Engines may utilize turbocharging to increase power density and/or increase engine fuel efficiency.  However, during transient conditions, such as a driver request for increased engine output, turbocharger inertia and flow dynamics may result in “turbo lag.”  Such lag may be reduced in some examples by reducing turbocharger size and weight, and/or taking various measures via engine control. 
One control approach to address turbo lag uses a late fuel injection into lean diesel combustion to generate exhaust heat, thereby maintaining spin-up of the turbine of the turbocharger.  Specifically, the late injection generates exhaust heat, which in turn increases the speed of the turbine.  Then, when a transient occurs, such as a request for an increase in engine output, the turbine is already spinning fast enough to provide the rapid increase in engine output. 
However, the inventors herein have recognized some issues with the above approach.  In particular, in gasoline applications, the excess fuel used to generate increased exhaust heat may degrade fuel efficiency.  This is especially true during idle, where significant spark reserve may be used for purposes of disturbance rejection.  The combined fuel economy degradation of the excess injection, in addition to the fuel economy degradation due to the spark reserve, can lead to significant overall fuel economy losses.  Additionally, in some applications, a lean exhaust air-fuel ratio may increase emissions. 
The above issues may be at least partially addressed by a method for controlling engine operation for an engine having a turbocharger and direction injection.  The method may comprise: performing at least a first and second injection during a cylinder cycle, the first injection generating a lean combustion and the second injection injected after combustion such that it exits the cylinder unburned into the exhaust upstream of a turbine of the turbocharger; and adjusting at least the first injection based on engine speed, where said at least first and second injection are performed responsive to turbocharger speed. 
In this way, less spark reserve may be used, at least in cylinders utilizing the first and second injection, since adjustments in the fuel injection may be used to manage torque and speed disturbances.  In other words, the lean combustion already has sufficient air to enable increased combustion torque from increased fuel injection.  However, the overall air-fuel ratio can be maintained about stoichiometry via the second injection, while also providing increased heat to maintain turbocharger speed.  In this way, increased fuel economy can be achieved.