Patent Publication Number: US-9845736-B2

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

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to Great Britain Patent Application No. 1501066.3, filed Jan. 22, 2015, which is incorporated herein by reference in its entirety. 
     TECHNICAL FIELD 
     The present disclosure pertains to a method of operating an internal combustion engine of a motor vehicle, such as a Diesel engine or a Gasoline engine, and more particularly a method of determining the fuel quantity and timing of fuel injection by an engine fuel injector into a combustion chamber. 
     BACKGROUND 
     It is known that an internal combustion engine of a motor vehicle generally includes a fuel injection system having a high pressure fuel pump, which delivers fuel at high pressure to a fuel rail, and a plurality of fuel injectors in fluid communication with the fuel rail. Each injector is provided for injecting metered quantities of fuel inside a corresponding combustion chamber of the engine. Conventionally, each fuel injector performs a plurality of injection pulses per engine cycle, according to a multi-injection pattern. This multi-injection pattern usually includes a main injection, which is executed to generate torque at the crankshaft, and several smaller injections, which may be executed before the main injection (e.g. pilot-injections and pre-injections) and/or after the main injection (e.g. after-injections and post-injections). Each of these small injection pulses is made to inject into the combustion chamber a small quantity of fuel, typically lower than 2.5 mm 3  (for example 1 mm 3 ), with the aim of reducing polluting emissions and/or combustion noise of the internal combustion engine. 
     The fuel injectors are essentially embodied as electromechanical valves having a needle, which is normally biased in a closed position by a spring, and an electro-magnetic actuator (e.g. solenoid), which moves the needle towards an open position in response of an energizing electrical current. The energizing electrical current is provided by an electronic control unit, which is generally configured to determine the fuel quantity to be injected by each single injection pulse, to calculate the duration of the energizing electrical current (i.e. the energizing time) needed for injecting the desired fuel quantity, and finally to energize the fuel injector accordingly. 
     However, it may happen that the fuel quantity actually injected during an injection pulse is different from the desired one. This undesirable condition may be caused by several factors, including drift of the injection characteristics and production spread of the fuel injectors. In particular, the correlation between the electrical command and the injector needle displacement can be affected by variables that are difficult to control during the injectors manufacturing, such as magnetic permeability drift of the actuator, tolerance of the needle spring coefficient, aging effect, and temperature dependency. Therefore, it is very likely that two fuel injectors (even of the same production slot) behave differently in response of the same electrical command. 
     As a result of these factors, for a given energizing time and a given fuel rail pressure, the fuel quantity actually injected into the combustion chambers of an internal combustion engine may be different injector-by-injector and/or vary with the aging of the injection system. This problem is particularly critical for the small injection pulses, whose accuracy and repetitiveness is important to achieve the expected improvements in terms of polluting emission and combustion noise. 
     To solve this drawback, while the internal combustion engine is running under cut-off conditions, the electronic control unit is conventionally configured to perform from time to time a procedure aimed to measure the actual fuel quantity which is injected by each fuel injector. According to the known solutions, the actually injected fuel quantity may be estimated on the basis of input signals deriving from different kinds of sensors such as knock sensors or on the basis of the crankshaft wheel signal. 
     A drawback of these prior solutions lies in the fact that such fuel quantity estimation is indirect and that the signals involved, for example the crankshaft wheel signal or other signals, are easily affected by noise and all sorts of disturbances coming from external environment such as rough roads, electric loads or other external or internal conditions, so that the resulting estimation may be not always reliable. Another drawback is that some of these known solutions cannot be performed during the execution of the so-called stop-start running strategies. 
     The stop-start running strategies are strategies that provide for disengaging the clutch and shutting off the engine when the motor vehicle is coasting, thereby saving fuel and reducing pollutant emissions. Under these circumstances, since the clutch is disengaged, some of the sensors involved in the conventional estimation of the fuel injected quantity cannot be used. 
     Still another drawback of the known solutions is that they are not able to measure the Start Of Injection (SOI). The SOI is a parameter that represents the instant when the injection pulse starts and is usually expressed in terms of an angular position of the engine crankshaft. 
     The SOI ideally coincides with the instant when the electronic control unit applies the energizing current to the fuel injector. However, due to fuel injector configurations (particularly solenoid injectors), there is always a certain delay between the application of the energizing current and the actual opening of the fuel injector. This delay is not the same for all the fuel injectors but is affected by the same factors that also affect the fuel injected quantity, such as for example the magnetic permeability drift of the actuator, the tolerance of the needle spring coefficient, aging effect, and temperature dependency. As a consequence, it may happen that two fuel injectors of the same kind (e.g. of the same production slot) open at different instants, even if the energizing current is applied at the very same time. 
     SUMMARY 
     In accordance with the present disclosure, strategies for determining the actual quantity of fuel injected by a fuel injection are provided, which are more reliable and less affected by external disturbances with respect to the known strategies. Furthermore, the strategies determine the instant when the fuel injection actually occurs. The strategies disclosed herein meet these goals by in a rational and relatively inexpensive solution. 
     An embodiment of the present disclosure provides a method of operating an internal combustion engine, wherein the internal combustion engine includes a fuel rail in fluid communication with a fuel pump and with a fuel injector. A fuel injector is operated to perform a fuel injection. A pressure signal representative of a fuel pressure within the fuel rail during the fuel injection is sampled and used as an input for a first integral transform yielding as output a value of a first function having as variables a fuel rail pressure drop caused by the fuel injection and a timing parameter indicative of an instant when the fuel injection started. The pressure signal is also used as input of a second integral transform yielding as output a value of a second function 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. The value of the first function and the value of the second function 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 calculated value of the fuel rail pressure drop. 
     This solution provides a reliable and effective strategy for determining both the actual injected fuel quantity and the actual timing of the fuel injection, with low computational effort and without requiring additional sensors, thereby representing a cost effective solution. The solution may be performed during strong transient and even during the execution of stop-start running strategies, because the pressure within the fuel rail is not affected by the clutch. 
     According to an aspect of the present disclosure, the fuel rail pressure signal may be sampled in a crankshaft angular domain (i.e. referred to the angular position of the engine crankshaft). The advantage of this aspect is that the determination of the fuel injected quantity becomes independent from the engine speed. 
     According to another aspect of the present disclosure, the value of the first function may be calculated with the following integral transform:
 
 L   α =∫ 0   2π   P (θ)·cos(θ) d (θ)≅ T   α (Δ P   inj ,γ inj )=Δ P   inj ·sin γ inj  
 
     Wherein:
         L α  is the value of the first function T α ;   P is the fuel rail pressure;   Θ is an angular position of the 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 inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is an angular distance of the fuel injection from the staring value 0 of the integration interval.       

     As can be understood from the equations, this integral transform is effectively able to yield a value L α  of first function T α  that, with good approximation, depends only on the fuel rail pressure drop ΔP inj  and on the instant when the fuel injection occurs, namely the angular distance γ inj . 
     According to another aspect of the present disclosure, the value of the second function may be calculated with the following integral transform:
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)≅ T   β (Δ P   inj ,γ inj )=Δ P   inj ·(1−cos γ inj )
 
     Wherein:
         L β  is the value of the second function T β ;   P is the fuel rail pressure;   Θ is the angular position of the 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 inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is an angular distance of the fuel injection from the starting value of the integration interval.       

     As can be understood from the equations, this integral transform is effectively able to yield a value L β  of a second function T β  that, with good approximation, depends only on the fuel rail pressure drop and on the instant when the fuel injection occurs, which is still represented by the angular distance γ inj . 
     An aspect of the present disclosure provides that the starting value of the integration interval may be an angular position of the crankshaft for which a piston of the fuel pump has already completed the compression stroke. This solution guarantees that, during the integration interval, the fuel rail pressure is not affected by the pump. 
     According to another aspect of the present disclosure, the value of the fuel quantity injected by the fuel injection may be calculated taking into account a value of a hydraulic capacitance of the fuel rail. This aspect of the present disclosure provides a reliable solution for calculating the fuel injected quantity starting from the pressure drop within the fuel rail. 
     An aspect of the present disclosure particularly provides that the value of the hydraulic capacitance may be varied on the basis of an average value of the pressure within the fuel rail. This aspect of the present disclosure increases the reliability of the strategy, since the hydraulic capacitance of the fuel rail generally depends on the pressure level. 
     According to another aspect of the present disclosure, the value of the hydraulic capacitance may be determined with a learning procedure, which is performed while the engine operates under a fuel cut-off condition (even during the execution of a stop-start running strategy). The fuel pump is operated to deliver a predetermined volume of fuel into the fuel rail per compression stroke. A value of a fuel rail pressure increment due to the delivery of said volume of fuel is measured. The value of the hydraulic capacitance are calculated as a function of the volume of fuel delivered into the fuel rail and the measured value of the fuel rail pressure increment. This solution provides a reliable and effective strategy for learning the hydraulic capacitance of the fuel rail. 
     An aspect of the present disclosure provides that the teaming procedure may further include calculating an average value of the fuel rail pressure during the delivery of said volume of fuel. The calculated value of the hydraulic capacitance is stored in memory, thereby correlating it to the calculated average value of the fuel rail pressure. This solution is used to create an array or map that correlates each value of the fuel rail pressure with a corresponding value of the hydraulic capacitance, which in turn may be effectively used to calculate the fuel injected quantity. 
     According to an aspect of the present disclosure, the fuel injection performed by the fuel injector may include a single injection pulse. This aspect of the present disclosure, which may be implemented while the internal combustion engine is running under cut-off conditions, can be reliably used to determine the fuel quantity that is actually injected by a single injection pulse. 
     According to another aspect of the present disclosure, the fuel injection performed by the fuel injector may include a plurality of injection pulses, for example according to a mufti-injection pattern. This aspect of the present disclosure, which may be implemented either under cut-off conditions or normal operating conditions, can be reliably used to determine the overall fuel quantity that is actually injected by the fuel injector per engine cycle. 
     The method may further include calculating a difference between the calculated value of the fuel injected quantity and a predetermined target value thereof. The calculated difference is used to correct an energizing time of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the fuel injected quantity. 
     Another aspect of the present disclosure provides that the method may include calculating a difference between the calculated value of the timing parameter and a predetermined target value thereof. The calculated difference is used to correct a start of injection of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the injection timing. 
     In accordance with the present disclosure, the method may be embodied in a non transitory computer readable medium including a program-code for carrying out all the steps of the method described above, and in the form of a computer program product including the computer program and a electronic control unit. The method can be also embodied as an electromagnetic signal carrying a sequence of data bits which represent a computer program to carry out all steps of the method. 
     Another embodiment of the present disclosure provides an internal combustion engine including a fuel rail in fluid communication with a fuel pump and with a fuel injector, and an electronic control unit configured to operate the fuel injector to perform a fuel injection, sample a signal representative of a fuel pressure within the fuel rail during the fuel injection, use the pressure signal as input of a first integral transform yielding as output a value of a first function having as variables 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 input of a second integral transform yielding as output a value of a second function having as variables the fuel rail pressure drop caused by the fuel injection and the tinting parameter indicative of the instant when the fuel injection started, use the value of the first function and the value of the second function 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 value of the fuel rail pressure drop. 
     This embodiment achieves basically the same effects mentioned before, in particular that of providing a reliable and effective strategy for determining both the actual injected fuel quantity and the actual timing of the fuel injection, with low computational effort and without requiring additional sensors, thereby representing a cost effective solution. 
     According to an aspect of the present disclosure, the electronic control unit may be configured to sample the fuel rail pressure signal in a crankshaft angular domain (i.e. referred to the angular position of the engine crankshaft). The advantage of this aspect is that the determination of the fuel injected quantity becomes independent from the engine speed. 
     According to another aspect of the present disclosure, the electronic control unit may be configured to calculate the value of the first function with the following integral transform:
 
 L   α =∫ 0   2π   P (θ)·cos(θ) d (θ)≅ T   α (Δ P   inj ,γ inj )=Δ P   inj ·sin γ inj  
 
     wherein:
         L α  is the value of the first function T α ;   P is the fuel rail pressure;   Θ is the angular position of the 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 inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is an angular distance of the fuel injection from the staring value 0 of the integration interval.
 
As can be understood from the equations, this integral transform is effectively able to yield a value L α  of first function T α  that, with good approximation, depends only on the fuel rail pressure drop ΔP inj  and on the instant when the fuel injection occurs, namely the angular distance γ inj .
       

     According to another aspect of the present disclosure, the electronic control unit may be configured to calculate the value of the second function with the following integral transform:
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)≅ T   β (Δ P   inj ,γ inj )=Δ P   inj ·(1−cos γ inj )
 
     wherein:
         L β  is the value of the second function T β ;   P is the fuel rail pressure;   Θ is the angular position of the 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 inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is the angular distance of the fuel injection from the starting value of the integration interval.
 
As can be understood from the equations, this integral transform is effectively able to yield a value L β  of a second function L β  that, with good approximation, depends only on the fuel rail pressure drop and on the instant when the fuel injection occurs, which is still represented by the angular distance γ inj .
       

     An aspect of the present disclosure provides that the starting value of the integration interval may be an angular position of the crankshaft for which a piston of the fuel pump has already completed the compression stroke. This solution guarantees that, during the integration interval, the fuel rail pressure is not affected by the pump. 
     According to another aspect of the present disclosure, the electronic control unit may be configured to calculate the value of the fuel quantity injected by the fuel injection taking into account a value of an hydraulic capacitance of the fuel rail. This aspect of the present disclosure provides a reliable solution for calculating the fuel injected quantity starting from the pressure drop within the fuel rail. 
     An aspect of the present disclosure particularly provides that the electronic control unit may be configured to vary the value of the hydraulic capacitance on the basis of an average value of the pressure within the fuel rail. This aspect of the present disclosure increases the reliability of the strategy, since the hydraulic capacitance of the fuel rail generally depends on the pressure level. 
     According to another aspect of the present disclosure, the electronic control unit may be configured to determine the value of the hydraulic capacitance with a teaming procedure, which is performed while the engine operates under a fuel cut-off condition (even during the execution of a stop-start running strategy) and which includes 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. This solution provides a reliable and effective strategy for learning the hydraulic capacitance of the fuel rail. 
     An aspect of the present disclosure provides that the leaning procedure may include calculating an average value of the fuel rail pressure during the delivery of said volume of fuel, storing the calculated value of the hydraulic capacitance, thereby correlating it to the calculated average value of the fuel rail pressure. This solution allows to create an array or map that correlates each value of the fuel rail pressure with a corresponding value of the hydraulic capacitance, which in turn may be effectively used to calculate the fuel injected quantity. 
     According to an aspect of the present disclosure, the fuel injection performed by the fuel injector may include a single injection pulse. This aspect of the present disclosure, which may be implemented while the internal combustion engine is running under cut-off conditions, can be reliably used to determine the fuel quantity that is actually injected by a single injection pulse. 
     According to another aspect of the present disclosure, the fuel injection performed by the fuel injector may include a plurality of injection pulses, for example according to a multi-injection pattern. This aspect of the present disclosure, which may be implemented either under cut-off conditions or normal operating conditions, can be reliably used to determine the overall fuel quantity that is actually injected by the fuel injector per engine cycle. 
     The electronic control unit may be further configured to calculate a difference between the calculated value of the fuel injected quantity and a predetermined target value thereof, and use the calculated difference to correct an energizing time of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the fuel injected quantity. 
     Another aspect of the present disclosure provides that the electronic control unit may be further configured calculate a difference between the calculated value of the timing parameter and a predetermined target value thereof, and use the calculated difference to correct a start of injection of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the injection timing. 
     Another embodiment of the present disclosure provides an apparatus for operating an internal combustion engine, wherein the internal combustion engine includes a fuel rail in fluid communication with a fuel pump and with a fuel injector. The apparatus includes means for operating the fuel injector to perform a fuel injection, means for sampling a signal representative of a fuel pressure within the fuel rail during the fuel injection, means using the pressure signal as input of a first integral transform yielding as output a value of a first function having as variables a fuel rail pressure drop caused by the fuel injection and a timing parameter indicative of an instant when the fuel injection started, means for using the pressure signal as input of a second integral transform yielding as output a value of a second function 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, means for using the value of the first function and the value of the second function to calculate a value of the fuel rail pressure drop caused by the fuel injection and a value of the timing parameter, and means for calculating a value of a fuel quantity injected by the fuel injection as a function of the value of the fuel rail pressure drop. 
     This embodiment achieves basically the same effects mentioned before, in particular that of providing a reliable and effective strategy for determining both the actual injected fuel quantity and the actual timing of the fuel injection, with low computational effort and without requiring additional sensors, thereby representing a cost effective solution. 
     According to an aspect of the present disclosure, the apparatus may include means for sampling the fuel rail pressure signal in a crankshaft angular domain (i.e. referred to the angular position of the engine crankshaft). The advantage of this aspect is that the determination of the fuel injected quantity becomes independent from the engine speed. 
     According to another aspect of the present disclosure, the means for calculating the value of the first function may include the following integral transform:
 
 L   α =∫ 0   2π   P (θ)·cos(θ) d (θ)≅ T   α (Δ P   inj ,γ inj )=Δ P   inj ·sin γ inj  
 
     wherein:
         L α  is the value of the first function T α ;   P is the fuel rail pressure;   Θ is the angular position of the crankshaft;   0 is a predetermined starting value of an integration interval [0, 2n] in the crankshaft angular domain;   2π is a predetermined final value of the integration interval [0, 2π] in the crankshaft angular domain;   ΔP inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is an angular distance of the fuel injection from the staring value 0 of the integration interval.       

     As can be understood from the equations, this integral transform is effectively able to yield a value L α  of first function T α  that, with good approximation, depends only on the fuel rail pressure drop ΔP inj  and on the instant when the fuel injection occurs, namely the angular distance γ inj . 
     According to another aspect of the present disclosure, the means for calculating the value of the second function including with the following integral transform:
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)≅ T   β (Δ P   inj ,γ inj )=Δ P   inj ·(1−cos γ inj )
 
     wherein:
         L β  is the value of the second function T β ;   P is the fuel rail pressure;   Θ is an angular position of the 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 inj  is the fuel rail pressure drop caused by the fuel injection; and   γ inj  is the angular distance of the fuel injection from the starting value of the integration interval.       

     As can be understood from the equations, this integral transform is effectively able to yield a value L β  of a second function T β  that, with good approximation, depends only on the fuel rail pressure drop and on the instant when the fuel injection occurs, which is still represented by the angular distance γ inj . 
     An aspect of the present disclosure provides that the starting value of the integration interval may be an angular position of the crankshaft for which a piston of the fuel pump has already completed the compression stroke. This solution guarantees that, during the integration interval, the fuel rail pressure is not affected by the pump. 
     According to another aspect of the present disclosure, the apparatus may include means for calculating the fuel quantity injected by the fuel injection taking into account a value of an hydraulic capacitance of the fuel rail. This aspect of the present disclosure provides a reliable solution for calculating the fuel injected quantity starting from the pressure drop within the fuel rail. 
     An aspect of the present disclosure particularly provides that the apparatus may include means for varying the value of the hydraulic capacitance on the basis of an average value of the pressure within the fuel rail. This aspect of the present disclosure increases the reliability of the strategy, since the hydraulic capacitance of the fuel rail generally depends on the pressure level. 
     According to another aspect of the present disclosure, the apparatus may include means for performing, while the engine operates under a fuel cut-off condition (even during the execution of a stop-start running strategy), a learning procedure to determine the hydraulic capacitance. The means for performing the learning procedure including means for operating the fuel pump to deliver a predetermined volume of fuel into the fuel rail per compression stroke, means for measuring a value of a fuel rail pressure increment due to the delivery of said volume of fuel, and means 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. This solution provides a reliable and effective strategy for learning the hydraulic capacitance of the fuel rail. 
     An aspect of the present disclosure provides that the means for performing the learning procedure may further include means for calculating an average value of the fuel rail pressure during the delivery of said volume of fuel, and means for memorizing the calculated value of the hydraulic capacitance, thereby correlating it to the calculated average value of the fuel rail pressure. This solution creates an array or map that correlates each value of the fuel rail pressure with a corresponding value of the hydraulic capacitance, which in turn may be effectively used to calculate the fuel injected quantity. 
     According to an aspect of the present disclosure, the fuel injection performed by the fuel injector may include a single injection pulse. This aspect of the present disclosure, which may be implemented while the internal combustion engine is running under cut-off conditions, can be reliably used to determine the fuel quantity that is actually injected by a single injection pulse. 
     According to another aspect of the present disclosure, the fuel injection performed by the fuel injector may include a plurality of injection pulses, for example according to a multi-injection pattern. This aspect of the present disclosure, which may be implemented either under cut-off conditions or normal operating conditions, can be reliably used to determine the overall fuel quantity that is actually injected by the fuel injector per engine cycle. 
     The apparatus may further include means for calculating a difference between the calculated value of the fuel injected quantity and a predetermined target value thereof, and means for using the calculated difference to correct an energizing time of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the fuel injected quantity. 
     Another aspect of the present disclosure provides that the apparatus may further include means for calculating a difference between the calculated value of the timing parameter and a predetermined target value thereof, and means for using the calculated difference to correct a start of injection of the fuel injector. This aspect of the present disclosure realizes a closed-loop control strategy for compensating possible errors of the injection timing. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements. 
         FIG. 1  schematically shows an automotive system; 
         FIG. 2  is the section A-A of  FIG. 1 ; 
         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; 
         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 ; 
         FIG. 5  shows in greater details a fuel injector of the automotive system of  FIG. 1 ; 
         FIG. 6  is a flowchart that represents a close-loop control strategy of the fuel injected quantity; 
         FIG. 7  is a flowchart that represents a closed-loop control strategy of the start of injection; and 
         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 
     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. 
     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 . 
     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 . 
     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 . 
     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. 
     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 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 . 
     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. 
     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 . 
     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. 
     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. 
     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. 
     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”. 
     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). 
     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. 
     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. 
     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 
     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. 
     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. 
     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. 
     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   α =∫ 0   2π   P (θ)·cos(θ) d (θ)
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)
             wherein:       L α  is the value yielded by the first integral transform;   L 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.       

     Looking at  FIG. 4 , the pressure P of the fuel rail may be considered as the sum of two contributions:
 
 P=P   eq   +δP   noise  
 
     wherein:
         P eq  represents an equivalent pressure (e.g. a mean pressure) of the fuel rail  170 ; and   δP noise  represents the pressure fluctuations due to the pressure waves and electronic noise of the sensor.       

     As a consequence, the preceding integral transforms may be rewritten as follows:
 
 L   α =∫ 0   2π   P (θ)·cos(θ) d (θ)=∫ 0   2π   [P   eq   +δP   noise ]·cos(θ) d (θ)
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)=∫ 0   2π   [P   eq   +δP   noise ]·sin(θ) d (θ)
 
     However, the frequency spectrum of the pressure fluctuations δP noise  is much higher than the frequency spectrum of the equivalent pressure P eg , so that the contribution of the pressure fluctuations to the integral transform is negligible:
 
∫ 0   2π   δP   noise ·cos(θ) d (θ)≅∫ 0   2π   δP   noise ·sin(θ) d (θ)≅0
 
     As a consequence, the preceding integral transforms may be rewritten as follows:
 
 L   α =∫ 0   2π   P (θ)·cos(θ) d (θ)≈∫ 0   2π   P   eq ·cos(θ) d (θ)= T   α (Δ P   inj ,γ inj )=Δ P   inj ·sin γ inj  
 
 L   β =∫ 0   2π   P (θ)·sin(θ) d (θ)≈∫ 0   2π   P   eq ·sin(θ) d (θ)= T   β (Δ P   inj ,γ inj )=Δ P   inj ·(1−cos γ inj )
 
     wherein:
         ΔP inj  is the fuel rail pressure drop caused by the fuel injection;   γ inj  is the angular distance of the fuel injection from the staring value 0 of the integration interval [0, 2π]; and,   T α  and T β  are two functions having as variables the fuel rail pressure drop ΔP inj  and the angular distance γ inj .       

     After having calculated the values L α  and L β , the ECU  450  may thus calculate (block  625 ) the fuel rail pressure drop ΔP inj  and the angular distance γ inj  with the following equations: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 P 
                 inj 
               
             
             = 
             
               - 
               
                 
                   
                     L 
                     α 
                     2 
                   
                   + 
                   
                     L 
                     β 
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   
                     L 
                     β 
                   
                 
               
             
           
         
       
       
         
           
             
               γ 
               inj 
             
             = 
             
                 
             
             ⁢ 
             
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 sin 
                 ( 
                 
                   - 
                   
                     
                       2 
                       ⁢ 
                       
                         L 
                         α 
                       
                       ⁢ 
                       
                         L 
                         β 
                       
                     
                     
                       
                         L 
                         α 
                         2 
                       
                       + 
                       
                         L 
                         β 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     In this way, the angular distance γ inj  provides a measurement of the Start of Injection (SOI), whereas the fuel rail pressure drop ΔP inj  can be used to calculate the fuel quantity actually injected by the fuel injection (block  630 ). 
     More particularly, the fuel rail pressure drop ΔP inj  can be used to calculate a dynamic fuel quantity q inlet  that actually flows through the fuel injector  160  according to the following equation:
 
= q   inlet   =C   hyd   ·ΔP   inj  
 
     wherein C hyd  is the value of the hydraulic capacitance of the fuel rail  170 , 
     As represented in  FIG. 5 , the dynamic fuel quantity q inlet  is the sum of two contributions, namely the fuel injected quantity g inj  and the dynamic leakage q dyn . The fuel injected quantity g inj  is the quantity of fuel that actually enters the combustion chamber  150 , whereas dynamic leakage q 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 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 inj  and the dynamic leakage q dyn :
 
 q   inlet   =q   inj   +q   dyn  
 
     However, q inlet , g inj  and q 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 inlet , the fuel pressure and the energizing time used to perform the fuel injection, it is possible to determine the value q inj  of the fuel injected quantity as a function of q inlet :
 
 q   inj =ƒ( q   inlet )
 
     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 inj  of the fuel injected quantity according to the method above, calculating a difference e between the calculated value q inj * and a predetermined target value of the fuel injected quantity, and then to use said difference to correct an energizing time ET 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 ET  to be added to the energizing time ET inj *, in order to obtain a corrected energizing time ET inj  that is finally used to operate the fuel injector  160 . 
     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 inj  of the SOI according to the method above, calculating a difference e between the calculated value γ inj  and a predetermined target value γ inj * of the SOI, and then to use said difference to correct the target value γ 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 γ  to be added to the target value γ inj *, in order to obtain a corrected value SOI inj  of the start of injection that is finally used to operate the fuel injector  160 . 
     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 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 . 
     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 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. 
     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 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·μ 
 
     wherein V is the pump displacement and μ is the pump volumetric efficiency. 
     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 0  to a predetermined maximum value P 1  as shown in  FIG. 8 . 
     For each step, the learning procedure may prescribe to calculate a value ΔP k  of a fuel rail pressure increment due to the delivery of the volume Q of fuel and an average value P k  of the fuel rail pressure, namely an average between the pressure values before and after the delivery of said volume Q of fuel. 
     The calculated value ΔP k  may then be used to calculate the value C hyd,k  of the hydraulic capacitance according to the following equation: 
     
       
         
           
             
               C 
               
                 hyd 
                 , 
                 k 
               
             
             = 
             
               Q 
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   P 
                   k 
                 
               
             
           
         
       
     
     The value C hyd,k  of the hydraulic capacitance may be finally memorized in the memory system  460 , thereby correlating it to the corresponding average value P 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 k  with a corresponding value of the hydraulic capacitance C hyd,k , which in turn may be effectively used to calculate the fuel injected quantity according to the method set forth above. 
     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.