Method of determining the timing and quantity of fuel injection to operate an internal combustion engine

Abstract

A method of determining the timing and quantity of fuel injection to operate an internal combustion engine is disclosed. While operating the fuel injector to perform a fuel injection; a signal of a fuel pressure within the fuel rail during the fuel injection is sampled. The signal is used to determine first and second integral transforms yielding as output a value of first and second functions having as variables the fuel rail pressure drop caused by the fuel injection and the timing parameter indicative of the instant when the fuel injection started. Values of the first and second functions are used to calculate a value of the fuel rail pressure drop caused by the fuel injection and a value of the timing parameter. A value of a fuel quantity injected by the fuel injection is calculated as a function of the value of the fuel rail pressure drop.

Claims

1. A method of operating an internal combustion engine having a fuel rail in fluid communication with a fuel pump and a fuel injector, the method comprising: operating the fuel injector to perform a fuel injection; sampling a pressure signal representative of a fuel pressure within the fuel rail during the fuel injection; using the pressure signal as a first input of a first integral transform yielding a first function value based on a fuel rail pressure drop caused by the fuel injection and a timing parameter indicative of an instant when the fuel injection started; using the pressure signal as a second input of a second integral transform yielding a second function value based on the fuel rail pressure drop caused by the fuel injection and the timing parameter indicative of the instant when the fuel injection started; using the first function value and the second function value to calculate a value of the fuel rail pressure drop caused by the fuel injection and a value of the timing parameter; and calculating a value of a fuel quantity injected by the fuel injection as a function of the calculated value of the fuel rail pressure drop.

2. The method according to claim 1, wherein the fuel rail pressure signal is sampled in a crankshaft angular domain.

3. The method according to claim 1, wherein the value of the first function is calculated with the following integral transform:
L.sub.α=∫.sub.0.sup.2πP(θ).Math.cos(θ)d(θ)≅T.sub.α(ΔP.sub.inj,γ.sub.inj)=ΔP.sub.inj.Math.sin γ.sub.inj wherein L.sub.α is the value of the first function T.sub.α, P is the fuel rail pressure, Θ is an angular position of a crankshaft, 0 is a predetermined starting value of an integration interval [0, 2π] in the crankshaft angular domain, 2π is a predetermined final value of the integration interval [0, 2π] in the crankshaft angular domain, ΔP.sub.inj is the fuel rail pressure drop caused by the fuel injection, γ.sub.inj is an angular distance of the fuel injection from the staring value 0 of the integration interval.

4. The method according to claim 1, wherein the value of the second function is calculated with the following integral transform:
L.sub.β=∫.sub.0.sup.2πP(θ).Math.sin(θ)d(θ)≅T.sub.β(ΔP.sub.inj,γ.sub.inj)=ΔP.sub.inj.Math.(1−cos γ.sub.inj) wherein L.sub.β is the value of the second function T.sub.β, P is the fuel rail pressure, Θ is an angular position of a crankshaft, 0 is a predetermined starting value of an integration interval [0, 2π] in the crankshaft angular domain, 2π is a predetermined final value of the integration interval [0, 2π] in the crankshaft angular domain, ΔP.sub.inj is the fuel rail pressure drop caused by the fuel injection, γ.sub.inj is the angular distance of the fuel injection from the starting value of the integration interval.

5. The method according to claim 3, wherein the starting value of the integration interval is an angular position of the engine crankshaft for which a piston of the fuel pump has already completed the compression stroke.

6. The method according to claim 1, wherein the fuel injection performed by the fuel injector includes a single injection pulse.

7. The method according to claim 1, wherein the fuel injection performed by the fuel injector includes a plurality of injection pulses.

8. The method according to claim 1 further comprising: calculating a difference between the calculated value of the fuel injected quantity and a predetermined target value thereof; and using the calculated difference to correct an energizing time of the fuel injector.

9. The method according to claim 1 further comprising: calculating a difference between the calculated value of the timing parameter and a predetermined target value thereof; and using the calculated difference to correct a start of injection of the fuel injector.

10. A non-transitory computer readable medium having computer program comprising a computer code that when executed on a processor performs the method according to claim 1.

11. A method of operating an internal combustion engine having a fuel rail in fluid communication with a fuel pump and a fuel injector, the method comprising: operating the fuel injector to perform a fuel injection; sampling a pressure signal representative of a fuel pressure within the fuel rail during the fuel injection; using the pressure signal as a first input of a first integral transform yielding a first function value based on a fuel rail pressure drop caused by the fuel injection and a timing parameter indicative of an instant when the fuel injection started; using the pressure signal as a second input of a second integral transform yielding a second function value based on the fuel rail pressure drop caused by the fuel injection and the timing parameter indicative of the instant when the fuel injection started; using the first function value and the second function value to calculate a value of the fuel rail pressure drop caused by the fuel injection and a value of the timing parameter; and calculating a value of a fuel quantity injected by the fuel injection as a function of the calculated value of the fuel rail pressure drop, wherein the value of the fuel quantity injected by the fuel injection is calculated taking into account a hydraulic capacitance of the fuel rail.

12. The method according to claim 11, wherein the value of the hydraulic capacitance is varied on the basis of an average value of the pressure within the fuel rail.

13. The method according to claim 11, wherein the value of the hydraulic capacitance is determined with a learning procedure, which is performed while the engine is in a fuel cut-off condition and further comprising: operating the fuel pump to deliver a predetermined volume of fuel into the fuel rail per compression stroke; measuring a value of a fuel rail pressure increment due to the delivery of said volume of fuel; and calculating the value of the hydraulic capacitance as a function of the volume of fuel delivered into the fuel rail and the measured value of the fuel rail pressure increment.

14. The method according to claim 13, wherein the learning procedure further comprises: calculating an average value of the fuel rail pressure during the delivery of said volume of fuel; and memorizing the calculated value of the hydraulic capacitance, thereby correlating it to the calculated average value of the fuel rail pressure.

15. An internal combustion engine comprising a fuel pump in fluid communication with a fuel injector through a fuel rail, and an electronic control unit configured to: operate the fuel injector to perform a fuel injection; sample a pressure signal representative of a fuel pressure within the fuel rail during the fuel injection; use the pressure signal as a first input of a first integral transform yielding a value of a first function value based on a fuel rail pressure drop caused by the fuel injection and a timing parameter indicative of an instant when the fuel injection started; use the pressure signal as second input of a second integral transform yielding a second function value based on the fuel rail pressure drop caused by the fuel injection and the timing parameter indicative of the instant when the fuel injection started; use the first function value and the second function value to calculate a value of the fuel rail pressure drop caused by the fuel injection and a value of the timing parameter; and calculate a value of a fuel quantity injected by the fuel injection as a function of the calculated value of the fuel rail pressure drop, wherein the value of the fuel quantity injected by the fuel injection is calculated taking into account a hydraulic capacitance of the fuel rail.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present disclosure will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements.

(2) FIG. 1 schematically shows an automotive system;

(3) FIG. 2 is the section A-A of FIG. 1;

(4) FIG. 3 is a flowchart that represents a method for determining the actual fuel quantity that is injected by an engine fuel injector and the instant when such fuel injection actually occurs;

(5) FIG. 4 is a diagram that represents the fuel rail pressure variation over the crankshaft angular position during the execution of the method of FIG. 3;

(6) FIG. 5 shows in greater details a fuel injector of the automotive system of FIG. 1;

(7) FIG. 6 is a flowchart that represents a close-loop control strategy of the fuel injected quantity;

(8) FIG. 7 is a flowchart that represents a closed-loop control strategy of the start of injection; and

(9) FIG. 8 is a diagram that represents the fuel rail pressure variation over the crankshaft angular position during the execution of a learning procedure of the hydraulic capacitance of the fuel rail.

DETAILED DESCRIPTION

(10) The following detailed description is merely exemplary in nature and is not intended to limit the invention or the application and uses of the invention. Furthermore, there is no intention to be bound by any theory presented in the preceding background, summary or description of drawings or the following detailed description.

(11) Some embodiments may include an automotive system 100 (e.g. a motor vehicle), as shown in FIGS. 1 and 2, that includes an internal combustion engine (ICE) 110 having an engine block 120 defining at least one cylinder 125 having a piston 140 coupled to rotate a crankshaft 145. A cylinder head 130 cooperates with the piston 140 to define a combustion chamber 150. A fuel and air mixture (not shown) is disposed in the combustion chamber 150 and ignited, resulting in hot expanding exhaust gasses causing reciprocal movement of the piston 140. The fuel is provided by at least one fuel injector 160 per combustion chamber and the air through at least one intake port 210. The fuel is provided at high pressure to the fuel injector 160 from a fuel rail 170 in fluid communication with a high pressure fuel pump 180 that increase the pressure of the fuel received from a fuel source 190.

(12) The high pressure fuel pump 180 may be embodied as a volumetric pump having a cylinder and a reciprocating piston which is accommodated inside the cylinder to define an operating chamber. The piston is driven by the engine crankshaft 145 through a timing system and moves between a Top Dead Center (TDC) position, which corresponds to a minimum volume of the operating chamber, and a Bottom Dead Center (BDC) position, which corresponds to a maximum volume of the operating chamber. Due to this reciprocating movement, the piston cyclically performs a suction stroke that fills the operating chamber with the fuel coming from the fuel source 190, followed by a compression stroke that delivers the fuel at high pressure inside the fuel rail 170.

(13) Each of the cylinders 125 has at least two valves 215, actuated by a camshaft 135 rotating in time with the crankshaft 145. The valves 215 selectively allow air into the combustion chamber 150 from the port 210 and alternately allow exhaust gases to exit through a port 220. In some examples, a cam phaser 155 may selectively vary the timing between the camshaft 135 and the crankshaft 145.

(14) The air may be distributed to the air intake port(s) 210 through an intake manifold 200. An air intake duct 205 may provide air from the ambient environment to the intake manifold 200. In other embodiments, a throttle body 330 may be provided to regulate the flow of air into the manifold 200. In still other embodiments, a forced air system such as a turbocharger 230, having a compressor 240 rotationally coupled to a turbine 250, may be provided. Rotation of the compressor 240 increases the pressure and temperature of the air in the duct 205 and manifold 200. An intercooler 260 disposed in the duct 205 may reduce the temperature of the air. The turbine 250 rotates by receiving exhaust gases from an exhaust manifold 225 that directs exhaust gases from the exhaust ports 220 and through a series of vanes prior to expansion through the turbine 250. The exhaust gases exit the turbine 250 and are directed into an exhaust system 270. This example shows a variable geometry turbine (VGT) with a VGT actuator 290 arranged to move the vanes to alter the flow of the exhaust gases through the turbine 250. In other embodiments, the turbocharger 230 may be fixed geometry and/or include a waste gate.

(15) The exhaust system 270 may include an exhaust pipe 275 having one or more exhaust aftertreatment devices 280. The aftertreatment devices may be any device configured to change the composition of the exhaust gases. Some examples of aftertreatment devices 280 include, but are not limited to, catalytic converters (two and three way), oxidation catalysts, lean NO.sub.x traps, hydrocarbon adsorbers, selective catalytic reduction (SCR) systems, and particulate filters. Other embodiments may include an exhaust gas recirculation (EGR) system 300 coupled between the exhaust manifold 225 and the intake manifold 200. The EGR system 300 may include an EGR cooler 310 to reduce the temperature of the exhaust gases in the EGR system 300. An EGR valve 320 regulates a now of exhaust gases in the EGR system 300.

(16) The automotive system 100 may further include an electronic control unit (ECU) 450 in communication with one or more sensors and/or devices associated with the ICE 110. The ECU 450 may receive input signals from various sensors configured to generate the signals in proportion to various physical parameters associated with the ICE 110. The sensors include, but are not limited to, a mass airflow and temperature sensor 340, a manifold pressure and temperature sensor 350, a combustion pressure sensor 360, coolant and oil temperature and level sensors 380, a fuel rail pressure sensor 400, a cam position sensor 410, a crank position sensor 420, exhaust pressure and temperature sensors 430, an EGR temperature sensor 440, and an accelerator pedal position sensor 445. Furthermore, the ECU 450 may generate output signals to various control devices that are arranged to control the operation of the ICE 110, including, but not limited to, the fuel injectors 160, the throttle body 330, the EGR Valve 320, the VGT actuator 290, and the cam phaser 155. Note, dashed lines are used to indicate communication between the ECU 450 and the various sensors and devices, but some are omitted for clarity.

(17) Turning now to the ECU 450, this apparatus may include a digital central processing unit (CPU) in communication with a memory system and an interface bus. The CPU is configured to execute instructions stored as a program in the memory system 460, and send and receive signals to/from the interface bus. The memory system 460 may include various non-transitory, computer-readable storage medium including optical storage, magnetic storage, solid state storage, and other non-volatile memory. The interface bus may be configured to send, receive, and modulate analog and/or digital signals to/from the various sensors and control devices. The program may embody the methods disclosed herein, allowing the CPU to carryout out the steps of such methods and control the ICE 110.

(18) The program stored in the memory system 460 is transmitted from outside via a cable or in a wireless fashion. Outside the automotive system 100 it is normally visible as a computer program product, which is also called computer readable medium or machine readable medium in the art, and which should be understood to be a computer program code residing on a carrier, said carrier being transitory or non-transitory in nature with the consequence that the computer program product can be regarded to be transitory or non-transitory in nature.

(19) An example of a transitory computer program product is a signal, e.g. an electromagnetic signal such as an optical signal, which is a transitory carrier for the computer program code. Carrying such computer program code can be achieved by modulating the signal by a conventional modulation technique such as QPSK for digital data, such that binary data representing said computer program code is impressed on the transitory electromagnetic signal. Such signals are e.g. made use of when transmitting computer program code in a wireless fashion via a WiFi connection to a laptop.

(20) In case of a non-transitory computer program product the computer program code is embodied in a tangible storage medium. The storage medium is then the non-transitory carrier mentioned above, such that the computer program code is permanently or non-permanently stored in a retrievable way in or on this storage medium. The storage medium can be of conventional type known in computer technology such as a flash memory, an Asic, a CD or the like.

(21) Instead of an ECU 450, the automotive system 100 may have a different type of processor to provide the electronic logic, e.g. an embedded controller, an onboard computer, or any processing module that might be deployed in the vehicle. One of the tasks of the ECU 450 is that of operating the fuel injectors 160 to inject fuel into the combustion chambers 150. In this regard, it should be observed that each fuel injector 160 is generally embodied as an electromechanical valve having a nozzle in fluid communication with the corresponding combustion chamber 150, a needle, which is normally biased by a spring in a closed position of the nozzle, and an electro-magnetic actuator (e.g. solenoid), which moves the needle towards an open position of the nozzle in response of an energizing electrical current. In this way, any time the electro-magnetic actuator is provided with the energizing electrical current (also named electrical command), a direct connection is opened between the fuel rail 170 and the cylinder 125, which let a certain quantity of fuel to be injected into the combustion chamber 150. Any one of these events is conventionally referred as “injection pulse”.

(22) During normal operations, the ECU 450 generally commands each fuel injector 160 to perform a “fuel injection” per engine cycle, wherein the fuel injection includes a plurality of injection pulses according to a multi-injection pattern. The timing of each single injection pulse generally depends on the instant when the electric command is applied to the actuator of the fuel injector 160. Therefore, the ECU 450 is generally configured to determine the Start Of Injection (SOI) of the injection pulse and then to start the application of the electric command accordingly. The SOI is generally expressed as the angular position of the engine crankshaft 145 when the fuel injection starts. This angular position is normally quantified as an angular displacement, namely a difference between the angular position of the crankshaft 145 at the time when the fuel injection starts and a predetermined angular position of the crankshaft 145, which is chosen as a reference. The reference angular position of the crankshaft 145 is usually chosen as the position for which the piston 140 reaches the Top Dead Center (TDC).

(23) The fuel quantity injected into the combustion chamber 150 by each single injection pulse generally depends on the pressure of the fuel in the fuel rail 170 and on the needle displacement, which is correlated with the duration of the electrical command (i.e. energizing time ET). Therefore, the ECU 450 is generally configured to determine the fuel quantity to be injected with each single injection pulse, to calculate the energizing time necessary for injecting, the desired fuel quantity, and finally to energize the fuel injector 160 accordingly.

(24) However, the SOI and/or the quantity of fuel actually injected by the fuel injector 160 may sometimes be different with respect to the desired ones, due to aging effect and/or production spread of the fuel injector 160. For this reason, the ECU 450 may be configured to perform a method for determining the real SOI and the real quantity of fuel injected by each of the fuel injector 160 in response to a given energizing time, for example in order to diagnose the efficiency of the injection system and/or to be able to correct the electric command with the aim of injecting exactly a desired fuel quantity and/or with the desired timing.

(25) This method may be performed while the engine is under a cut-off condition, for example but not exclusively during the execution of a stop-start running strategy, and may require that the ECU 450 operates one fuel injector 160 at the time, while keeping the other inactive

(26) As shown in the flowchart of FIG. 3, the method prescribes to energize the fuel injector 160 for a predetermined energizing time to perform a fuel injection (block 600). This fuel injection may include a single (i.e. only one) injection pulse or a plurality of injection pulses according to a predetermined multi-injection pattern. While executing the fuel injection, the strategy also prescribes to sample the pressure within the fuel rail 170 (block 605). The fuel rail pressure may be sampled by means of the fuel rail pressure sensor 400. In particular, the pressure may be sampled in an angular domain (i.e. referred to the crankshaft angular position), in order to make it independent from the engine speed.

(27) Under these prescribed conditions, variation of pressure within the fuel rail 170 is generally affected by the fuel injection and by the fuel delivered by the high pressure fuel pump 180, so that the graph of the pressure P over the crankshaft angular position Θ should be of the kind shown in FIG. 4. As a matter of fact, the fuel rail pressure P has an increment, indicated by the ellipses 610, which is caused by the compression stroke of the high pressure fuel pump 180, and a drop, indicated by the ellipses 615, which is caused by the fuel injection.

(28) As can be seen from FIG. 4, it is possible to determine an angular interval that contains the pressure drop caused by the fuel injection but not the increment caused by the pump. To this angular interval can be assigned an extension ranging from 0 to 2π, even if this angular interval does not actually correspond to a full rotation of the crankshaft 145, but actually to a selected portion of it. In this way, during the selected angular interval [0, 2π] the fuel rail pressure is affected by the fuel injection and not by the pump 180. To achieve this effect, at least the staring value 0 of the angular interval [0, 2π] should be chosen as an angular position of the crankshaft 145 that corresponds to a position of the piston of the fuel pump 180 included between its Top Dead Center (TDC) position and its Bottom Dead Center (BDC) position. More particularly, when the crankshaft 145 is in the starting angular position 0, the piston of the fuel pump 180 should be performing the suction stroke, after having passed the IDC position and thus competed the compression stroke.

(29) Using the angular interval [0, 2π] as interval of integration, the strategy may prescribe that the ECU 450 calculates the following integral transforms (block 620):
L.sub.α=∫.sub.0.sup.2πP(θ).Math.cos(θ)d(θ)
L.sub.β=∫.sub.0.sup.2πP(θ).Math.sin(θ)d(θ) wherein: L.sub.α is the value yielded by the first integral transform; L.sub.β is the value yielded by the second integral transform; P is the fuel rail pressure; Θ is the angular position of the crankshaft; θ is the predetermined starting value of the integration interval [0, 2π] in the crankshaft angular domain; and 2π is the predetermined final value of the integration interval [0, 2π] in the crankshaft angular domain.

(30) Looking at FIG. 4, the pressure P of the fuel rail may be considered as the sum of two contributions:
P=P.sub.eq+δP.sub.noise

(31) wherein: P.sub.eq represents an equivalent pressure (e.g. a mean pressure) of the fuel rail 170; and δP.sub.noise represents the pressure fluctuations due to the pressure waves and electronic noise of the sensor.

(32) As a consequence, the preceding integral transforms may be rewritten as follows:
L.sub.α=∫.sub.0.sup.2πP(θ).Math.cos(θ)d(θ)=∫.sub.0.sup.2π[P.sub.eq+δP.sub.noise].Math.cos(θ)d(θ)
L.sub.β=∫.sub.0.sup.2πP(θ).Math.sin(θ)d(θ)=∫.sub.0.sup.2π[P.sub.eq+δP.sub.noise].Math.sin(θ)d(θ)

(33) However, the frequency spectrum of the pressure fluctuations δP.sub.noise is much higher than the frequency spectrum of the equivalent pressure P.sub.eg, so that the contribution of the pressure fluctuations to the integral transform is negligible:
∫.sub.0.sup.2πδP.sub.noise.Math.cos(θ)d(θ)≅∫.sub.0.sup.2πδP.sub.noise.Math.sin(θ)d(θ)≅0

(34) As a consequence, the preceding integral transforms may be rewritten as follows:
L.sub.α=∫.sub.0.sup.2πP(θ).Math.cos(θ)d(θ)≈∫.sub.0.sup.2πP.sub.eq.Math.cos(θ)d(θ)=T.sub.α(ΔP.sub.inj,γ.sub.inj)=ΔP.sub.inj.Math.sin γ.sub.inj
L.sub.β=∫.sub.0.sup.2πP(θ).Math.sin(θ)d(θ)≈∫.sub.0.sup.2πP.sub.eq.Math.sin(θ)d(θ)=T.sub.β(ΔP.sub.inj,γ.sub.inj)=ΔP.sub.inj.Math.(1−cos γ.sub.inj)

(35) wherein: ΔP.sub.inj is the fuel rail pressure drop caused by the fuel injection; γ.sub.inj is the angular distance of the fuel injection from the staring value 0 of the integration interval [0, 2π]; and, T.sub.α and T.sub.β are two functions having as variables the fuel rail pressure drop ΔP.sub.inj and the angular distance γ.sub.inj.

(36) After having calculated the values L.sub.α and L.sub.β, the ECU 450 may thus calculate (block 625) the fuel rail pressure drop ΔP.sub.inj and the angular distance γ.sub.inj with the following equations:

(37) $\Delta P_{\mathrm{inj}}=-\frac{L_{\alpha }^2+L_{\beta }^2}{2L_{\beta }}$${\gamma }_{\mathrm{inj}}=\mathrm{arc}\sin \left(-\frac{2L_{\alpha }{L}_{\beta }}{L_{\alpha }^2+L_{\beta }^2}\right)$

(38) In this way, the angular distance γ.sub.inj provides a measurement of the Start of Injection (SOI), whereas the fuel rail pressure drop ΔP.sub.inj can be used to calculate the fuel quantity actually injected by the fuel injection (block 630).

(39) More particularly, the fuel rail pressure drop ΔP.sub.inj can be used to calculate a dynamic fuel quantity q.sub.inlet that actually flows through the fuel injector 160 according to the following equation:
=q.sub.inlet=C.sub.hyd.Math.ΔP.sub.inj

(40) wherein C.sub.hyd is the value of the hydraulic capacitance of the fuel rail 170,

(41) As represented in FIG. 5, the dynamic fuel quantity q.sub.inlet is the sum of two contributions, namely the fuel injected quantity g.sub.inj and the dynamic leakage q.sub.dyn. The fuel injected quantity g.sub.inj is the quantity of fuel that actually enters the combustion chamber 150, whereas dynamic leakage q.sub.dyn is a quantity of fuel that, when the injector needle is moved in the open position, flows through a backflow outlet of the fuel injector 160 and returns into the fuel source 190. As a consequence, the dynamic fuel quantity q.sub.inlet that globally flows through the fuel injector 160 during a fuel injection (in addition to the static leakage that is always present) may be considered as the sum of the fuel injected quantity g.sub.inj and the dynamic leakage q.sub.dyn:
q.sub.inlet=q.sub.inj+q.sub.dyn

(42) However, q.sub.inlet, g.sub.inj and q.sub.dyn are parameters that depend only on the fuel pressure at the inlet of the fuel injector 160 and on the energizing time (which determines the needle lift), Therefore, knowing q.sub.inlet, the fuel pressure and the energizing time used to perform the fuel injection, it is possible to determine the value q.sub.inj of the fuel injected quantity as a function of q.sub.inlet:
q.sub.inj=ƒ(q.sub.inlet)

(43) The method disclosed above may be involved in a closed-loop control strategy of the fuel injected quantity. As shown in FIG. 6, this strategy may provide for determining the value q.sub.inj of the fuel injected quantity according to the method above, calculating a difference e between the calculated value q.sub.inj* and a predetermined target value of the fuel injected quantity, and then to use said difference to correct an energizing time ET.sub.inj* to be applied to the fuel injector 160, in order to minimize the error. In particular, the calculated difference e may be used as input of a controller, for example a proportional-integrative (PI) controller, that yields as output a correction value δ.sub.ET to be added to the energizing time ET.sub.inj*, in order to obtain a corrected energizing time ET.sub.inj that is finally used to operate the fuel injector 160.

(44) At the same time or as an alternative, the method disclosed above may be involved in a closed-loop control strategy of the SOI. As shown in FIG. 7, this strategy may provide for determining the value γ.sub.inj of the SOI according to the method above, calculating a difference e between the calculated value γ.sub.inj and a predetermined target value γ.sub.inj* of the SOI, and then to use said difference to correct the target value γ.sub.inj* before using it to operate to the fuel injector 160, in order to minimize the error. In particular, the calculated difference e may be used as input of a controller, for example a proportional-integrative (PI) controller, that yields as output a correction value δ.sub.γ to be added to the target value γ.sub.inj*, in order to obtain a corrected value SOI.sub.inj of the start of injection that is finally used to operate the fuel injector 160.

(45) Turning now to the hydraulic capacitance of the fuel rail 170, this parameter depends on constructional and geometrical characteristics of the fuel rail 170. For this reason, the value C.sub.hyd of hydrodynamic capacitance may be a calibration parameter, which can be determined by means of an experimental activity and then stored in the memory system 460.

(46) However, the hydraulic capacitance depends also on the fuel properties and on the fuel pressure within the fuel rail 170, so that the value C.sub.hyd determined by means of the experimental activity may not always be reliable. For this reason, a dedicated learning procedure may be executed from time to time, in order to determine the actual value of the hydraulic capacitance.

(47) This learning procedure may be performed while the engine 110 operates under a fuel cut-off condition (even during the execution of a stop-start running strategy). When the engine 110 operates under a fuel cut-off condition, the pressure within the fuel rail 170 is conventionally decreased to a minimum allowable value thereof, which is indicated with P.sub.0 in FIG. 8. Under these conditions, the learning phase may prescribe to operate the fuel pump 180 to deliver a predetermined volume Q of fuel into the fuel rail 170 per compression stroke. By way of example, the fuel pump 180 may be arranged to deliver its maximum fuel quantity, so that the fuel volume Q may be calculated with the following equation:
Q=v.Math.μ

(48) wherein V is the pump displacement and μ is the pump volumetric efficiency.

(49) While the fuel pump is operated in this way, the strategy may provide for monitoring the fuel rail pressure, which is expected to increase step by step from the minim value P.sub.0 to a predetermined maximum value P.sub.1 as shown in FIG. 8.

(50) For each step, the learning procedure may prescribe to calculate a value ΔP.sub.k of a fuel rail pressure increment due to the delivery of the volume Q of fuel and an average value P.sub.k of the fuel rail pressure, namely an average between the pressure values before and after the delivery of said volume Q of fuel.

(51) The calculated value ΔP.sub.k may then be used to calculate the value C.sub.hyd,k of the hydraulic capacitance according to the following equation:

(52) $C_{\mathrm{hyd},k}=\frac{Q}{\Delta P_k}$

(53) The value C.sub.hyd,k of the hydraulic capacitance may be finally memorized in the memory system 460, thereby correlating it to the corresponding average value P.sub.k of the fuel rail pressure. In this way, it is possible to generate an array or map that correlates each value of the fuel rail pressure P.sub.k with a corresponding value of the hydraulic capacitance C.sub.hyd,k, which in turn may be effectively used to calculate the fuel injected quantity according to the method set forth above.

(54) While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing an exemplary embodiment, it being understood that various changes may be made in the function and arrangement of elements described in an exemplary embodiment without departing from the scope of the invention as set forth in the appended claims and their legal equivalents.