Patent Publication Number: US-9885310-B2

Title: System and methods for fuel pressure control

Description:
FIELD 
     The present description relates generally to methods and systems for operating a fuel lift pump. 
     BACKGROUND/SUMMARY 
     Engine fuel may be pumped out of a fuel tank by a lift pump. The lift pump propels fuel towards a fuel rail before being injected by fuel injectors. A check valve may be included between the lift pump and the fuel rail to maintain fuel rail pressure and prevent fuel in the fuel rail from flowing back towards the lift pump. Operation of the lift pump is typically feedback controlled by an engine controller based on outputs from a pressure sensor coupled in the fuel rail. The controller attempts to maintain the pressure in the fuel rail to a desired pressure by adjusting an amount of power supplied to the lift pump based on a difference, or error, between the desired fuel pressure and a measured fuel pressure obtained from the pressure sensor. 
     However, the inventors herein have recognized potential issues with such systems. As one example, when the fuel injectors are turned off, such as during deceleration fuel shut-off (DFSO), power to the lift pump may be reduced. Turning off the fuel injectors may cause fuel rail pressure to increase while the lift pump is on and spinning. Thus, power to the lift pump, and therefore lift pump speed may be reduced in an attempt to reduce fuel rail pressure. However, since fuel is prevented from flowing backwards through the check valve, reducing power to the fuel pump may have no effect on the fuel pressure of fuel included between the check valve and the fuel rail. Further, when fuel injection is commanded back on, it may take time for the fuel pump to spin up. Due to the delay of the fuel pump spin-up time, and/or integrator wind-up of the controller, transient fuel pressure drops may occur when exiting DFSO, leading to fuel metering errors that may degrade engine thermal efficiency and increase regulated emissions. 
     Further, in examples where the fuel rail pressure is variable, closed loop control of the lift pump may command for a decrease in lift pump voltage when fuel injection is insufficient to lower the fuel rail pressure at a desired rate. However, since decreasing lift pump voltage may have little to no effect on fuel rail pressure, such closed loop control of the lift pump may result in wind-up of the integral term and transient pressure undershoots. 
     As one example, the issues described above may be addressed by a method comprising closed loop operating a lift pump of a fuel system based on a difference between a desired fuel rail pressure and an estimated fuel rail pressure, and open loop operating the lift pump to the desired fuel rail pressure in response to a fuel flow rate in a direction of a fuel rail through a check valve positioned between the lift pump and the fuel rail decreasing to a threshold. 
     During the closed loop operating the lift pump, an amount of power supplied to the lift pump may be adjusted based on outputs from a pressure sensor coupled in the fuel rail. Specifically, the closed loop operating the lift pump may comprise adjusting an amount of power supplied to the lift pump based on one or more of a proportional term, integral term, and derivative term. Updating and computing the proportional term and integral term may comprise calculating an error based on a current difference between the desired fuel rail pressure and a most recently estimated fuel rail pressure obtained from the pressure sensor. However, open loop operating the lift pump may comprise adjusting the amount of power supplied to the lift pump based only on the desired fuel rail pressure and not based on outputs from the pressure sensor. Specifically, open loop operating the lift pump may comprise freezing the integral term and clipping the proportional term to non-negative values. 
     In another example, a method for an engine may comprise adjusting an amount of power supplied to a lift pump of a fuel system based on a difference between a desired fuel rail pressure and an estimated fuel rail pressure of a fuel rail, and regulating the amount of power supplied to the lift pump based on a desired lift pump outlet pressure in response to a fuel flow rate in a direction of the fuel rail through a check valve positioned between the lift pump and the fuel rail decreasing to a threshold. 
     In yet another example, an engine system may comprise a lift pump, a fuel rail including one or more fuel injectors for injecting liquid fuel, a check valve positioned between the lift pump and the fuel rail, a pressure sensor coupled to the fuel rail, and a controller including non-transitory memory with instruction for: switching from closed loop control of the lift pump to open loop control in response to a fuel flow rate through the check valve decreasing to a threshold, and resuming closed loop control of the lift pump in response to the fuel flow rate through the check valve increasing above the threshold. 
     In this way, transient pressure drops in the fuel rail may be reduced. Specifically, by open loop operating the lift pump during DFSO, lift pump speed may be maintained at a higher level than it would be under closed loop control during DFSO. As such, lift pump spin-up time when exiting DFSO may be reduced, and pressure drops in the fuel rail may be reduced. Thus, fluctuations in fuel rail pressure may be reduced and fuel rail pressure consistency may be increased. 
     It should be understood that the summary above is provided to introduce in simplified form a selection of concepts that are further described in the detailed description. It is not meant to identify key or essential features of the claimed subject matter, the scope of which is defined uniquely by the claims that follow the detailed description. Furthermore, the claimed subject matter is not limited to implementations that solve any disadvantages noted above or in any part of this disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a schematic diagram of an example engine system including a fuel system that may comprise one or more of direct injection and port injection. 
         FIG. 2  shows a block diagram of a first example embodiment of a fuel system that may be included in the engine system of  FIG. 1 . 
         FIG. 3  shows a schematic diagram of an example control system that may be used by a controller of the fuel system of  FIG. 2 . 
         FIG. 4  shows a flow chart of a first example routine for operating a fuel lift pump of the fuel system of  FIG. 2 . 
         FIG. 5  shows a first graph depicting example fuel lift pump operation under varying engine operating conditions. 
         FIG. 6  shows a block diagram of a second example embodiment of a fuel system that may be included in the engine system of  FIG. 1 . 
         FIG. 7  shows a flow chart of a second example routine for operating a fuel lift pump of the fuel system of  FIG. 6 . 
     
    
    
     DETAILED DESCRIPTION 
     The following description relates to systems and methods for operating a lift pump. The lift pump may be included in a fuel system of an engine system, such as the engine system shown in  FIG. 1 . As shown in the example fuel system of  FIG. 2 , the lift pump pumps fuel from a fuel tank where the fuel is stored, to a fuel rail where the fuel is injected by fuel injectors. In some examples, the fuel system may be a direct injection (DI) system and fuel may be injected directly into one or more engine cylinders from a direct injection fuel rail. In such examples, a direct injection pump may be positioned between the lift pump and the direct injection fuel rail to further pressurize the fuel prior to injection into the one or more engine cylinders. However, in other examples, the fuel system may be a port fuel injection (PFI) system, and fuel may be injected into an intake port, upstream of the engine cylinders, by a port injection fuel rail. In such examples, fuel may be supplied directly to the port injection fuel rail by the lift pump. In still further examples, the fuel system may include both port fuel injection and direct injection, and as such may be referred to as port fuel direct injection (PFDI). Operation of the lift pump may be feedback controlled by an engine controller based on a fuel pressure at the fuel rail provided by a fuel rail pressure sensor, as is shown in the example fuel control system of  FIG. 3 . Thus, power supplied to the lift pump may be adjusted to maintain a desired fuel rail pressure. 
     The volume of fuel in the fuel rail, and thus the fuel rail pressure, may be determined by an amount of fuel entering the fuel rail, an amount of fuel leaving the fuel rail via one or more fuel injectors, and a temperature of the fuel. Thus, the fuel rail pressure may increase with increasing lift pump speeds, and therefore increased fuel flow rates into the fuel rail. Further, the fuel rail pressure may increase with decreasing fuel injection rates, and increasing fuel temperatures of fuel included in the fuel rail. In some examples, fuel temperature may increase at a higher rates when injection flow rates are lower or near zero. When the fuel injection rate is high, and the fuel rail pressure is greater than desired, a reduction in applied lift pump power may result in the desired fuel rail pressure drop. 
     However, when fuel injection is minimal and/or off, such as during deceleration fuel shut-off (DFSO), reducing power to the lift pump may be ineffective in decreasing fuel rail pressure. That is, in order for the fuel rail pressure to decrease, the rate at which fuel exits the rail via the injectors may need to exceed the rate at which fuel enters the fuel rail from the lift pump. When the injectors are off however, the rate at which fuel exits the fuel rail via the injectors may be approximately zero. Thus, in order for the fuel rail pressure to decrease, fuel flow in the fuel system must reverse direction and flow from the fuel rail to the fuel pump. However, since the fuel system may include a check valve that prevents the flow of fuel from the fuel rail to the fuel pump, no amount of power reduction to the fuel pump may bring about a reduction in fuel rail pressure when the fuel injectors are off. When exiting DFSO, and an increase in fuel rail pressure is desired, there may be a delay to deliver the desired increase in fuel rail pressure. For example, it may take time for the lift pump to spin up to a speed sufficient to deliver the desired pressure. Integrator wind-up of the engine controller may further exacerbate the delay. 
     Thus, closed loop feedback control of the lift pump during DFSO may lead to pressure drops at the fuel rail under certain engine operating conditions, such as when exiting DFSO. As such, the lift pump may not be feedback controlled and may instead be open loop controlled under certain engine operating conditions, such as when the rate at which fuel exits the fuel rail decreases below a threshold, as shown in the example routine of  FIG. 4 .  FIG. 5  shows example closed loop and open loop lift pump operation under varying engine operating conditions. By open loop operating the lift pump when fuel injection is minimal and/or off, such as during deceleration fuel shut-off (DFSO), the lift pump speed may be maintained at a higher level than it would otherwise be adjusted to during closed loop feedback control. In this way, lift pump spin-up time may be reduced, and pressure drops in the fuel rail when exiting DFSO may be reduced. Thus, fluctuations in fuel rail pressure may be reduced and fuel rail pressure consistency may be increased. 
     In other examples, where the fuel system includes a second pressure sensor near an outlet of the lift pump, such as in the example fuel system shown in  FIG. 6 , the lift pump may be feedback controlled based on outputs from the second pressure instead of being open loop controlled. Thus, the lift pump may be closed loop feedback controlled based on outputs from the fuel rail pressure sensor when fuel injection is on, since the fuel rail pressure sensor may provide a more accurate estimate of the actual fuel rail pressure than the second pressure sensor. Then, under certain engine operating conditions, such as when the fuel flow rate from the lift pump to the fuel rail decreases below a threshold, the lift pump may switch to being feedback controlled based on outputs from the second pressure sensor as shown in the example routine of  FIG. 7 . 
     Thus, in examples where the second pressure sensor is included in the fuel system, the lift pump may be continuously feedback controlled, and may not engage and/or enter into open loop control. In such examples, the operation of the lift pump may be adjusted based on outputs from the second pressure sensor. A pressure drop between the first and second pressure sensors may be learned based on outputs from the first and second pressure sensors, and this learned pressure drop may be used to correct lift pump operation. 
     Regarding terminology used throughout this detailed description, the higher pressure pump, or direct injection fuel pump, may be abbreviated as a HP pump (alternatively, HPP) or a DI fuel pump respectively. As such, DI fuel pump may also be termed DI pump. Accordingly, HPP and DI fuel pump may be used interchangeably to refer to the higher pressure direct injection fuel pump. Similarly, the lift pump may also be referred to as a lower pressure pump. Further, the lower pressure pump may be abbreviated as LP pump or LPP. Port fuel injection may be abbreviated as PFI while direct injection may be abbreviated as DI. Additionally, fuel systems including both port fuel injection and direct injection may be referred to herein as port fuel direct injection and may be abbreviated as PFDI. Also, fuel rail pressure, or the value of pressure of fuel within a fuel rail may be abbreviated as FRP. A direct injection fuel rail may also be referred to as a higher pressure fuel rail, which may be abbreviated as HP fuel rail. Further, a port fuel injection rail may also be referred as a lower pressure fuel rail, which may be abbreviated as LP fuel rail. 
     It will be appreciated that in the example port fuel direct injection (PFDI) systems shown in the present disclosure, the direct injectors or the port injectors may be deleted without departing from the scope of this disclosure. 
       FIG. 1  depicts an example of a combustion chamber or cylinder of internal combustion engine  10 . Engine  10  may be controlled at least partially by a control system including controller  12  and by input from a vehicle operator  130  via an input device  132 . In this example, input device  132  includes an accelerator pedal and a pedal position sensor  134  for generating a proportional pedal position signal PP. Cylinder  14  (herein also termed combustion chamber  14 ) of engine  10  may include combustion chamber walls  136  with piston  138  positioned therein. Piston  138  may be coupled to crankshaft  140  so that reciprocating motion of the piston is translated into rotational motion of the crankshaft. Crankshaft  140  may be coupled to at least one drive wheel of the passenger vehicle via a transmission system (not shown). Further, a starter motor (not shown) may be coupled to crankshaft  140  via a flywheel (not shown) to enable a starting operation of engine  10 . 
     Cylinder  14  can receive intake air via a series of intake air passages  142 ,  144 , and  146 . Intake air passages  142 ,  144 , and  146  can communicate with other cylinders of engine  10  in addition to cylinder  14 . In some examples, one or more of the intake air passages may include a boosting device such as a turbocharger or a supercharger. For example,  FIG. 1  shows engine  10  configured with a turbocharger including a compressor  174  arranged between intake air passages  142  and  144 , and an exhaust turbine  176  arranged along exhaust passage  158 . Compressor  174  may be at least partially powered by exhaust turbine  176  via a shaft  180  where the boosting device is configured as a turbocharger. However, in other examples, such as where engine  10  is provided with a supercharger, exhaust turbine  176  may be optionally omitted, where compressor  174  may be powered by mechanical input from a motor or the engine. 
     A throttle  162  including a throttle plate  164  may be arranged between intake air passages  144  and  146  of the engine for varying the flow rate and/or pressure of intake air provided to the engine cylinders. As shown in  FIG. 1 , throttle  162  may be positioned downstream of compressor  174 , or alternatively may be provided upstream of compressor  174 . 
     Exhaust manifold  148  can receive exhaust gases from other cylinders of engine  10  in addition to cylinder  14 . Exhaust gas sensor  128  is shown coupled to exhaust passage  158  upstream of emission control device  178 . Sensor  128  may be selected from among various suitable sensors for providing an indication of exhaust gas air/fuel ratio such as a linear oxygen sensor or UEGO (universal or wide-range exhaust gas oxygen), a two-state oxygen sensor or EGO (as depicted), a HEGO (heated EGO), a NOx, HC, or CO sensor, for example. Emission control device  178  may be a three way catalyst (TWC), NOx trap, various other emission control devices, or combinations thereof. 
     Each cylinder of engine  10  may include one or more intake valves and one or more exhaust valves. For example, cylinder  14  is shown including at least one intake poppet valve  150  and at least one exhaust poppet valve  156  located at an upper region of cylinder  14 . In some examples, each cylinder of engine  10 , including cylinder  14 , may include at least two intake poppet valves and at least two exhaust poppet valves located at an upper region of the cylinder. 
     Intake valve  150  may be controlled by controller  12  via actuator  152 . Similarly, exhaust valve  156  may be controlled by controller  12  via actuator  154 . During some conditions, controller  12  may vary the signals provided to actuators  152  and  154  to control the opening and closing of the respective intake and exhaust valves. The position of intake valve  150  and exhaust valve  156  may be determined by respective valve position sensors (not shown). The valve actuators may be of the electric valve actuation type or cam actuation type, or a combination thereof. The intake and exhaust valve timing may be controlled concurrently or any of a possibility of variable intake cam timing, variable exhaust cam timing, dual independent variable cam timing or fixed cam timing may be used. Each cam actuation system may include one or more cams and may utilize one or more of cam profile switching (CPS), variable cam timing (VCT), variable valve timing (VVT) and/or variable valve lift (VVL) systems that may be operated by controller  12  to vary valve operation. For example, cylinder  14  may alternatively include an intake valve controlled via electric valve actuation and an exhaust valve controlled via cam actuation including CPS and/or VCT. In other examples, the intake and exhaust valves may be controlled by a common valve actuator or actuation system, or a variable valve timing actuator or actuation system. 
     Cylinder  14  can have a compression ratio, which is the ratio of volumes when piston  138  is at bottom dead center position or top dead center position. In one example, the compression ratio is in the range of 9:1 to 10:1. However, in some examples where different fuels are used, the compression ratio may be increased. This may happen, for example, when higher octane fuels or fuels with higher latent enthalpy of vaporization are used. The compression ratio may also be increased if direct injection is used due to its effect on engine knock. 
     In some examples, each cylinder of engine  10  may include a spark plug  192  for initiating combustion. Ignition system  190  can provide an ignition spark to combustion chamber  14  via spark plug  192  in response to spark advance signal SA from controller  12 , under select operating modes. However, in some embodiments, spark plug  192  may be omitted, such as where engine  10  may initiate combustion by auto-ignition or by injection of fuel as may be the case with some diesel engines. 
     In some examples, each cylinder of engine  10  may be configured with one or more fuel injectors for providing fuel thereto. As a non-limiting example, cylinder  14  is shown including first fuel injector  166 . Fuel injector  166  is shown coupled directly to cylinder  14  for injecting fuel directly therein in proportion to the pulse width of signal FPW- 1  received from controller  12  via electronic driver  168 . In this manner, fuel injector  166  provides what is known as direct injection (hereafter referred to as “DI”) of fuel into cylinder  14 . Thus, first fuel injector  166 , may also be referred to herein as DI fuel injector  166 . While  FIG. 1  shows injector  166  positioned to one side of cylinder  14 , it may alternatively be located overhead of the piston, such as near the position of spark plug  192 . Such a position may improve mixing and combustion when operating the engine with an alcohol-based fuel due to the lower volatility of some alcohol-based fuels. Alternatively, the injector may be located overhead and near the intake valve to improve mixing. Fuel may be delivered to fuel injector  166  from a fuel tank of fuel system  8  via a higher pressure fuel pump  73 , and a fuel rail. Further, the fuel tank may have a pressure transducer providing a signal to controller  12 . 
     Additionally or alternatively, engine  10  may include second fuel injector  170 . Fuel injector  166  and  170  may be configured to deliver fuel received from fuel system  8 . Specifically, fuel may be delivered to fuel injector  170  from a fuel tank of fuel system  8  via a lower pressure fuel pump  75 , and a fuel rail. As elaborated later in the detailed description, fuel system  8  may include one or more fuel tanks, fuel pumps, and fuel rails. 
     Fuel system  8  may include one fuel tank or multiple fuel tanks. In embodiments where fuel system  8  includes multiple fuel tanks, the fuel tanks may hold fuel with the same fuel qualities or may hold fuel with different fuel qualities, such as different fuel compositions. These differences may include different alcohol content, different octane, different heat of vaporizations, different fuel blends, and/or combinations thereof etc. In one example, fuels with different alcohol contents could include gasoline, ethanol, methanol, or alcohol blends such as E85 (which is approximately 85% ethanol and 15% gasoline) or M85 (which is approximately 85% methanol and 15% gasoline). Other alcohol containing fuels could be a mixture of alcohol and water, a mixture of alcohol, water and gasoline etc. In some examples, fuel system  8  may include a fuel tank holding a liquid fuel, such as gasoline, and also include a fuel tank holding a gaseous fuel, such as CNG. 
     Fuel injectors  166  and  170  may be configured to inject fuel from the same fuel tank, from different fuel tanks, from a plurality of the same fuel tanks, or from an overlapping set of fuel tanks. Fuel system  8  may include the lower pressure fuel pump  75  (such as a lift pump) and a higher pressure fuel pump  73 . The lower pressure fuel pump  75  may be a lift pump that pumps fuel out of the one or more fuel tanks towards the one or more injectors  166  and  170 . As detailed below with reference to the fuel system of  FIG. 2 , fuel provided to the first fuel injector  166  may be further pressurized by higher pressure fuel pump  73 . Thus, the lower pressure fuel pump  75  may provide fuel directly to one or more of a port injection fuel rail and the higher pressure fuel pump  73 , while higher pressure fuel pump  73  may deliver fuel to a direct injection fuel rail. 
     Fuel injector  170  is shown arranged in intake air passage  146 , rather than in cylinder  14 , in a configuration that provides what is known as port injection of fuel into the intake port upstream of cylinder  14 . Second fuel injector  170  may inject fuel, received from fuel system  8 , in proportion to the pulse width of signal FPW- 2  received from controller  12  via electronic driver  171 . Note that a single electronic driver  168  or  171  may be used for both fuel injection systems, or multiple drivers, for example electronic driver  168  for fuel injector  166  and electronic driver  171  for optional fuel injector  170 , may be used, as depicted. 
     In an alternate example, each of fuel injectors  166  and  170  may be configured as direct fuel injectors for injecting fuel directly into cylinder  14 . In another example, each of fuel injectors  166  and  170  may be configured as port fuel injectors for injecting fuel upstream of intake valve  150 . In yet other examples, cylinder  14  may include only a single fuel injector that is configured to receive different fuels from the fuel systems in varying relative amounts as a fuel mixture, and is further configured to inject this fuel mixture either directly into the cylinder as a direct fuel injector or upstream of the intake valves as a port fuel injector. In still another example, cylinder  14  may be fueled solely by optional fuel injector  170 , or solely by port injection (also termed, intake manifold injection). As such, it should be appreciated that the fuel systems described herein should not be limited by the particular fuel injector configurations described herein by way of example. 
     Fuel may be delivered by both injectors to the cylinder during a single cycle of the cylinder. For example, each injector may deliver a portion of a total fuel injection that is combusted in cylinder  14 . Further, the distribution and/or relative amount of fuel delivered from each injector may vary with operating conditions, such as engine load, knock, and exhaust temperature, such as described herein below. The port injected fuel may be delivered during an open intake valve event, closed intake valve event (e.g., substantially before the intake stroke), as well as during both open and closed intake valve operation. Similarly, directly injected fuel may be delivered during an intake stroke, as well as partly during a previous exhaust stroke, during the intake stroke, and partly during the compression stroke, for example. As such, even for a single combustion event, injected fuel may be injected at different timings from the port and direct injector. Furthermore, for a single combustion event, multiple injections of the delivered fuel may be performed per cycle. The multiple injections may be performed during the compression stroke, intake stroke, or any appropriate combination thereof. 
     As described above,  FIG. 1  shows only one cylinder of a multi-cylinder engine. As such, each cylinder may similarly include its own set of intake/exhaust valves, fuel injector(s), spark plug, etc. It will be appreciated that engine  10  may include any suitable number of cylinders, including 2, 3, 4, 5, 6, 8, 10, 12, or more cylinders. Further, each of these cylinders can include some or all of the various components described and depicted by  FIG. 1  with reference to cylinder  14 . 
     Fuel injectors  166  and  170  may have different characteristics. These include differences in size, for example, one injector may have a larger injection hole than the other. Other differences include, but are not limited to, different spray angles, different operating temperatures, different targeting, different injection timing, different spray characteristics, different locations etc. Moreover, depending on the distribution ratio of injected fuel among fuel injectors  170  and  166 , different effects may be achieved. 
     Controller  12  is shown in  FIG. 1  as a microcomputer, including microprocessor unit  106 , input/output ports  108 , an electronic storage medium for executable programs and calibration values shown as non-transitory read only memory chip  110  in this particular example for storing executable instructions, random access memory  112 , keep alive memory  114 , and a data bus. Controller  12  may receive various signals from sensors coupled to engine  10 , in addition to those signals previously discussed, including measurement of inducted mass air flow (MAF) from mass air flow sensor  122 ; engine coolant temperature (ECT) from temperature sensor  116  coupled to cooling sleeve  118 ; a profile ignition pickup signal (PIP) from Hall effect sensor  120  (or other type) coupled to crankshaft  140 ; throttle position (TP) from a throttle position sensor; and absolute manifold pressure signal (MAP) from sensor  124 . Engine speed signal, RPM, may be generated by controller  12  from signal PIP. Manifold pressure signal MAP from a manifold pressure sensor  124  may be used to provide an indication of vacuum, or pressure, in the intake manifold. 
     The controller  12  receives signals from the various sensors of  FIG. 1  and employs the various actuators of  FIG. 1  (e.g., throttle  162 , fuel injector  166 , fuel injector  170 , higher pressure fuel pump  73 , lower pressure fuel pump  75  etc.) to adjust engine operation based on the received signals and instructions stored on a memory of the controller. Specifically, the controller  12  may adjusting operation of the lower pressure fuel pump  75  based on a desired fuel injection amount and/or a pressure of a fuel rail as described in greater detail below with reference to  FIG. 2 . 
       FIG. 2  schematically depicts an example embodiment of a fuel system  200 , which may be the same or similar to fuel system  8  of  FIG. 1 . Thus, fuel system  200  may be operated to deliver fuel to an engine, such as engine  10  of  FIG. 1 . Fuel system  200  may be operated by a controller  222 , which may be the same or similar to controller  12  described above with reference to  FIG. 1 , to perform some or all of the operations described below with reference to the flow charts of  FIGS. 4 and 7 . 
     Fuel system  200  includes a fuel tank  210 , a lift pump  212 , a check valve  213 , one or more fuel rails, a low pressure passage  218  providing fluidic communication between the pump  212  and the one or more fuel rails, fuel injectors, one or more fuel rail pressure sensors, and engine block  202 . Lift pump  212  may also be referred to herein as lower pressure pump (LPP)  212 . 
     As depicted in the example of  FIG. 2 , the fuel system  200  may be configured as a port fuel direction injection (PFDI) system that includes both a direct injection (DI) fuel rail  250 , and a port fuel injection (PFI) fuel rail  260 . Lift pump  212  may be operated by the controller  222  to pump fuel from the fuel tank  210  towards one or more of the DI fuel rail  250  and PFI fuel rail  260  via the low pressure passage  218 . Check valve  213  may be positioned in the low pressure passage  218 , more proximate the fuel pump  212  than the fuel rails  250  and  260 , to facilitate fuel delivery and maintain fuel line pressure in passage  218 . Specifically, in some examples, check valve  213  may be included in the fuel tank  210 . The check valve  213  may be included proximate an outlet  251  of the lift pump  212 . As such, flow in the low pressure passage  218  may be unidirectional from the lift pump  212  towards the fuel rails  250  and  260 . Said another way, the check valve  213  may prevent bidirectional fuel flow in passage  218  since fuel does not flow backwards through the check valve  213  towards the lift pump  212  and away from the fuel rails  250  and  260 . Thus, fuel may only flow away from the lift pump  212  towards one or more of the fuel rails  250  and  260  in the fuel system  200 . In the description of fuel system  200  herein, upstream flow therefore refers to fuel flow traveling from fuel rails  250 ,  260  towards LPP  212  while downstream flow refers to the nominal fuel flow direction from the LPP towards the HPP  214  and thereon to the fuel rails. 
     After being pumped out of the fuel tank  210  by the lift pump  212 , fuel may flow along passage  218  to either the DI fuel rail  250 , or the PFI fuel rail  260 . Thus, passage  218  may branch into DI supply line  278  and port injection supply line  288 , where DI supply line  278  provides fluidic communication with the DI fuel rail  250  and port injection supply line  288  provides fluidic communication with the PFI fuel rail  260 . Before reaching the DI fuel rail  250  via the low pressure passage  218 , fuel may be further pressurized by a DI pump  214 . DI pump  214  may also be referred to in the description herein as higher pressure pump (HPP)  214 . Pump  214  may increase the pressure of the fuel prior to direct injection into one or more engine cylinders  264  by direct injectors  252 . Thus, fuel pressurized by DI pump  214 , may flow through DI supply line  278  to the DI fuel rail  250 , where it may await direct injection to the engine cylinders  264  via the direct injectors  252 . Direct injectors  252  may be the same or similar to fuel injector  166  described above with reference to  FIG. 1 . Further, direct injectors  252  may also be referred to in the description herein as direct injectors  252 . DI fuel rail  250  may include a first fuel rail pressure sensor  248  for providing an indication of the fuel pressure in the fuel rail  250 . Thus, controller  222  may estimate and/or determine the fuel rail pressure (FRP) of the DI fuel rail  250  based on outputs received from the first fuel rail pressure sensor  248 . 
     In some examples, fuel flowing to the PFI fuel rail  260  may not be further pressurized after being pumped out of the fuel tank  210  by the lift pump  212 . However, in other examples, fuel flowing to the PFI fuel rail  260  may be further pressurized by DI pump  214  before reaching the PFI fuel rail  260 . Thus, fuel may flow from the lift pump  212  to the PFI fuel rail  260 , prior to injection into an intake port, upstream of the engine cylinders  264  via port injectors  262 . Specifically, fuel may flow through the low pressure passage  218 , and then on to port injection supply line  288  before reaching the PFI fuel rail  260 . Port injectors  262  may be the same or similar to injector  170  described above with reference to  FIG. 1 . Further, port injectors  262  may also be referred to in the description herein as port injectors  262 . PFI fuel rail  260  may include a second fuel rail pressure sensor  258  for providing an indication of the fuel pressure in the fuel rail  260 . Thus, controller  222  may estimate and/or determine the FRP of the PFI fuel rail  260  based on outputs received from the second fuel rail pressure sensor  258 . 
     Although depicted as a PFDI system in  FIG. 2 , it should be appreciated that fuel system  200  may also be configured as a DI system, or as a PFI system. When configured as a DI system, fuel system  200  may not include PFI fuel rail  260 , port injectors  262 , pressure sensor  258 , and port injection supply line  288 . Thus, in examples where the fuel system  200  is configured as a DI fuel system, substantially all fuel pumped from the fuel tank  210  by the lift pump  212  may flow to the DI pump  214 , en route to the DI fuel rail  250 . As such, the DI fuel rail  250  may receive approximately all of the fuel pumped from the fuel tank  210  by the lift pump  212 . 
     Further, it should also be appreciated that in examples where the fuel system  200  is configured as a PFI system, DI pump  214 , DI supply line  278 , DI fuel rail  250 , pressure sensor  248 , and direct injectors  252  may not be included in the fuel system  200 . Thus, in examples where the fuel system  200  is configures as a PFI system, substantially all fuel pumped from the fuel tank  210  by the lift pump  212  may flow to the PFI fuel rail  260 . As such the PFI fuel rail  260  may receive approximately all of the fuel pumped from the fuel tank  210  by the lift pump  212 . 
     Continuing with the description of the fuel system  200 , fuel tank  210  stores the fuel on-board the vehicle. Fuel may be provided to fuel tank  210  via fuel filling passage  204 . LPP  212  may be disposed at least partially within the fuel tank  210 , and may be an electrically-powered fuel pump. LPP  212  may be operated by controller  222  (e.g., controller  12  of  FIG. 1 ) to provide fuel to HPP  214  via low pressure passage  218 . As one example, LPP  212  may be a turbine (e.g., centrifugal) pump including an electric (e.g., DC) pump motor, whereby the pressure increase across the pump and/or the volumetric flow rate through the pump may be controlled by varying the electrical power provided to the pump motor, thereby increasing or decreasing the motor speed. For example, as the controller  222  may send signals to the lift pump  212 , and/or to a power supply of the lift pump  212 , to reduce the electrical power that is provided to lift pump  212 . By reducing the electrical power provided to the lift pump  212 , the volumetric flow rate and/or pressure increase across the lift pump may be reduced. Conversely, the volumetric flow rate and/or pressure increase across the lift pump may be increased by increasing electrical power provided to the lift pump  212 . 
     As one example, the electrical power supplied to the lower pressure pump motor can be obtained from an alternator or other energy storage device on-board the vehicle (not shown), whereby the control system can control the electrical load that is used to power the lower pressure pump. Thus, by varying the voltage and/or current provided to the lower pressure fuel pump, the flow rate and pressure of the fuel provided at the inlet of the higher pressure fuel pump  214  is adjusted. 
     A filter  217  may be disposed downstream of the lift pump  212 , and may remove small impurities contained in the fuel that could potentially damage fuel handling components. In some examples, the filter  217  may be positioned downstream of the check valve  213 . However, in other examples, filter  217  may be positioned upstream of the check valve  213 , between the fuel pump  212  and the check valve  213 . Furthermore, a pressure relief valve  219  may be employed to limit the fuel pressure in low pressure passage  218  (e.g., the output from lift pump  212 ). Relief valve  219  may include a ball and spring mechanism that seats and seals at a specified pressure differential, for example. The pressure differential set-point at which relief valve  219  may be configured to open may assume various suitable values; as a non-limiting example the set-point may be set to anywhere between 6.4 bar and 5 bar (g). An orifice  223  may be utilized to allow for air and/or fuel vapor to bleed out of the lift pump  212 . This bleed at orifice  223  may also be used to power a jet pump used to transfer fuel from one location to another within the tank  210 . In one example, an orifice check valve (not shown) may be placed in series with orifice  223 . In some embodiments, fuel system  200  may include one or more (e.g., a series) of check valves fluidly coupled to low-pressure fuel pump  212  to impede fuel from leaking back upstream of the valves. 
     Fuel lifted by LPP  212  may be supplied at a lower pressure into low pressure passage  218 . From low pressure passage  218 , fuel may flow to an inlet  203  of HPP  214 . More specifically, in the example depicted in  FIG. 2 , supply line  288  may be coupled on a first end to downstream of check valve  234 , proximate or at an outlet  203  of the DI pump  214 , and on a second end to the PFI fuel rail  260  to provide fluidic communication there-between. As such, substantially all fuel pumped out of the tank  210  by the lift pump  212  may be further pressurized by HPP  214  before reaching either of the fuel rails  250  and  260 . In such examples, HPP  214  may be operated to raise the pressure of fuel delivered to each of the fuel rails  250  and  260  above the lift pump pressure, where the DI fuel rail  250  coupled to the direct injectors  252  may operate with a variable high pressure while the PFI fuel rail  260  coupled to the port injectors  262 , may operate with a fixed high pressure. Thus, high-pressure fuel pump  214  may be in communication with each of fuel rail  260  and fuel rail  250 . As a result, high pressure port and direct injection may be enabled. 
     In such examples, supply line  288  may include valves  244  and  242 . Valves  244  and  242  may work in conjunction to keep the PFI fuel rail  260  pressurized to a threshold pressure (e.g., 15 bar) during the compression stroke of piston  228  of DI pump  214 . Pressure relief valve  242  may limit the pressure that can build in fuel rail  260  due to thermal expansion of fuel. In some examples, the pressure relief valve  242  may open and allow fuel to flow upstream from the fuel rail  260  towards the passage  218 , when the pressure between the valve  242  and the PFI fuel rail  260  increases above a threshold (e.g., 15 bar). 
     Alternatively, fuel may flow directly from low pressure passage  218  to PFI fuel rail  260  without passing through and/or being pressurized by DI pump  214 . In such examples, supply line  288  may be coupled directly to low pressure passage  218 , upstream of check valve  234 . That is, the supply line  288  may be coupled on one end to upstream of the check valve  234  and downstream of the check valve  213 , and on the opposite end to the PFI fuel rail  260 , for providing fluidic communication there-between. Thus, no additional pumping and/or pressurization of the fuel may occur between lift pump  212  and the PFI fuel rail  260 . Thus, in some examples, DI pump  214  may only be in communication with DI fuel rail  250  and may only pressurize fuel supplied to the DI pump  214 . Thus, although the PFI fuel rail  260  is depicted in  FIG. 2 , to be coupled to downstream of check valve  234  via supply line  288 , the supply line  288  may alternatively be coupled to upstream of the check valve  234 . 
     As such, PFI fuel rail  260  may be supplied fuel at a lower pressure than the DI fuel rail  250 . Specifically, PFI fuel rail  260  may be supplied with fuel at a pressure approximately the same as the fuel pressure at an outlet of the lift pump  212 . 
     The pressure of each of the fuel rails  250  and  260 , may depend on the mass fuel flow rate into the rails  250  and  260  via supply lines  218  and  288 , respectively, and the mass fuel flow rates out of the rails  250  and  260  via the injectors  248  and  258 , respectively. For example, the fuel rail pressures may increase when the mass flow rate into the fuel rail is greater than the mass flow rate out of the fuel rail. Similarly, the pressure may decrease when the mass flow rate out of the fuel rail is greater than the mass flow rate in to the fuel rail. Thus, when the injectors are off, and fuel is not exiting the fuel rail, the fuel rail pressure may increase while the lift pump  212  is on and spinning, so long as the pressure at the outlet of the fuel pump is greater than the pressure in the fuel rail, and the fuel pump  212  is therefore pushing fuel into the fuel rail. 
     While each of the DI fuel rail  250  and PFI fuel rail  260  are shown dispensing fuel to four fuel injectors of the respective injectors  252 ,  262 , it will be appreciated that each fuel rail  250  and  260  may dispense fuel to any suitable number of fuel injectors. As one example, DI fuel rail  250  may dispense fuel to one fuel injector of first injectors  252  for each cylinder of the engine while PFI fuel rail  260  may dispense fuel to one fuel injector of second injectors  262  for each cylinder of the engine. Controller  222  can individually actuate each of the port injectors  262  via a port injection driver  237  and actuate each of the direct injectors  252  via a direct injection driver  238 . The controller  222 , drivers  237  and  238 , and other suitable engine system controllers can comprise a control system. While the drivers  237 ,  238  are shown external to the controller  222 , it should be appreciated that in other examples, the controller  222  can include the drivers  237 ,  238  or can be configured to provide the functionality of the drivers  237 ,  238 . Controller  222  may include additional components not shown, such as those included in controller  12  of  FIG. 1 . 
     Controller  222  may be a proportional integral (PI) or proportional integral derivative (PID) controller. As described above, controller  22  may receive an indication of fuel rail pressure via one or more of the first and second fuel rail pressure sensors  248  and  258 . More specifically, the controller  222  may estimate the fuel rail pressure in one or more of the DI fuel rail  250  based on outputs from the first fuel rail pressure sensor  248  and in the PFI fuel rail  260  based on outputs from the second fuel rail pressure sensor  258 . Based on a difference between a desired fuel rail pressure, and the actual measured fuel rail pressure provided by the one or more of the pressure sensors  248  and  258 , the controller  222 , may calculate an error. Thus, the error may represent the current difference between the desired fuel rail pressure and the fuel rail pressure estimated based on outputs from the one or more pressure sensors  248  and  258 . The error may be multiplied by a proportional gain factor (K p ) to obtain a proportional term. Further, the sum of the error over a duration may be multiplied by an integral gain factor (K i ) to obtain an integral term. In examples, where the controller  222  is configured as a PID controller, the controller may further calculate a derivative term based on the rate of change of the error and a derivative gain factor (K d ). 
     One or more of the proportional term, integral term, and derivative term may then be incorporated into an output signal (e.g., voltage) sent from the controller  222  to pump  212  and/or a power source providing power to the pump  212 , to adjust an amount of power supplied to the pump  212 . Specifically, a voltage and/or current supplied to the pump  212  may be adjusted by the controller  222  to match the fuel rail pressure to the desired fuel rail pressure based on one or more of the proportional, integral, and derivative terms. A driver (not shown) electronically coupled to controller  222  may be used to send a control signal to the lift pump  212 , as required, to adjust the output (e.g., speed) of the lift pump  212 . Thus, based on a difference between the estimated fuel rail pressure obtained from one or more of the pressure sensors  248  and  258  and the desired fuel rail pressure, the controller  222  may adjust an amount of electrical power supplied to the pump  212 , to match the actual fuel rail pressure more closely to the desired fuel rail pressure. Generally, the controller  222  may therefore increase power supply to the pump  212  when the fuel rail pressure is less than desired, and may decrease power supply to the pump  212  when the fuel rail pressure is greater than desired. This control scheme, where the controller  222  adjusts its output based on input received from one or more of the pressure sensors  248  and  258  may be referred to herein as closed loop, or feedback control. However, in some examples, as described below with reference to  FIG. 4 , the controller  222  may operate open loop under certain engine operating conditions. 
     During open loop control, the controller  222  may not adjust its output and/or the electrical power supplied to the pump  212  based on signals received from one or more of the pressure sensors  248  and  258 . Thus, during open loop control, the controller  222  may adjust operation of pump  212  based on the desired fuel rail pressure only. Specifically, the controller  222  may stop updating or freeze the integral term during open loop control. Thus, the controller  222  may not calculate an integral term during open loop control. Additionally or alternatively, the controller  222  may prevent the proportional term from decreasing below a threshold. In some examples, the threshold may be zero. However, in other examples, the threshold may be greater or less than zero. Said another way, the controller  222  may clip the proportional term to only positive values. As such, the proportional term may be set to the threshold (e.g., zero) whenever the proportional term drops below the threshold. In still further examples, the controller  222  may additionally stop updating and/or freeze the proportional term during open loop control. Thus, the controller  222  may in some examples, not calculate a proportional term during open loop control. 
     HPP  214  may be an engine-driven, positive-displacement pump. As one non-limiting example, HPP  214  may be a BOSCH HDP5 HIGH PRESSURE PUMP. The HPP  214  may utilize a solenoid activated control valve (e.g., fuel volume regulator, magnetic solenoid valve, etc.)  236  to vary the effective pump volume of each pump stroke. The outlet check valve of HPP is mechanically controlled and not electronically controlled by an external controller. HPP  214  may be mechanically driven by the engine in contrast to the motor driven LPP  212 . HPP  214  includes a pump piston  228 , a pump compression chamber  205  (herein also referred to as compression chamber), and a step-room  227 . Pump piston  228  receives a mechanical input from the engine crank shaft or cam shaft via cam  230 , thereby operating the HPP according to the principle of a cam-driven single-cylinder pump. A sensor (not shown in  FIG. 2 ) may be positioned near cam  230  to enable determination of the angular position of the cam (e.g., between 0 and 360 degrees), which may be relayed to controller  222 . 
     Continuing with the description of fuel system  200 , it may optionally further include accumulator  215 . When included, accumulator  215  may be positioned downstream of lower pressure fuel pump  212  and upstream of higher pressure fuel pump  214 , and may be configured to hold a volume of fuel that reduces the rate of fuel pressure increase or decrease between fuel pumps  212  and  214 . For example, accumulator  215  may be coupled in low pressure passage  218 , as shown, or in a bypass passage  211  coupling low pressure passage  218  to the step-room  227  of HPP  214 . The volume of accumulator  215  may be sized such that the engine can operate at idle conditions for a predetermined period of time between operating intervals of lower pressure fuel pump  212 . For example, accumulator  215  can be sized such that when the engine idles, it takes one or more minutes to deplete pressure in the accumulator to a level at which higher pressure fuel pump  214  is incapable of maintaining a sufficiently high fuel pressure for fuel injectors  252 ,  262 . Accumulator  215  may thus enable an intermittent operation mode (or pulsed mode) of lower pressure fuel pump  212 . By reducing the frequency of LPP operation, power consumption may be reduced. In other embodiments, accumulator  215  may inherently exist in the compliance of fuel filter  217  and low pressure passage  218 , and thus may not exist as a distinct element. Alternatively, the accumulator may be sized to be the approximate size of the pump displacement. In other words, as fluid is expelled upstream from chambers  227  or  205 , the fluid may collect in accumulator  215  while minimizing the pressure change in lines  218 ,  211 , and/or  203 . 
     An engine speed sensor  233  can be used to provide an indication of engine speed to the controller  222 . The indication of engine speed can be used to identify the speed of higher pressure fuel pump  214 , since the pump  214  may be mechanically driven by the engine  202 , for example, via the crankshaft or camshaft. 
     DI fuel rail  250  is coupled to an outlet  208  of HPP  214  along DI supply line  278 . In comparison, PFI fuel rail  260  may be coupled to the inlet  203  of HPP  214  via port injection supply line  288  in examples, where the HPP  214  is configured to pressurize fuel supplied to the PFI fuel rail  260 . In other examples, PFI fuel rail  260  may not be coupled to the inlet  203  of the HPP  214  and may instead be coupled directly to the passage  218 , upstream of check valve  234 . A check valve  274  and/or a pressure relief valve  272  may be positioned between the outlet  208  of the HPP  214  and the DI fuel rail  250 . Pressure relief valve  272  may be arranged parallel to check valve  274  in bypass passage  279  and may limit the pressure in DI supply line  278 , located downstream of HPP  214  and upstream of DI fuel rail  250 . For example, pressure relief valve  272  may limit the pressure in DI supply line  278  to an upper threshold pressure (e.g., 200 bar). As such, pressure relief valve  272  may limit the pressure that would otherwise be generated in DI supply line  278  if control valve  236  were (intentionally or unintentionally) open and while high pressure fuel pump  214  were pumping. 
     One or more check valves and pressure relief valves may also be coupled to low pressure passage  218 , downstream of LPP  212  and upstream of HPP  214 . For example, check valve  234  may be provided in low pressure passage  218  to reduce or prevent back-flow of fuel from high pressure pump  214  to low pressure pump  212  and fuel tank  210 . In addition, pressure relief valve  232  may be provided in a bypass passage, positioned parallel to check valve  234 . Pressure relief valve  232  may limit the pressure downstream of the check valve  234  to a threshold amount (e.g., 10 bar) higher than the pressure upstream of the check valve  234 . Said another way, pressure relief valve  232  may allow fuel flow upstream, around the check valve  234 , and towards LPP  212  when pressure the pressure increase across the relief valve  232  is greater than the threshold (e.g., 10 bar). 
     Controller  222  may be configured to regulate fuel flow into HPP  214  through control valve  236  by energizing or de-energizing the control valve  236  (based on the solenoid valve configuration) in synchronism with the driving cam. Accordingly, the solenoid activated control valve  236  may be operated in a first mode where the valve  236  is positioned within HPP inlet  203  to limit (e.g., inhibit) the amount of fuel traveling through the solenoid activated control valve  236 . Depending on the timing of the solenoid valve actuation, the volume transferred to the fuel rail  250  may be varied. The control valve  236  may also be operated in a second mode where the solenoid activated control valve  236  is effectively disabled and fuel can travel upstream and downstream of the valve, and in and out of HPP  214 . 
     As such, solenoid activated control valve  236  may be configured to regulate the mass (or volume) of fuel compressed into the DI pump  214 . In one example, controller  222  may adjust a closing timing of the solenoid pressure control check valve to regulate the mass of fuel compressed. For example, a late pressure control valve closing may reduce the amount of fuel mass ingested into compression chamber  205 . The solenoid activated check valve opening and closing timings may be coordinated with respect to stroke timings of the direct injection fuel pump. 
     Piston  228  may reciprocate up and down. HPP  214  is in a compression stroke when piston  228  is traveling in a direction that reduces the volume of compression chamber  205 . HPP  214  is in a suction stroke when piston  228  is traveling in a direction that increases the volume of compression chamber  205 . 
     Controller  222  may also control the operation of DI pump  214  to adjust an amount, pressure, flow rate, etc., of a fuel delivered to the DI fuel rail  250 . As one example, controller  222  can vary a pressure setting, a pump stroke amount, a pump duty cycle command, and/or fuel flow rate of the fuel pumps to deliver fuel to different locations of the fuel system. A driver (not shown) electronically coupled to controller  222  may be used to send a control signal to the low pressure pump, as required, to adjust the output (e.g., speed) of the low pressure pump. In some examples, the solenoid valve may be configured such that high pressure fuel pump  214  delivers fuel only to DI fuel rail  250 , and in such a configuration, PFI fuel rail  260  may be supplied fuel at the lower outlet pressure of lift pump  212 . 
     Controller  222  may control the operation of each of the injectors  252  and  262 . For example, controller  222  may control the distribution and/or relative amount of fuel delivered from each injector, which may vary with operating conditions, such as engine load, knock, and exhaust temperature. Specifically, controller  222  may adjust a direct injection fuel ratio by sending appropriate signals to port fuel injection driver  237  and direct injection  238 , which may in turn actuate the respective port fuel injectors  262  and direct injectors  252  with desired pulse-widths for achieving the desired injection ratios. Additionally, controller  222  may selectively enable and disable (i.e., activate or deactivate) one or more of the injectors  252  and  262  based on fuel pressure within each rail. An example control scheme of the controller  222  is shown below with reference to  FIG. 3 . 
     Turning now to  FIG. 3 , it shows an example PID control scheme  300  that may be implemented by a controller (e.g., controller  222  shown in  FIG. 2  and controller  12  shown in  FIG. 1 ) to regulate fuel rail pressure in a fuel system (e.g., fuel system  200  shown in  FIG. 2 ). Thus, the control scheme  300  shown in  FIG. 3 , may be used and/or may be incorporated into the controller  222  shown in  FIG. 2 , to regulate fuel pressure in one or more of a PFI fuel rail (e.g., PFI fuel rail  260  shown in  FIG. 2 ), and a DI fuel rail (e.g., DI fuel rail  250  shown in  FIG. 2 ). It should be appreciated that in the description herein, a signal may refer to an electrical signal such as an electric current, and that modification of a signal may refer to a change in voltage of the electric current. 
     A pressure scheduler  308  may first determine a desired fuel rail pressure, which may be a desired pressure of the PFI fuel rail and/or a desired pressure of the DI fuel rail, based on one or more of an intake manifold pressure, fuel injection rate, fuel volatility  302 , engine speed  304 , and fuel temperature  306 . Thus, as inputs, the pressure scheduler  308  may receive a first signal  302  corresponding to a fuel volatility, a second signal corresponding to engine speed  304 , and a third signal  306  corresponding to fuel temperature. However, the pressure scheduler  308  may determine the desired fuel rail pressure based on additional engine operating conditions such as a position of an engine throttle (e.g., throttle  162  shown in  FIG. 1 ), engine load, alternator torque, exhaust pressure, speed of a turbocharger (e.g., compressor  174  shown in  FIG. 1 ), intake temperature, intake pressure, etc. The pressure scheduler may determine the desired fuel rail pressure based on the received signals and send a fourth signal  310  corresponding to the desired fuel rail pressure to one or more of a subtractor  312  and a feed-forward scheduler  318 . Fuel rail pressure may be an absolute pressure, gauge pressure, or a differential pressure between rail and intake manifold pressure. 
     The feed-forward scheduler  318  may receive as an input, a fifth signal  316  corresponding to an injector flow rate. Based on the injector flow rate received via the fifth signal  316 , the feed-forward scheduler  318  may modify the desired fuel rail pressure to a corrected desired fuel rail pressure, and send a sixth signal  320 , to a summer  334 . Thus, the feed-forward scheduler  318 , may correct the desired fuel rail pressure based on the injector flow rate, and may send a fifth signal  316  to the summer  334 , where the fifth signal  316  may represent the corrected desired fuel rail pressure. 
     The subtractor  312  may receive as inputs the desired fuel rail pressure, and an estimate of the actual fuel rail pressure from a pressure sensor  340  via a sixth signal  342  sent from the pressure sensor  340  to the subtractor  312 . Thus, the subtractor  312  may determine an estimate of the actual fuel rail pressure based on outputs received from the pressure sensor  340 . Pressure sensor  340  may be the same or similar to pressure sensors  248  and  258  shown in  FIG. 2 . The subtractor  312  may compute a difference between the desired fuel rail pressure received via the fourth signal  310 , and the estimated fuel rail pressure received from the sixth signal  342 . Based on the difference, the subtractor  312  may compute an error, represented by seventh signal  322  in  FIG. 2 . In some examples, the error may be approximately the same as the difference between the desired fuel rail pressure and the estimated fuel rail pressure. Thus, the seventh signal  322 , corresponding to the error, may be generated by the subtractor  312 . The seventh signal  322  may be processed and/or modified separately by a proportional gain (K p )  328  and by both of an integrator  324  and integral gain (K i )  326 . Thus, the seventh signals  322  be modified by a proportional gain (K p )  328  to generate a proportional term sent as input to the summer  334  via eighth signal  330 . Further, the seventh signal  322  corresponding to the error may be integrated by an integrator block  324  in parallel with the modification by the proportional gain (K p ). The integrated error signal may then be modified by an integral gain (K i )  326  to generate an integral term. Thus, the seventh signal  322  may be processed separately by the integrator block  324  and proportional gain (K p ). Said another way, an eighth signal  330  representing the proportional term and a ninth signal  332  corresponding to the integral term may be used as inputs for the summer  334 . 
     In total, the summer  334  may receive the proportional term via signal (e.g., voltage)  330 , integral term via signal  332 , and feed-forward term via the fifth signal  320 . Based on the received signals, the summer  334  may output a voltage or tenth signal  336  to a lift pump  338  (e.g., lift pump  212  shown in  FIG. 2 ). The tenth signal  336  may be sent to the lift pump  338  to adjust lift pump operation. Specifically, the tenth signal may correspond to a power to be supplied to the lift pump  338 . In this way, power supplied to the pump  338  may be adjusted based on changes in the tenth signal  336 . However, it is important to note that one or more of a voltage, current, duty cycle, and/or speed or torque command supplied to the pump  338  may be adjusted based on changes in the tenth signal  336 . 
     During closed loop, or feedback control, the pressure sensor may continue to monitor the pressure in the fuel rail and send an estimate of the fuel rail pressure to the subtractor  312 . As such, the proportional and integral terms may be affected by the output from the pressure sensor  340 , since the error calculated by the subtractor  312  may fluctuate as the estimated fuel rail pressure changes. Thus, during closed loop or feedback control, the output or tenth signal  336  generated by summer  334  may be modified and/or affected by the output from the pressure sensor  340 . In this way power supplied to the lift pump  338  may be adjusted based on outputs from the pressure sensor  340 . 
     However, as described in greater detail below with reference to  FIG. 4 , the controller may periodically switch to open loop control of the lift pump  338 . During open loop control, the output  336  generated by the summer  334 , and therefore the power supplied to the lift pump  338  may not be adjusted based on outputs from the pressure sensor  340 . Specifically, in some example the integral term may be frozen and/or not updated. As such, a most recent integral term obtained during closed loop control may continue to be used as input to the summer  334 . However, in other examples, the tenth signal  336  output by the summer  334  may not be modified and/or adjusted based on the signal  332  corresponding to the integral term. More simply, the integral term may not be used as input by the summer  334 , and the output signal  336  to the lift pump  338  may be unaffected by the integral term. Thus, signal  332  may not be used to modify and/or adjust the signal  336  output by the summer  334 . In yet further examples, the summing block  334  may generate output  336  based only on input  320  received from feed-forward scheduler  318 . Additionally or alternatively, the proportional term may be clipped to zero during open loop control. Thus, during open loop control the proportional term may not drop below zero. Any values for the proportional term that are below zero may therefore be set to zero. However, in other examples, the signal  330  corresponding to the proportional term may not be used to modify and/or adjust the signal  336  output by the summer  334 . Thus, the summing block  334  may not use the signal  330  as input when generating the signal  336 . 
     Turning now to  FIG. 4 , it shows a flow chart of an example method  400  for adjusting operation of lift pump (e.g., lift pump  212  shown in  FIG. 2 ) of an engine fuel system (e.g., fuel system  200  shown in  FIG. 2 ). During engine operation an amount of power supplied to the lift pump may be adjusted to achieve a desired fuel pressure in a fuel rail (e.g., fuel rails  250  and  260  shown in  FIG. 2 ). Thus, the lift pump may be closed loop of feedback controlled by an engine controller (e.g., controller  222  shown in  FIG. 2 ) based on outputs from a pressure sensor (e.g., pressure sensors  248  and  258  shown in  FIG. 2 ) positioned in the fuel rail. However, the controller may switch to open loop control of the lift pump in response to a fuel flow through a check valve (e.g., check valve  213  shown in  FIG. 2 ) positioned between the lift pump and the fuel rail decreasing below a threshold. 
     Instructions for executing method  400  may be stored in the memory of the controller. Therefore method  400  may be executed by the controller based on the instructions stored in the memory of the controller and in conjunction with signals received from sensors of the engine system, such as the sensors described above with reference to  FIGS. 1 and 2 . The controller may send signals to the lift pump and/or to a power source supplying power to the lift pump, to adjust an amount of power supplied to the lift pump, and therefore an output of the lift pump. 
     Method  400  begins at  402  which comprises estimating and/or measuring engine operating conditions. Engine operating conditions may include a fuel rail pressure, a current lift pump speed, an engine speed, a throttle position, an engine load, an operator commanded torque, an intake mass airflow, a fuel injection amount or flow rate, etc. 
     After estimating and/or measuring engine operating conditions at  402 , method  400  may continue to  404  which comprises determining a desired fuel rail pressure based on engine operating conditions. For example, as described above with reference to  FIG. 3 , the desired fuel rail pressure may be determined based on one or more of an estimated fuel volatility, fuel temperature, and engine speed. However, the desired fuel rail pressure may additionally be determined based on the engine load, alternator torque, fuel injection flow rate, lift pump speed, etc. The desired fuel rail pressure may be determined from a look-up table stored in memory of the controller based on one or more of the fuel volatility, fuel temperature, and engine speed. 
     Method  400  may then proceed to  406  which comprises determining a current fuel flow rate through the check valve. The check valve may be positioned more proximate an outlet of the lift pump than the fuel rail, as depicted for check valve  213  above in  FIG. 2 . The current fuel flow rate through the check valve may be computed based on a current injection flow rate, a rate of pressure increase in a fuel line coupling the lift pump to the fuel rail (e.g., passage  218  shown in  FIG. 2 ), and a known or estimated fuel density. Specifically, the flow rate may be computed from the equation below: 
     
       
         
           
             
               Fuel 
               ⁢ 
               
                   
               
               ⁢ 
               Mass 
               ⁢ 
               
                   
               
               ⁢ 
               Flow 
               ⁢ 
               
                   
               
               ⁢ 
               Rate 
             
             = 
             
               
                 F 
                 ⁡ 
                 
                   ( 
                   i 
                   ) 
                 
               
               + 
               
                 
                   dP 
                   dt 
                 
                 * 
                 k 
                 * 
                 ρ 
               
             
           
         
       
       
         
           
             
               Fuel 
               ⁢ 
               
                   
               
               ⁢ 
               Volume 
               ⁢ 
               
                   
               
               ⁢ 
               Flow 
               ⁢ 
               
                   
               
               ⁢ 
               Rate 
             
             = 
             
               
                 F 
                 ⁡ 
                 
                   ( 
                   i 
                   ) 
                 
               
               + 
               
                 
                   dP 
                   dt 
                 
                 * 
                 k 
               
             
           
         
       
     
     In the above equations, F(i) may represent a volumetric injection flow rate, or a mass flow rate of fuel flowing through one or more injectors (e.g., injectors  252  and  262  shown in  FIG. 2 ) in a PFI fuel system. In a DI fuel system, F(i) may represent the fuel flow rate through a high pressure pump (e.g., HPP  214  shown in  FIG. 2 ). In a PFDI fuel system, F(i) may represent the sum of injection flow rate and HPP flow rate. Thus, F(i) may represent the mass flow rate of fuel exiting one or more fuel rails. 
     The 
             dP   dt         
term may represent the rate of change of pressure in the fuel line, k represents compliance, and ρ is the fuel density. Fuel line pressure may be obtained by an engine controller (e.g., controller  222  shown in  FIG. 2 ) sampling the fuel line pressure sensor (e.g., pressure sensors  248  and  258  shown in  FIG. 2 ). The rate of change of fuel line pressure may be obtained by differentiating fuel line pressure with respect to time. The engine controller may perform this task by computing the difference in fuel line pressure of successive samples and dividing by the time between samples. However, a more sophisticated processing such as the use of the Savitzky-Golay filter could be used for increased accuracy.
 
     Fuel line compliance may be obtained by observing the change in pressure of the fuel line after a known decrease in fuel line volume. When the lift pump is commanded off (e.g., 0V, 0 W, 0 Nm, etc.), a check valve included between the lift pump and the fuel rail (e.g., check valve  213  shown in  FIG. 2 ) prevents fuel from exiting the fuel line into the fuel tank. Thus, the change in volume of the fuel line may be due solely to F(i), the flow rate of fuel exiting the fuel line. The engine controller may integrate F(i) over a known span of time to obtain a volume. During the same span of time, the engine controller may also calculate the initial and final pressure of the fuel line using the fuel line pressure sensor. The engine controller may use this change in pressure and volume to infer the compliance of the fuel line. It is important to note that this procedure may be executed in steady-state periods of engine operation for consistent, more accurate measurements. For example, the procedure may not be executed during DFSO operation so as to avoid changes in fuel line volume due to heating. Such an effect may be negligible while fuel is being injected into a running engine. 
     Thus, the flow rate through the check valve may be affected by the pressure difference between the outlet of the lift pump and the fuel rail, and the injection flow rate of fuel exiting the fuel rail. However, in some examples, the flow rate may additionally be adjusted based on a temperature of the fuel. Specifically, the pressure in the fuel rail may change due to changes in the temperature of the fuel included in the fuel rail. The pressure in the fuel rail may increase as the temperature of the fuel increases, since the density of the fuel may decrease and therefore, the volume of the fuel may increase with increasing fuel temperatures. For example the fuel density may decrease 0.095% for each 1° C. of temperature increase. After estimating the current fuel flow rate through the check valve at  406 , method  400  may proceed to  408  which comprises determining if the fuel flow rate is less than a threshold flow rate. In some examples, the threshold flow rate may be approximately zero. However, in other examples the threshold flow rate may be greater or less than zero. If the flow rate through the check valve is greater than the threshold flow rate, then method  400  may continue from  408  to  410  which comprises continuing to feedback control the lift pump based on output from the pressure sensor positioned in the fuel rail. In other examples, the method  400  at  408  may additionally or alternatively comprise determining if the fuel injection flow rate is less than a threshold. In some examples the fuel injection flow rate threshold may be zero. However, in other examples, the fuel injection flow rate threshold may be greater than zero. Thus, in some examples, the method  400  at  408  may comprise determining if deceleration fuel shut off (DFSO) conditions exit. If it is determined that DFSO conditions do not exist and fuel is being injected by the fuel injectors, and/or the fuel injection flow rate is greater than a threshold, method  400  may continue from  408  to  410 . 
     At,  410  the controller may continue to compute an error based on the difference between the desired fuel rail pressure and the estimated fuel rail pressure obtained from outputs of the pressure sensor, as described above with reference to  FIGS. 2 and 3 . Thus, outputs from the pressure sensor may be used to estimate the current fuel rail pressure. Based on the difference between the current fuel rail pressure and the desired fuel rail pressure, the controller may adjust an amount of power supplied to the lift pump to more closely align the actual fuel rail pressure to the desired fuel rail pressure. Specifically, the controller may compute and/or update a proportional term and an integral term based on the error. In some examples, the controller may additionally compute and/or update a derivative term based on the error. The proportional and integral terms, and in some the examples the derivative term may be used to adjust a voltage output by the controller, and thus an amount of power supplied to the lift pump. Generally, the controller may signal for a reduction in lift pump power when the estimated fuel rail pressure exceeds the desired fuel rail pressure in an attempt to reduce fuel rail pressure, and may signal for an increase in lift pump power when the estimated when the desired fuel rail pressure exceeds the estimated fuel rail pressure to increase fuel rail pressure. Method  400  may then return. 
     However, if at  408  it is determined that one or more of the fuel flow rate through the check valve is less than the threshold, the injection flow rate is less than the injection flow rate threshold, and/or DFSO conditions do exist and fuel is not being injected by the fuel injectors, then method  400  may proceed from  408  to optional step  411  which comprises determining if the fuel rail pressure error is less than zero. When the fuel rail pressure error is less than zero, the current/instantaneous estimated fuel rail pressure obtained from a most recent output from the pressure sensor positioned in the fuel rail, may be greater than the desired fuel rail pressure, therefore signaling for a decrease in fuel rail pressure and/or lift pump power, voltage, current, etc. If the fuel rail pressure error is not less than zero, (e.g., measured fuel rail pressure is not greater than desired) then method  400  may continue from  411  to  410  and continue to feedback control the lift pump based on outputs from the fuel rail pressure sensor. However, if the fuel rail pressure error is less than zero at  411 , method  400  may proceed from  411  to  412 , which comprises open loop operating the fuel lift pump based on a the desired fuel rail pressure. Thus, in some examples, the controller may only switch to open loop control of the fuel lift pump when the fuel flow rate through the check valve is less than the threshold, and the current fuel rail pressure is greater than desired (e.g., fuel rail pressure error is less than zero). 
     However, in some examples, method  400  may proceed directly from  408  to  412 , and may not execute  411 . Thus, in other examples, the controller may switch to open loop operating the lift pump anytime the fuel flow rate through the check valve is less than the threshold at  408 . The method  400  at  412  may comprise not adjusting the power supplied to the lift pump based on outputs from the pressure sensor. Said another way, the power supplied to the lift pump may be adjusted based on the desired fuel pressure only, and may not be adjusted based on the estimated pressure in the fuel rail. In some examples, the method at  412  may therefore comprise maintaining the power supplied to the lift pump at an approximately constant level. Thus, lift pump speed may be kept approximately consistent. 
     More specifically, the method  400  at  412  may include the additional steps of freezing the integral term at  414 , and/or clipping the proportional term to non-negative values at  416 . Thus, open loop operating the lift pump may comprise freezing and/or not updating the integral term at  416 . The integral term therefore, may not be used to adjust lift pump operation. In some examples however, freezing the integral term may comprise not updating the integral term, but using a most recently computed value for the integral term for continued lift pump control. Additionally or alternatively, the method  400  may additionally comprise clipping the proportional term to non-negative values at  416 . Thus the method at  416 , may comprise preventing the proportional term from decreasing below a threshold (e.g., 0). In some examples, the method at  416  may comprise not updating and/or freezing the proportional term. Thus, the proportional term may not be calculated and or updated during open loop operation of the lift pump and may not be used to adjust lift pump operation. 
     Method  400  may then continue from  412  to  418  which comprises determining if the fuel flow rate is greater than the threshold in the same or similar manner to that described at  408 . If one or more of DFSO conditions still exist, fuel injection flow rate is less than the threshold, and/or flow rate through the check valve is less than the threshold, method  400  may return to  412  and may continue to open loop operate the fuel lift pump. However, if it is determined that one or more of fuel injection has been turned on, the injection flow rate has increased above the threshold, and/or the flow rate through the check valve has increased above the threshold, then method  400  may continue to  420  which comprises resuming closed loop feedback control of the lift pump based on outputs from the pressure sensor positioned in the fuel rail. 
     Thus, at  420 , the controller may resume adjusting an amount of power supplied to the lift pump based on outputs from the pressure sensor. As such, the controller may update the integral and proportional terms, and may allow the proportional term to go negative. More simply, the controller may operate the lift pump in the same or similar closed loop manner described above at  410 . In some examples, the method  400  at  420  may include optional step  422  which may comprise closed loop controlling the lift pump to a set point less than the desired pressure for a duration before resuming the same closed loop control described above at  410 , and then gradually bringing the set point to the desired pressure. 
     Thus, when exiting DFSO, or when the flow rate through the check valve increases above the threshold, the controller may compute the error based on a difference between the estimated fuel rail pressure and a fuel rail pressure that is less than the desired fuel rail pressure. In other words, the set point to which the estimated fuel rail pressure is compared may be set to lower than the desired fuel rail pressure when exiting DFSO, and/or when the flow rate through the check valve increases above a threshold. In this way, overshoots in the fuel rail pressure may be reduced. Specifically, when fuel injection is turned back on, fuel rail pressure may decrease significantly. As such, switching directly back to closed loop control may cause overshoots in fuel rail pressure due to attempts by the controller to increase fuel rail pressure to compensate for the drop that occurs when exiting DFSO. Thus, when exiting DFSO, and/or when the flow rate through the check valve increases above the threshold, the controller may closed loop control the lift pump to a set point less than the desired pressure for a first duration, and then may gradually bring the set point to the desired pressure over a second duration. After the second duration, the controller may closed loop control the lift pump to the desired fuel rail pressure. However, it should be appreciated that in other examples, the controller may not execute  422  and may switch to closed loop feedback control of the lift pump to achieve the desired fuel rail pressure when DFSO ends and/or the fuel flow rate through the check valve increases above the threshold. Method  400  may then return. 
     Turning now to  FIG. 5 , it shows a graph  500  depicting example operation of a lift pump (e.g., lift pump  212  shown in  FIG. 2 ) under varying engine operating conditions. Power supplied to the lift pump, and therefore lift pump speed, may be adjusted by an engine controller (e.g., controller  222  shown in  FIG. 2 ). When fuel is being injected by one or more fuel injectors (e.g., injectors  252  and  262  shown in  FIG. 2 ) the lift pump may be feedback controlled by the controller based on outputs from a pressure sensor (e.g., pressure sensors  248  and  258  shown in  FIG. 2 ) positioned in a fuel rail. Thus, lift pump operation may be closed loop feedback controlled based on a fuel pressure in a fuel rail (e.g., fuel rails  250  and  260  shown in  FIG. 2 ) inferred from the pressure sensor. However, during DFSO, and/or when flow through a check valve (e.g., check valve  213  shown in  FIG. 2 ) positioned in a fuel line (e.g., passage  218  shown in  FIG. 2 ) between the lift pump and the fuel rail decreases below a threshold, the controller may switch to open loop operating the lift pump. 
     Graph  500  shows changes in the fuel injection mass flow rate at plot  502 . Fuel injection mass flow rate may be determined based on a commanded fuel injection amount from the controller. Changes in the flow rate through the check valve are shown at plot  504 . The flow rate through the check valve may be inferred based on one or more of the injection flow rate, a rate of change in pressure in the fuel line, and a temperature of the fuel as described in more detail above with reference to step  408  in  FIG. 4 . The check valve may be positioned near an outlet of the lift pump, and may restrict and/or prevent flow back towards the lift pump. When the pressure at the outlet of the lift pump is greater than the pressure downstream of the check valve (e.g., at the fuel rail), fuel may flow through the check valve in the direction of the fuel rail. However, when the pressure at the outlet of the lift pump is less than the pressure downstream of the check valve, the check valve may restrict fuel from flowing back through the check valve towards the lift pump. Thus, the check valve may effectively maintain fuel rail pressure, when the pressure in the fuel rail is greater than the pressure at the outlet of the lift pump. 
     First threshold  505 , may represent substantially zero flow through the check valve. Thus, the threshold  505  may represent a condition where the pressure in the fuel rail is approximately the same as the pressure at the outlet of the lift pump. As such, flow through the check valve may not decrease below the threshold, since flow rates below the threshold may represent flow reversing direction and flowing towards the lift pump, which is prevented by the check valve. However, in other examples, the threshold  505  may represent a flow rate through the check valve greater than zero. Fuel rail pressure is shown at plot  506  and may be estimated based on outputs from the pressure sensor. The second threshold  507 , represents a fuel rail pressure level that is substantially the same as the pressure at the outlet of the lift pump. Thus, for fuel rail pressures above the threshold, the fuel rail may be at a higher pressure than the outlet of the lift pump. In such situations, the check valve may preventing fuel from flowing back towards the lift pump. Further, for fuel rail pressures below the threshold, the fuel rail may be at a lower pressure than the outlet of the lift pump and fuel may flow from the lift pump towards the fuel rail. It is important to note that the second threshold  507  is dependent on the pressure at the outlet of the lift pump. Thus, although depicted as constant in  FIG. 5 , the threshold  507  may fluctuate as lift pump speed fluctuates. For example, at greater lift pump speeds, and therefore greater lift pump outlet pressures, the second threshold  507  may be higher than at lower lift pump speeds and/or lift pump outlet pressures. In some examples, as shown below with reference to  FIGS. 6 and 7 , pressure at the outlet of the lift pump may be estimated based on outputs from a pressure sensor position at the lift pump outlet. Changes in the amount of power supplied to the lift pump are shown at plot  508 . Control of the lift pump in either open loop or closed loop control by the controller is shown at plot  510 . 
     Starting before t 1 , fuel injection may be on (plot  502 ), and the fuel injectors may be injecting fuel. Fuel may be flowing through the check valve from the lift pump towards the fuel rail (plot  504 ) to maintain the fuel rail pressure (plot  506 ) at a desired pressure. However, fuel rail pressure is below the threshold  507 . Further, before t 1  the operation of the lift pump may be closed loop controlled by the controller based on outputs from the pressure sensor (plot  510 ). Thus, the lift pump may be provided with enough power to maintain the fuel rail pressure at the desired pressure, which before t 1  may be around a higher first level P 1  (plot  508 ). 
     At t 1  fuel injection may be turned off, and the fuel injectors may stop injecting fuel. However, fuel may still flow through the check valve since the pressure at the outlet of the fuel pump may still be greater than the fuel rail pressure. However, the flow rate through the check valve may begin to decrease at t 1 , and may continue to decrease until the pressure at the fuel rail reaches the lift pump outlet pressure. Due to the closing of the fuel injectors, the fuel rail pressure may begin to increase at t 1 . Power to the lift pump may be reduced at t 1 , since the lift pump may continue to be operated closed loop. In response to the increase in fuel rail pressure, closed loop operation of the lift pump may signal for a decrease in power supplied to the lift pump. 
     Between t 1  and t 2 , fuel injection remains off, the fuel rail pressure continues to increase, and the flow rate through the check valve continues to decrease. As such, power to the lift pump continues to be reduced, as the lift pump continues to be operated in a closed loop, feedback controlled manner by the controller. 
     At t 2 , the fuel rail pressure may reach the lift pump outlet pressure, and flow through the check may reach the threshold  505  (e.g., zero). Thus, the fuel rail pressure may reach the threshold  507 , and flow through the check valve may substantially stop. In response to the flow through the check valve reaching the threshold  505  at t 2 , the controller may switch to open loop operating the lift pump. Thus, closed loop control of the lift pump may stop at t 2 . As such, power to the lift pump may be adjusted based on the desired fuel rail pressure, which may be dependent on the fuel injection rate, engine speed, etc., as explained above with reference to  FIGS. 3 and 4 . 
     Between t 2  and t 3  fuel injection may remain off, fuel may continue to not flow through the check valve, and the lift pump may continue to be operated based on the desired fuel rail pressure. Since fuel injection may remain off between t 2  and t 3 , power to the lift pump may continue to be held approximately constant at lower second level P 2 . Due to thermal heating of the fuel in the fuel rail, the fuel rail pressure may continue to increase between t 2  and t 3 . 
     At t 3 , the fuel injectors may be turned back on, the fuel may begin to flow out of the fuel rail. As such, the fuel rail pressure may begin to decrease. However, since the fuel rail pressure may still be higher than the lift pump outlet pressure, fuel may not flow through the check valve, and as such the flow rate through the check valve may remain at the threshold  505 . In some examples, the lift pump may continue to be operated open loop by the controller at t 3 , since the flow rate through the check valve is still at the threshold  505 . As such, power to the lift pump may be supplied at around the lower second level P 2 . 
     Between t 3  and t 4 , the fuel rail pressure may continue to decrease as fuel injection remains on. However, fuel rail pressure may remain above lift pump outlet pressure, and as such, fuel may not flow through the check valve. As such, the lift pump may continue to be open loop controlled, and power supplied to the lift pump may be adjusted based on the desired fuel rail pressure only, and not based on the estimated fuel rail pressure. 
     However, at t 4 , fuel may continue to be injected by the fuel injectors, and the fuel rail pressure may decrease below the pressure at the outlet of the lift pump. As such, fuel may begin flowing through the check valve, and the flow rate through the check valve may increase above the threshold  505 . In response to one or more of the pressure at the outlet of the lift pump increasing above the pressure at the fuel rail and/or the flow rate through the check valve increasing above the threshold  505 , the controller may switch back to closed loop control of the lift pump at t 4 . Due to the decreasing fuel rail pressure at t 4 , closed loop control of the lift pump may signal for an increase in lift pump power to match the fuel rail pressure to the desired fuel rail pressure. 
     Between t 4  and t 5 , the lift pump may continue to be closed loop controlled, and power to the lift pump may be varied depending on differences between the desired fuel rail pressure and the estimated fuel rail pressure. Fuel injection remains on, and the fuel rail pressure may remain below the threshold  507 . As such, fuel may continue to flow through the check valve, and the flow rate through the check valve may continue to fluctuate above the threshold  505 . 
     At t 5 , fuel injection may be turned off, and thus DFSO conditions may resume at t 5 , similar at time t 1 . Although the flow rate through the check valve may remain above the threshold  505  at t 5 , the controller may switch to open loop control of the lift pump. Thus, in some examples, the controller may switch to open loop control of the lift pump in response to the flow rate through the check valve reaching the threshold  505 , as is shown at t 2 . However, in other examples, the controller may switch to open loop operating the lift pump in response to the fuel injectors being turned off and/or initiation of DFSO. In yet further examples, the controller may switch to open loop operating the lift pump in response to whichever occurs first: either the fuel injectors being turned off, or the flow through the check valve reaching the threshold  505 . The fuel rail pressure may begin to increase at t 5  since the fuel injectors are off. Further, power to the lift pump may be reduced to approximately earlier levels around P 2  due to the open loop control of the lift pump. 
     Between t 5  and t 6  fuel injection may remain off, and the lift pump may continue to be open loop operated by the controller. As such, power to the lift pump may fluctuate around P 2  depending on changes in the desired fuel rail pressure. The fuel rail pressure may remain above the threshold  507 . The flow rate through the check valve may remain around the threshold  505  due to the fuel rail pressure remaining above the threshold  507 . 
     At t 6 , fuel injection may resume, and the fuel may exit the fuel rail. In response to exiting DFSO conditions at t 6 , the controller may resume closed loop operation of the lift pump. As such, power to the lift pump may increase at t 6  in response to the drop in fuel rail pressure at t 6  resulting from the fuel injectors being turned back on. The fuel rail pressure may being to decrease at t 6  but may remain above the threshold  507  and as such fuel flow may remain at the threshold  505 . 
     However, after t 6 , the fuel rail pressure may decrease below the threshold  507 , and flow rate through the check valve may increase above the threshold  505 . Fuel injection remains on, and power to the lift pump may continue to be adjusted based on outputs from the pressure sensor, in a closed loop manner. 
     Moving on to  FIG. 6 , it shows an example fuel system  600  that may be the same or similar to fuel system  200  of  FIG. 2 , except that fuel system  600  may include an additional pressure sensor at an outlet of the lift pump. Thus, fuel system  600  may include the same components as fuel system  200  shown in  FIG. 2  and may be numbered similarly in  FIG. 6 . As such, components of the fuel system  600  already described in  FIG. 2 , may not be reintroduced or described again in the description of  FIG. 6  herein. 
     As described above, fuel system  600  may be the same as fuel system  200 . However, fuel system  600  may include a pressure sensor  631  between the lift pump  212  and check valve  213 . Thus, the pressure sensor  631  may be configured to measure a pressure of the fuel included between the lift pump  212  and the check valve  213 . Said another way, outputs from the pressure sensor  631  may be used to estimate a pressure at the outlet  251  of the lift pump  212 . The controller  222  may under certain engine operating conditions, adjust an amount of power supplied to the lift pump  212  based on outputs from the pressure sensor  631  as described below with reference to  FIG. 6 . Thus, the controller  222  may switch between adjusting the power supplied to the lift pump  212  based on outputs from the pressure sensor  631 , and based on outputs from one or more of the fuel rail pressure sensor  248  and  258 . However, in other examples, the controller  222  may switch between adjusting the power supplied to the lift pump  212  based on outputs from both the pressure sensor  631  and one or more of the fuel rail pressure sensors  248  and  258 , and based only on outputs from one or more of the fuel rail pressure sensors  248  and  258 . 
     Turning now to  FIG. 7 , it shows a flow chart of an example method  700  for operating a lift pump (e.g., lift pump  212  shown in  FIGS. 2 and 6 ) of an engine fuel system (e.g., fuel system  200  shown in  FIG. 2 ) that includes a pressure sensor (e.g., pressure sensor  631  shown in  FIG. 6 ) on or proximate the lift pump outlet, and upstream of any check valve (e.g., check valve  213  shown in  FIGS. 2 and 6 ). The method  700  shown in  FIG. 7  describes a system where at low fuel flow rates (e.g., injection flow rates), outputs from the pressure sensor positioned at the outlet of the lift pump may be used to feedback control operation of the lift pump. Further, during higher flow rates, lift pump operation may be feedback controlled based on outputs from a pressure sensor (e.g., pressure sensor  248  shown in  FIGS. 2 and 6 ) positioned in a fuel rail (e.g., fuel rails  250  and  260  shown in  FIGS. 2 and 6 ). 
     Method  700  may be therefore be the same or similar to method  400  described above with reference to  FIG. 4 , except that instead of open loop operating the lift pump when fuel flow rates decrease below a threshold, as described at  412  in  FIG. 4 , method  700  may comprise closed loop operating the lift pump based on outputs from the lift pump outlet pressure sensor (e.g., pressure sensor  631  shown in  FIG. 6 ). Thus, during higher injection fuel flow rates, an amount of power supplied to the lift pump may be adjusted to achieve a desired fuel pressure in a fuel rail (e.g., fuel rails  250  and  260  shown in  FIG. 2 ). The lift pump may therefore be closed loop feedback controlled by an engine controller (e.g., controller  222  shown in  FIGS. 2 and 6 ) based on outputs from one or more fuel rail pressure sensors (e.g., pressure sensors  248  and  258  shown in  FIGS. 2 and 6 ) positioned in the fuel rail. However, the controller may switch to closed loop control of the lift pump based on outputs from the lift pump outlet pressure in response to a fuel flow through a check valve (e.g., check valve  213  shown in  FIGS. 2 and 6 ) positioned between the lift pump and the fuel rail decreasing below a threshold. 
     Instructions for executing method  700  may be stored in the memory of the controller. Therefore method  700  may be executed by the controller based on the instructions stored in the memory of the controller and in conjunction with signals received from sensors of the engine system, such as the sensors described above with reference to  FIGS. 1-2 and 6 . The controller may send signals to the lift pump and/or to a power source supplying power to the lift pump, to adjust an amount of power supplied to the lift pump, and therefore an output of the lift pump. 
     Method  700  begins at  702  which comprises estimating and/or measuring engine operating conditions in the same or similar manner to that described above with reference to  402  in  FIG. 4 . 
     After estimating and/or measuring engine operating conditions at  702 , method  700  may continue to  704  which comprises determining a desired fuel rail pressure based on engine operating conditions in the same or similar manner to that described above with reference to  404  in  FIG. 4 . 
     Method  700  may then proceed to  706  which comprises determining a current fuel flow rate through the check valve in the same or similar manner to that described above with reference to  406  in  FIG. 4 . 
     After estimating the current fuel flow rate through the check valve at  706 , method  700  may proceed to  708  which comprises determining if the fuel flow rate is less than a threshold flow rate in the same or similar manner to that described above with reference to  408  in  FIG. 4 . 
     If one or more of the flow rate through the check valve is greater than the threshold flow rate, and/or it is determined that DFSO conditions do not exist and fuel is being injected by the fuel injectors, and/or the fuel injection flow rate is greater than a threshold, method  700  may continue from  708  to  710 . 
     At,  710  the controller may continue to compute an error based on the difference between the desired fuel rail pressure and the estimated fuel rail pressure obtained from outputs of the fuel rail pressure sensor, in the same or similar manner described above with reference to  410  in  FIG. 4 . 
     However, if at  708  it is determined that one or more of the fuel flow rate through the check valve is less than the threshold, the injection flow rate is less than the injection flow rate threshold, and/or DFSO conditions do exist and fuel is not being injected by the fuel injectors, then method  700  may proceed from  708  to  711  which comprises determining if the fuel rail pressure error is less than zero, in the same or similar manner described above with reference to  411  in  FIG. 4 . If the fuel rail pressure is greater than desired and the fuel rail pressure error is therefore less than zero, method  700  may continue from  708  to  712 , where the method  700  at  712  comprises determining a desired fuel pressure in a volume included between the lift pump and the check valve. Thus, in some examples, method  700  may only proceed to  712  if the fuel flow rate through the check valve is less than threshold, and the fuel rail pressure error is less than zero. However, in other examples, method  700  may not execute  711  and may proceed directly from  708  to  712  if the fuel flow rate through the check valve is less than the threshold. 
     The method  700  at  712  may comprise determining a desired lift pump outlet pressure. In some examples, the desired lift pump outlet pressure may be a pre-set or threshold amount lower than the desired fuel rail pressure determined at  704  and/or the fuel rail pressure measured via outputs from the fuel rail pressure sensor. The desired lift pump outlet pressure may in some examples be 5 kPA below the desired fuel rail pressure and/or estimated fuel rail pressure. However, in other examples, the desired lift pump outlet pressure may be determined based on engine operating conditions, such as a fuel injection amount, a flow rate through one or more check valves positioned between the lift pump and the fuel rail, a fuel rail pressure, a desired fuel rail pressure, etc. For example, when the fuel injectors are turned on, and fuel is flowing out of the fuel rail at higher rates, the desired lift pump outlet pressure may be greater than the desired pressure at the fuel rail. Specifically, the desired lift pump outlet pressure may be 20 kPa greater than the desired fuel rail pressure, to facilitate the flow of fuel from the lift pump to the fuel rail. However, when fuel injection flow rates are lower, and/or when fuel injection is off, and the fuel rail pressure exceeds the desired fuel rail pressure, the desired lift pump outlet pressure may be slightly less than the fuel rail pressure (e.g., 1-10 kPa less than fuel rail pressure) to reduce and/or prevent any pressure being added to the fuel rail. 
     Thus, the lift pump outlet pressure may be kept just below the fuel rail pressure when fuel flow through the check valve is less than the threshold at  708 , and/or the fuel rail pressure is greater than desired, so that substantially no additional fuel flows from the lift pump to the fuel rail. In this way, substantially no additional pressure may be added to the fuel rail by the lift pump, while the speed of the lift pump may be increased relative to what it would be under feedback control from the fuel rail pressure sensor. Thus, the lift pump may remain on, and the speed of the pump remain sufficiently high enough to maintain the lift pump outlet pressure approximately at, or just below the fuel rail pressure when fuel flow rate through the check valve is less than the threshold and fuel rail pressure is greater than desired. 
     After determining the desired lift pump outlet pressure at  712 , method  700  may continue from  712  to  714  which comprises closed loop feedback controlling the lift pump based on outputs from the lift pump outlet pressure sensor to achieve the desired lift pump outlet pressure. In some examples, such as where the desired lift pump outlet pressure is determined based on the desired fuel rail pressure, and is not dependent on the estimated fuel rail pressure obtained from the fuel rail pressure sensor, the method  700  at  714  may comprise not adjusting the lift pump operation based on the fuel rail pressure sensor. That is, the method  700  at  714  may comprise closed loop operating the lift pump based only on outputs from the lift pump outlet pressure sensor and not based on outputs from the fuel rail pressure sensor, to maintain the lift pump outlet pressure at the desired lift pump outlet pressure. Thus, power supplied to the lift pump may be adjusted to maintain the fuel pressure of fuel included between the lift pump and the check valve to a threshold difference of the desired fuel rail pressure. 
     However, in other examples, such as where the desired lift pump outlet pressure is determined based on the estimated fuel rail pressure, the method  700  at  714  may comprise adjusting the lift pump operation based on both the fuel rail pressure sensor and the lift pump outlet pressure sensor. More specifically, the controller may adjust the amount of power supplied to the lift pump to maintain the fuel pressure of fuel included between the lift pump and the check valve to within a threshold difference of the estimated fuel rail pressure. Based on the difference between the estimated fuel rail pressure obtained from the fuel rail pressure sensor, and the lift pump outlet pressure obtained from the lift pump outlet pressure sensor, the controller may adjust the amount of power supplied to the lift pump to maintain the desired lift pump outlet pressure. Thus, the controller may increase the amount of power supplied to the lift pump in response to the lift pump outlet pressure decreasing below the estimated fuel rail pressure by more than the threshold amount. In other examples, the controller may decrease the amount of power supplied to the lift pump in response to the lift pump outlet pressure increasing such that the difference between the lift pump outlet pressure and the estimated fuel rail pressure is less than the threshold amount. 
     Method  700  may then continue from  714  to  716  which comprises determining if the fuel flow rate is greater than the threshold in the same or similar manner to that described above with reference to  418  in  FIG. 4 . If one or more of DFSO conditions still exist, fuel injection flow rate is less than the threshold, and/or flow rate through the check valve is less than the threshold, method  700  may return to  714  and may continue to adjust lift pump operation based on the lift pump outlet pressure sensor. However, if it is determined that one or more of fuel injection has been turned on, the injection flow rate has increased above the threshold, and/or the flow rate through the check valve has increased above the threshold, then method  700  may continue to  718  which comprises resuming closed loop feedback control of the lift pump based on outputs from the fuel rail pressure sensor in the same or similar manner to that described above with reference to  420  in  FIG. 4 . Method  700  then returns. 
     It should be appreciated that in other examples at  710  and  718 , the lift pump may be closed loop feedback controlled based on outputs from the lift pump outlet pressure sensor (e.g., pressure sensor  631  shown in  FIG. 6 ) and may not be controlled based on outputs from the fuel rail pressure sensor. Thus, in some examples, the lift pump may be closed loop feedback controlled based on the lift pump outlet pressure sensor under all engine operating conditions, and may not be feedback controlled based on outputs from the fuel rail pressure sensor. In such examples, a slow adaptive correction factor for the desired lift pump outlet pressure may be learned based on the difference between outputs from the lift pump outlet pressure sensor and the fuel rail pressure sensor. Thus, the desired lift pump outlet pressure may be corrected over time based on differences between outputs from the lift pump outlet pressure sensor and the fuel rail pressure sensor. In some examples, this correction factor may be highly correlated to fuel flow rate (e.g., injection flow rate). 
     In this way, a technical effect of reducing the frequency and intensity of pressure drops in a fuel rail may be reduced by open loop operating a lift pump in response to one or more of a fuel flow rate through a check valve coupled between the lift pump and the fuel rail decreasing to a threshold, entering DFSO, and an injection flow rate decreasing below a threshold. Specifically, by open loop operating the lift pump during DFSO, lift pump speed may be maintained at a higher level than it would be under closed loop control during DFSO. As such, lift pump spin-up time when exiting DFSO may be reduced, and pressure drops in the fuel rail may be reduced. Thus, fluctuations in fuel rail pressure may be reduced and fuel rail pressure consistency may be increased. 
     Note that the example control and estimation routines included herein can be used with various engine and/or vehicle system configurations. The control methods and routines disclosed herein may be stored as executable instructions in non-transitory memory and may be carried out by the control system including the controller in combination with the various sensors, actuators, and other engine hardware. The specific routines described herein may represent one or more of any number of processing strategies such as event-driven, interrupt-driven, multi-tasking, multi-threading, and the like. As such, various actions, operations, and/or functions illustrated may be performed in the sequence illustrated, in parallel, or in some cases omitted. Likewise, the order of processing is not necessarily required to achieve the features and advantages of the example embodiments described herein, but is provided for ease of illustration and description. One or more of the illustrated actions, operations and/or functions may be repeatedly performed depending on the particular strategy being used. Further, the described actions, operations and/or functions may graphically represent code to be programmed into non-transitory memory of the computer readable storage medium in the engine control system, where the described actions are carried out by executing the instructions in a system including the various engine hardware components in combination with the electronic controller. 
     It will be appreciated that the configurations and routines disclosed herein are exemplary in nature, and that these specific embodiments are not to be considered in a limiting sense, because numerous variations are possible. For example, the above technology can be applied to V-6, I-4, I-6, V-12, opposed 4, and other engine types. The subject matter of the present disclosure includes all novel and non-obvious combinations and sub-combinations of the various systems and configurations, and other features, functions, and/or properties disclosed herein. 
     The following claims particularly point out certain combinations and sub-combinations regarded as novel and non-obvious. These claims may refer to “an” element or “a first” element or the equivalent thereof. Such claims should be understood to include incorporation of one or more such elements, neither requiring nor excluding two or more such elements. Other combinations and sub-combinations of the disclosed features, functions, elements, and/or properties may be claimed through amendment of the present claims or through presentation of new claims in this or a related application. Such claims, whether broader, narrower, equal, or different in scope to the original claims, also are regarded as included within the subject matter of the present disclosure.