t} = \frac{\frac{\partial i (t)}{\partial t} - R_{0} (x (t)) \cdot \frac{\partial ϕ (t)}{\partial t} - \frac{\partial H_{fe} (ϕ (t))}{\partial ϕ} \cdot \frac{\partial ϕ (t)}{\partial t}}{\frac{\partial R_{0} (x (t))}{\partial x} \cdot ϕ (t)} \end{matrix}$
reading from left to right it can be seen that: the time derivative of the current i(t) can be calculated easily by deriving the measurement signal of the ammeter [0041] 20 (this signal is generally very clean (i.e. free from noise) and free from abrupt variations and, therefore, can be time-derived with no particular problems);
the partial derivative of the reluctance R[0042] ₀of the air gap 19 with respect to the position x can be calculated as a simple derivation of the equation R₀=R₀(x) described above;
the time derivative of the position x(t) is the speed v(t); [0043]
the flux φ(t) can be calculated by using one of the two methods described above; [0044]
the reluctance R[0045] ₀of the air gap 19 can easily be calculated from the equation R₀=R₀(x) described above if the value of the position x is known;
the partial derivative of the quantity of ampere-turns Hfe of the iron with respect to the flux φ can be obtained easily if the structural properties of the [0046] magnetic circuit 18 are known; and
the time derivative of the flux φ(t) can be calculated with one of the two methods described above. [0047]
Assuming that we are starting from a conventional moment t=0 in which both the value of the flux φ and the value of the position x are known (as described above, this moment 0 is normally chosen at the moment in which the swinging [0048] arm 4 is in a given position, typically abutting against the pole pieces 10 of the electromagnet 8).
Starting from the moment t=0, the value of the reluctance R[0049] ₀of the air gap 19 is calculated at the moment t=0 using the value of the position x(0) at the moment 0; inserting this value into the last equation described above (and previously also calculating the other values in this equation by the method indicated earlier), it is possible to calculate very easily the value of the speed v(0) at the moment t=0.
If a substantially negligible error is committed, it may be assumed that the speed v remains substantially constant for a period of time dt (of a very small amplitude and depending on the desired accuracy); on the basis of this hypothesis, after the time dt, the position x(0+dt) at the moment 0+dt will be: [0050]
x(0+dt)=x(0+v(0)*dt
in this way the value of the position x(0+dt) at the moment 0+dt is calculated, and the operations described above are repeated until the value of the speed v(0+dt) at the moment 0+dt is determined, and so on. [0051]
The method described above has the merit of supplying accurately and quickly either the value of the position x, or the value of the speed v. [0052]
In the description given above two methods have been provided for estimating the time derivative of the flux φ(t) (hence the value of the flux φ(t) can be calculated), and two methods for calculating the position x(t) and the speed v(t). According to one embodiment a choice is made to use only one method for calculating the time derivative of the flux φ(t) and one method for calculating the position x(t) and the speed v(t). According to another embodiment the choice is made to use both methods for calculating the time derivative of the flux φ(t) and/or both the methods for calculating the position x(t) and the speed v(t), and to use an average (weighted if necessary with respect to the estimated accuracy) of the results of the two methods used, or to use one result in order to verify the other (if there is a notable inconsistency between the two results it is likely that an error in estimating will be verified). [0053]
Finally, it is useful to observe that the methods described above for estimating the position x(t) and the speed v(t) can be used only when there is a current passing through the [0054] coil 17 of an electromagnet 8. For this reason, as described above, the estimation block 15 works with both the electromagnets 8, so as to use the estimation performed with one electromagnet 8 when the other is switched off. When both the electromagnets are on, the estimation block 15 performs an average of the two values x(t) calculated with both electromagnets 8, weighted if necessary on the basis of the accuracy attributed to each value x(t) (generally the estimation of the position x made with respect to an electromagnet 8 is more accurate when the swinging arm 4 is relatively close to the pole piece 10 of said electromagnet 8.

Claims

1. Method for estimating the position (x) and the speed (v) of an actuator body (4) in an electromagnetic actuator (1) for controlling a valve (2) of an engine; the actuator body (4) being made at least partly of ferromagnetic material and being displaced towards at least one electromagnet (8) through the effect of the force of magnetic attraction generated by said electromagnet (8); the method being characterised by the fact that starting from a known value of the position (x) and a first moment (T1), a value is calculated at the first moment (T1) of the magnetic flux (p) passing through a magnetic circuit (18) constituted by the electromagnet (8) and by the actuator body (4), the value of the speed (v) at the first moment (T1) is estimated as a function of the magnetic flux (φ) and the position (x) at the first moment (T1), and the value is calculated at a second moment (T2) following the first moment (T1) and separated from said first moment (T1) by an interval of time (dt) determined by adding to the value of the position (x) at the first moment (T1) the product of the speed (v) at the first moment (T1) for the interval of time (dt).

2. Method according to claim 1, in which said electromagnet (8) defines, together with said actuator body (4), a magnetic circuit (18) influenced by a magnetic flux (φ) produced by a coil (17) through which an electric current (i) passes; said magnetic circuit (18) having a total reluctance (R), which is assumed to be composed of the sum of a first reluctance (R₀) arising from an air gap (19) in the magnetic circuit (18) and a second reluctance (R_fe) arising from the part of the magnetic circuit (18) made of ferromagnetic material (4, 16); the first reluctance (R₀) depending on the structural properties of the magnetic circuit (18) and on the value of the position (x), while the second reluctance (R_fe) depending on the structural properties of the magnetic circuit (18) and on the value of the magnetic flux (φ) passing through the magnetic circuit (18).

3. Method according to claim 2, in which the value for said first reluctance (R₀) and the value for said position (x) are connected by the following equation:

R ₀(x(t))=K ₁[1−e ^−k ^₂ ^x(t) +k ₃ ·x(t)]+K ₀

in which R₀is said first reluctance (R₀), x(t) is said position (x) and K₀, K₁, K₂, K₃are four constants.

4. Method according to claim 2, in which the relationship between the speed (v), the magnetic flux (φ) and the position (x) is supplied by the following equation:

\begin{matrix} \frac{\partial i (t)}{\partial t} = \frac{\partial R_{0} (x (t))}{\partial x} \cdot \frac{\partial x (t)}{\partial t} \cdot ϕ (t) + R_{0} (x (t)) \cdot \frac{\partial ϕ (t)}{\partial t} + \frac{\partial H_{fe} (ϕ (t))}{\partial ϕ} \cdot \frac{\partial ϕ (t)}{\partial t} \\ \frac{\partial x (t)}{\partial t} = \frac{\frac{\partial i (t)}{\partial t} - R_{0} (x (t)) \cdot \frac{\partial ϕ (t)}{\partial t} - \frac{\partial H_{fe} (ϕ (t))}{\partial ϕ} \cdot \frac{\partial ϕ (t)}{\partial t}}{\frac{\partial R_{0} (x (t))}{\partial x} \cdot ϕ (t)} \end{matrix}

in which i is the electric current (i) circulating within the coil (17), R₀is said first reluctance (R₀), x is the position (x) of the actuator body (4), φ is the magnetic flux (φ) and H_feis the quantity of ampere-turns acted on by the iron part (4, 16) of the magnetic circuit (18).

5. Method according to claim 1, in which the value of the magnetic flux (φ) is estimated by measuring the value assumed from some electric parameters (i, v; va) of an electric circuit (17; 22) coupled to the magnetic circuit (18), calculating the time derivative of the magnetic flux (φ) as a linear combination of the values of the electrical parameters (i, v; va), and integrating in time the derivative of the magnetic flux (φ).

6. Method according to claim 5, in which the current (i) circulating through a coil (17) of the electromagnet (8) and the voltage (v) applied to the terminals of said coil (17) are measured; the time derivative of the magnetic flux (φ) and the magnetic flux (φ) itself being calculated by applying the following formulae:

\begin{matrix} \frac{\partial ϕ (t)}{\partial t} = v (t) - RES \cdot i (t) \\ ϕ (T) = \int_{0}^{T} (v (t) - RES \cdot i (t)) \partial t + ϕ (0) \end{matrix}

in which φ is the magnetic flux (φ), v is the voltage (v) applied to the terminals of the coil (17), RES is the resistance of the coil (17) and i is the current (i) circulating through the coil (17).

7. Method according to claim 6, in which the voltage (v_aux) at the terminals of an auxiliary turn (22) coupled to the magnetic circuit (18) and concatenating the magnetic flux (φ) is measured; the auxiliary turn (22) being substantially open electrically; and the time derivative of the magnetic flux (φ) and the magnetic flux (φ) itself being calculated by applying the following formulae:

\begin{matrix} \frac{\partial ϕ (t)}{\partial t} = v_{aus} (t) \\ ϕ (T) = \int_{0}^{T} v_{aus} (t) \partial t + ϕ (0) \end{matrix}

in which φ is the magnetic flux (φ) and v_auxis the voltage (v_aux) present at the terminals of the auxiliary turn (22).