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Methods and systems are provided for closed loop operation of a high pressure fuel pump connected to the direct injectors of an internal combustion engine. During operation of the high pressure pump a dead zone may exist where a substantial change in the pump duty cycle does not correspond to a subs

Methods and systems are provided for closed loop operation of a high pressure fuel pump connected to the direct injectors of an internal combustion engine. During operation of the high pressure pump a dead zone may exist where a substantial change in the pump duty cycle does not correspond to a substantial change in the fuel rail pressure. To operate outside the dead zone, a relationship between the pump duty cycle and fuel rail pressure is learned upon completion of several pump and engine conditions, thereby improving high pressure pump operation and reducing pump degradation.

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1. A method, comprising: via a controller of an engine control system, decreasing fuel rail pressure below a threshold; thenwhile not direct injecting fuel into an engine, learning a dead zone for a high pressure fuel pump based on a change in pump duty cycle relative to a resulting change in fuel r

1. A method, comprising: via a controller of an engine control system, decreasing fuel rail pressure below a threshold; thenwhile not direct injecting fuel into an engine, learning a dead zone for a high pressure fuel pump based on a change in pump duty cycle relative to a resulting change in fuel rail pressure via the controller; andwhile direct injecting fuel into the engine and during closed-loop control of the fuel rail pressure, adjusting the pump duty cycle to stay above the learned dead zone via the controller. 2. The method of claim 1, wherein learning the dead zone based on the change in pump duty cycle relative to the resulting change in fuel rail pressure includes, via the controller: commanding a first duty cycle and determining a first fuel rail pressure;then commanding a second, higher duty cycle and determining a second fuel rail pressure; andlearning the dead zone based on a difference between the first and second fuel rail pressures relative to a difference between the commanded first and second duty cycles. 3. The method of claim 1, wherein the high pressure fuel pump is coupled to a direct fuel injector of the engine, the engine further including a port fuel injector coupled to a low pressure fuel pump, and wherein not direct injecting fuel into the engine includes only port injecting fuel into the engine. 4. The method of claim 1, wherein the high pressure fuel pump is coupled to a direct fuel injector of the engine, and wherein not direct injecting fuel into the engine includes one of an engine-off condition and a deceleration fuel shut-off condition. 5. The method of claim 1, further comprising, via the controller, commanding a fixed pump duty cycle inside the dead zone, the fixed pump duty cycle based on a desired fuel rail pressure. 6. A method for an engine fuel system, comprising: via a controller of an engine control system, learning an affine relationship between a duty cycle for a high pressure fuel pump and a fuel rail pressure for a direct fuel injector based on a change in the duty cycle relative to a resulting change in the fuel rail pressure during selected conditions when not direct injecting fuel into an engine; andadjusting the duty cycle of the high pressure fuel pump during closed-loop control of the fuel rail pressure based on the learned affine relationship to operate outside a dead zone of the high pressure fuel pump via the controller, wherein adjusting the duty cycle of the high pressure fuel pump during closed-loop control includes adjusting the duty cycle of the high pressure fuel pump while direct injecting fuel into the engine. 7. The method of claim 6, wherein the direct fuel injector is coupled to the high pressure fuel pump, and wherein the engine further includes a port fuel injector, and wherein the selected conditions include engine idling conditions where the fuel rail pressure is below a threshold, and the engine is fueled via the port injection only. 8. The method of claim 6, wherein the direct fuel injector is coupled to the high pressure fuel pump, and wherein the selected conditions include one of an engine-off condition and a deceleration fuel shut-off condition where the fuel rail pressure is below a threshold. 9. The method of claim 6, wherein learning the affine relationship includes, via the controller: changing the duty cycle from a first, lower duty cycle to a second, higher duty cycle;determining a first fuel rail pressure at the first duty cycle and a second fuel rail pressure at the second duty cycle;determining a slope based on a difference between the first and second fuel rail pressures relative to the change in duty cycle; andlearning an affine transfer function based on the determined slope. 10. The method of claim 9, wherein the learning includes calculating an offset based on the determined slope and learning the affine transfer function based on each of the determined slope and the calculated offset via the controller. 11. The method of claim 6, wherein the dead zone of the high pressure fuel pump is a region where an actual change in fuel rail pressure responsive to a change in pump duty cycle is lower than an expected change in fuel rail pressure. 12. An engine system, comprising: an engine;a direct fuel injector configured to direct inject fuel into the engine;a high pressure fuel pump;a fuel rail;a pressure sensor configured to estimate a fuel rail pressure;a controller with computer readable instructions stored in non-transitory memory for:direct injecting fuel into the engine during engine idling conditions until the fuel rail pressure is below a threshold; then, while not direct injecting fuel into the engine, commanding a change in duty cycle to the high pressure fuel pump and estimating a corresponding change in fuel rail pressure;learning a dead zone of the high pressure fuel pump based on the change in fuel rail pressure relative to the change in commanded duty cycle; andupon learning the dead zone of the high pressure fuel pump, executing a programmed pump operating scheme. 13. The system of claim 12, wherein the controller includes further instructions for, while direct injecting fuel into the engine, adjusting the duty cycle of the high pressure fuel pump to operate outside the dead zone of the high pressure fuel pump, the dead zone being a zone where changes in pump duty cycle do not substantially change pump outlet pressure by more than a threshold. 14. The system of claim 13, wherein not direct injecting fuel into the engine includes operating the engine in a deceleration fuel shut-off mode. 15. The system of claim 13, further comprising a port fuel injector configured to port inject fuel into the engine, wherein not direct injecting fuel into the engine includes port injecting fuel into the engine. 16. The system of claim 15, wherein the dead zone of the high pressure fuel pump is a region where an actual change in fuel rail pressure responsive to a change in pump duty cycle is lower than an expected change in fuel rail pressure. 17. The system of claim 12, wherein the programmed pump operating scheme includes freezing an integral term of the controller.