Patent Publication Number: US-8116963-B2

Title: Fuel injection pin displacement profile interpolation

Description:
TECHNICAL FIELD 
     The present invention relates generally to internal combustion engine fuel injection, and more particularly, some embodiments relate to determination of fuel injection pin displacement profiles. 
     DESCRIPTION OF THE RELATED ART 
     By precisely injecting fuel into an internal combustion engine, the efficiency of the engine can be increased. The optimum operation of the engine is affected by how the fuel injector is opened and closed. The fuel injection is controlled by the lifting and lowering of an injector pin. The function of the pin lift with respect to the timing of the engine&#39;s piston around top dead center directly affects the operation and efficiency of an internal combustion engine. This function is an injection profile. A profile comprises a set of points; pairs of time and pin displacement values.  FIG. 1  illustrates a conventional example injection profile comprising points  50 ,  51 ,  52 ,  53 , and  54 . In  FIG. 1 , and in the other injection profiles illustrated herein, the horizontal axis is the time axis while the vertical axis is the displacement axis. Contour  55  illustrates the injection pin displacement as a function of time, assuming that the injection pin displaces at a constant rate between injection profile points. For ease of explanation only, injector and profiles herein will be illustrated under this assumption. 
     The injection profile can be determined experimentally at different combinations of an engine&#39;s torque and speed, known as operating points. Since an engine can operate over a continuous range of operating points in the operating plane, the number of points to determine experimentally is infinite. 
     BRIEF SUMMARY OF EMBODIMENTS OF THE INVENTION 
     According to various embodiments of the invention, a map of injection pin profiles is experimentally determined at various locations spanning an engine operating plane. Injector pin profiles at points within the continuum spanned by the experimentally determined profiles are determined by interpolating between surrounding experimentally determined injector pin profiles. Various methods are used to adjust the interpolation procedure in cases where one injector pin profile has more or fewer points than the other injector pin profile. 
     According to one embodiment of the invention, a method is presented for determining an interpolated injector pin profile for a fuel injector at an engine operating point in an engine operating plane, the engine operating plane comprising operating points having speed values and torque values. In this embodiment the method comprises obtaining a first injector pin profile associated with a first operating point and comprising a first set of profile points comprising displacement values and time values; obtaining a second injector pin profile associated with a second operating point and comprising a second set of profile points, the second injector pin profile comprising at least as many profile points as the first injector pin profile; connecting a profile point of the second injector pin profile with a profile point of the first injector pin profile to form connected injector pin profiles; and using the connected injector profiles to determine an interpolated injector pin profile at an operating point between the first operating point and the second operating point. 
     According to another embodiment of the invention, a fuel injection system comprises a fuel injector configured to inject fuel into an internal combustion engine and comprising a pin configured to be displaced using an actuator; the actuator configured to displace the pin according to an actuator driving signal; and an engine control unit configured to generate the actuator driving signal according to an engine operating point in an engine operating plane, the engine operating plane comprising operating points having speed values and torque values; wherein the engine control unit is configured to determine the engine operating point using a firing signal and a throttle position signal; and wherein the engine control unit is further configured to generate the actuator driving signal using an interpolated injector pin profile, the interpolated injector pin profile determined using the steps of: obtaining a first injector pin profile associated with a first operating point and comprising a first set of profile points comprising displacement values and time values; obtaining a second injector pin profile associated with a second operating point and comprising a second set of profile points, the second injector pin profile comprising at least as many profile points as the first injector pin profile; connecting a profile point of the second injector pin profile with a profile point of the first injector pin profile to form connected injector pin profiles; and using the connected injector profiles to determine an interpolated injector pin profile at an operating point between the first operating point and the second operating point. 
     According to a further embodiment of the invention, the engine control unit is further configured to perform the step of adding profile points to the first injector pin profile such that the first injector pin profile and the second injector pin profile comprise an equal number of points; and wherein the step of forming the connected injector pin profiles comprises connecting the profile points of the first injector pin profile with the profile points of the second injector pin profile according to increasing time value. 
     Other features and aspects of the invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the features in accordance with embodiments of the invention. The summary is not intended to limit the scope of the invention, which is defined solely by the claims attached hereto. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention, in accordance with one or more various embodiments, is described in detail with reference to the following figures. The drawings are provided for purposes of illustration only and merely depict typical or example embodiments of the invention. These drawings are provided to facilitate the reader&#39;s understanding of the invention and shall not be considered limiting of the breadth, scope, or applicability of the invention. It should be noted that for clarity and ease of illustration these drawings are not necessarily made to scale. 
         FIG. 1  (prior art) is an illustration of an injector pin profile at a point of an engine operating plane. 
         FIG. 2  is a graphic representation of an engine operating plane. 
         FIG. 3  is a graphic representation of an interpolation procedure according to an embodiment of the invention. 
         FIG. 4  is a graphic representation of a method of determining corresponding subsets of points between injector pin profiles according to an embodiment of the invention. 
         FIG. 5  illustrates the results of the interpolation procedure described with respect to  FIG. 4 . 
         FIG. 6  illustrates an example of interpolation between two profiles according to an embodiment of the invention. 
         FIG. 7  illustrates an example of interpolation between an injector profile and a zero lift profile according to an embodiment of the invention. 
         FIG. 8  is a graphic representation of an alternative method of point addition according to an embodiment of the invention. 
         FIG. 9  is a graphic representation of a method for evaluating the characteristics of an injection pin profile according to an embodiment of the invention. 
         FIG. 10  illustrates an alternative result of the method described with respect to  FIG. 9  and further illustrates a method of injection pin profile point determination according to the results of the method. 
         FIG. 11  illustrates an environment in which embodiments of the invention might be implemented. 
         FIG. 12  illustrates a method of operation for an engine control unit according to an embodiment of the invention. 
     
    
    
     The figures are not intended to be exhaustive or to limit the invention to the precise form disclosed. It should be understood that the invention can be practiced with modification and alteration, and that the invention be limited only by the claims and the equivalents thereof. 
     DETAILED DESCRIPTION OF THE EMBODIMENTS OF THE INVENTION 
     Before describing the invention in detail, it is useful to describe an example environment with which the invention can be implemented. One such example is that of a fuel injector for use in an internal combustion engine employing direct injection. Such fuel injectors may be of the types described in U.S. Pat. No. 7,444,230, “FUEL INJECTOR HAVING ALGORITHM CONTROLLED LOOK AHEAD TIMING FOR INJECTOR-IGNITION”; U.S. patent application Ser. No. 12/237,302, “FUEL INJECTOR HAVING ALGORITHM CONTROLLED LOOK AHEAD TIMING FOR INJECTOR-IGNITION”; and U.S. patent application Ser. No. 11/692,111, “HEATED CATALYZED FUEL INJECTOR FOR INJECTION IGNITION ENGINES”; the contents of which are hereby incorporated by reference in their entirety. Additionally, such fuel injectors may employ piezoelectric actuators of the types described in U.S. Provisional Patent Application No. 61/081,326, “A PIEZO ACTUATED FUEL INJECTOR WITH CATALYTIC SECTION”; U.S. Provisional Patent Application No. 61/117,897, “DUAL SOLENOID FUEL INJECTOR WITH CATALYTIC ACTIVATOR SECTION”; U.S. Provisional Patent Application No. 61/144,274, “MULTI-ELEMENT PIEZOELECTRIC ACTUATOR DRIVER”; U.S. Provisional Patent Application No. 61/144,265, “PIEZOELECTRIC ACTUATOR FAULT RECOVERY SYSTEM AND METHOD”; U.S. Provisional Patent Application No. 61/144,260, “SERIALLY OPERATING MULTI-ELEMENT PIEZOELECTRIC ACTUATOR DRIVER”; U.S. Provisional Patent Application No. 61/144,270, “SYSTEM AND METHOD FOR DEFINING PIEZOELECTRIC ACTUATOR WAVEFORM”; U.S. Provisional Patent Application No. 61/144,254, “PIEZOELECTRIC ACTUATOR EMPLOYING SWITCH”; and U.S. Provisional Patent Application No. 61/159,044, “REVERSE OPERATING NONLINEAR SPRING.” 
       FIG. 11  illustrates an example environment in which the present invention may be implemented. Engine  500  may comprise, for example, a gasoline direct injection engine, a diesel engine, or any other fuel injected internal combustion engine. Sensors such as cam censor  501  and crank sensor  502  provide engine operating data to the engine control unit (ECU)  503 . The ECU  503  uses this data to determine where on the operating plane the engine is currently operating. As described herein, using this information and predetermined injection pin profiles spanning the engine operating plane, the ECU determines an injection pin profile for the engine&#39;s  500  fuel injectors at the operating point. The fuel injector  504  is in connection with the ECU  503 , for example via a fuel injector driver, and is caused to inject fuel into the engine  500  according to the injection pin profile determined for the current operating point. 
     From time-to-time, the present invention is described herein in terms of these example environments. Description in terms of these environments is provided to allow the various features and embodiments of the invention to be portrayed in the context of an exemplary application. After reading this description, it will become apparent to one of ordinary skill in the art how the invention can be implemented in different and alternative environments. 
       FIG. 2  is a graphic representation of an engine operating plane. The operating plane comprises a plane spanned by an operating range of engine torque values  110  and an operating range of engine speed values  115 . According to various embodiments of the present invention, methods for determining an injector pin profile at an engine operating point may comprise determining the injector pin profile by interpolating between predetermined injector pin profiles. Experimentally derived injector pin profiles at a plurality of predetermined operating points  130  and one or more predefined boundary conditions in the boundary region  120  may be used during the interpolation procedure. For example, if an engine is required to operate at operating point  140   a , the engine control unit (ECU) may use a predefined look up table to determine that points  130   a ,  130   b ,  130   c , and  130   d  immediately surrounding operating point  140   a  and having associated predetermined injector pin profiles. Interpolating along contours of constant speed, the ECU then determines two additional injector pin profiles at operating points  135   a  and  135   b  having the same torque value as operating point  140   a . Then, interpolating along a contour of constant torque, the ECU determines the injector pin profile for operating point  140   a.    
     In some instances, the engine may operate at a point outside of the operating points having experimentally determined profiles. Such a case is illustrated as operating point  140   b . In such instances, some embodiments utilize a default boundary condition in at least a portion of border region  120  as a predetermined injector profile for a step of the interpolation procedure. For example, to determine the injector pin profile for operating point  140   b , the ECU first interpolates between experimentally derived points  130   f  and  130   e  to determine an injection profile at operating point  135   c . The ECU determines the profile at point  140   b  using the previously determined injection profile at point  135   c  and a predetermined border injection profile, such as a zero lift profile comprising a number of displacements/time pairs, where each displacement is zero. 
       FIG. 3  is a graphical representation of an interpolation procedure according to an embodiment of the invention. A computing device that is performing the interpolation procedure, such as an ECU, is presented with two injector profiles at two distinct points in the operating plane,  167 , and  169 . The operating points will typically be located on the same contour of a constant dimension of the operating plane. For example, the two operating points may have the same engine speed values and vary only in their engine torque values, or vice versa. In this embodiment, points in the provided injection pin profiles  169  and  167  are connected to allow interpolation of an injection profile  168  at an operating point between the operating points associated with profiles  169  and  167 . 
     The illustrated interpolation procedure proceeds in an ordered fashion across provided injector profiles  167  and  169 . According to an algorithm used in some embodiments of the invention, the internal representation of the injection pin profiles are modified such that each of the two provided profiles has the same number of points. For example, in  FIG. 3  the profile  167  has three points while profile  169  has four points; accordingly, one point is added to profile  167 . Various embodiments may employ different rules for adding points to profiles, based on for example, the number of points for each profile, the location of the points on each profile, or the location of each profile in the operating plane. By way of example, in a situation where one profile has three points while the other profile has more than three points, the peak point of the first profile is duplicated until each profile has the same number of points. In the illustrated interpolation, after point  161  is duplicated, profile  167  may be represented as the set of points { 160 ,  161 ,  161 ,  162 } and profile  169  remains the set of points { 163 ,  164 ,  165 ,  166 }. Once both profiles have an equal number of points, the connection proceeds using the first point of the first set and the first point of the second set, and so on. As illustrated, this results in two connection lines originating from point  161 , a first line between point  161  and point  164 , and a second line between point  161  and point  165 . An interpolated injection pin profile  168  can then be determined at any operating point between the operating points of profile  167  and profile  169 . In some cases, the interpolated injection pin profile  168  can serve as the injection pin profile used during engine operation, or in other cases, the interpolated injection pin profile  168  can serve as one of two further injection pin profiles provided to the ECU for interpolation in the other operating plane dimension. 
       FIG. 4  is a graphic representation of a method of determining corresponding subsets of points between injector pin profiles according to an embodiment of the invention. In this embodiment, a corresponding subset of points of the larger profile  218  is determined for each point in the smaller profile  217 . The corresponding subsets are determined using a divide and conquer algorithm executed recursively on the smaller pin profile. 
     The set of nonzero points in the smaller profile  217  { 199 ,  200 ,  210 } is divided  170  into two distinct subsets including subset/point { 199 } and subset { 200 ,  210 }. Correspondingly, the set of nonzero points in the larger profile  218  { 197 ,  198 ,  205 ,  215 ,  216 } is divided into two subsets { 197 ,  198 } and { 205 ,  215 ,  216 }, corresponding to the subset/point { 199 } and subset { 200 ,  210 } of the smaller profile  217 , respectively. This procedure proceeds recursively, where each subset having more than one point of the smaller profile  217  and its corresponding subset in the larger profile  218 , are divided into two subsets. Accordingly, in the illustrated profiles, the corresponding subsets { 200 ,  210 } and { 205 ,  215 ,  216 } are each divided at  180  and  185 , respectively. Afterwards, subset { 200 } corresponds with subset { 205 }, while subset { 210 } corresponds with subset { 215 ,  216 }. In these embodiments, when there are an odd number of points in a subset to be divided, the extra point is placed in the subset that is closer to the peak of the profile. However, other methods of partitioning the injector profile may be employed. 
     In the illustrated embodiment, once recursive division of the smaller profile results in a subset comprising a single point, connection proceeds between the single point and each of the elements of the corresponding subset. Similar to the case described with respect to  FIG. 3 , when the larger profile&#39;s corresponding subset has more than one point, connection lines are formed between each element of the corresponding subset and the single point. This is illustrated in  FIG. 4  with respect to the subset comprising points { 215 ,  216 }, corresponding to the subset comprising point  210 . In some embodiments, the connection  191  and  192  may occur independently between each point of the corresponding subset and the single point. In other embodiments, the single point may be duplicated to form a duplicate point set having a number of duplicated points equal to the number of elements of the corresponding subset. In these embodiments, each duplicated point is connected to one and only one element of the corresponding subset. 
       FIG. 5  illustrates the final results of a connection procedure as described with respect to  FIG. 4 . After a corresponding subset of the larger profile  218  is determined for each point of the profile  217 , the resultant connections are as follows: (1) connection  190  between { 212 } and { 214 }; (2) connection  196  between { 199 } and { 197 ); (3) connection  194  between { 199 } and { 198 }; (4) connection  193  between { 200 } and { 205 }; (5) connection  191  between { 210 } and { 215 }; (6) connection  192  between { 210 } and { 216 }; and (7) connection  195  between { 211 } and { 219 }. Accordingly, after the profiles have been connected in this manner, an injection pin profile may be interpolated at any operating point between the operating points associated with the connected profiles  217  and  218 . 
       FIG. 6  illustrates an example of interpolation between two profiles according to an embodiment of the invention. In this example, after the starting points  250  and  270  and ending points  253  and  276  of the profiles  290  and  291  are matched, two points remain in the smaller profile  290 , while five points remain in the larger profile  291 . Accordingly, as described with respect to  FIG. 4 , the sets of points { 251 ,  252 } and { 271 ,  272 ,  273 ,  274 ,  275 } are each divided once, at  260  and  261 , respectively. This results in the set { 271 ,  272 } corresponding to point  251  and the set, { 273 ,  274 ,  275 } corresponding to point  252 . Accordingly, interpolation proceeds by first duplicating point  251  and tripling point  252  such that connection proceeds as follows: (1) connection  287  between { 250 } and { 270 }; (2) connection  286  between { 251 } and { 271 }; (3) connection  285  between { 251 } and { 272 }; (4) connection  284  between { 252 } and { 273 }; (5) connection  282  between { 252 } and { 274 }; (6) connection  281  between { 252 } and { 275 }; and (7) connection  280  between { 253 } and { 276 }. In this embodiment, after the profiles have been connected an interpolated injection profile may be determined at any point between the profiles. 
       FIG. 7  illustrates an example of interpolation between an injector profile and a zero lift profile according to an embodiment of the invention. In some embodiments, it may be necessary to determine an injection pin profile for an engine operating outside the part of the operating plane covered by the experimentally determined injection pin profile. In these cases, a predetermined injector pin profile may be used as a boundary condition when necessary. For example, a zero displacement injection pin profile  351  may be used as such a boundary condition. Determining an injection pin profile between a predetermined profile  350  and a boundary profile  351  may comprise determining an appropriate set of points, each with zero displacement, that corresponds to the predetermined injection pin profile  350 . For example, for each point in the injection pin profile  350 , a corresponding point having the same time value but a zero pin displacement is determined, e.g. point  361  and point  371  at time value t 1 . An interpolated injection pin profile for operating points between the zero pin profile and the predetermined injection pin profile  350  may then be determined as described herein. In other embodiments, the placement of the points may be determined in other manners. For example, it may be desired to have sharper shaped injection pin profiles where the points of the zero displacement profile are placed closer to the center, or it may be desired to have broader shaped injection pin profiles where the points of the zero displacement profile are placed towards the ends. 
       FIG. 8  is a graphic representation of an alternative method of point addition according to an embodiment of the invention. In the illustrated embodiment, after recursive application of a divide and conquer application, the subset of the larger pin profile  381  corresponding to point  390  is { 400 } and the subset corresponding to point  391  is { 401 ,  402 }. In this embodiment, rather than duplicating point  391  in the smaller injection pin profile  380 , an additional point  393  is added to the injection pin profile  380  between points  390  and  391 . In such embodiments, the additional point may be added at a predetermined ratio of the length between the two points. For example if one point is to be added, it can be placed halfway along the line, and if two points are to be added, they can be placed at one third intervals along the line. In other embodiments, different methods of determining point placement may be used. For example, additional points may be placed at the same time value as the corresponding points of the larger profile. In  FIG. 8 , this would result in point  393  having a time value halfway between point  390  and point  391 . 
       FIGS. 9 and 10  illustrate methods of dividing injection pin profiles according to some embodiments of the invention. In these embodiments, before dividing the profile to determine where additional points will be added, the profile is inspected to determine its peak characteristics. This may be accomplished by determining the profile point having the maximal displacement value, and the displacement values of the points surrounding this point. For example, in the profile illustrated in  FIG. 9 , point  411  has the maximal displacement value and points  410  and  412  surround it. In one embodiment, different profiles are classified as sharp peak profiles and broad peak profiles according to these displacement values. A sharp peak profile is one in which the displacement amplitude of the nearest neighboring points of the maximal point differs from the maximal point by a predetermined amount. For example, this predetermined amount may be a certain percentage or may be a certain displacement length.  FIG. 9  illustrates a sharp peak, where the nearest neighboring point  412  of the maximal point  411  is outside the predetermined displacement length  414 . When the profiles used to derive an interpolated injection pin profile contain sharp peaks, superior fidelity in the interpolated profile may be achieved by dividing the sharp peak profile at the location of the sharp peak. In an embodiment employing a divide and conquer algorithm, as described herein, this may be achieved by adding a duplicate point to the sharp peak profile at the location of the maximal point. For example in  FIG. 9 , a second point  411 ′ may be added to the injection pin profile. The addition of such a duplicate point allows the profile to be divided  413  at the location of the maximal point such that each portion of the divided injection pin profile has a copy of the maximal point for use in interpolation. This results in the interpolated profiles retaining some of the sharp peak characteristics of the source profile. As described herein, after this initial step of analyzing the profile and duplicating a sharp peak point, the algorithm proceeds to add points to the profile according to the number of corresponding points in the other profile used for interpolation. 
       FIG. 10  illustrates a point addition procedure in the case of a broad peak profile according to this embodiment. In this case, the neighboring point  440  is within the predetermined threshold distance  445  of the maximal point  450 . Accordingly, this injection pin profile is classified as a broad peak profile. In some instances, superior fidelity in interpolated profiles may be accomplished by dividing a broad peak such that each divided portion retains one of the points making up the peak. In one embodiment, the ECU may add points to the injection pin profile on the side having the fewer number of points such that the divide and conquer algorithm divides the profile between the points making up the peak. For example, in  FIG. 10  side  460  has fewer points than side  461 . Accordingly, two points  472  and  473  may be added to the profile as illustrated such that the resulting profile has an equal number of points on each side, so that division results in equal numbers of points in each divided portion. In some embodiments, the locations  470  and  471  of the points to be added may be equally spaced on the lines to which they are added, as described herein. For example, in  FIG. 10  locations  470  and  471  divided the line between point  440  and the beginning of the profile into equal thirds. 
     In addition to determining a plurality of pin displacement and timing pairs that make up an interpolated injection pin profile, some embodiments determine other characteristics associated with an injection pin profile. For example, an injection pin firing delay may also be associated with the points of the operating plane having predetermined injector pin profiles. In these embodiments, during the determination of the injection pin profiles for the experimentally derived points  130  of  FIG. 2 , firing delays may also be determined for each of the points. These firing delays may be determined to appropriately time the beginning of injection pin displacement at the various points of the operating plane. This allows a firing delay time for an engine operating point be determined through interpolation along with the interpolation of the injection profile for operating point. For example, in the interpolation procedure described with respect to the point  140   a , an interpolated firing delay may be determined for the profiles at points  135   a  and  135   b . Then, along with the determination of the injection pin profile at point  140   a , a firing delay for point  104   a  may be determined from the derived firing delays for points  135   a  and  135   b.    
       FIG. 12  illustrates a method of ECU operation according to an embodiment of the invention. In this embodiment, ECU  549  is configured to generate a fuel injection control signal using two input control signals. The first input control signal is a firing signal  550  produced by a top dead center detector disposed in the engine. The top dead center detector generates firing signal  550  when an engine piston reaches top dead center in the cylinder. The second input control signal is a throttle valve signal  551  that is generated from the accelerator pedal position. Firing signal  550  is used by the ECU  549  along with a clock signal provided by clock  552  to determine the operating engine speed  553 . ECU  549  then determines the operating engine torque  554  using engine speed  553  and throttle signal  551 . Next, ECU  549  uses a map  555  of the operating plane to determine the points  556  surrounding operating torque  554  and speed  553 . In this embodiment, surrounding points  556  may comprise points on the engine operating plane  555  having higher and lower torque values than operating torque  554  and higher and lower engine speeds than operating speed  553 . As described herein, surrounding points  556  may have predetermined injection profiles associated therewith. 
     In the illustrated embodiment, after determining surrounding points  556 , ECU  549  uses the surrounding points  556  to determine an interpolated firing delay  557  and an interpolated injection profile  568 . As described herein, determining interpolated injection profile  568  comprises obtaining the predetermined injection profiles at surrounding points  556 . This comprises (1) obtaining the predetermined injection profile  559  associated with an operating point having a lower speed value and a higher torque value than the current operating point; (2) obtaining the predetermined injection profile  560  associated with an operating point having a higher speed value and a higher torque value than the current operating point; (3) obtaining the predetermined injection profile  561  associated with an operating point having a lower speed value and a lower torque value than the current operating point; and (4) obtaining the predetermined injection profile  562  associated with an operating point having a higher speed value and a lower torque value than the current operating point. 
     In this embodiment, a second step of determining an interpolated injection pin profile  568  comprises connecting the predetermined injection pin profiles to allow intermediate injection pin profiles  565  and  566  to be interpolated. In the illustrated embodiment, intermediate connection  563  and  564  proceeds along the speed axis of the operating plane. Accordingly, predetermined injection pin profiles  559  and  560  are connected  563  and predetermined injection pin profiles  561  and  562  are connected  564 . As described herein, the steps of connecting the profiles  563  and  564  may proceed by way of a divide and conquer algorithm that may preferentially connect the peaks or peak areas of the predetermined profiles. After the predetermined profiles are connected, intermediate profiles  565  and  566  are interpolated at the speed values corresponding to operating speed  553 . Thus, an interpolated profile  565  is determined for an operating point having a higher torque value than operating torque value  554  but having the same speed value as operating speed  553 ; and a second interpolated profile  566  is determined for an operating point having a lower torque value than operating torque value  554  but having the same speed value as operating speed  553 . 
     In this embodiment, a third step of determining final injection profile  568  comprises connecting  567  intermediate profiles  565  and  566 . In some embodiments, connection step  567  proceeds substantially similar to connection steps  563  and  564 , for example through the use of a divide and conquer algorithm described herein. The connection of intermediate profiles  565  and  566  allows interpolation to proceed in the torque direction of the operating plane. Accordingly, the final injection pin profile  568  is interpolated for the current operating point comprising operating engine speed  553  and torque value  554 . 
     As described herein, interpolating  557  the firing delay may comprise obtaining predetermined firing delays for the surrounding points  556  and performing a two-step interpolation procedure. For example, the two-step interpolation procedure may proceed as follows: first, the firing delays having the same torque values are connected, for example as described with respect to connection steps  563  and  564 ; and second, the determined intermediate firing delays are connected to determine an interpolated firing delay  558  at the engine operating point comprising speed value  553  and torque value  554 . The actual determined firing delay  558  may be determined using the clock signal from clock  552  and the results of the interpolation procedure  557 . 
     After determining an appropriate firing delay  558  and injection profile  560 , the ECU  549  uses this information to obtain data  569  for use by a digital to analog converter  570  in generating an analog signal to drive a piezoelectric actuator  571 . In this embodiment, firing delay  558  may be used to generate data  569  that takes into account the physical characteristics of the piezoelectric actuator  571  and fuel injector  573  to generate an appropriate digital signal from the interpolated profile  568 . Additionally, firing delay  558  may also be used by the digital to analog converter  570  for timing the analog signal. In this embodiment, the digital to analog converter  570  provides the analog piezoelectric driving signal to the piezoelectric actuator  571  to cause an injector pin disposed in fuel injector  573  to be displaced according to the determined to interpolated profile  568 . 
     While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not of limitation. Likewise, the various diagrams may depict an example architectural or other configuration for the invention, which is done to aid in understanding the features and functionality that can be included in the invention. The invention is not restricted to the illustrated example architectures or configurations, but the desired features can be implemented using a variety of alternative architectures and configurations. Indeed, it will be apparent to one of skill in the art how alternative functional, logical or physical partitioning and configurations can be implemented to implement the desired features of the present invention. Also, a multitude of different constituent module names other than those depicted herein can be applied to the various partitions. Additionally, with regard to flow diagrams, operational descriptions and method claims, the order in which the steps are presented herein shall not mandate that various embodiments be implemented to perform the recited functionality in the same order unless the context dictates otherwise. 
     Although the invention is described above in terms of various exemplary embodiments and implementations, it should be understood that the various features, aspects and functionality described in one or more of the individual embodiments are not limited in their applicability to the particular embodiment with which they are described, but instead can be applied, alone or in various combinations, to one or more of the other embodiments of the invention, whether or not such embodiments are described and whether or not such features are presented as being a part of a described embodiment. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments. 
     Terms and phrases used in this document, and variations thereof, unless otherwise expressly stated, should be construed as open ended as opposed to limiting. As examples of the foregoing: the term “including” should be read as meaning “including, without limitation” or the like; the term “example” is used to provide exemplary instances of the item in discussion, not an exhaustive or limiting list thereof; the terms “a” or “an” should be read as meaning “at least one,” “one or more” or the like; and adjectives such as “conventional,” “traditional,” “normal,” “standard,” “known” and terms of similar meaning should not be construed as limiting the item described to a given time period or to an item available as of a given time, but instead should be read to encompass conventional, traditional, normal, or standard technologies that may be available or known now or at any time in the future. Likewise, where this document refers to technologies that would be apparent or known to one of ordinary skill in the art, such technologies encompass those apparent or known to the skilled artisan now or at any time in the future. 
     The presence of broadening words and phrases such as “one or more,” “at least,” “but not limited to” or other like phrases in some instances shall not be read to mean that the narrower case is intended or required in instances where such broadening phrases may be absent. The use of the term “module” does not imply that the components or functionality described or claimed as part of the module are all configured in a common package. Indeed, any or all of the various components of a module, whether control logic or other components, can be combined in a single package or separately maintained and can further be distributed in multiple groupings or packages or across multiple locations. When used to describe injector pin profiles, the terms “smaller” and “larger” refer to having fewer points than another injector pin profile and having more points than another injector pin profile, respectively. 
     Additionally, the various embodiments set forth herein are described in teens of exemplary block diagrams, flow charts and other illustrations. As will become apparent to one of ordinary skill in the art after reading this document, the illustrated embodiments and their various alternatives can be implemented without confinement to the illustrated examples. For example, block diagrams and their accompanying description should not be construed as mandating a particular architecture or configuration.