Patent Publication Number: US-8121765-B2

Title: System constraints method of controlling operation of an electro-mechanical transmission with two external input torque ranges

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 60/984,830 filed on Nov. 2, 2007 which is hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This disclosure pertains to control systems for electro-mechanical transmissions. 
     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 can include multiple torque-generative devices, including internal combustion engines and non-combustion machines, e.g., electric machines, which transmit 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 transmitting tractive torque thereto. Machines, operative as motors or generators, can generate torque inputs to the transmission independently of a torque input from the internal combustion engine. The machines may transform vehicle kinetic energy transmitted through the vehicle driveline to energy that is storable in an 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 power interchange among the energy storage device and the machines to manage outputs of the transmission, including torque and rotational speed. 
     SUMMARY 
     A powertrain includes an electro-mechanical transmission mechanically-operatively coupled to an internal combustion engine and first and second electric machines to transmit power to an output member. A method for controlling the electro-mechanical transmission includes determining motor torque constraints for the first and second electric machines, and determining battery power constraints for an electrical energy storage device electrically connected to the first and second electric machines. A range for a first torque input to the electro-mechanical transmission is determined and a range for a second torque input to the electro-mechanical transmission is determined. A preferred output torque to the output member of the electro-mechanical transmission is determined that is achievable within the motor torque constraints, is achievable within the range for the first torque input, is achievable within the range for the second torque input, and is based upon the battery power 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 powertrain, in accordance with the present disclosure; 
         FIG. 2  is a schematic diagram of an exemplary architecture for a control system and powertrain, in accordance with the present disclosure; and 
         FIGS. 3 ,  4 ,  5  (including  5 A and  5 B) are graphical diagrams, 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 electro-mechanical hybrid powertrain. The exemplary electro-mechanical 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 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 power which can be transferred to the transmission  10 . 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  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 torques 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 torque commands and control inverter states therefrom for providing motor drive or regeneration 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 signally 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 torques T A  and T B  for the first and second electric machines  56  and  72 . The TCM  17  is operatively connected to the hydraulic control circuit  42  and provides various functions including monitoring various pressure sensing devices (not shown) and generating and communicating control signals to various solenoids (not shown) thereby controlling pressure switches and control valves contained within the hydraulic control circuit  42 . 
     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 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 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 serial peripheral interface 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 operating range states that can be described in terms of an engine state comprising one of an engine-on state (‘ON’) and an engine-off state (‘OFF’), and a transmission state 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 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, a commanded output torque between the transmission  10  and 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. The commanded output torque can be a tractive torque wherein torque flow originates in the engine  14  and the first and second electric machines  56  and  72  and is transferred through the transmission  10  to the driveline  90 , and can be a reactive torque wherein torque flow originates in the vehicle wheels  93  of the driveline  90  and is transferred through the transmission  10  to first and second electric machines  56  and  72  and the engine  14 . 
     Final vehicle acceleration can be affected by other factors including, e.g., road load, road grade, and vehicle mass. The operating range state is determined for the transmission  10  based upon a variety of 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 operating range 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 operating range 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  required in response to the desired output torque at output member  64  to meet the operator torque request. 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 . 
     Operation of the engine  14  and transmission  10  is constrained by power, torque and 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 one or two of the torque-transfer clutches. Torque constraints for each of the engine  14  and the first and second electric machines  56  and  72  and speed constraints for each of the engine  14 , the first and second electric machines  56  and  72 , and the output shaft  64  of the transmission  10  are determined. Battery power constraints for the ESD  74  are determined, and are applied to further limit the motor torque constraints for the first and second electrical machines  56  and  72 . The preferred operating region for the powertrain is determined using the system constraint equation, based upon the battery power constraints, the motor torque constraints, and the speed 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 the following linear equations:
 
 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]
 
wherein, in this embodiment,
         T M1  represents the output torque T O  at output member  64 ,   T M2  represents the input torque T I  at input shaft  12 ,   T M3  represents the reactive 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 ,   T A toT M1 , T A toT M2 , T A toT M3  are contributing factors of T A  to T M1 , T M2 , T M3 , respectively,   T B toT M1 , T B toT M2 , T B toT M3  are contributing factors of T B  to T M1 , T M2 , T M3 , respectively,   Misc_T M1 , Misc_T M2 , and Misc_T M3  are constants which contribute to T M1 , T M2 , T M3  by N I     —     DOT , N O     —     DOT , and N C     —     DOT  (time-rate changes in the input speed, output speed and clutch slip speed) respectively, and   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 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 . The speed constraints for the first and second electric machines  56  and  72  can be as follows:
 
−10,500 rpm≦N A ≦+10,500 rpm, and
 
−10,500 rpm≦N B ≦+10,500 rpm.
 
     The torque constraints include engine torque constraints including T I     —     MIN &lt;T I &lt;T I     —     MAX , and motor torque constraints for the first and second electric machines including T A     —     MIN &lt;T A &lt;T A     —     MAX  and T B     —     MIN &lt;T B &lt;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-generative motor and an electrical generator, 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-generative motor and an electrical generator, respectively. The maximum and minimum motor torque constraints T A     —     MAX , T A     —     MIN , T B     —     MAX , 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 are empirically derived from conventional dynamometer testing of the combined motor and power electronics (e.g., power inverter) 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. Battery power is defined as positive when discharging and negative when charging. 
     Minimum and maximum values for T M1  are determined within the speed constraints, the motor torque constraints, clutch torque constraints, and the battery power constraints during ongoing operation, in order to control operation of the engine  14 , the first and second electric machines  56  and  72 , also referred to hereinafter as Motor A  56  and Motor B  72 , and the transmission  10  to meet the operator torque request and the commanded output torque. 
     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 Motor A  56 ,
         P B,ELEC  comprises electrical power from Motor B  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:
 
 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 Motor A  56 ,
         P A,LOSS  comprises power losses from Motor A  56 ,   P B,MECH  comprises mechanical power from Motor B  72 , and   P B,LOSS  comprises power losses from Motor B  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     )         )       )     +     P     DC   ⁢   _   ⁢   LOAD                 [   6   ]               
wherein N A , N B  comprise speeds of Motors A and B  56  and  72 ,
         T A , T B  comprise torques of Motors A and B  56  and  72 , and   a 1 , a 2 , a 3 , b 1 , b 2 , b 3  each comprise quadratic coefficients which are a function of respective motor speeds, N A , N B .       

     This can be restated as Eq. 7 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 Eq. 8 as follows. 
     
       
         
           
             
               
                 
                   
                     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 Eq. 9 as follows. 
     
       
         
           
             
               
                 
                   
                     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 Eq. 10 as follows. 
     
       
         
           
             
               
                 
                   
                     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 Eq. 11 as follows.
 
 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 )       

     The motor torques T A  and T B  can be transformed to 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 can thus be further reduced as follows.
 
 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 battery power range between maximum and minimum battery power 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 transformed space T X /T Y , designated by the letter K as shown with reference to  FIG. 3 , wherein:
 
 R   Min   =SQRT ( P   BAT     —     MIN   −C ), and
 
 R   Max   =SQRT ( P   BAT     —     MAX   −C ).
 
     The minimum and maximum battery powers, P BAT     —     MIN  and P BAT     —     MAX , are preferably correlated to battery physics, 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  and N B , within the motor torque limits. Physically, when T A =0 and T B =0 the 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 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 maximum battery charging power. 
     The forgoing transformations to the T X /T Y  space are shown in  FIG. 3 , with representations of the battery power constraints as concentric circles having radii of R Min  and R Max  (‘Battery Power Constraints’) and linear representations of the motor torque constraints (‘Motor Torque Constraints’) 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 comprising the minimum and maximum battery powers identified by R Min  and R Max  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 torques in the T X /T Y  space is shown with reference to  FIG. 3 , wherein points A, B, C, D, and E represent the bounds, and lines and radii are defined. 
     A constant torque line can be defined in the T X /T Y  space, and depicted in  FIG. 3  (‘T M1 =C 1 ’), comprising the limit torque T M1 , described in Eq. 1, above. The limit torque T M1  comprises the output torque T O  in this embodiment, Eqs. 1, 2, and 3 restated in the T X /T Y  space are as follows.
 
 T   M1   =T   A to T   M1 *( T   X   −A   2 )/ A   1   +T   B to T   M1 *( T   Y   −B   2 )/ B   1 +Misc —   T   M1   [15]
 
 T   M2   =T   A to T   M2 *( T   X   −A   2 )/ A   1   +T   B to T   M2 *( T   Y   −B   2 )/ B   1 +Misc —   T   M2   [16]
 
 T   M3   =T   A to T   M3 *( T   X   −A   2 )/ A   1   +T   B to T   M3 *( T   Y   −B   2 )/ B   1 +Misc —   T   M3   [17]
 
     Defining T M1     —     XY , T M2     —     XY , and T M3     —     XY  as parts of T M1 , T M2 , and T M3,  contributed by T A  and T B  only, then:
 
 T   M1     —     XY   =T   A to T   M1 *( T   X   −A   2 )/ A   1   +T   B to T   M1 *( T   Y   −B   2 )/ B   1   [18]
 
 T   M2     —     XY   =T   A to T   M2 *( T   X   −A   2 )/ A   1   +T   B to T   M2 *( T   Y   −B   2 )/ B   1   [19]
 
 T   M3     —     XY   =T   A to T   M3 *( T   X   −A   2 )/ A   1   +T   B to T   M3 *( T   Y   −B   2 )/ B   1    [20]
 
     The following coefficients can be defined:
     T X toT M1 =T A toT M1 /A 1 ,   T Y toT M1 =T B toT M1 /B 1 ,   T M1     —   Intercept=T A toT M1 *A 2 /A 1 +T B toT M1 *B 2 /B 1 ,   T X toT M2 =T A toT M2 /A 1 ,   T Y toT M2 =T B toT M2 /B 1 ,   T M2     —   Intercept=T A toT M2 *A 2 /A 1 +T B toT M2 *B 2 /B 1 ,   T X toT M3 =T A toT M3 /A 1 ,   T Y toT M3 =T B toT M3 /B 1 , and   T M3     —   Intercept=T A toT M3 *A 2 /A 1 +T B toT M3 *B 2 /B 1 .   

     Thus, Eqs. 1, 2, and 3 are transformed to T X /T Y  space as follows.
 
 T   M1     —     XY   =T   X to T   M1   *T   X   +T   Y to T   M1   *T   Y   +T   M1     —   Intercept  [21]
 
 T   M2     —     XY   =T   X to T   M2   *T   X   +T   Y to T   M2   *T   Y   +T   M2     —   Intercept  [22]
 
 T   M3     —     XY   =T   X to T   M3   *T   X   +T   Y to T   M3   *T   Y   +T   M3     —   Intercept  [23]
 
     The speed constraints, motor torque constraints, and battery power constraints can be determined during ongoing operation and expressed in linear equations which are transformed to T X /T Y  space. Eq. 21 comprises a limit torque function describing the output torque constraint T M1 , e.g., T O . 
     The torque limit of the transmission  10 , in this embodiment the output torque T O , can be determined by using Eq. 21 subject to the T M2  and T M3  constraints defined by Eqs. 22 and 23 to determine a transformed maximum or minimum limit torque in the T X /T Y  space, comprising one of T M1     —     XY Max and T M1     —     XY Min, e.g., maximum and minimum output torques T O     —     Max  and T O     —     Min  that have been transformed. Subsequently the transformed maximum or minimum limit torque in the T X /T Y  space can be retransformed out of the T X /T Y  space to determine maximum or minimum limit torques T M1     —   Max and T  M1     —   Min for managing control and operation of the transmission  14  and the first and second electric machines  56  and  72 . 
       FIG. 4  shows motor torque constraints comprising the minimum and maximum motor torques for T A  and T B  transformed to T X /T Y  space (‘Tx_Min’, ‘Tx_Max’, ‘Ty_Min’, ‘Ty_Max’). Battery power constraints are transformed to the T X /T Y  space (‘R_Min’, ‘R_Max’) and have a center locus point K comprising (Kx, Ky)=(0,0). The output torque constraint (Tm 1 =−Tx+Ty) is shown, having a tangent point with the battery power constraint. 
     Constraints comprising maximum and minimum limits for a first torque input to the transmission  10  are depicted (‘Tm 2 =Tm 2 _High_Lmt’ and ‘Tm 2 =Tm 2 _Low_Lmt’), and preferably comprise the range of input torques T I  at input shaft  12  transformed to T X /T Y  space in this embodiment and can be mathematically represented by the line T M2     —     XY  described with reference to Eq. 22, above. The lines T M2     —     XY  described in Eq. 22 include the T M2     —   Intercept having two different values corresponding to the maximum limit and the minimum limit for the engine input torque T I . Alternatively, the second input torque T M2     —     XY  can comprise a range of clutch torques or another torque input. 
     Constraints comprising maximum and minimum limits for a second torque input to the transmission  10  are depicted (‘Tm 3 _High_Lmt’) and a low limit (‘Tm 3 _Low_Lmt’), and preferably comprise the range of applied clutch torques transformed to T X /T Y  space in this embodiment and can be mathematically represented by the line T M3     —     XY  described with reference to Eq. 23, above. The lines T M3     —     XY  described in Eq. 23 include the T M3     —   Intercept having two different values corresponding to the maximum limit and the minimum limit for the applied one of the torque-transfer clutches C 1   70 , C 2   62 , C 3   73 , C 4   75  of the transmission  10 . Alternatively, the second input torque T M2     —     XY  can comprise a range of engine input torque or another torque input. 
     The output torque line (‘Tm 1 =−Tx+Ty’) representing line T M1     —     XY  has a positive slope of a/b of the general form in Eq. 24:
 
 Tm 1 =a*Tx+b*Ty+C   [24]
 
wherein a&lt;0 and b&gt;0 and C is a constant term. In the ensuing descriptions, the line T M1     —     XY  has a positive slope of 1:1 for purposes of illustration. The x-intercept C of Eq. 24 can change. The output torque line comprises the limit torque function describing the output torque.
 
       FIG. 5  (including  5 A and  5 B) depicts a process for determining one of the maximum and minimum output torques T O     —     Max  and T O     —     Min  based upon the speed constraints, motor torque constraints, and battery power constraints, and the first torque input range and the second torque input range, with reference back to  FIG. 4 . Equations for the maximum and minimum output torques, the speed constraints, the motor torque constraints, the battery power constraints, and the first torque input range and the second torque input range are transformed into T X /T Y  space. The maximum and minimum output torques T O     —     MAX  and T O     —     MIN  comprise one of T M1     —     XY Max and T M1     —     XY Min. It is determined whether the preferred solution is a maximum value for the output torque, i.e., T M1     —     XY Max as indicated by setting a flag (‘Tm 1 _Max_Flag’), or alternatively whether the preferred solution is a minimum value for the output torque, i.e., T M1     —     XY Min as indicated by not setting the Tm 1 _Max_Flag flag. A maximum (or minimum) value for the output torque Tm 1  is calculated based upon the motor torque constraints and battery power constraints in T X /T Y  space, comprising one of T M1     —     XY Max and T M1     —     XY Min, and having coordinates of (Tx, Ty) ( 505 ). 
     A value for the second input torque T M2     —     XY  (‘Tm 2 _Value’) is calculated at the maximum (or minimum) value for the output torque Tm 1  and having coordinates of (Tx, Ty), using Eq. 22 ( 530 ). It is determined whether the calculated value for the second input torque T M2     —     XY  (‘Tm 2 _Value’) is within the operating range of the second input torque, shown in  FIG. 4  as the lines representing the maximum and minimum limits for the first torque input to the transmission  10  (‘Tm 2 =Tm 2 _High_Lmt’ and ‘Tm 2 =Tm 2 _Low_Lmt’) ( 532 ). 
     When it is determined that the calculated value for the second input torque T M2     —     XY  (‘Tm 2 _Value’) is within the operating range of the second input torque at the maximum (or minimum) value for the first torque Tm 1 , an initially achievable first torque point having coordinates of (Tx 1 , Ty 1 ) is set equal to (Tx, Ty) ( 536 ,  540 ). When it is determined that the calculated value for the second input torque T M2     —     XY  (‘Tm 2 _Value’) is outside the operating range of the second input torque at the maximum (or minimum) value for the output torque Tm 1 , the (Tx, Ty) point is modified to an initially achievable first torque point having coordinates of (Tx 1 , Ty 1 ) ( 534 ,  538 ). The initially achievable first torque point lies on one of the lines representing the maximum and minimum limits for the first torque input to the transmission  10 , depending on the calculated value for the second input torque T M2     —     XY  at (Tx, Ty). In either instance, a maximum (or minimum) value for the output torque Tm 1  is calculated at the initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) using Eq. 21 ( 542 ). 
     A value for the third input torque T M3     —     XY  (‘Tm 3 _Value’) is calculated at the maximum (or minimum) value for the first torque Tm 1  and having coordinates of (Tx, Ty), using Eq. 23 ( 510 ). It is determined whether the calculated value for the third input torque T M3     —     XY  (‘Tm 3 _Value’) is within the operating range of the third input torque, shown in  FIG. 4  as the lines representing the high limit (‘Tm 3 _High_Lmt’) and the low limit (‘Tm 3 _Low_Lmt’) ( 512 ). When it is determined that the calculated value for the third input torque T M3     —     XY  (‘Tm 3 _Value’) is within the operating range of the third input torque at the maximum (or minimum) value for the output torque Tm 1 , an initially achievable second torque point having coordinates of (Tx 2 , Ty 2 ) is set equal to (Tx, Ty) ( 516 ,  520 ). When it is determined that the calculated value for the third input torque T M3     —     XY  (‘Tm 3 _Value’) is outside the operating range of the third input torque at the maximum (or minimum) value for the output torque Tm 1 , the (Tx, Ty) point is modified to an initially achievable second torque point having coordinates of (Tx 2 , Ty 2 ) ( 514 ,  518 ). The initially achievable first torque point lies on one of the lines representing the maximum and minimum limits for the second torque input to the transmission  10 , depending on the calculated value for the third input torque T M3     —     XY  at (Tx, Ty). In either instance, a maximum (or minimum) value for the output torque Tm 1  is calculated at the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ) using Eq. 21 ( 522 ). 
     The initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) and the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ) are compared ( 550 ). When the preferred solution is a maximum value for the output torque, i.e., T M1     —     XY Max as indicated by setting the flag (‘Tm 1 _Max_Flag’) and the initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) is greater than or equal to the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ), then the final solution (Tx_Final, Ty_Final) is the output torque at the achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) ( 550 ,  570 ). 
     When the preferred solution is a maximum value for the output torque, i.e., T M1     —     XY Max as indicated by setting the flag (‘Tm 1 _Max_Flag’) and the initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) is less than the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ), then the final solution (Tx_Final, Ty_Final) is the output torque at the achievable second torque point having the coordinates of (Tx 2 , Ty 2 ) ( 550 ,  560 ). 
     When the preferred solution is a minimum value for the output torque, i.e., T M1     —     XY Min as indicated by not setting the flag (‘Tm 1 _Max_Flag’) and the initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) is less than or equal to the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ), then the final solution (Tx_Final, Ty_Final) is the achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) ( 550 ,  570 ). When the preferred solution is a minimum value for the output torque, i.e., T M1     —     XY Min as indicated by not setting the flag (‘Tm 1 _Max_Flag’) and the initially achievable first torque point having the coordinates of (Tx 1 , Ty 1 ) is greater than the initially achievable second torque point having the coordinates of (Tx 2 , Ty 2 ), then the final solution (Tx_Final, Ty_Final) is the output torque at the achievable second torque point having the coordinates of (Tx 2 , Ty 2 ) ( 550 ,  560 ). 
     The (Tx_Final, Ty_Final) point represents the preferred solution for controlling operation that can be retransformed to motor torques (T A , T B ) to control operation of the first and second electric machines  56  and  72  to achieve the output torque ( 580 ). Thus, the preferred minimum or maximum output torque is constrained based upon the speed constraints, the motor torque constraints, the battery power constraints, and the first torque input range, e.g., the input torque T I , and the second torque input range, e.g., the clutch torque. 
     The embodiment described hereinabove is based upon the output torque line T M1     —     XY  having a positive slope of a/b of the general form in Eq. 24 (as above):
 
 Tm 1 =a*Tx+b*Ty+C   [24]
 
wherein a&lt;0 and b&gt;0 and C is a constant term, with a slope of a/b=1:1 for purposes of illustration with the x-intercept C being changeable, and is indicative of a maximum battery power discharge. The description is applicable to combinations of a&gt;0, b&lt;0, and the slope of a/b being less than 1:1 and being greater than 1:1.
 
     The disclosure has described certain preferred embodiments and modifications thereto. Further modifications and alterations may occur to others upon reading and understanding the specification. Therefore, it is intended that the disclosure not be limited to the particular embodiment(s) disclosed as the best mode contemplated for carrying out this disclosure, but that the disclosure will include all embodiments falling within the scope of the appended claims.