Patent Publication Number: US-8112207-B2

Title: Method and apparatus to determine a preferred output torque for operating a hybrid transmission in a continuously variable mode

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 60/985,382, filed on Nov. 5, 2007 which is hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This disclosure pertains to control systems for hybrid powertrain systems. 
     BACKGROUND 
     The statements in this section merely provide background information related to the present disclosure and may not constitute prior art. 
     Known hybrid powertrain architectures include torque-generative devices, including internal combustion engines and electric machines, which transfer torque through a transmission device to an output member. One exemplary hybrid powertrain includes a two-mode, compound-split, electro-mechanical transmission which utilizes an input member for receiving tractive torque from a prime mover power source, preferably an internal combustion engine, and an output member. The output member can be operatively connected to a driveline for a motor vehicle for transferring tractive torque thereto. Electric machines, operative as motors or generators, generate a torque input to the transmission, independently of a torque input from the internal combustion engine. The electric machines may transform vehicle kinetic energy, transferred through the vehicle driveline, to electrical energy that is storable in an electrical energy storage device. A control system monitors various inputs from the vehicle and the operator and provides operational control of the hybrid powertrain, including controlling transmission operating state and gear shifting, controlling the torque-generative devices, and regulating the electrical power interchange among the electrical energy storage device and the electric machines to manage outputs of the transmission, including torque and rotational speed. 
     SUMMARY 
     A hybrid transmission includes a torque machine and an energy storage device connected thereto. The hybrid transmission is operative to transfer power between an input member and an output member and the torque machine in a continuously variable operating range state. A method for controlling the hybrid transmission includes determining a preferred output torque, determining a relationship between power from the energy storage device and an output torque of the transmission, determining power constraints from the energy storage device, determining motor torque constraints for the torque machine, determining linear torque constraints to the output torque based upon the motor torque constraints for the torque machine, determining quadratic output torque constraints based upon the power constraints from the energy storage device and the relationship between the power from the energy storage device and the output torque of the transmission device, and determining an output torque to the output member responsive to the preferred output torque and achievable based upon the linear output torque constraints and the quadratic output torque constraints. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       One or more embodiments will now be described, by way of example, with reference to the accompanying drawings, in which: 
         FIG. 1  is a schematic diagram of an exemplary hybrid powertrain, in accordance with the present disclosure; 
         FIG. 2  is a schematic diagram of an exemplary architecture for a control system and hybrid powertrain, in accordance with the present disclosure; 
         FIG. 3  is a schematic flow diagram of a control system architecture for controlling and managing torque in a hybrid powertrain system, in accordance with the present disclosure, 
         FIG. 4  is a graphical depiction, in accordance with the present disclosure; 
         FIGS. 5A and 5B  are schematic flow diagrams of a control system architecture for controlling and managing torque in a hybrid powertrain system, in accordance with the present disclosure, 
         FIGS. 6A and 6B  are graphical depictions, in accordance with the present disclosure; 
         FIGS. 7 ,  8 , and  9  are algorithmic flowcharts, in accordance with the present disclosure; and 
         FIG. 10  is a graphical depiction, in accordance with the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Referring now to the drawings, wherein the showings are for the purpose of illustrating certain exemplary embodiments only and not for the purpose of limiting the same,  FIGS. 1 and 2  depict an exemplary hybrid powertrain. The exemplary hybrid powertrain in accordance with the present disclosure is depicted in  FIG. 1 , comprising a two-mode, compound-split, electro-mechanical hybrid transmission  10  operatively connected to an engine  14  and torque machines comprising first and second electric machines (‘MG-A’)  56  and (‘MG-B’)  72 . The engine  14  and first and second electric machines  56  and  72  each generate mechanical power which can be transferred to the transmission  10 . The engine  14 , transmission  10  and the torque machines comprising the first and second electric machines in this embodiment comprise torque actuators. The power generated by the engine  14  and the first and second electric machines  56  and  72  and transferred to the transmission  10  is described in terms of input and motor torques, referred to herein as T I , T A , and T B  respectively, and speed, referred to herein as N I , N A , and N B , respectively. 
     The exemplary engine  14  comprises a multi-cylinder internal combustion engine selectively operative in several states to transfer torque to the transmission  10  via an input shaft  12 , and can be either a spark-ignition or a compression-ignition engine. The engine  14  includes a crankshaft (not shown) operatively coupled to the input shaft  12  of the transmission  10 . A rotational speed sensor  11  monitors rotational speed of the input shaft  12 . Power output from the engine  14 , comprising rotational speed and engine torque, can differ from the input speed N I  and the input torque T I  to the transmission  10  due to placement of torque-consuming components on the input shaft  12  between the engine  14  and the transmission  10 , e.g., a hydraulic pump (not shown) and/or a torque management device (not shown). 
     The exemplary transmission  10  comprises three planetary-gear sets  24 ,  26  and  28 , and four selectively engageable torque-transferring devices, i.e., clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75 . As used herein, clutches refer to any type of friction torque transfer device including single or compound plate clutches or packs, band clutches, and brakes, for example. A hydraulic control circuit (‘HYD’)  42 , preferably controlled by a transmission control module (hereafter ‘TCM’)  17 , is operative to control clutch states. Clutches C 2   62  and C 4   75  preferably comprise hydraulically-applied rotating friction clutches. Clutches C 1   70  and C 3   73  preferably comprise hydraulically-controlled stationary devices that can be selectively grounded to a transmission case  68 . Each of the clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75  is preferably hydraulically applied, selectively receiving pressurized hydraulic fluid via the hydraulic control circuit  42 . 
     The first and second electric machines  56  and  72  preferably comprise three-phase AC machines, each including a stator (not shown) and a rotor (not shown), and respective resolvers  80  and  82 . The motor stator for each machine is grounded to an outer portion of the transmission case  68 , and includes a stator core with coiled electrical windings extending therefrom. The rotor for the first electric machine  56  is supported on a hub plate gear that is operatively attached to shaft  60  via the second planetary gear set  26 . The rotor for the second electric machine  72  is fixedly attached to a sleeve shaft hub  66 . 
     Each of the resolvers  80  and  82  preferably comprises a variable reluctance device including a resolver stator (not shown) and a resolver rotor (not shown). The resolvers  80  and  82  are appropriately positioned and assembled on respective ones of the first and second electric machines  56  and  72 . Stators of respective ones of the resolvers  80  and  82  are operatively connected to one of the stators for the first and second electric machines  56  and  72 . The resolver rotors are operatively connected to the rotor for the corresponding first and second electric machines  56  and  72 . Each of the resolvers  80  and  82  is signally and operatively connected to a transmission power inverter control module (hereafter ‘TPIM’)  19 , and each senses and monitors rotational position of the resolver rotor relative to the resolver stator, thus monitoring rotational position of respective ones of first and second electric machines  56  and  72 . Additionally, the signals output from the resolvers  80  and  82  are interpreted to provide the rotational speeds for first and second electric machines  56  and  72 , i.e., N A  and N B , respectively. 
     The transmission  10  includes an output member  64 , e.g. a shaft, which is operably connected to a driveline  90  for a vehicle (not shown), to provide output power to the driveline  90  that is transferred to vehicle wheels  93 , one of which is shown in  FIG. 1 . The output power at the output member  64  is characterized in terms of an output rotational speed N O  and an output torque T O . A transmission output speed sensor  84  monitors rotational speed and rotational direction of the output member  64 . Each of the vehicle wheels  93  is preferably equipped with a sensor  94  adapted to monitor wheel speed, the output of which is monitored by a control module of a distributed control module system described with respect to  FIG. 2 , to determine vehicle speed, and absolute and relative wheel speeds for braking control, traction control, and vehicle acceleration management. 
     The input torque from the engine  14  and the motor torques from the first and second electric machines  56  and  72  (T I , T A , and T B  respectively) are generated as a result of energy conversion from fuel or electrical potential stored in an electrical energy storage device (hereafter ‘ESD’)  74 . The ESD  74  is high voltage DC-coupled to the TPIM  19  via DC transfer conductors  27 . The transfer conductors  27  include a contactor switch  38 . When the contactor switch  38  is closed, under normal operation, electric current can flow between the ESD  74  and the TPIM  19 . When the contactor switch  38  is opened electric current flow between the ESD  74  and the TPIM  19  is interrupted. The TPIM  19  transmits electrical power to and from the first electric machine  56  by transfer conductors  29 , and the TPIM  19  similarly transmits electrical power to and from the second electric machine  72  by transfer conductors  31  to meet the torque commands for the first and second electric machines  56  and  72  in response to the motor torque commands T A  and T B . Electrical current is transmitted to and from the ESD  74  in accordance with whether the ESD  74  is being charged or discharged. 
     The TPIM  19  includes the pair of power inverters (not shown) and respective motor control modules (not shown) configured to receive the motor torque commands and control inverter states therefrom for providing motor drive or regenerative functionality to meet the commanded motor torques T A  and T B . The power inverters comprise known complementary three-phase power electronics devices, and each includes a plurality of insulated gate bipolar transistors (not shown) for converting DC power from the ESD  74  to AC power for powering respective ones of the first and second electric machines  56  and  72 , by switching at high frequencies. The insulated gate bipolar transistors form a switch mode power supply configured to receive control commands. There is typically one pair of insulated gate bipolar transistors for each phase of each of the three-phase electric machines. States of the insulated gate bipolar transistors are controlled to provide motor drive mechanical power generation or electric power regeneration functionality. The three-phase inverters receive or supply DC electric power via DC transfer conductors  27  and transform it to or from three-phase AC power, which is conducted to or from the first and second electric machines  56  and  72  for operation as motors or generators via transfer conductors  29  and  31  respectively. 
       FIG. 2  is a schematic block diagram of the distributed control module system. The elements described hereinafter comprise a subset of an overall vehicle control architecture, and provide coordinated system control of the exemplary hybrid powertrain described in  FIG. 1 . The distributed control module system synthesizes pertinent information and inputs, and executes algorithms to control various actuators to meet control objectives, including objectives related to fuel economy, emissions, performance, drivability, and protection of hardware, including batteries of ESD  74  and the first and second electric machines  56  and  72 . The distributed control module system includes an engine control module (hereafter ‘ECM’)  23 , the TCM  17 , a battery pack control module (hereafter ‘BPCM’)  21 , and the TPIM  19 . A hybrid control module (hereafter ‘HCP’)  5  provides supervisory control and coordination of the ECM  23 , the TCM  17 , the BPCM  21 , and the TPIM  19 . A user interface (‘UI’)  13  is operatively connected to a plurality of devices through which a vehicle operator controls or directs operation of the electro-mechanical hybrid powertrain. The devices include an accelerator pedal  113  (‘AP’), an operator brake pedal  112  (‘BP’), a transmission gear selector  114  (‘PRNDL’), and a vehicle speed cruise control (not shown). The transmission gear selector  114  may have a discrete number of operator-selectable positions, including the rotational direction of the output member  64  to enable one of a forward and a reverse direction. 
     The aforementioned control modules communicate with other control modules, sensors, and actuators via a local area network (hereafter ‘LAN’) bus  6 . The LAN bus  6  allows for structured communication of states of operating parameters and actuator command signals between the various control modules. The specific communication protocol utilized is application-specific. The LAN bus  6  and appropriate protocols provide for robust messaging and multi-control module interfacing between the aforementioned control modules, and other control modules providing functionality including e.g., antilock braking, traction control, and vehicle stability. Multiple communications buses may be used to improve communications speed and provide some level of signal redundancy and integrity. Communication between individual control modules can also be effected using a direct link, e.g., a serial peripheral interface (‘SPI’) bus (not shown). 
     The HCP  5  provides supervisory control of the hybrid powertrain, serving to coordinate operation of the ECM  23 , TCM  17 , TPIM  19 , and BPCM  21 . Based upon various input signals from the user interface  13  and the hybrid powertrain, including the ESD  74 , the HCP  5  determines an operator torque request, an output torque command, an engine input torque command, clutch torque(s) for the applied torque-transfer clutches C 1   70 , C 2   62 , C 3   73 , C 4   75  of the transmission  10 , and the motor torque commands T A  and T B  for the first and second electric machines  56  and  72 . 
     The ECM  23  is operatively connected to the engine  14 , and functions to acquire data from sensors and control actuators of the engine  14  over a plurality of discrete lines, shown for simplicity as an aggregate bi-directional interface cable  35 . The ECM  23  receives the engine input torque command from the HCP  5 . The ECM  23  determines the actual engine input torque, T I , provided to the transmission  10  at that point in time based upon monitored engine speed and load, which is communicated to the HCP  5 . The ECM  23  monitors input from the rotational speed sensor  11  to determine the engine input speed to the input shaft  12 , which translates to the transmission input speed, N I . The ECM  23  monitors inputs from sensors (not shown) to determine states of other engine operating parameters including, e.g., a manifold pressure, engine coolant temperature, ambient air temperature, and ambient pressure. The engine load can be determined, for example, from the manifold pressure, or alternatively, from monitoring operator input to the accelerator pedal  113 . The ECM  23  generates and communicates command signals to control engine actuators, including, e.g., fuel injectors, ignition modules, and throttle control modules, none of which are shown. 
     The TCM  17  is operatively connected to the transmission  10  and monitors inputs from sensors (not shown) to determine states of transmission operating parameters. The TCM  17  generates and communicates command signals to control the transmission  10 , including controlling the hydraulic control circuit  42 . Inputs from the TCM  17  to the HCP  5  include estimated clutch torques for each of the clutches, i.e., C 1   70 , C 2   62 , C 3   73 , and C 4   75 , and rotational output speed, N O , of the output member  64 . Other actuators and sensors may be used to provide additional information from the TCM  17  to the HCP  5  for control purposes. The TCM  17  monitors inputs from pressure switches (not shown) and selectively actuates pressure control solenoids (not shown) and shift solenoids (not shown) of the hydraulic control circuit  42  to selectively actuate the various clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75  to achieve various transmission operating range states, as described hereinbelow. 
     The BPCM  21  is signally connected to sensors (not shown) to monitor the ESD  74 , including states of electrical current and voltage parameters, to provide information indicative of parametric states of the batteries of the ESD  74  to the HCP  5 . The parametric states of the batteries preferably include battery state-of-charge, battery voltage, battery temperature, and available battery power, referred to as a range P BAT     —     MIN  to P BAT     —     MAX . 
     A brake control module (hereafter ‘BrCM’)  22  is operatively connected to friction brakes (not shown) on each of the vehicle wheels  93 . The BrCM  22  monitors the operator input to the brake pedal  112  and generates control signals to control the friction brakes and sends a control signal to the HCP  5  to operate the first and second electric machines  56  and  72  based thereon. 
     Each of the control modules ECM  23 , TCM  17 , TPIM  19 , BPCM  21 , and BrCM  22  is preferably a general-purpose digital computer comprising a microprocessor or central processing unit, storage mediums comprising read only memory (‘ROM’), random access memory (‘RAM’), electrically programmable read only memory (‘EPROM’), a high speed clock, analog to digital (‘A/D’) and digital to analog (‘D/A’) circuitry, and input/output circuitry and devices (‘I/O’) and appropriate signal conditioning and buffer circuitry. Each of the control modules has a set of control algorithms, comprising resident program instructions and calibrations stored in one of the storage mediums and executed to provide the respective functions of each computer. Information transfer between the control modules is preferably accomplished using the LAN bus  6  and SPI buses. The control algorithms are executed during preset loop cycles such that each algorithm is executed at least once each loop cycle. Algorithms stored in the non-volatile memory devices are executed by one of the central processing units to monitor inputs from the sensing devices and execute control and diagnostic routines to control operation of the actuators, using preset calibrations. Loop cycles are executed at regular intervals, for example each 3.125, 6.25, 12.5, 25 and 100 milliseconds during ongoing operation of the hybrid powertrain. Alternatively, algorithms may be executed in response to the occurrence of an event. 
     The exemplary hybrid powertrain selectively operates in one of several states that can be described in terms of engine states comprising one of an engine-on state (‘ON’) and an engine-off state (‘OFF’), and transmission operating range states comprising a plurality of fixed gears and continuously variable operating modes, described with reference to Table 1, below. 
     
       
         
           
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                 Engine 
                 Transmission Operating 
                 Applied 
               
               
                 Description 
                 State 
                 Range State 
                 Clutches 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 M1_Eng_Off 
                 OFF 
                 EVT Mode 1 
                 C1 70 
                   
               
               
                 M1_Eng_On 
                 ON 
                 EVT Mode 1 
                 C1 70 
               
               
                 G1 
                 ON 
                 Fixed Gear Ratio 1 
                 C1 70 
                 C4 75 
               
               
                 G2 
                 ON 
                 Fixed Gear Ratio 2 
                 C1 70 
                 C2 62 
               
               
                 M2_Eng_Off 
                 OFF 
                 EVT Mode 2 
                 C2 62 
               
               
                 M2_Eng_On 
                 ON 
                 EVT Mode 2 
                 C2 62 
               
               
                 G3 
                 ON 
                 Fixed Gear Ratio 3 
                 C2 62 
                 C4 75 
               
               
                 G4 
                 ON 
                 Fixed Gear Ratio 4 
                 C2 62 
                 C3 73 
               
               
                   
               
            
           
         
       
     
     Each of the transmission operating range states is described in the table and indicates which of the specific clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75  are applied for each of the operating range states. A first continuously variable mode, i.e., EVT Mode  1 , or M 1 , is selected by applying clutch C 1   70  only in order to “ground” the outer gear member of the third planetary gear set  28 . The engine state can be one of ON (‘M 1 _Eng_On’) or OFF (‘M 1 _Eng_Off’). A second continuously variable mode, i.e., EVT Mode  2 , or M 2 , is selected by applying clutch C 2   62  only to connect the shaft  60  to the carrier of the third planetary gear set  28 . The engine state can be one of ON (‘M 2 _Eng_On’) or OFF (‘M 2 _Eng_Off’). For purposes of this description, when the engine state is OFF, the engine input speed is equal to zero revolutions per minute (‘RPM’), i.e., the engine crankshaft is not rotating. A fixed gear operation provides a fixed ratio operation of input-to-output speed of the transmission  10 , i.e., N I /N O . A first fixed gear operation (‘G 1 ’) is selected by applying clutches C 1   70  and C 4   75 . A second fixed gear operation (‘G 2 ’) is selected by applying clutches C 1   70  and C 2   62 . A third fixed gear operation (‘G 3 ’) is selected by applying clutches C 2   62  and C 4   75 . A fourth fixed gear operation (‘G 4 ’) is selected by applying clutches C 2   62  and C 3   73 . The fixed ratio operation of input-to-output speed increases with increased fixed gear operation due to decreased gear ratios in the planetary gears  24 ,  26 , and  28 . The rotational speeds of the first and second electric machines  56  and  72 , N A  and N B  respectively, are dependent on internal rotation of the mechanism as defined by the clutching and are proportional to the input speed measured at the input shaft  12 . 
     In response to operator input via the accelerator pedal  113  and brake pedal  112  as captured by the user interface  13 , the HCP  5  and one or more of the other control modules determine torque commands to control the torque-generative devices comprising the engine  14  and the first and second electric machines  56  and  72  to meet the operator torque request at the output member  64  and transferred to the driveline  90 . Based upon input signals from the user interface  13  and the hybrid powertrain including the ESD  74 , the HCP  5  determines the operator torque request, the output torque commanded from the transmission  10  to the driveline  90 , an input torque from the engine  14 , clutch torques for the torque-transfer clutches C 1   70 , C 2   62 , C 3   73 , C 4   75  of the transmission  10 ; and the motor torques for the first and second electric machines  56  and  72 , respectively, as is described hereinbelow. 
     Final vehicle acceleration can be affected by other factors including, e.g., road load, road grade, and vehicle mass. The engine state and the transmission operating range state are determined based upon operating characteristics of the hybrid powertrain. This includes the operator torque request communicated through the accelerator pedal  113  and brake pedal  112  to the user interface  13  as previously described. The transmission operating range state and the engine state may be predicated on a hybrid powertrain torque demand caused by a command to operate the first and second electric machines  56  and  72  in an electrical energy generating mode or in a torque generating mode. The transmission operating range state and the engine state can be determined by an optimization algorithm or routine which determines optimum system efficiency based upon operator demand for power, battery state of charge, and energy efficiencies of the engine  14  and the first and second electric machines  56  and  72 . The control system manages torque inputs from the engine  14  and the first and second electric machines  56  and  72  based upon an outcome of the executed optimization routine, and system efficiencies are optimized thereby, to manage fuel economy and battery charging. Furthermore, operation can be determined based upon a fault in a component or system. The HCP  5  monitors the torque-generative devices, and determines the power output from the transmission  10  at output member  64  that is required to meet the operator torque request while meeting other powertrain operating demands, e.g., charging the ESD  74 . As should be apparent from the description above, the ESD  74  and the first and second electric machines  56  and  72  are electrically-operatively coupled for power flow therebetween. Furthermore, the engine  14 , the first and second electric machines  56  and  72 , and the electro-mechanical transmission  10  are mechanically-operatively coupled to transfer power therebetween to generate a power flow to the output member  64 . 
       FIG. 3  shows a control system architecture for controlling and managing signal flow in a hybrid powertrain system having multiple torque generative devices, described hereinbelow with reference to the hybrid powertrain system of  FIGS. 1 and 2 , and residing in the aforementioned control modules in the form of executable algorithms and calibrations. The control system architecture is applicable to alternative hybrid powertrain systems having multiple torque generative devices, including, e.g., a hybrid powertrain system having an engine and a single electric machine, a hybrid powertrain system having an engine and multiple electric machines. Alternatively, the hybrid powertrain system can utilize non-electric torque machines and energy storage systems, e.g., hydraulic-mechanical hybrid transmissions using hydraulically powered torque machines (not shown). 
     In operation, the operator inputs to the accelerator pedal  113  and the brake pedal  112  are monitored to determine the operator torque request (‘To_req’). Operation of the engine  14  and the transmission  10  are monitored to determine the input speed (‘Ni’) and the output speed (‘No’). A strategic optimization control scheme (‘Strategic Control’)  310  determines a preferred input speed (‘Ni_Des’) and a preferred engine state and transmission operating range state (‘Hybrid Range State Des’) based upon the output speed and the operator torque request, and optimized based upon other operating parameters of the hybrid powertrain, including battery power limits and response limits of the engine  14 , the transmission  10 , and the first and second electric machines  56  and  72 . The strategic optimization control scheme  310  is preferably executed by the HCP  5  during each 100 ms loop cycle and each 25 ms loop cycle. The desired operating range state for the transmission  10  and the desired input speed from the engine  14  to the transmission  10  are inputs to the shift execution and engine start/stop control scheme  320 . 
     The shift execution and engine start/stop control scheme  320  commands changes in the transmission operation (‘Transmission Commands’) including changing the operating range state based upon the inputs and operation of the powertrain system. This includes commanding execution of a change in the transmission operating range state if the preferred operating range state is different from the present operating range state by commanding changes in application of one or more of the clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75  and other transmission commands. The present operating range state (‘Hybrid Range State Actual’) and an input speed profile (‘Ni_Prof’) can be determined. The input speed profile is an estimate of an upcoming input speed and preferably comprises a scalar parametric value that is a targeted input speed for the forthcoming loop cycle. The engine operating commands and the operator torque request are based upon the input speed profile during a transition in the operating range state of the transmission. A clutch torque (‘Tcl’) for each clutch is estimated in the TCM  17 , including the presently applied clutches and the non-applied clutches. 
     A tactical control scheme (‘Tactical Control and Operation’)  330  is repeatedly executed during one of the control loop cycles to determine engine commands (‘Engine Commands’) for operating the engine, including a preferred input torque from the engine  14  to the transmission  10  based upon the output speed, the input speed, and the operator torque request and the present operating range state for the transmission. The engine commands also include engine states including one of an all-cylinder operating state and a cylinder deactivation operating state wherein a portion of the engine cylinders are deactivated and unfueled, and engine states including one of a fueled state and a fuel cutoff state. A present engine input torque (‘Ti’) reacting with the input member  12  is determined in the ECM  23 . 
     An output and motor torque determination scheme (‘Output and Motor Torque Determination’)  340  is executed to determine the preferred output torque from the powertrain (‘To_cmd’). This includes determining motor torque commands (‘T A ’, ‘T B ’) to transfer a net output torque to the output member  64  of the transmission  10  that meets the operator torque request, by controlling the first and second electric machines  56  and  72  in this embodiment. The immediate accelerator output torque request, the immediate brake output torque request, the present input torque from the engine  14  and the estimated applied clutch torque(s), the present operating range state of the transmission  10 , the input speed, the input speed profile, and the axle torque response type are inputs. The output and motor torque determination scheme  340  executes to determine the motor torque commands during each iteration of one of the loop cycles. The output and motor torque determination scheme  340  includes algorithmic code which is regularly executed during the 6.25 ms and 12.5 ms loop cycles to determine the preferred motor torque commands. 
     The hybrid powertrain is controlled to transfer the output torque to the output member  64  and thence to the driveline  90  to generate tractive torque at wheel(s)  93  to forwardly propel the vehicle in response to the operator input to the accelerator pedal  113  when the operator selected position of the transmission gear selector  114  commands operation of the vehicle in the forward direction. Similarly, the hybrid powertrain is controlled to transfer the output torque to the output member  64  and thence to the driveline  90  to generate tractive torque at wheel(s)  93  to propel the vehicle in a reverse direction in response to the operator input to the accelerator pedal  113  when the operator selected position of the transmission gear selector  114  commands operation of the vehicle in the reverse direction. Preferably, propelling the vehicle results in vehicle acceleration so long as the output torque is sufficient to overcome external loads on the vehicle, e.g., due to road grade, aerodynamic loads, and other loads. 
     Operation of the engine  14  and transmission  10  is constrained by power, torque and rotational speed limits of the engine  14 , the first and second electric machines  56  and  72 , the ESD  74  and the clutches C 1   70 , C 2   62 , C 3   73 , and C 4   75 . The operating constraints on the engine  14  and transmission  10  can be translated to a set of system constraint equations executed as one or more algorithms in one of the control modules, e.g., the HCP  5 . 
     Referring again to  FIG. 1 , in overall operation, the transmission  10  operates in one of the operating range states through selective actuation of the torque-transfer clutches in one embodiment. Torque constraints for each of the engine  14  and the first and second electric machines  56  and  72  and speed constraints for the engine  14  and the first and second electric machines  56  and  72  are determined. Battery power constraints for the ESD  74 , i.e., the available battery power is determined, and applied to further limit operation of the first and second electric machines  56  and  72 . The preferred operating region for the powertrain is determined using the system constraint equations that are based upon the battery power constraints, the motor torque constraints, the speed constraints, and clutch reactive torque constraints. The preferred operating region comprises a range of permissible operating torques or speeds for the engine  14  and the first and second electric machines  56  and  72 . By deriving and simultaneously solving dynamics equations of the transmission  10 , the torque limit, in this embodiment the output torque T O , can be determined using linear equations including: the following.
 
 T   M1   =T   A to T   M1   *T   A   +T   B to T   M1   *T   B +Misc —   T   M1 ;  [1]
 
 T   M2   =T   A to T   M2   *T   A   +T   B to T   M2   *T   B +Misc —   T   M2 ;  [2]
 
 T   M3   =T   A to T   M3   *T   A   +T   B to T   M3   *T   B +Misc —   T   M3   [3]
 
     In one embodiment, the torque values include: T M1  representing the output torque T O  at output member  64 , T M2  representing the input torque T I  at input shaft  12 , and T M3  representing the clutch torque(s) for the applied torque-transfer clutches C 1   70 , C 2   62 , C 3   73 , C 4   75  of the transmission  10 ; 
     The factors T A toT M1 , T A toT M2 , and T A toT M3  are contributing factors of T A  to T M1 , T M2 , T M3 , respectively. The factors T B toT M1 , T B toT M2 , and T B toT M3  are contributing factors of T B  to T M1 , T M2 , T M3 , respectively. The factors Misc_T M1 , Misc_T M2 , and Misc_T M3  are constants which contribute to T M1 , T M2 , T M3  by non T A , T B , T M1 , T M2  and T M3  parameters such as time-rate changes in speed of the input member  12 , time-rate changes in speed of the output member  64 , and slip speed(s) of the torque-transfer clutches C 1   70 , C 2   62 , C 3   73 , C 4   75  depending on the applications, and are described herein. The torque parameters T A  and T B  are the motor torques from the first and second electric machines  56  and  72 . The torque parameters T M1 , T M2 , T M3  can be any three independent parameters, depending upon the operating range state and the application. 
     The engine  14  and transmission  10  and the first and second electric machines  56  and  72  have speed constraints, torque constraints, and battery power constraints due to mechanical and system limitations. The speed constraints can include engine speed constraints of N I =0 (engine off state), and N I  ranging from 600 rpm (idle) to 6000 rpm for the engine  14 . Exemplary speed constraints for the first and second electric machines  56  and  72  for this embodiment can be:
 
−10,500 rpm≦ N   A ≦+10,500 rpm, and,
 
−10,500 rpm≦ N   B ≦+10,500 rpm,
 
and can vary based upon operating conditions. The torque constraints include motor torque constraints for the first and second electric machines  56  and  72  including T A     —     MIN ≦T A ≦T A     —     MAX  and T B     —     MIN≦T   B ≦T B     —     MAX . The motor torque constraints T A     —     MAX  and T A     —     MIN  comprise torque limits for the first electric machine  56  when working as a torque motor and an electric generator at positive rotational speeds, respectively. The motor torque constraints T B     —     MAX  and T B     —     MIN  comprise torque limits for the second electric machine  72  when working as a torque motor and an electric generator at positive rotational speeds, respectively. The maximum and minimum motor torques T A     —     MAX , T A     —     MIN , T B     —     MAX , and T B     —     MIN  are preferably obtained from data sets stored in tabular format within one of the memory devices of one of the control modules. Such data sets can be empirically derived from conventional dynamometer testing of the combined motor and power electronics, e.g., the TPIM  19 , at various temperature and voltage conditions. Battery power constraints comprise the available battery power within the range of P BAT     —     MIN  to P BAT     —     MAX , wherein P BAT     —     MIN  is maximum allowable battery charging power and P BAT     —     MAX  is the maximum allowable battery discharging power. The operation of the system described hereinbelow is determined at known engine input speeds and torques, and thus the derivation of the equations is based upon torque transfer within the transmission  14 .
 
     An operating range, comprising a torque output range is determinable based upon the battery power constraints of the ESD  74 . Calculation of battery power usage, P BAT  is as follows:
 
 P   BAT   =P   A,ELEC   +P   B,ELEC   +P   DC     —     LOAD   [4]
 
wherein P A,ELEC  comprises electrical power from the first electric machine  56 ,
         P B,ELEC  comprises electrical power from the second electric machine  72 , and   P DC     —     LOAD  comprises known DC load, including accessory loads.
 
Substituting equations for P A,ELEC  and P B,ELEC , yields the following equation:
 
 P   BAT =( P   A,MECH   +P   A,LOSS )+( P   B,MECH   +P   B,LOSS )+ P   DC LOAD   [5]
 
wherein P A,MECH  comprises mechanical power from the first electric machine  56 ,
   P A,LOSS  comprises power losses from the first electric machine  56 ,   P B,MECH  comprises mechanical power from the second electric machine  72 , and   P B,LOSS  comprises power losses from the second electric machine  72 .       

     Eq. 5 can be restated as Eq. 6, below, wherein speeds, N A  and N B , and torques, T A  and T B , are substituted for powers P A  and P B . This includes an assumption that motor and inverter losses can be mathematically modeled as a quadratic equation based upon torque, as follows: 
                     P   BAT     =       (         N   A     ⁢     T   A       +     (           a   1     ⁡     (     N   A     )       ⁢     T   A   2       +         a   2     ⁡     (     N   A     )       ⁢     T   A       +       a   3     ⁡     (     N   A     )         )       )     +     (         N   B     ⁢     T   B       +     (           B   1     ⁡     (     N   B     )       ⁢     T   B   2       +         B   2     ⁡     (     N   B     )       ⁢     T   B       +       B   3     ⁡     (     N   B     )         )       )     +     PDC_   LOAD               [   6   ]               
wherein N A , N B  comprise motor speeds for the first and second electric machines  56  and  72 ,
         T A , T B  comprise the motor torques for the first and second electric machines  56  and  72 , and   a1, a2, a3, b1, b2, b3 each comprise quadratic coefficients which are a function of respective motor speeds, N A , N B .       

     This can be restated as follows: 
     
       
         
           
             
               
                 
                   
                     P 
                     BAT 
                   
                   = 
                   
                     
                       
                         a 
                         1 
                         * 
                       
                       ⁢ 
                       
                         T 
                         A 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             N 
                             A 
                           
                           + 
                           
                             a 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         T 
                         A 
                       
                     
                     + 
                     
                       
                         b 
                         1 
                       
                       * 
                       
                         T 
                         B 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             N 
                             B 
                           
                           + 
                           
                             b 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         T 
                         B 
                       
                     
                     + 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       P 
                       
                         DC 
                         ⁢ 
                         _ 
                         ⁢ 
                         LOAD 
                       
                     
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
           
         
       
     
     This reduces to: 
     
       
         
           
             
               
                 
                   
                     P 
                     BAT 
                   
                   = 
                   
                     
                       
                         a 
                         1 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             T 
                             A 
                             2 
                           
                           + 
                           
                             
                               
                                 T 
                                 A 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     N 
                                     A 
                                   
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             / 
                             
                               a 
                               1 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       N 
                                       A 
                                     
                                     + 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 / 
                                 
                                   ( 
                                   
                                     2 
                                     * 
                                     
                                       a 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         b 
                         1 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             T 
                             B 
                             2 
                           
                           + 
                           
                             
                               
                                 T 
                                 B 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     N 
                                     B 
                                   
                                   + 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             / 
                             
                               b 
                               1 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       N 
                                       B 
                                     
                                     + 
                                     
                                       b 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 / 
                                 
                                   ( 
                                   
                                     2 
                                     * 
                                     
                                       b 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       P 
                       
                         DC 
                         ⁢ 
                         _ 
                         ⁢ 
                         LOAD 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               A 
                             
                             + 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             a 
                             1 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               B 
                             
                             + 
                             
                               b 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             b 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   8 
                   ] 
                 
               
             
           
         
       
     
     This reduces to: 
     
       
         
           
             
               
                 
                   
                     P 
                     BAT 
                   
                   = 
                   
                     
                       
                         
                           a 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               T 
                               A 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     N 
                                     A 
                                   
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                               / 
                               
                                 ( 
                                 
                                   2 
                                   * 
                                   
                                     a 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       2 
                     
                     + 
                     
                       
                         
                           b 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               T 
                               B 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     N 
                                     B 
                                   
                                   + 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               / 
                               
                                 ( 
                                 
                                   2 
                                   * 
                                   
                                     b 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       2 
                     
                     + 
                     
                       a 
                       3 
                     
                     + 
                     
                       b 
                       3 
                     
                     + 
                     
                       P 
                       
                         DC 
                         ⁢ 
                         _ 
                         ⁢ 
                         LOAD 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               A 
                             
                             + 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             a 
                             1 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               B 
                             
                             + 
                             
                               b 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             b 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   9 
                   ] 
                 
               
             
           
         
       
     
     This reduces to: 
     
       
         
           
             
               
                 
                   
                     P 
                     BAT 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               SQRT 
                               ⁡ 
                               
                                 ( 
                                 
                                   a 
                                   1 
                                 
                                 ) 
                               
                             
                             * 
                             
                               T 
                               A 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   A 
                                 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                             / 
                             
                               ( 
                               
                                 2 
                                 * 
                                 
                                   SQRT 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       a 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       2 
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               SQRT 
                               ⁡ 
                               
                                 ( 
                                 
                                   b 
                                   1 
                                 
                                 ) 
                               
                             
                             * 
                             
                               T 
                               B 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   B 
                                 
                                 + 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             / 
                             
                               ( 
                               
                                 2 
                                 * 
                                 
                                   SQRT 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       b 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       2 
                     
                     + 
                     
                       a 
                       3 
                     
                     + 
                     
                       b 
                       3 
                     
                     + 
                     
                       P 
                       
                         DC 
                         ⁢ 
                         _ 
                         ⁢ 
                         LOAD 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               A 
                             
                             + 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             a 
                             1 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               B 
                             
                             + 
                             
                               b 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       / 
                       
                         ( 
                         
                           4 
                           * 
                           
                             b 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   10 
                   ] 
                 
               
             
           
         
       
     
     This reduces to:
 
 P   BAT =( A   1   *T   A   +A   2 ) 2 +( B   1   *T   B   +B   2 ) 2   +C   [11]
 
wherein A 1 =SQRT(a 1 ),
 
     B 1 =SQRT(b 1 ), 
     A 2 =(N A +a 2 )/(2*SQRT(a 1 )), 
     B 2 =(N B +b 2 )/(2*SQRT(b 1 )), and 
     C=a 3 +b 3 +P DC     —     LOAD −(N A +a 2 ) 2 /(4*a 1 )−(N B +b 2 ) 2 /(4*b 1 ). 
     This can be restated as:
 
 P   BAT   =P   A     —     ELEC   +P   B     —     ELEC   +P   DC     —     LOAD   [11A]
 
wherein P A     —     ELEC =( A   1 *T A +A 2 ) 2 +C A , and
 
     P B     —     ELEC =(B 1 *T B +B 2 ) 2 +C B    
     wherein C A =a 3 −(N A +a 2 ) 2 /(4*a 1 ) and 
     C B =b 3 −(N B +b 2 ) 2 /(4*b 1 ), and 
     C=C A +C B +P DC     —     LOAD    
     The motor torques T A  and T B  can be transformed to the T X  and T Y  as follows: 
                     [           T   X               T   Y           ]     =         [           A   1         0           0         B   1           ]     *     [           T   A               T   B           ]       +     [           A   2               B   2           ]               [   12   ]               
wherein T X  is the transformation of T A ,
 
     T Y  is the transformation of T B , and 
     A 1 , A 2 , B 1 , B 2  comprise application-specific scalar values. 
     Eq. 11 further reduces to the following.
 
 P   BAT =( T   X   2   +T   Y   2 )+ C   [13]
 
 P   BAT   =R   2   +C   [14]
 
     Eq. 12 specifies the transformation of motor torque, T A  to T X  and the transformation of motor torque T B  to T Y . Thus, a new coordinate system referred to as T X /T Y  space is defined, and Eq. 13 comprises battery power P BAT  transformed into T X /T Y  space. Therefore, the available battery power between maximum and minimum battery powers P BAT     —     MAX  and P BAT     —     MIN  can be calculated and graphed as radii (‘R MAX ’ and ‘R MIN ’) with a center at locus (0, 0) in the T X /T Y  space, and designated by the letter K, wherein:
 
 R   MIN =SQRT( P   BAT     —     MIN   −C )
 
 R   MAX =SQRT( P   BAT     —     MAX   −C )
 
     The minimum and maximum battery powers, P BAT     —     MIN  and P BAT     —     MAX , are preferably correlated to various conditions, e.g. state of charge, temperature, voltage and usage (amp-hour/hour). The parameter C, above, is defined as the absolute minimum possible battery power at given motor speeds, N A , N B , ignoring motor torque limits. Physically, when T A =0 and T B =0 the mechanical output power from the first and second electric machines  56  and  72  is zero. Physically T X =0 and T Y =0 corresponds to a maximum charging power condition for the ESD  74 . The positive sign (‘+’) is defined as discharging power from the ESD  74 , and the negative sign (‘−’) is defined as charging power into the ESD  74 . R MAX  defines a maximum battery power typically a discharging power, and R MIN  defines a minimum battery power, typically a charging power. 
     The forgoing transformations to the T X /T Y  space, designated by a second coordinate system K, are shown in  FIG. 6 , with representations of the battery power constraints as concentric circles having radii of R Min  and R Max  and linear representations of the motor torque constraints (‘T A     —     MIN ’, ‘T A     —     MAX ’, ‘T B     —     MIN ’, and ‘T B     —     MAX ’) circumscribing an allowable operating region. Analytically, the transformed vector [T X /T Y ] determined in Eq. 12 is solved simultaneously with the vector defined in Eq. 13 to identify a range of allowable torques in the T X /T Y  space which are made up of motor torques T A  and T B  constrained by the minimum and maximum battery powers P BAT     —     MIN  to P BAT     —     MAX . The range of allowable motor torques in the T X /T Y  space is shown with reference to  FIG. 4 , wherein points A, B, C, D, and E represent the bounds, and lines and radii are defined. A first coordinate system L depicts T A /T B  space in relationship to the T X /T Y  space. 
       FIGS. 5A and 5B  schematically show a control scheme and  FIGS. 6A and 6B  graphically shows operation of the control scheme of  FIGS. 5A and 5B  for determining a preferred output torque from a powertrain system having multiple torque-generative devices, described hereinbelow with reference to the powertrain system depicted in  FIGS. 1 and 2 , residing in the aforementioned control modules in the form of executable algorithms and calibrations, and preferably used in the control system architecture of  FIG. 3 . 
       FIG. 5A  depicts operation in one of the continuously variable mode operating range states. During ongoing operation in one of the mode operating range states, the operator inputs to the accelerator pedal  113  and the brake pedal  112  are monitored to determine the operator torque request. The system calculates offset motor torques based upon inputs including the operating range state of the transmission  10 , the input torque and terms based upon system inertias, system damping, and clutch slippage (‘T A  Misc Opt’, ‘T B  Misc Opt’, ‘TCL 1  Misc Opt’) described hereinbelow with reference to Eqs. 17, 18, and 19 ( 510 ). The term ‘CL 1 ’ designates a first applied clutch, i.e., one of clutches C 1   70  and C 2   62  in the embodiment described, and ‘TCL 1 ’ is the torque transferred across CL 1 . The offset motor torques and clutch torque are inputs to calculate linear torque constraints to the output torque ( 520 ) and calculate an unconstrained quadratic solution to the output torque ( 530 ). 
     The unconstrained quadratic solution to the output torque ( 530 ) is calculated using the power limits for the ESD  74 , i.e., the available battery power P BAT     —     MIN  to P BAT     —     MAX , the offset motor torque constraints, motor torque characteristics defined in terms of coefficients for converting between mechanical torque and electric power in terms of efficiency and power loss in the first and second electric machines  56  and  72  (‘Donut Space Coefficients’), and other scalar terms related to the torque outputs from the first and second electric machines  56  and  72  and the reactive torque of the applied clutch (‘K TA from To ’, ‘K TB from To ’, ‘K TCL1 from To ’). The aforementioned inputs are used to calculate a first constraint on the output torque from the transmission  10 , comprising the unconstrained quadratic solution, which includes an optimized output torque (‘To*’) and an optimized battery power (‘P* BAT ’) for operating the system without concern for other constraints on the system. The power for the energy storage device  74  can be represented mathematically as a function of the transmission output torque To as shown in Eq. 15.
 
 P   BAT ( T   O )=( a   1   2   +b   1   2 )( T   O   −T   O *) 2   +P   BAT *  [15]
 
wherein a 1  and b 1  represent scalar values determined for the specific application. Eq. 15 can be solved for the output torque, as shown in Eq. 16:
 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       O 
                     
                     ⁡ 
                     
                       ( 
                       
                         P 
                         BAT 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       T 
                       O 
                       * 
                     
                     ± 
                     
                       
                         
                           
                             P 
                             BAT 
                           
                           - 
                           
                             P 
                             BAT 
                             * 
                           
                         
                         
                           
                             a 
                             1 
                             2 
                           
                           + 
                           
                             b 
                             1 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   16 
                   ] 
                 
               
             
           
         
       
     
     For the available battery power range P BAT     —     MIN  to P BAT     —     MAX , four distinct output torques can be determined from Eq. 16, including maximum and minimum quadratic output torque constraints for the positive root case (‘To@P BAT  Max Opt (Pos Root)’ and ‘To@P BAT  Min Opt (Pos Root)’), and minimum and maximum quadratic output torque constraints for the negative root case (‘To@P BAT  Max Opt (Neg Root)’ and ‘To@P BAT  Min Opt (Neg Root)’), plotted with reference to  FIG. 6A .  FIG. 6A  shows valid, i.e., achievable ranges of output torque determined based the battery power constraints. 
     The linear torque constraints to the output torque, i.e., minimum and maximum linear output torques (‘To Min Linear’, ‘To Max Linear’) ( 520 ) are determined based upon the offset motor torques, the scalar terms related to the motor torques from the first and second electric machines  56  and  72  and the reactive torque of the applied clutch (‘K TA from To ’, ‘K TB from To ’, ‘K TCL1 from To ’), the motor torque constraints comprising minimum and maximum motor torque limits of the first and second electric machines  56  and  72 . Minimum and maximum clutch reactive torques for applied clutch(es) CL 1  (and CL 2 ) are graphed relative to the motor torque constraints, for first and, as shown (where necessary) second applied clutches (‘T CL1  MIN’, ‘T CL1  MAX’) and (‘T CL2  MIN’, ‘T CL2  MAX’). 
       FIG. 6B  graphically shows determining minimum and maximum linear output torques (‘To Min Lin’, ‘To Max Lin’) based upon the offset motor torques, the minimum and maximum achievable motor torques for the first and second electric machines  56  and  72  and the minimum and maximum clutch reactive torque(s) for the applied clutch(es) ( 520 ). The minimum and maximum linear output torques are the minimum and maximum output torques that meet the motor torque constraints and also meets the applied clutch torque constraints. An operating region for the exemplary powertrain system is graphically shown, including the motor torque constraints (‘Motor Torque Constraints’) comprising maximum and minimum achievable motor torques (‘T A     —     MAX’, ‘T   A     —     MIN’, ‘T   B     —     MAX’ and ‘T   B     —     MIN ’) for the first and second electric machines  56  and  72  in one embodiment. Minimum and maximum clutch reactive torques for applied clutch(es) CL 1  and CL 2  are graphed relative to the motor torque constraints, for first and, as shown (where necessary), second applied clutches (‘T CL1  MIN’, ‘T CL1  MAX’) and (‘T CL2  MIN’, ‘T CL2  MAX’). Minimum and maximum linear output torques (‘To Min Lin’, ‘To Max Lin’) can be determined based upon the offset motor torques, the minimum and maximum achievable motor torques for the first and second electric machines  56  and  72  and the minimum and maximum clutch reactive torques for the applied clutch(es). The minimum and maximum linear output torques are the minimum and maximum output torques that meet the motor torque constraints and also meet the applied clutch torque constraints. In the example shown, the minimum and maximum clutch reactive torques for the second applied clutch CL 2  are less restrictive and outside the motor torque constraints, and thus do not constrain the output torque. Operation is bounded by the region defined by the minimum and maximum clutch reactive torques for the first applied clutch CL 1  and the maximum and minimum motor torque constraints for the second electric machine  72 , i.e., T B     —     MAX  and T B     —MIN   . The maximum linear output torque is the maximum output torque in this region, i.e., the output torque at the intersection between the maximum motor torque constraint for the second electric machine  72  and minimum clutch reactive torque for the first applied clutch (‘T CL1  Min’). The minimum linear output torque is the minimum output torque in this region, i.e., the output torque at the intersection between the minimum motor torque command for the second electric machine  72  and maximum clutch reactive torque for the first applied clutch (‘T CL1  Max’). 
       FIG. 5B  depicts operation in one of the fixed gear operating range states. During ongoing operation in one of the fixed gear operating range states, the operator inputs to the accelerator pedal  113  and the brake pedal  112  are monitored to determine the operator torque request. The system calculates offset torques based upon inputs including the operating range state of the transmission  10 , the input torque and terms based upon system inertias, system damping, and clutch slippage (‘To Misc Opt’, ‘TCL 1  Misc Opt’, ‘TCL 2  Misc Opt’) described hereinbelow with reference to Eqs. 20, 21, 22, and 23 ( 510 ′). The term ‘CL 1 ’ designates a first applied clutch, i.e., one of clutches C 1   70  and C 2   62  in the embodiment described and ‘TCL 1 ’ is the torque transferred across CL 1 . The term ‘CL 2 ’ designates a second applied clutch, when applied, and includes one of C 2   62 , C 3   73 , and C 4   75  in the embodiment described and ‘TCL 2 ’ is the torque transferred across CL 2 . The offset torques are inputs to calculate linear output torque constraints ( 520 ′) and calculate an unconstrained quadratic solution to the output torque ( 530 ′). 
     The unconstrained quadratic solution to the output torque ( 530 ′) is calculated using the power limits for the ESD  74 , i.e., the available battery power P BAT     —     MIN  to P BAT     —     MAX , the offset motor torque constraints, motor torque characteristics defined in terms of coefficients for converting between mechanical torque and electric power in terms of efficiency and power loss in the first and second electric machines  56  and  72  (‘Donut Space Coefficients’) which are detailed in Eq. 11, and other scalar terms related to the torque outputs from the first and second electric machines  56  and  72  and the reactive torque of the applied clutch (‘K TA from To ’, ‘K TB from To ’, ‘K TCL1 from To ’ ‘K TCL2 from To ’), that are preferably determined along an optimal motor torque split line. The aforementioned inputs are used to calculate the first constraint on the output torque from the transmission  10 , comprising the unconstrained quadratic solution, which includes the optimized output torque (‘To*’) and an optimized battery power (‘P* BAT ’) for operating the system without concern for other constraints on the system ( 530 ), and described hereinabove with reference to Eqs. 15 and 16, and plotted with reference to  FIG. 6A . 
     The linear output torque constraints, i.e., the minimum and maximum linear output torques (‘To Min Linear’, ‘To Max Linear’) ( 520 ′) are determined based upon the offset torques, the scalar terms related to the output torque and the reactive torque of the applied clutches (‘k To From TA ’, ‘k To From TB ’, ‘k TA From TCL1 ’, ‘k TB From TCL1 ’, ‘k TA From TCL2 , ‘k TB From TCL2 ’), motor torque ranges comprising minimum and maximum motor torque limits of the first and second electric machines  56  and  72  (‘T A  Min’, ‘T A  Max’, ‘T B  Min’, ‘T B  Max’). Minimum and maximum clutch reactive torque constraints of the applied torque transfer clutches (‘TCL 1  Min’, ‘TCL 1 Max’, ‘TCL 2  Min’, ‘TCL 2 Max’), comprising the specifically applied clutches for the selected operating range state, are determined. The constraints comprising the minimum and maximum linear output torques (‘To Min Linear’, ‘To Max Linear’) are determined, with the minimum linear output torque preferably comprising a maximum value of the aforementioned minimum torque values and the maximum linear output torque preferably comprising a minimum value of the aforementioned maximum torque values. 
     In both the mode operating range state and the fixed gear operating range state, the unconstrained quadratic solution to the output torque and the maximum and minimum linear output torques are combined with a preferred output torque to calculate a preferred output torque (‘To Opt’) and output torque constraints (‘To Min Raw’, ‘To Max Raw’) ( 540 ). The preferred output torque is described in terms of a search range (‘To Min Search’, ‘To Max Search’), which preferably comprises the operator torque request, or another allowable torque constraint. The preferred output torque can comprise an output torque which minimizes battery power consumption within the output torque range and meets the operator torque request in this embodiment. 
     The output torque constraints (‘To Min Raw’, ‘To Max Raw’) comprise maximum and minimum unfiltered output torques that are determined based upon inputs including the input speed, output speed, motor torque constraints, reactive clutch torque constraints for the applied clutches, engine input torque, and input and output accelerations. The preferred output torque is subject to the output torque constraints and is determined based upon the range of allowable output torques, which can vary, and may include the immediate accelerator output torque request. The preferred output torque may comprise an output torque corresponding to a minimum battery discharge power or an output torque corresponding to a maximum battery charge power. The preferred output torque is based upon a capacity of the powertrain to transmit and convert electric power to mechanical torque through the first and second electric machines  56  and  72 , and the immediate or present torque, speed, and reactive clutch torque constraints, and electric power inputs thereto. 
     The output torque constraints including the maximum and minimum unfiltered output torques (‘To Min Raw’, ‘To Max Raw’) and the preferred output torque (‘To Opt’) can be determined by executing and solving an optimization function in one of the operating range states for neutral, mode and fixed gear operation. The output torque constraints comprise a preferred output torque range at the present input torque, within the available battery power (‘P BAT Min/Max’) and within the motor torque constraints comprising the ranges of available motor torques (‘T A  Min/Max’, ‘T B  Min/Max’), subject to the reactive clutch torques of the applied torque transfer clutch(es) (‘TCL 1  Min’, ‘TCL 1  Max’, ‘TCL 2  Min’, ‘TCL 2  Max’). Under non-braking operation, the output torque request is constrained to a maximum output torque capacity. 
     The inputs used to determine the maximum and minimum constraints on the output torque and the preferred output torque include power output capacity of the ESD  74 , including the available battery power and any DC load on the ESD  74 , the motor torque characteristics defined in terms of coefficients for converting between mechanical torque and electric power in terms of efficiency and power loss in the first and second electric machines  56  and  72  (‘Donut Space Coefficients’). Furthermore the present operating range state of the transmission (‘Hybrid Range State’), the input torque, the input speed (‘N I ’), output speed (‘N O ’), clutch speed (‘N C ’), acceleration of slipping clutches (‘Ncsdot’), acceleration of the output member  64  (‘Nodot’), and acceleration of the input member  12  (‘Nidot’) and maximum and minimum reactive clutch torques for the applied clutch(es) (‘TCL 1  Min’, ‘TCL 1  Max’, ‘TCL 2  Min’, ‘TCL 2  Max’) and estimated torques of the non-applied, slipping clutches (‘Tcs’) are monitored. The aforementioned accelerations are preferably based upon acceleration profiles with targeted acceleration rates, but can be actual accelerations. The ranges of available motor torques for the first and second electric machines  56  and  72  are also monitored and used as described hereinabove. 
     The optimization function preferably comprises a linear equation that is implemented in an executable algorithm and solved during ongoing operation of the system to determine the preferred output torque range to minimize battery power consumption and meet the operator torque request. The linear equation takes into account the input torque (‘Ti’), system inertias and linear damping. Preferably, there is a linear equation for each of the operating range states for mode operation. 
     When the transmission  14  is in one of the mode operating range states the linear equation for the system is Eq. 17: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             T 
                             A 
                           
                         
                       
                       
                         
                           
                             T 
                             B 
                           
                         
                       
                       
                         
                           
                             T 
                             
                               CL 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     A 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 
                                   
                                     T 
                                     B 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         T 
                         O 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     A 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     I 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 
                                   
                                     T 
                                     B 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     I 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     I 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         T 
                         I 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 11 
                               
                             
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                           
                           
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 21 
                               
                             
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                           
                           
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 31 
                               
                             
                             
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                           
                         
                         ] 
                       
                       * 
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               Nidot 
                             
                           
                           
                             
                               Nodot 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 11 
                               
                             
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                           
                           
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 21 
                               
                             
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                           
                           
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 31 
                               
                             
                             
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                           
                         
                         ] 
                       
                       * 
                       
                         [ 
                         
                           
                             
                               
                                 N 
                                 I 
                               
                             
                           
                           
                             
                               
                                 N 
                                 O 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                         
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                           ] 
                         
                         * 
                         
                           [ 
                           
                             
                               
                                 
                                   Tcs 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   Tcs 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Tcs 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   17 
                   ] 
                 
               
             
           
         
       
     
     Eq. 17 can be solved to determine a preferred output torque which minimizes the battery power and meets the operator torque request. The TCL 1  term represents reactive torque transfer across the applied clutch for the mode operation, i.e., clutch C 1   62  in Mode  1  and clutch C 2   70  in Mode  2 . The terms Tcs1, Tcs2, Tcs3 represent torque transfer across the non-applied, slipping clutches for the specific mode operation. 
     The term 
               [           k       T   A     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   I                   k       T   B     ⁢           ⁢   From   ⁢           ⁢     T   I                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   I               ]     ⁢     T   I           
represents contributions to the motor torques (T A , T B ) and the reactive torque transfer across the applied clutch T CL1  due to the input torque T I . The scalar terms are based upon the torque outputs from the first and second electric machines  56  and  72  and the reactive torque of the applied clutch related to the input torque (‘k TA from TI ’, ‘k TB from TI ’, ‘k TCL1 from TI ’) determined for the specific system application.
 
     The term 
               [           k       T   A     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   O                   k       T   B     ⁢           ⁢   From   ⁢           ⁢     T   O                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   O               ]     ⁢     T   O           
represents contributions to the motor torques (T A , T B ) and the reactive torque transfer across the applied clutch T CL1  due to the output torque T O . The scalar terms are based upon the torque outputs from the first and second electric machines  56  and  72  and the reactive torque of the applied clutch related to the input torque (‘k TA from To ’, ‘k TB from To ’, ‘k TCL1 from To ’) determined for the specific system application.
 
     The term 
               [           a   ⁢           ⁢   11           a   ⁢           ⁢   12               a   ⁢           ⁢   21           a   ⁢           ⁢   22               a   ⁢           ⁢   31           a   ⁢           ⁢   32           ]     *     [         Nidot           Nodot         ]           
represents contributions to the motor torques (T A , T B ) and the reactive torque transfer across the applied clutch T CL1  due to system inertias, having two degrees of freedom. The input acceleration term and the output acceleration term are selected as two linearly independent system accelerations which can be used to characterize the inertias of the components of the powertrain system. The a11-a32 terms are system-specific scalar values determined for the specific system application.
 
     The term 
               [           b   ⁢           ⁢   11           b   ⁢           ⁢   12               b   ⁢           ⁢   21           b   ⁢           ⁢   22               b   ⁢           ⁢   31           b   ⁢           ⁢   32           ]     *     [           N   I               N   O           ]           
represents contributions to the motor torques (T A , T B ) and the reactive torque transfer across the applied clutch T CL1  due to linear damping, having two degrees of freedom, selected as two linearly independent system speeds, i.e., the input and output speeds, which can be used to characterize the damping of the components of the powertrain system. The b11-b32 terms are system-specific scalar values determined for the specific system application. The term
 
               [           c   ⁢           ⁢   11           c   ⁢           ⁢   12           c   ⁢           ⁢   13               c   ⁢           ⁢   21           c   ⁢           ⁢   22           c   ⁢           ⁢   23               c   ⁢           ⁢   31           c   ⁢           ⁢   32           c   ⁢           ⁢   33           ]     *     [           Tcs   ⁢           ⁢   1               Tcs   ⁢           ⁢   2               Tcs   ⁢           ⁢   3           ]           
represents contributions to the motor torques (T A , T B ) and the reactive torque transfer across the applied clutch T CL1  due to non-applied, slipping clutch torques. The Tcs1, Tcs2, and Tcs3 terms represent clutch torques across the non-applied, slipping torque transfer clutches. The c11-c33 terms are system-specific scalar values determined for the specific system application.
 
     Eq. 17 can be rewritten as Eq. 18: 
                     [           T   A               T   B               T     CL   ⁢           ⁢   1             ]     =         [           k       T   A     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   O                   k       T   B     ⁢           ⁢   From   ⁢           ⁢     T   O                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   O               ]     ⁢     T   O       +       [           k       T   A     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   I                   k       T   B     ⁢           ⁢   From   ⁢           ⁢     T   I                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   I               ]     ⁢     T   I       +     [             T   A     ⁢   Misc                 T   B     ⁢   Misc                 T     CL   ⁢           ⁢   1       ⁢   Misc           ]               [   18   ]               
with the offset motor torques based upon inputs including the operating range state of the transmission  10 , the input torque and terms based upon system inertias, system damping, and clutch slippage (‘T A  Misc’, ‘T B  Misc’, ‘T CL1  Misc’) combined into a single vector.
 
     For an input torque T I , Eq. 18 reduces to Eq. 19 as follows. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             T 
                             A 
                           
                         
                       
                       
                         
                           
                             T 
                             B 
                           
                         
                       
                       
                         
                           
                             T 
                             
                               CL 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     A 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                         
                                     
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     B 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     O 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         T 
                         O 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 A 
                               
                               ⁢ 
                               Offset 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 B 
                               
                               ⁢ 
                               Offset 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 
                                   CL 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               Offset 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   19 
                   ] 
                 
               
             
           
         
       
     
     Eq. 19 can be solved using the preferred output torque (‘To Opt’) to determine preferred motor torques from the first and second electric machines  56  and  72  (‘T A  Opt’, ‘T B  Opt’) ( 550 ). Preferred battery powers (‘P BAT  Opt’, ‘P A  Opt’, ‘P B  Opt’) can be calculated based thereon ( 560 ). 
     When the transmission  14  is in one of the fixed gear operating range states the linear equation for the system is Eq. 20. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             T 
                             O 
                           
                         
                       
                       
                         
                           
                             T 
                             
                               CL 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             T 
                             
                               CL 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     O 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                         
                                     
                                   
                                   ⁢ 
                                   
                                     T 
                                     A 
                                   
                                 
                               
                             
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     O 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     A 
                                   
                                 
                               
                             
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     A 
                                   
                                 
                               
                             
                             
                               
                                 k 
                                 
                                   
                                     T 
                                     
                                       CL 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   From 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     B 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       * 
                       
                         [ 
                         
                           
                             
                               
                                 T 
                                 A 
                               
                             
                           
                           
                             
                               
                                 T 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 
                                   
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                   [ 
                   20 
                   ] 
                 
               
             
           
         
       
     
     Eq. 20 can be solved to determine a preferred output torque which minimizes the battery power and meets the operator torque request. The T CL1  and T CL2  terms represent reactive torque transfer across the applied clutches for the fixed gear operation. The terms Tcs1 and Tcs2 represent torque transfer across the non-applied, slipping clutches for the specific fixed gear operation. 
     The term 
               [           k       T   O     ⁢           ⁢   From   ⁢           ⁢     T   I                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   I                   k       T   CL2     ⁢           ⁢   From   ⁢           ⁢     T   I               ]     *     T   I           
represents contributions to the output torque To and the reactive torque transfer across the applied clutches T CL1  and T CL2  due to the input torque T I . The scalar terms are based upon the output torque and the reactive torques of the applied clutches related to the input torque (‘k To from TI ’, ‘k TCL1 from TI ’, ‘k TCL2 from TI ’) determined for the specific system application.
 
     The term 
               [           k       T   O     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   A               k       T   O     ⁢           ⁢   From   ⁢           ⁢     T   B                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   A               k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   B                   k       T     CL   ⁢           ⁢   2       ⁢           ⁢   From   ⁢           ⁢     T   A               k       T     CL   ⁢           ⁢   2       ⁢           ⁢   From   ⁢           ⁢     T   B               ]     *     [           T   A               T   B           ]           
represents contributions to the output torques and the reactive torque transfer across the applied clutches due to the motor torques T A  and T B . The scalar terms are based upon the output torque and the reactive torque of the applied clutches related to the torque outputs from the first and second electric machines  56  and  72  determined for the specific system application.
 
     The term 
               [           b   ⁢           ⁢   11               b   ⁢           ⁢   21               b   ⁢           ⁢   31           ]     *   Nidot         
represents contributions to the output torques and the reactive torque transfer across the applied clutches (TCL 1 , TCL 2 ) due to system inertias, having a single degree of freedom. The input acceleration term is selected as a linearly independent system acceleration which can be used to characterize the inertias of the components of the powertrain system. The b11-b31 terms are system-specific scalar values determined for the specific system application.
 
     The term 
               [           a   ⁢           ⁢   11               a   ⁢           ⁢   21               a   ⁢           ⁢   31           ]     *     N   I           
represents contributions to the output torques and the reactive torque transfer across the applied clutches T CL1  and T CL2  due to linear damping, having a single degree of freedom, selected as a linearly independent system speed which can be used to characterize the damping of the components of the powertrain system. The a11-a31 terms are system-specific scalar values determined for the specific system application.
 
     The term 
               [           c   ⁢           ⁢   11           c   ⁢           ⁢   12               c   ⁢           ⁢   21           c   ⁢           ⁢   22               c   ⁢           ⁢   31           c   ⁢           ⁢   32           ]     *     [           Tcs   ⁢           ⁢   1               Tcs   ⁢           ⁢   2           ]           
represents contributions to the output torque and the reactive torque transfer across the applied clutches T CL1  and T CL2  due to non-applied, slipping clutch torques. The Tcs1 and Tcs2 terms represent clutch torques across the non-applied, slipping torque transfer clutches. The c11-c32 terms are system-specific scalar values determined for the specific system application.
 
     Eq. 20 can be rewritten as Eq. 21: 
     
       
         
           
             
               
                 
                   
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                   [ 
                   21 
                   ] 
                 
               
             
           
         
       
     
     For an input torque T I , Eq. 21 can be rewritten as Eq. 22: 
                     [           T   O               T     CL   ⁢           ⁢   1                 T     CL   ⁢           ⁢   2             ]     =         [           k       T   O     ⁢           ⁢   From   ⁢             ⁢             ⁢     T   A               k       T   O     ⁢           ⁢   From   ⁢           ⁢     T   B                   k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   A               k       T     CL   ⁢           ⁢   1       ⁢           ⁢   From   ⁢           ⁢     T   B                   k       T     CL   ⁢           ⁢   2       ⁢           ⁢   From   ⁢           ⁢     T   A               k       T     CL   ⁢           ⁢   2       ⁢           ⁢   From   ⁢           ⁢     T   B               ]     ⁡     [           T   A               T   B           ]       +     [             T   O     ⁢   Offset                 T     CL   ⁢           ⁢   1       ⁢   Offset                 T     CL   ⁢           ⁢   2       ⁢   Offset           ]               [   22   ]               
with the output torque and the reactive torque transfer across the applied clutches T CL1  and T CL2  based upon the motor torques with the operating range state of the transmission  10 , and terms based upon input torque, system inertias, system damping, and clutch slippage (‘T O  Offset’, ‘T CL1  Offset’, ‘T CL2     —   Offset’) combined into a single vector. Eq. 22 can be solved using the preferred output torque (‘To Opt’) determined in Eq. 20 to determine preferred motor torques from the first and second electric machines  56  and  72 , including determining preferred motor torque split (‘T A  Opt’, ‘T B  Opt’) ( 550 ′).
 
     The motor torque commands can be used to control the first and second electric machines  56  and  72  to transfer output torque to the output member  64  and thence to the driveline  90  to generate tractive torque at wheel(s)  93  to propel the vehicle in response to the operator input to the accelerator pedal  113 . Preferably, propelling the vehicle results in vehicle acceleration so long as the output torque is sufficient to overcome external loads on the vehicle, e.g., due to road grade, aerodynamic loads, and other loads. 
       FIG. 7  shows a process ( 700 ) for determining a preferred output torque (‘To Opt’) including preferred motor torques for controlling the first and second electric machines  56  and  72  (‘T A  Opt’, ‘T B  Opt’) and a preferred battery power (‘P BAT  Opt’) based thereon. This includes determining the minimum and maximum linear output torques (‘To Min Linear’, ‘To Max Linear’) ( 710 ), and thus determining minimum and maximum output torques (‘To Min’) ( 720 ) and (‘To Max’) ( 730 ). A search is executed to determine a minimum output torque, and the preferred output torque can be calculated (‘To Opt’) ( 740 ). This includes selecting a temporary output torque comprising a minimum value of the minimum of the search range for the output torque (‘To Min Search’) and the maximum output torque (‘To Max’). The preferred output torque is selected as the maximum of the temporary output torque, the minimum output torque, and the minimum linear output torque. Preferred motor torques and battery powers (‘Ta Opt’, Tb Opt’, and ‘P BAT  Opt’) can be determined based upon the preferred output torque ( 750 ), and used to control operation of the powertrain system. 
       FIG. 8  shows a flowchart  720  for determining the minimum output torque (‘To Min’).  FIGS. 10 and 11  show results illustrative of the concept, when operating in a fixed gear operating range state. The preferred output torque at maximum charging battery power (‘To@P BAT  Min Pos Root’) can be determined (‘D OPT ’), as described herein with reference to  FIG. 6  and Eqs. 15 and 16 ( 802 ). The preferred output torque at maximum charging battery power is compared to the minimum linear output torque (‘To Min Lin’) ( 804 ). When the minimum linear output torque is greater than or equal to the preferred output torque at maximum charging battery power, the output torque is set equal to the minimum linear output torque ( 806 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min. 
     When the minimum linear output torque is less than the preferred output torque at minimum charging battery power, clutch torques for the applied clutches are determined for operating the powertrain system at the preferred output torque at maximum charging battery power ( 808 ). For purposes of the figures, the applied clutches are referred to as ‘CL 1 ,’ and ‘CL 2 ’, wherein the applied clutches are specific to the selected transmission operating range state. When the transmission  10  is operating in one of the mode operating range states, i.e., M 1  and M 2  in this embodiment, torques and forces related to clutch CL 2  are ignored. When the clutch torques for the first and second applied clutches CL 1  and CL 2  are between respective minimum and maximum clutch reactive torques and thus achievable ( 810 ), the output torque is set equal to the preferred output torque at the maximum charging battery power ( 812 ). 
     When the clutch torque for the first applied clutch CL 1  is within respective minimum and maximum clutch reactive torques and thus achievable ( 814 ), the clutch torque for the second applied clutch CL 2  is compared to a maximum achievable clutch torque ( 816 ), and when greater, the preferred output torque (‘To Min’) is determined to be a maximum charging battery power (‘To@P BAT Min’) at a point (‘D CL2MAX ’) at which the output torque meets the maximum achievable clutch torque for the second applied clutch CL 2  (‘T CL2 MAX’) and is within the battery power constraints and the motor torque constraints ( 818 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min ( 819 ). 
     When the second applied clutch CL 2  is less than the maximum achievable clutch torque ( 816 ), i.e., less than T CL2 Min, the preferred output torque (‘To Min’) is determined to be a maximum charging battery power (‘To@P BAT Min’) at a point (‘D CL2MIN ’) at which the output torque meets the minimum achievable clutch torque for the second applied clutch CL 2  (‘T CL2 MIN’) and is within the battery power constraints and the motor torque constraints ( 820 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min ( 821 ). 
     When the clutch torque for the first applied clutch CL 1  is not within respective minimum and maximum clutch reactive torques ( 814 ), it is determined whether the clutch torque for the second applied clutch CL 2  is within respective minimum and maximum clutch reactive torques and is thus achievable ( 822 ). When the clutch torque for the second applied clutch CL 2  is within respective minimum and maximum clutch reactive torques, a slope of the output torque line (‘To’) is compared to the slope of the clutch torque for the first applied clutch CL 1  ( 824 ). When the output torque line (‘To’) is parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), the minimum output torque (‘To Min’) is determined to be the minimum linear output torque (‘To Min Lin’) ( 826 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min. 
     When the clutch torque for the first applied clutch CL 1  (‘T CL1 ’) is greater than the maximum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) ( 828 ,  830 ), the preferred output torque (‘To Min’) is determined to be the maximum charging battery power (‘To@P BAT Min’) at a point (‘D CL1MAX ’) at which the output torque which meets the maximum achievable clutch torque for CL 1  (‘T CL1MAX ’) and is within the battery power constraints and the motor torque constraints ( 830 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min ( 831 ). 
     When the output torque line (‘To’) is not parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), and when the clutch torque for the first applied clutch CL 1  (‘T CL1 ’) is less than the minimum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) ( 828 ,  834 ), i.e., less than T CL2 Min, the preferred output torque (‘To Min’) is determined to be the maximum charging battery power (‘To@P BAT Min’) at a point (‘D CL1MIN ’) at which the output torque which meets the minimum achievable clutch torque for CL 1  (‘T CL1 MIN’) and is within the battery power constraints and the motor torque constraints ( 834 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min ( 835 ). 
     When the clutch torque for the second applied clutch CL 2  is outside the respective minimum and maximum clutch reactive torques, a clutch torque limit (‘T CL1 Limit’) for the first applied clutch CL 1  is set to a minimum of the maximum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) and a maximum of the first clutch torque (‘T CL1 ’) and a minimum clutch torque of the first applied clutch CL 1  (‘T CL1 MIN’). A clutch torque limit (‘T CL2 Limit’) for the second applied clutch CL 2  is set to a minimum of the maximum clutch torque of the second applied clutch CL 2  (‘T CL2 MAX’) and a maximum of the second clutch torque (‘T CL2 ’) and a minimum clutch torque of the second applied clutch CL 2  (‘T CL2 MIN’) ( 836 ). 
     Subsequently, a slope of the output torque line (‘To’) is compared to the slope of the clutch torque line for the first applied clutch CL 1  ( 838 ). When the output torque line (‘To’) is parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), the output torque (‘Return To’) at the maximum charging battery power (‘To@P BAT Min’) at a point that is within the battery power constraints and the motor torque constraints and meeting limits of the clutch torque of the second applied clutch CL 2  (‘T CL2 Limit’) is determined ( 840 ). The minimum output torque (‘To min’) is determined to be the maximum of the maximum linear output torque (‘To Max Lin’) and the return output torque (‘Return To’) ( 841 ). This is the preferred output torque returned to the process  700  at  720  as the minimum output torque To Min. 
     When the output torque line (‘To’) is not parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), a first output torque is determined at the maximum charging battery power (‘To@P BAT Min’) at a point that is within the battery power constraints and the motor torque constraints and that meets limits of the clutch torque of the first applied clutch CL 1  (‘T CL1 Limit’). A second output torque is determined at the maximum charging battery power (‘To@P BAT Min’) at a point that is within the battery power constraints and the motor torque constraints and that meets limits of the clutch torque of the second applied clutch CL 2  (‘T CL2 Limit’) ( 842 ). The preferred output torque is a maximum of the first and second output torque, which is returned to the process  700  at  720  as the minimum output torque To Min ( 844 ). 
       FIG. 9  shows a flowchart  730  for determining the maximum output torque (‘To Min’).  FIGS. 10 and 11  show results illustrative of the concept. The preferred output torque at maximum discharging battery power (‘To@P BAT  Max Opt’), which comprises ‘To@P BAT  Max Pos Root’ of  FIG. 6 , can be determined (‘C OPT ’) as described herein with reference to  FIG. 6  and Eqs. 15 and 16 ( 902 ). The preferred output torque at maximum discharging battery power is compared to the maximum linear output torque (‘To Max Lin’) ( 904 ). When the preferred output torque at the maximum discharging battery power is greater than the maximum linear output torque, the output torque (‘C OPT ’) is set equal to the maximum linear output torque ( 906 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max. 
     When the preferred output torque at maximum discharging battery power is less than or equal to the maximum linear output torque, clutch torques for the applied clutches are determined for operating the powertrain system at the preferred output torque at maximum discharging battery power ( 908 ). For purposes of this figure, the applied clutches are referred to as ‘CL 1 ,’ and ‘CL 2 ’, wherein the applied clutches are specific to the selected transmission operating range state. When the transmission  10  is operating in one of the mode operating range states, i.e., M 1  and M 2  in this embodiment, torques and forces related to clutch CL 2  are ignored. When the clutch torques for the first and second applied clutches CL 1  and CL 2  are between respective minimum and maximum clutch reactive torques and thus achievable ( 910 ), the output torque is set equal to the preferred output torque at the maximum discharging battery power ( 912 ). 
     When the clutch torque for the first applied clutch CL 1  is within respective minimum and maximum clutch reactive torques and thus achievable ( 914 ), the clutch torque for the second applied clutch CL 2  is compared to a maximum achievable clutch torque ( 916 ), and when greater, the preferred output torque (‘To Max’) is determined to be a maximum discharging battery power (‘To@P BAT Max’) at a point (‘C CL2MAX ’) at which the output torque meets the maximum achievable clutch torque for the second applied clutch CL 2  (‘T CL2 MAX’) and is within the battery power constraints and the motor torque constraints ( 918 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max ( 919 ). 
     When the second applied clutch CL 2  is less than the maximum achievable clutch torque ( 916 ), the preferred output torque (‘To Max’) is determined to be a maximum discharging battery power (‘To@P BAT Max’) at a point (‘C CL2MIN ’) at which the output torque meets the minimum achievable clutch torque for the second applied clutch CL 2  (‘T CL2 MIN’) and is within the battery power constraints and the motor torque constraints ( 920 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max ( 921 ). 
     When the clutch torque for the first applied clutch CL 1  is not within respective minimum and maximum clutch reactive torques ( 914 ), it is determined whether the clutch torque for the second applied clutch CL 2  is within respective minimum and maximum clutch reactive torques and is thus achievable ( 922 ). When the clutch torque for the second applied clutch CL 2  is within respective minimum and maximum clutch reactive torques, a slope of the output torque line (‘To’) is compared to the slope of the clutch torque for the first applied clutch CL 1  ( 924 ). When the output torque line (‘To’) is parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), the preferred output torque (‘To Max’) is determined to be the maximum linear output torque (‘To Max Lin’) ( 926 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max. 
     When the output torque line (‘To’) is not parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), when the clutch torque for the first applied clutch CL 1  (‘T CL1 ’) is greater than the maximum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) ( 928 ,  930 ), the preferred output torque (‘To Max’) is determined to be the maximum discharging battery power (‘To@P BAT Max’) at a point (‘C CL1MAX ’) at which the output torque which meets the maximum achievable clutch torque for CL 1  (‘T CL1 MAX’) and is within the battery power constraints and the motor torque constraints ( 930 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max ( 931 ). 
     When the clutch torque for the first applied clutch CL 1  (‘T CL1 ’) is less than the maximum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) ( 928 ,  934 ), the preferred output torque (‘To Max’) is determined to be the maximum discharging battery power (‘To@P BAT Max’) at a point (‘C CL1MIN ’) at which the output torque which meets the minimum achievable clutch torque for CL 1  (‘T CL1 MIN’) and is within the battery power constraints and the motor torque constraints ( 934 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max ( 935 ). 
     When the clutch torque for the second applied clutch CL 2  is outside the respective minimum and maximum clutch reactive torques, a clutch torque limit (‘T CL1  Limit’) for the first applied clutch CL 1  is set to a minimum of the maximum clutch torque of the first applied clutch CL 1  (‘T CL1 MAX’) and a maximum of the first clutch torque (‘T CL1 ’) and a minimum clutch torque of the first applied clutch CL 1  (‘T CL1 MIN’). A clutch torque limit (‘T CL2  Limit’) for the second applied clutch CL 2  is set to a minimum of the maximum clutch torque of the second applied clutch CL 2  (‘T CL2 MAX’) and a maximum of the second clutch torque (‘T CL2 ’) and a minimum clutch torque of the second applied clutch CL 2  (‘T CL2 MIN’) ( 932 ). Subsequently, a slope of the output torque line (‘To’) is compared to the slope of the clutch torque for the first applied clutch CL 1  ( 938 ). When the output torque line (‘To’) is parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), the output torque (‘Return To’) at the maximum discharging battery power (‘To@P BAT Max’) at a point that is within the battery power constraints and the motor torque constraints and meeting limits of the clutch torque of the second applied clutch CL 2  (‘T CL2 Limit’) is determined ( 940 ). The maximum output torque (‘To Max’) is determined to be the minimum of the maximum linear output torque (‘To Max Lin’) and the return output torque (‘Return To’) ( 941 ). This is the preferred output torque returned to the process  700  at  720  as the maximum output torque To Max. 
     When the output torque line (‘To’) is not parallel to the clutch torque for the first applied clutch CL 1  (‘T CL1 ’), a first output torque is determined at the maximum discharging battery power (‘To@P BAT Max’) at a point that is within the battery power constraints and the motor torque constraints and that meets limits of the clutch torque of the first applied clutch CL 1  (‘T CL1  Limit’). A second output torque is determined at the maximum discharging battery power (‘To@P BAT Max’) at a point that is within the battery power constraints and the motor torque constraints and that meets limits of the clutch torque of the second applied clutch CL 2  (‘T CL2  Limit’) ( 942 ). The preferred output torque is a minimum of the first and second output torques, which is returned to the process  700  at  720  as the maximum output torque To Max ( 944 ). 
       FIG. 10  graphically shows operation of the control scheme to control the powertrain system described with reference to  FIGS. 1 ,  2 , and  3 , utilizing the graphical mathematical constructs described with reference to  FIGS. 4 and 6 . The first coordinate system L depicts T A /T B  space, and is based upon motor torques T A  and T B . The second coordinate system K depicts motor torques T A  and T B  transformed to the T X /T Y  space, described with reference to  FIG. 4  and Eqs. 1-12, and is plotted in relationship to the first coordinate system L and the T A /T B  space. The third coordinate system M depicts P BAT /T O  space, and is based upon battery power P BAT  in relation to output torque T O , described with reference to  FIG. 6A  and Eqs. 1-12. In the illustrative operation shown, the transmission  10  is operating in G 2  with CL 1 =C 1   70  and CL 2 =C 2   62 . The independently determined parameters include P BAT  Min and P BAT  Max, plotted as R MIN  and R MAX  in T X /T Y  space relative to the second coordinate system K. A net zero battery power line R 0  is also shown. The transformed motor torque constraints (‘Motor torque constraints’) can be determined and plotted in the T X /T Y  space, or in the T A /T B  space. The first coordinate system L shows the T A /T B  space with T A =0 and T B =0, which can be determined in the T X /T Y  space and plotted relative to the K coordinate system. The two points T A =0, T B =0 and T X =0, T Y =0 define the preferred motor torque split line (‘Optimal Motor Torque Split Line’), which comprises a torque split between the first and second electric machines  56  and  72  that achieves minimum power losses and can be determined based upon the motor torque constraints and minimizes battery power for an output torque. Clutch reactive torque ranges comprising minimum, maximum, and zero clutch torques for the first clutch (‘T CL1 MIN’, ‘T CL1 MAX’, ‘T CL1 =0’) and for the second clutch (‘T CL2 MIN’, ‘T CL2 MAX’, ‘T CL2 =0’) can be determined relative to the motor torque constraints and battery power constraints and plotted in the T X /T Y  space relative to the second coordinate system K or the T A /T B  space relative to the first coordinate system L. The battery powers P BAT  Min and P BAT  Max are plotted relative to the third coordinate system M which shows P BAT  in relationship to output torque T O . The third coordinate system M shows P BAT  in relationship to output torque T O , from zero output torque (‘T O =0’) with increasing output torque, including lines representing the minimum and maximum linear output torques (‘T O  Min Lin’, ‘T O  Max Lin’). The maximum and minimum output torques are plotted in the third coordinate system (‘To@P BAT  MAX Opt’ and ‘To@P BAT  MAX Opt’), and are shown for the positive root case, the derivation of which is described with reference to  FIG. 6A . The maximum and minimum output torques translate to the T X /T Y  space, as shown. 
       FIG. 10  depicts operation in one of the continuously variable mode operating range states, e.g., Mode  2  for the embodiment described with reference to  FIGS. 1 and 2 . The preferred output torque at maximum charging battery power (‘To@P BAT  MIN Opt’) is greater than the minimum linear output torque (‘T O  Min Lin’), and therefore, the preferred minimum output torque (‘D’) comprises the output torque at the maximum charging battery power (‘To@P BAT  MIN Opt’), i.e., the output torque that satisfies the battery power constraints (‘P BAT  MIN’). The preferred output torque at maximum discharging battery power (‘To@P BAT  MAX Opt’) is less than the minimum linear output torque (‘T O  Max Lin’), and therefore, the preferred maximum output torque (‘C’) comprises the output torque at the maximum discharging battery power (‘To@P BAT  MAX Opt’), i.e., the output torque that satisfies the battery power constraints (‘P BAT  MAX’). 
     The motor torque commands can be used to control the first and second electric machines  56  and  72  to generate tractive torque and to react tractive torque at the output member  64 . Reacting tractive torque comprises reacting torque originating at the wheel(s)  93  that is transferred through the driveline  90  for regenerative operation as a part of vehicle braking or vehicle coastdown. The reactive torque at the output member  64  is transferred to one or both of the first and second electric machines  56  and  72  to generate electric power which can be stored in the ESD  74  or consumed by the other of the first and second electric machines  56  and  72 . The motor torque control scheme  340  executes to determine the motor torque commands during each iteration of one of the loop cycles. 
     It is understood that modifications are allowable within the scope of the disclosure. The disclosure has been described with specific reference to the preferred embodiments and modifications thereto. Further modifications and alterations may occur to others upon reading and understanding the specification. It is intended to include all such modifications and alterations insofar as they come within the scope of the disclosure.