Abstract:
A method for improving turbocharger waste-gate control is presented. The method can reduce turbocharger flow oscillation, at least during some conditions.

Description:
FIELD 
       [0001]    The present description relates to a method for controlling a turbocharger waste-gate that operates as part of an internal combustion engine. 
       BACKGROUND 
       [0002]    Turbocharged internal combustion engines often have a waste-gate that can be opened to allow a portion of exhaust gasses to bypass the turbocharger&#39;s turbine. One way to actuate the bypass valve (also known as the waste-gate) is by an actuator that is comprised of a flexible diaphragm and spring. The spring applies a force to the diaphragm and lever connected to the diaphragm to keep the bypass valve closed when atmospheric pressure is present on both sides of the diaphragm. The valve can be opened by applying a gas pressure to the diaphragm. When the gas pressure overcomes the closing spring force, the waste-gate opens and a portion of exhaust gases bypass the turbocharger. One example of controlling a waste-gate in this way is described in U.S. Pat. No. 5,729,980. This patent describes a spring return pneumatically actuated waste-gate that is controlled based on engine speed. In particular, this turbocharger waste-gate actuator has two ports wherein gas pressure can act on the waste-gate diaphragm from either direction. This arrangement, when coupled with a throttle position sensor, is claimed to allow the waste-gate to remain closed for a longer period of time during full-throttle acceleration. 
         [0003]    The above-mentioned method can also have several disadvantages. For example, the method simply controls the boost pressure depending on engine speed. In this arrangement, it may be possible to cause the spring assisted waste-gate valve to flutter or oscillate causing exhaust flow to the turbocharger to increase and decrease in an undesirable manner. This is especially true when the gas pressure used to open the bypass valve approaches the spring force that is applied to the diaphragm when the waste-gate is closed. In addition, this approach does not recognize or compensate part-to-part variations that can occur during manufacturing (e.g., spring rate variations) and that may result in uneven flow in dual turbocharger applications. 
         [0004]    The inventors herein have recognized the above-mentioned disadvantages and have developed a method that offers substantial improvements. 
       SUMMARY 
       [0005]    One embodiment of the present description includes a method to operate a turbocharger having a spring assisted waste-gate closing mechanism, the method comprising: applying a first force to operate a waste-gate actuator; applying a second force by a spring acting in opposition to said first force, said second force being the force applied when said waste-gate is in a closed position; commanding said first force to a level that is less than said second force when a desired force is less than said second force plus a predetermined force; and commanding said first force to a level that is greater than said second force plus said predetermined force when said desired force is greater than said second force plus said predetermined force. This method overcomes at least some disadvantages of the above-mentioned method. 
         [0006]    Turbocharger waste-gate flutter can be decreased by ingenuously controlling waste-gate actuator forces. Specifically, when a spring return waste-gate actuator is used to regulate boost pressure, gas forces applied to the diaphragm side opposite the spring can be controlled to reduce the possibility of generating forces that are proximate to the spring force applied to a closed waste-gate. In other words, a control pressure (i.e., a pressure applied to the diaphragm side opposite the spring side) can be applied such that the force created by the control pressure acting on the diaphragm is not substantially equal to the closing spring force applied to the opposite diaphragm side. For example, if a spring applies X force to the waste-gate actuator diaphragm, control pressure acting opposite the spring is allowed to exert forces less than X minus a predetermined force or greater than X plus a predetermined force. In this way, control pressures acting in opposition to the spring force can be generated to provide a positive opening and closing of the waste-gate, thereby avoiding waste-gate flutter. 
         [0007]    When the above method is applied to an engine having two parallel turbochargers, a further enhancement over previous waste-gate control methods can also be realized. Namely, by setting the force that can be applied by gases to a level that is offset from the nominal spring force, two turbocharger waste-gates may be opened simultaneously so that compressor flow is better equalized between the two turbochargers. In other words, when waste-gate opening is desired, applying a force greater than the closing force applied by either waste-gate closing spring promotes more positive waste-gate operation. 
         [0008]    The present description can provide several advantages. For example, the approach can provide more uniform turbocharger compressor flow when a waste-gate is initially opened. Further, as mentioned above, the method can reduce turbocharger waste-gate flutter or flow oscillations when a waste-gate force balance is near zero. Further still, when the method is applied to two waste-gates of a twin parallel turbocharged engine, the waste-gates will open at substantially the same time, thereby providing improved boost regulation. 
         [0009]    The above advantages and other advantages, and features of the present description will be readily apparent from the following Detailed Description when taken alone or in connection with the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    The advantages described herein will be more fully understood by reading an example of an embodiment, referred to herein as the Detailed Description, when taken alone or with reference to the drawings, wherein: 
           [0011]      FIG. 1  is a schematic diagram of an engine configured to operate with two turbochargers; 
           [0012]      FIG. 2  is a flowchart of an example waste-gate control method; 
           [0013]      FIG. 3  is a schematic diagram of a spring assisted diaphragm operated waste-gate actuator control system; 
           [0014]      FIG. 4  is a plot illustrating waste-gate pressure regulator control of the prior art; 
           [0015]      FIG. 5  is a plot illustrating waste-gate pressure regulator control of the present method; 
           [0016]      FIG. 6  is a plot illustrating waste-gate based boost pressure regulation; and 
           [0017]      FIG. 7  is a plot illustrating one way to determine waste-gate spring forces acting on a waste-gate actuator diaphragm. 
       
    
    
     DETAILED DESCRIPTION 
       [0018]    Referring to  FIG. 1 , internal combustion engine  10  is controlled by electronic engine controller  12 . Engine  10  includes a plurality of cylinders in a “V” configuration that are similar to cylinder  50 . Cylinder banks  13  and  14  are comprised of three cylinders each. Intake cams (not shown) operate intake valves (not shown) to regulate airflow into the cylinders of banks  13  and  14 . Exhaust cams (not shown) operate exhaust valves (not shown) to regulate exhaust flow out of cylinder banks  13  and  14 . Timing of intake and exhaust cams relative to crankshaft position can be varied by adjusting phasers (not shown). Further, intake and/or exhaust valves may be configured to vary valve lift. Each cylinder surrounds a piston that transfers combustion energy to mechanical energy through crankshaft  40 . Intake manifold  44  is in communication with electronically controlled throttle  125  and directs air to cylinder banks  13  and  14 . Intake air is routed through duct  42  and mass airflow sensor  60  before being compressed by first turbocharger  30 . A second turbocharger  34 , also compresses air from duct  42 . Compressed air is routed through duct  43  to intercooler  50  and proceeds to the inlet of electronic throttle  125 . Combusted gases exit cylinder banks  13  and  14  through exhaust manifolds  52  and  54 . Exhaust gases rotate turbines  31  and  36  of turbochargers  30  and  34 , turbines  31  and  36  cause compressors  32  and  35  to rotate compressing fresh air. Waste-gates  33  and  37  (i.e., turbocharger control actuators) allow exhaust gases to bypass turbines  31  and  36  so that turbine work of each turbocharger can be controlled. Also, if desired, a single turbocharger can be used to compress fresh air for both cylinder banks. 
         [0019]    Distributor-less ignition system (not shown) provides ignition spark to the cylinders of banks  13  and  14 . Universal Exhaust Gas Oxygen (UEGO) sensors  85  and  86  are shown coupled to exhaust manifolds  52  and  54  upstream of catalytic converters  70  and  71 . Alternatively, two-state exhaust gas oxygen sensors may be substituted for UEGO sensors  85  and  86 . Two-state exhaust gas oxygen sensor  98  is shown coupled to exhaust pipe  49  downstream of catalytic converter  70 . Alternatively, sensor  98  can also be a UEGO sensor. A second two-state oxygen sensor  99  is shown similarly situated. Catalytic converter temperature is measured by temperature sensor  77 , and/or estimated based on operating conditions such as engine speed, load, air temperature, engine temperature, and/or airflow, or combinations thereof. 
         [0020]    Converter  70  can include multiple catalyst bricks, in one example. In another example, multiple emission control devices, each with multiple bricks, can be used. Converter  70  can be a three-way type catalyst in one example. A second catalytic converter  71  processes exhaust gases on the opposite cylinder bank. 
         [0021]    Controller  12  is shown in  FIG. 1  as a conventional microcomputer including: microprocessor unit  102 , input/output ports  104 , and read-only-memory  106 , random-access-memory  108 ,  110  Keep-alive-memory, and a conventional data bus. Controller  12  is shown receiving various signals from sensors coupled to engine  10 , in addition to those signals previously discussed, including: engine coolant temperature (ECT) from temperature sensor  112 ; a position sensor  119  coupled to an accelerator pedal; a measurement of engine manifold pressure (MAP) from pressure sensor  122  coupled to intake manifold  44 ; crankshaft position  118 ; exhaust manifold pressure  62 - 63 ; throttle inlet air pressure  115 ; throttle inlet air temperature  117 , and a measurement (ACT) of engine air amount temperature or manifold temperature from temperature sensor  124 . 
         [0022]    The engine described in  FIG. 1  may be the primary means of generating motive force in a vehicle or it may be part of a vehicle having more than one means for generating motive force, a hybrid vehicle for example. The engine may generate wheel torque in conjunction with an electric motor when in a hybrid configuration. Alternatively, the engine may generate wheel torque in conjunction with a hydraulic motor. Thus, there are many configurations whereby features of the present description may be used to advantage. 
         [0023]    Controller  12  storage medium read-only memory  106  can be programmed with computer readable data representing instructions executable by processor  102  for performing the methods described below as well as other variants that are anticipated but not specifically listed. 
         [0024]    Referring now to  FIG. 2 , a flow chart of an example waste-gate control method is shown. The method of  FIG. 2  can be used to control a system with one or two turbochargers having spring assisted waste-gate actuators (see  FIG. 3  for an example waste-gate actuator). The method begins to open waste-gate when boost pressure generated by the turbocharger compressors reaches a level that can provide sufficient force to overcome the valve closing spring plus a predetermined offset force. This forms a pneumatic control system that can limit boost to a fixed level. The predetermined offset force can be used to represent manufacturing variation in spring force and valve seating frictional forces. In one embodiment, the offset force is selected to be a force that exceeds manufacturing and waste-gate seating forces by 0.6 N. 
         [0025]    The method also describes a way to determine the force at which the waste-gate begins to open. This feature allows the controller to change the estimated spring closing force (i.e., the amount of force the waste-gate closing spring applies to the waste-gate diaphragm when the waste-gate is in the closed position) so that the predetermined force does not have to be large to promote positive opening and closing control. In addition, when spring closing forces are determined for two turbochargers, the method can determine a control pressure that delivers a force at which both waste-gate actuators positively open. 
         [0026]    At step  201 , operating conditions are determined. Operating conditions may include engine operating conditions (e.g., engine speed, driver torque demand, and/or cylinder air charge), turbocharger operating conditions, ambient temperature, ambient pressure, and engine operating time, for example. After determining operating conditions the routine proceeds to step  203 . 
         [0027]    At step  203 , the routine determines if waste-gate valve opening parameters are to be determined. In one example, waste-gate actuator spring closing force may be determined after a predetermined number of engine starts. In another example, waste-gate actuator spring closing force may be determined at specific operating temperatures or at predetermined operating intervals. In still another example, waste-gate actuator spring closing force may be determined in response to a boost pressure. For example, if a waste-gate is commanded open during engine acceleration, and boost pressure is adjusted earlier or later than anticipated, the routine may initiate a procedure to determine the waste-gate actuator opening parameters. Logic or state machines may be used with operating parameters determined in step  201  to determine if it is desirable to learn valve opening parameters. If valve opening parameters are to be determined the routine proceeds to step  211 . If not, the routine proceeds to step  205 . 
         [0028]    In step  205 , the routine determines desired boost pressure. In one example, desired boost pressure is determined as a function of engine speed and engine torque demand. Further, desired boost may be adjusted for operating conditions such as atmospheric pressure and intake manifold pressure request (e.g., vacuum requests for brakes and vapor purge). The routine proceeds to step  207 . 
         [0029]    At step  207 , the routine determines if the waste-gate pressure regulator valve can deliver a control pressure that when acting on the actuator diaphragm is capable of providing a force that is greater than the spring force applied to the closed waste-gate actuator diaphragm plus a predetermined force offset. The predetermined force may be an empirically determined constant or it may be varied. In one embodiment, operating conditions described in step  201  may be used to adjust the predetermined force by indexing tables or functions that add or subtract from the empirically predetermined force. 
         [0030]    A waste-gate pressure regulator valve is typically comprised of a solenoid that regulates an outlet pressure based on a command signal and pressures at two inlet ports. The command signal instructs the valve to provide an outlet pressure that ranges between the two input pressures. Typically, the two pressures in a waste-gate actuator control system are boost pressure and atmospheric pressure. Thus, a waste-gate actuator control pressure is produced by a controller commanding the waste-gate pressure regulator valve to regulate a pressure between atmospheric pressure and boost pressure. The boost pressure corresponds to the high side of the available control pressure range while atmospheric pressure identifies the low side of the control pressure range. In one example, when the waste-gate control solenoid valve&#39;s duty cycle is zero, the boost pressure is limited to a fixed value (e.g., 6″Hg). When its duty cycle is 1, the boost pressure is unregulated (waste-gate closed). In between these limits, boost pressure is controllable by duty cycle selection. 
         [0031]    In one embodiment, a table of boost pressure versus waste-gate command duty cycle outputs a command pressure. In addition, a function relates command pressure to the amount of force applied to the opening side of the waste-gate actuator diaphragm. Alternatively, a formula that converts a pressure exerted over the diaphragm area can be used to estimate the amount of force applied to the diaphragm opening side. A second function (i.e., the force function) relates opening force to an open loop estimate of valve position. 
         [0032]    The force function may contain a plurality of entries. The force function is initially populated with predetermined entries based on nominal waste-gate closing spring rates. If desired, the force function can be adjusted during engine operation as described in steps  211 - 221  and the description of  FIG. 7 . 
         [0033]    The closing spring force can be determined from the force function by interpolating between the highest force applied to the opening side of the waste-gate actuator diaphragm when the valve remains closed and the amount of force applied to the waste-gate actuator diaphragm when the valve first opens. By determining the amount of force required to open the waste-gate valve, the waste-gate controller can directly command the waste-gate valve to a position without having to start at a zero command and ramp the command until the desired boost pressure is reached. This allows the controller to achieve the desired boost pressure more quickly. In addition, waste-gate position can be determined when a force equal to a predetermined force plus the spring closing force is applied to the waste-gate actuator diaphragm opening side by interpolating between entries in the force function. 
         [0034]    The spring closing force plus the above-mentioned predetermined force can be compared to the opening force that is produced by applying the available control pressure to the actuator diaphragm. If the force produced by applying the available control pressure to the opening side of the waste-gate actuator diaphragm is greater than the closing spring force plus the predetermined force, the routine proceeds to step  209 . Otherwise, the routine proceeds to step  223 . 
         [0035]    In step  209 , the routine commands the waste-gate pressure regulator valve to regulate boost. In one embodiment, a closed-loop controller is provided for controlling the waste-gate actuator position. The controller combines open-loop commands from the tables and functions described in steps  201 - 207  with feedback from a boost pressure sensor. Boost pressure is controlled to desired boost pressure, step  205 , by commanding the waste-gate to a position, and then boost pressure feedback is used to further match the actual boost pressure to the desired boost pressure. Typically, a proportional/integral controller is used to perform this task. After adjusting the waste-gate actuator to substantially match desired boost pressure (e.g., ±0.1 bar), the routine exits. 
         [0036]    At step  211 , learning of valve opening parameters is commenced. In particular, a control pressure is applied to a waste-gate actuator by commanding the valve pressure regulator to a level above atmospheric pressure when boost pressure is present at one of the pressure regulator inlet ports. Control pressure is applied to the waste-gate actuator diaphragm opening side when boost pressure is high enough to promote waste-gate opening. In one example, the turbo charger controller begins to ramp up the waste-gate command from one to zero (commanding 1 acts to close the waste-gate; commanding 0 acts to open the waste-gate). The routine proceeds to step  213 . 
         [0037]    At step  213 , the routine monitors boost pressure and determines if a boost pressure infection point is present. Boost pressure can be monitored using a pressure transducer located upstream from the turbocharger compressor. If boost pressure is increasing as the valve opening command is increased, the routine determines if the boost pressure slope has changed. If boost pressure is substantially constant when the valve begins to open boost pressure should begin to be reduced. 
         [0038]    If an inflection point is determined (e.g., by a change in boost slope), the control pressure and force at which the inflection is observed are stored in memory. Storing the variables to memory acts to update and adjust the spring closing force in the force function described in step  207 .  FIG. 7  describe one example way to determine waste-gate spring closing force. The routine then proceeds to step  215  after storing parameters to memory unless the system only incorporates a single turbo charger. If a single turbo charger is present the routine proceeds to step  205 . 
         [0039]    If an infection of boost pressure is not observed, the routine returns to step  211  where the control pressure command is increased. In this way, the control pressure command is ramped until the waste-gate spring closing force is determined. 
         [0040]    At step  215 , the routine begins learning waste-gate valve opening parameters for a second turbo charger. Like the operation described in step  211 , the turbo charger controller begins to ramp the turbo charger control pressure command and then proceeds to step  215 . 
         [0041]    At step  217 , the routine determines if an inflection point in the boost pressure is observed for the second turbocharger. If not, the routine proceeds to step  215  where the waste-gate control command is increased further. If so, the control pressure and force at which the inflection is observed are adjusted and stored in memory and the routine proceeds to step  205 . Storing the variables to memory acts to update and adjust the spring closing force in the force function described in step  207 . 
         [0042]    At step  223 , the routine commands waste-gates to the closed position. Commanding the waste-gate to the closed position can be accomplished in a variety of ways. For example, for operating conditions where boost pressure is low, a command may be issued that allows the full boost pressure to be routed to the opening side of the waste-gate actuator diaphragm. This method may be used until the force produced by the boost approaches the amount of force produced by the waste-gate closing spring. If the force produced by the boost pressure acting on the waste-gate diaphragm approaches the closing spring force, the control pressure applied to the waste-gate diaphragm can be reduced by regulating the pressure control valve output. That is, the pressure developed at the control valve output can be set between the boost pressure and atmospheric pressure. The pressure on the waste-gate diaphragm can be held to a level such that the force acting on the opening side of the waste-gate diaphragm is less than the closing spring force by a predetermined amount of force. 
         [0043]    In another example, a command can be sent to the waste-gate pressure control valve that regulates the valve output such that the pressure on the waste-gate opening diaphragm is between boost pressure and atmospheric pressure anytime the control pressure force is less than closing spring force plus a predetermined force. The control pressure in this mode of operation is maintained at a level that produces less force on the opening side of the waste-gate actuator diaphragm than the closing spring force. 
         [0044]      FIG. 5  provides an example plot of control pressure regulation described in  FIG. 2 . After the commands to close the waste-gates are delivered the routine proceeds to exit. 
         [0045]    Referring now to  FIG. 3 , a schematic diagram of a spring assisted diaphragm operated waste-gate actuator control system is shown. The figure illustrates a single turbocharger but a system having two turbochargers can be configured likewise. 
         [0046]    The turbocharger is comprised of an exhaust impeller  31  that is attached to a compressor  32  via shaft  340 . Exhaust gases from engine cylinders provide the energy that causes the impeller to rotate. The turbocharger is configured with a waste-gate  33  that reduces turbocharger efficiency when opened. Waste-gate  33  is operated by spring assisted actuator  301 . Spring  318  normally holds the waste-gate in the closed position by applying force to diaphragm  310 . The spring side of actuator  301  is also vented to atmosphere by vent  308 . Solenoid  305  can be used to regulate the position of diaphragm  310  by applying pressure to chamber  309 . Waste-gate control solenoid  305  connects chamber  309  to atmospheric pressure by channel  325  or to boost pressure by channel  323 . The solenoid is shown in the vent to atmospheric pressure position. Controller  12  can provide a modulated signal to solenoid  305  so that pressure in chamber  309  is regulated between atmospheric pressure and boost pressure. When pressure is increased to chamber  309 , the spring closing force applied to diaphragm  310  by spring  318  can be overcome, thereby opening the waste-gate  33 . 
         [0047]    Controller  12  adjusts the pressure in chamber  309  by adjusting the duty cycle supplied to solenoid control valve  305 . The duty cycle is varied in response to feedback from pressure sensor  115 . In one example, as described above, control pressure is related to waste-gate position and waste-gate position at selected engine operating conditions is related to a predetermined boost pressure. In this way, a control pressure can be related to a desired amount of boost. If the actual boost pressure, which is determined from pressure sensor  15 , does not match the desired boost pressure, adjustments can be made to the control signal that is applied to regulator valve  305  in order to adjust waste-gate position. 
         [0048]    Referring now to  FIG. 4 , a plot that illustrates prior art waste-gate control commands is shown. The Y axis represents duty cycle applied to the waste-gate pressure control valve. In this example, zero corresponds to no voltage being applied to the valve. One (at label  401 ) represents full voltage being applied to the valve. And a value of 0.5 represents 50% voltage on time at a predetermined control frequency. The X axis represents desired boost pressure generated by the turbocharger. Desired boost pressure increases from left to right. 
         [0049]    Also note that higher boost pressures are typically available at higher engine speeds because higher engine mass flow rates deliver more exhaust energy to the turbocharger. 
         [0050]    This example plot illustrates a pressure regulator that connects boost pressure to the waste-gate actuator diaphragm opening side when no voltage is applied to the pressure control valve. The control command is held at zero until desired boost pressure reaches a level that calls for lowering the diaphragm opening pressure below boost pressure so that boost pressure can be regulated by adjusting the waste-gate opening amount, see  402 . At this point, the force produced by boost pressure acting on the waste-gate actuator diaphragm is close to the closing spring force and flow through the waste-gate is minimal. This is the operating point at which system dynamics and adjusting the control pressure can cause the waste-gate closing apparatus to flutter or rapidly open and close in an undesirable manner. Thus, applying a duty cycle to the pressure regulator allows the turbocharger to develop a desired boost pressure, but the system can exhibit waste-gate flutter under some conditions. 
         [0051]    The control command duty cycle is shown increasing from  402  to  403 . The waste-gate moves from a full closed position to a full open position so that some exhaust gas energy bypasses the turbocharger at higher engine mass flow conditions. This action allows controller  12  to regulate turbocharger boost pressure. 
         [0052]    Referring now to  FIG. 5 , a plot that illustrates an example of the present method is shown. The X and Y axis of  FIG. 5  are the same as illustrated in  FIG. 4 . Label  501  represents full voltage applied to the waste-gate pressure control valve. Label  504  represents the position at which a duty cycle is applied in  FIG. 4 . 
         [0053]    In this example, the pressure command can occupy region  505 , a small region is excluded near label  504  where the duty cycle may not be sufficient to ensure positive waste-gate opening and closing. The size of this region is for illustration purposes only and not intended to limit the description. Label  502  represents a desired boost pressure at which the waste-gate is allowed to begin to open. This pressure is also shown for illustration purposes and is not intended to limit the scope or breadth of this description. Notice that this boost pressure is higher than the boost pressure at  504  (and higher than the desired boost pressure at  402 ) and that the duty cycle is non-zero. The higher boost pressure and non-zero duty cycle increase the possibility of positive waste-gate opening and closing. Higher boost pressure increases the amount of force that is available to open the waste-gate and the duty cycle opens the waste-gate to a position that has a higher flow rate. These actions can decrease the possibility of waste-gate flutter. In an example, the duty cycle is set to one until waste-gate opening is allowed. Waste-gate opening is not allowed until the expected waste-gate flowrate is non-zero. 
         [0054]    Like the control command described in  FIG. 4 , the control command duty cycle increases from  502  to  503 . As explained above, this causes the waste-gate actuator to move from a closed position to an open position as mass flow through the engine increases. 
         [0055]    Referring now to  FIG. 6 , a plot of different turbocharger operating modes is shown. The Y axis represents boost pressure developed by a turbocharger while the X axis represents air mass flow through the engine. 
         [0056]    Curve  603  represents the amount of turbocharger boost developed when the turbocharger waste-gate is held in a closed position. This curve essentially represents the turbocharger&#39;s capacity to generate boost. Curve  601  illustrates boost generated by a turbocharger having boost pressure in direct communication with a spring return waste-gate actuator. At  604 , the boost pressure reaches a level that overcomes the waste-gate actuator closing spring force and opens the waste-gate. This allows engine exhaust gases to bypass the turbocharger, thereby limiting the boost from  604  to the end of  601 . By controlling pressure on the opening side of the waste-gate actuator diaphragm, boost pressures between curve  603  and  601  may be produced. 
         [0057]    Opening of a waste-gate can be characterized by holding both waste-gates of a twin parallel turbocharged engine closed then subsequently varying the control of a single waste-gate to determine at what duty cycle command the waste-gate begins to open. Waste-gate opening can be noted by a reduction in boost pressure. 
         [0058]    Referring now to  FIG. 7 , a plot of an example way to determine waste-gate actuator spring closing forces for a twin turbocharger system is shown. The X represents boost pressure at substantially constant engine operating conditions. The Y axis represents time. Curve  701  represents the amount of boost pressure developed at different waste-gate operating conditions. At  702 , both turbocharger waste-gates are held in a closed position and the boost pressure is substantially constant. Before  703 , a commanded duty cycle is ramped to a first waste-gate actuator control pressure regulation valve that causes the waste-gate to open. The force created by the control pressure acting on the waste-gate actuator diaphragm opening side when the boost pressure begins to decrease (i.e., the inflection point) is the valve closing force used to update the force function described in the method of  FIG. 2 . After the first waste-gate at least partially opens, it is maintained in a partially open position while a command to open the second waste-gate is issued. Positioning the waste-gate in a partially open location allows both turbochargers to continue to pump while the waste-gate actuator closing spring force is determined. Alternatively, the first waste-gate can be closed while a command to open the second waste-gate is issued. When the opening force applied to the second waste-gate by commanding the pressure control valve exceeds the closing spring force the second waste-gate begins to open and the boost pressure begins to decrease at  705 . This method allows for the determination of spring closing forces for each turbocharger of a multiple turbocharged engine. 
         [0059]    As an alternative, closing spring forces may be determined when mass flow through an engine is changing (e.g., during acceleration) by monitoring a change in boost pressure slope as the waste-gate opening command is ramped from a full close command to an open command. Once the first turbocharger reaches an open condition, the second turbocharger command begins to be ramped from a close command to an open command. The waste-gate closing spring force can again be determined by monitoring the boost pressure for a change in slope. As mentioned above, the force function described in the method of  FIG. 2  can be adjusted and stored to memory when the waste-gate actuator spring closing force is determined. 
         [0060]    This concludes the description. The reading of it by those skilled in the art would bring to mind many alterations and modifications without departing from the spirit and the scope of the description. For example, 13, 14, 15, V6, V8, V10, and V12 engines operating in natural gas, gasoline, diesel, or alternative fuel configurations could use the present description to advantage.