Patent Publication Number: US-2023140485-A1

Title: Architecture and model predictive control-based methodology to manage chassis and driveline actuators

Description:
INTRODUCTION 
     The present disclosure relates to control systems for motor vehicles, and more specifically to systems and methods for accurately modeling automotive tire characteristics. 
     Static and dynamic motor vehicle control systems are increasingly being used to manage a wide variety of static and dynamic motor vehicle performance characteristics. This is especially true with respect to challenging driving scenarios involving tire slip. In many challenging driving scenarios, control actions such as wheel and/or axle torques should be allocated in an optimal manner such that tire capacity is fully utilized in longitudinal and lateral directions. Typical tire capacity management is performed within on-board computing platforms or controllers and sensors, including inertial measurement units (IMUs) to measure how the motor vehicle moves in space, referred to as vehicle dynamics. IMUs measure vehicle acceleration in three axes: x (forward/back), y (side-to-side), and z (up/down). IMUs additionally measure how quickly the motor vehicle is turning about the three axes, referred to as the pitch rate (about y), yaw rate (about z), and roll rate (about x). On-board computing platforms or controllers use the measurement data to estimate forces acting on the vehicle. 
     While current systems and methods distributing tire forces operate for their intended purpose, there is a need for new and improved systems and methods of tire force distribution that allow enhance vehicle stability in complex driving scenarios and which provide increased force generation at the tire/road interface or contact patch while maintaining or reducing cost and complexity, reducing calibration efforts, and improving simplicity, and while also providing increased redundancy and robustness. 
     SUMMARY 
     According to several aspects of the present disclosure, a system for managing chassis and driveline actuators of a motor vehicle includes one or more sensors disposed on the motor vehicle, the one or more of sensors measuring real-time static and dynamic data about the motor vehicle. The system further includes one or more actuators disposed on the motor vehicle, the one or more actuators altering static and dynamic characteristics of the motor vehicle. The system further includes a control module having a processor, a memory, and input/output (I/O ports, the processor executing program code portions stored in the memory. The program code portions include a first program code portion that causes the one or more sensors to obtain vehicle state information, a second program code portion that receives a driver input and generates a desired dynamic output based on the driver input and the vehicle state information, and a third program code portion that estimates actions of the one or more actuators based on the vehicle state information. The program code portions further include a fourth program code portion that generates one or more control action constraints based on the first and third program code portions, a fifth program code portion that generates a reference control action based on the vehicle state information, the estimated actions of the one or more actuators and the control action constraints, and a sixth program code portion that integrates the vehicle state information, the estimated actions of the one or more actuators, the desired dynamic output, the reference control action and the control action constraints to generate an optimal control action. The optimal control action defines a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle that falls within a range of predefined actuator capacities and maximizes driver control of the vehicle in complex driving scenarios. 
     In another aspect of the present disclosure the one or more sensors further comprise at least one of: inertial measurement units (IMUs) capable of measuring orientation in three dimensions, acceleration, and velocity. The one or more sensors further include Semi Active Damping Suspension (SADS) sensors capable of measuring orientation, position, velocity, acceleration in both linear and rotational aspects, and global positioning system (GPS) sensors capable of measuring a physical location of the motor vehicle. The one or more sensors further include wheel speed sensors, throttle position sensors, accelerator position sensors, steering position sensors, and tire pressure monitoring sensors. 
     In still another aspect of the present disclosure the real-time static and dynamic data further includes lateral velocity, longitudinal velocity, yaw rate, wheel angular velocity, and longitudinal, lateral, and normal forces on each tire of the motor vehicle. 
     In still another aspect of the present disclosure the second program code portion receives one or more of: a torque request from the driver; and a steering input from the driver. Based on measurements from the one or more sensors and an estimation of capacities of the one or more actuators, the torque request and the steering input from the driver are converted into a desired dynamic output that approximates vehicle dynamics that the driver input indicates. 
     In still another aspect of the present disclosure the third program code portion produces the estimation of capacities of the one or more actuators based on the vehicle state information and predetermined actuator capacity ranges. 
     In still another aspect of the present disclosure the fourth program code portion generates the one or more control action constraints from the vehicle state information and the estimation of capacities of the one or more actuators such that the control action constraints limit control signals to the one or more actuators to control actions possible within physical limitations or capacities of the actuators and fall within grip capacity of tires of the motor vehicle. 
     In still another aspect of the present disclosure the reference control action generated by the fifth program code portion further includes one or more of: an output command to one or more electronic limited slip differential (eLSD) actuators, an output command to one or more electronic all-wheel-drive (eAWD) actuators, and an output command to one or more active aerodynamic actuators. The output commands to the one or more eLSD, eAWD, and active aerodynamic actuators are calculated to achieve specific transient response characteristics of the motor vehicle which are linearized. 
     In still another aspect of the present disclosure the optimal control action generated by the sixth program code portion further includes a control signal to the one or more actuators of the motor vehicle that achieves increases motor vehicle stability, handling, maneuverability, and steerability from a first level to a second level greater than the first level for a driver input and vehicle state information at a given point in time. 
     In still another aspect of the present disclosure the optimal control action defines a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle increases motor vehicle performance from a first level to a second level greater than the first level while reducing or substantially eliminating control interventions, such as: a traction control system (TCS) input, a stability control system input, an antilock braking system (ABS) input. 
     In still another aspect of the present disclosure a method for managing chassis and driveline actuators of a motor vehicle includes processing a driver input through a control module having a processor, a memory, and input/output (I/O) ports, the processor executing program code portions stored in the memory. The program code portions: obtain vehicle state information from one or more sensors equipped to the motor vehicle, the one or more sensors measuring real-time static and dynamic data about the vehicle, receive the driver input and generating a desired dynamic output based on the driver input and the vehicle state information, and estimate actions of one or more actuators equipped to the motor vehicle based on the vehicle state information, the one or more actuators altering static and dynamic characteristics of the motor vehicle. The program code portions further generate one or more control action constraints based on the vehicle state information and estimated actions of the one or more actuators equipped to the motor vehicle, generate a reference control action based on the vehicle state information, estimated actions of the one or more actuators and the control action constraints, and integrate the vehicle state information, the estimated actions of the one or more actuators, the desired dynamic output, the reference control action and the control action constraints and generating an optimal control action based thereon. The optimal control action defines a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle that falls within a range of predefined actuator capacities and maximizes driver control of the vehicle in complex driving scenarios. 
     In still another aspect of the present disclosure obtaining vehicle state information further includes measuring orientation of the motor vehicle with inertial measurement units (IMUs) capable of measuring orientation, acceleration, and velocity in three or more degrees of freedom, measuring orientation of motor vehicle suspension componentry with Semi Active Damping Suspension (SADS) sensors capable of measuring orientation, position, velocity, acceleration in both linear and rotational aspects, and measuring a physical location of the motor vehicle with global positioning system (GPS) sensors. Obtaining vehicle state information further includes measuring motor vehicle wheel speed using wheel speed sensors, measuring a throttle position of the motor vehicle with throttle position sensors, measuring an accelerator pedal position with accelerator position sensors, measuring steering movement with steering position sensors, and measuring tire information with tire pressure monitoring sensors. 
     In still another aspect of the present disclosure obtaining vehicle state information further includes measuring a lateral velocity of the motor vehicle, measuring a longitudinal velocity of the motor vehicle, measuring a yaw rate of the motor vehicle, measuring wheel angular velocity of the wheels of the motor vehicle, and measuring longitudinal, lateral, and normal forces on each tire of the motor vehicle. 
     In still another aspect of the present disclosure receiving the driver input and generating a desired dynamic output based on the driver input and the vehicle state information further includes receiving a torque request from the driver, receiving a steering input from the driver; and based on measurements from the one or more sensors and an estimation of capacities of the one or more actuators, and converting the torque request and the steering input from the driver into a desired dynamic output that approximates vehicle dynamics that the driver input indicates. 
     In still another aspect of the present disclosure the method for managing chassis and driveline actuators of a motor vehicle further includes estimating capacities of the one or more actuators based on the vehicle state information and predetermined actuator capacity ranges. 
     In still another aspect of the present disclosure the method for managing chassis and driveline actuators of a motor vehicle further includes generating the one or more control action constraints from the vehicle state information and the estimation of capacities of the one or more actuators such that the control action constraints limit control signals to the one or more actuators to control signals possible within physical limitations or capacities of the actuators and so that the control signals fall within grip capacity of tires of the motor vehicle. 
     In still another aspect of the present disclosure generating the reference control action further includes generating an output command to one or more electronic limited slip differential (eLSD) actuators, generating an output command to one or more electronic all-wheel-drive (eAWD) actuators, and generating an output command to one or more active aerodynamic actuators. The output commands to the one or more eLSD, eAWD, and active aerodynamic actuators are calculated to achieve specific transient response characteristics of the motor vehicle which are linearized. 
     In still another aspect of the present disclosure the method for managing chassis and driveline actuators of a motor vehicle further includes generating the optimal control action generated further includes generating a control signal to the one or more actuators of the motor vehicle that achieves increases motor vehicle stability, handling, maneuverability, and steerability from a first level to a second level greater than the first level for a driver input and vehicle state information at a given point in time. 
     In still another aspect of the present disclosure the method for managing chassis and driveline actuators of a motor vehicle further includes generating the optimal control action further includes generating a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle that increases motor vehicle performance from a first level to a second level greater than the first level, and reducing or substantially eliminating control interventions from control systems including: a traction control system (TCS) input, a stability control system input, an antilock braking system (ABS) input. 
     In still another aspect of the present disclosure a method for managing chassis and driveline actuators of a motor vehicle includes processing a driver input through a control module having a processor, a memory, and input/output (I/O) ports, the processor executing program code portions stored in the memory. The program code portions obtain static and dynamic vehicle state information from one or more sensors equipped to the motor vehicle, the one or more sensors measuring orientation of the motor vehicle with inertial measurement units (IMUs) capable of measuring orientation, acceleration, and velocity in three or more degrees of freedom, and measure orientation of motor vehicle suspension componentry with Semi Active Damping Suspension (SADS) sensors capable of measuring orientation, position, velocity, acceleration in both linear and rotational aspects. The program code portions measure a physical location of the motor vehicle with global positioning system (GPS) sensors, and measure motor vehicle wheel speed using wheel speed sensors. The program code portions further measure a throttle position of the motor vehicle with throttle position sensors, measure an accelerator pedal position with accelerator position sensors, measure steering movement with steering position sensors, and measure tire information with tire pressure monitoring sensors. The program code portions also receive the driver input including receiving a torque request from the driver, and receiving a steering input from the driver. Based on measurements the one or more sensors and an estimation of capacities of the one or more actuators, the program code portions convert the torque request and the steering input from the driver into a desired dynamic output that approximates vehicle dynamics that the driver input indicates. The program code portions further estimate actions of one or more actuators equipped to the motor vehicle based on the vehicle state information, and estimating capacities of the one or more actuators based on the vehicle state information and predetermined actuator capacity ranges, the one or more actuators altering static and dynamic characteristics of the motor vehicle. The program code portions further generate one or more control action constraints from the vehicle state information and the estimation of capacities of the one or more actuators such that the control action constraints limit control signals to the one or more actuators to control signals possible within physical limitations or capacities of the actuators and so that the control signals fall within grip capacity of tires of the motor vehicle. The program code portions further generate one or more control action constraints from the vehicle state information and the estimation of capacities of the one or more actuators such that the control action constraints limit control signals to the one or more actuators to control signals possible within physical limitations or capacities of the actuators and so that the control signals fall within grip capacity of tires of the motor vehicle. The program code portions further generate a reference control action including: generating an output command to one or more electronic limited slip differential (eLSD) actuators, generating an output command to one or more electronic all-wheel-drive (eAWD) actuators, and generating an output command to one or more active aerodynamic actuators. The output commands to the one or more eLSD, eAWD, and active aerodynamic actuators are calculated to achieve specific transient response characteristics of the motor vehicle which are linearized. The program code portions integrate the vehicle state information, the estimated actions of the one or more actuators, the desired dynamic output, the reference control action and the control action constraints and generating an optimal control action based thereon. The optimal control action defines a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle that falls within a range of predefined actuator capacities and maximizes driver control of the vehicle in complex driving scenarios. 
     In still another aspect of the present disclosure obtaining vehicle state information further includes measuring a lateral velocity of the motor vehicle, measuring a longitudinal velocity of the motor vehicle, measuring a yaw rate of the motor vehicle, measuring wheel angular velocity of the wheels of the motor vehicle, and measuring longitudinal, lateral, and normal forces on each tire of the motor vehicle. Generating an optimal control action further includes generating a control signal to the one or more actuators of the motor vehicle that achieves increases motor vehicle stability, handling, maneuverability, and steerability from a first level to a second level greater than the first level for a driver input and vehicle state information at a given point in time. Generating the optimal control action further includes generating a modified driver desired dynamic output control signal to the one or more actuators of the motor vehicle that increases motor vehicle performance from a first level to a second level greater than the first level, and reducing or substantially eliminating control interventions from control systems including: a traction control system (TCS) input, a stability control system input, an antilock braking system (ABS) input. 
     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIG.  1    is a schematic diagram of a motor vehicle having an architecture and method for model predictive control based methodology to manage chassis and driveline actuators according to an according to an aspect of the present disclosure; 
         FIG.  2    is a block diagram of the system and method for model predictive control based management of chassis and driveline actuators of a motor vehicle of  FIG.  1    according to an aspect of the present disclosure; 
         FIG.  3    is a perspective side view of actual motions and control actions of a motor vehicle utilizing the system and method for model predictive control based management of chassis and driveline actuators according to an aspect of the present disclosure; 
         FIG.  4    is a schematic diagram of a drivetrain of a motor vehicle using an architecture and method for model predictive control based methodology to manage chassis and driveline actuators of  FIG.  1    according to an aspect of the present disclosure; and 
         FIG.  5    is a flowchart depicting a method for model predictive control based management of chassis and driveline actuators according to an aspect of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     Referring to  FIG.  1   , a system  10  for model-predictive control (MPC) based management of chassis and driveline actuators for a motor vehicle  12  is shown. The system  10  includes the motor vehicle  12  and one or more controllers  14 . The motor vehicle  12  is shown as a car, but it should be appreciated that the motor vehicle  12  may be a van, bus, tractor-trailer, semi, Sport-Utility Vehicle (SUV), all-terrain vehicle (ATV), truck, tricycle, motorcycle, airplane, amphibious vehicle, or any other such vehicle which makes contact with the ground without departing from the scope or intent of the present disclosure. The motor vehicle  12  includes one or more wheels having tires  18  and a drivetrain  20 . The drivetrain may include a variety of components such as internal combustion engines (ICE)  22  and/or electric motors  24 , and transmissions  26  capable of transmitting motive power developed by the ICEs  22  and/or electric motors  24  to the wheels  27 , and ultimately to the tires  18  affixed thereto. In one example, the motor vehicle  12  may include an ICE  22  acting on a rear axle  28  of the motor vehicle  12  as well as one or more electric motors  24  acting on a front axle  30  of the motor vehicle  12 . It should be appreciated, however, that the motor vehicle  12  may use one or more ICEs  22 , and/or one or more electric motors  24  disposed in other configurations without departing from the scope or intent of the present disclosure. For example, the motor vehicle  12  may have an ICE  22  acting only on the front axle  30 , while one or more electric motors  24  acts solely on the rear axle  28 . In further examples, the ICE  22  may act on both front and rear axles  30 ,  28 , and an electric motor may act on both front and rear axles  30 ,  28 . 
     In several aspects, the drivetrain  20  includes one or more in-plane actuators  32 . In-plane actuators  32  may include all-wheel drive (AWD) systems including electronically-controlled or electric AWD (eAWD)  34  systems as well as limited slip differentials (LSDs) including electronically-controlled or electric LSD (eLSD)  36  systems. In-plane actuators  32  including eAWD  34  and eLSD  36  systems can generate and/or modify force generation in X and/or Y directions at a tire  18  to road surface contact patch  38  within a certain predetermined capacity. An eAWD  34  system may transfer torque from front to rear of a motor vehicle  12  and/or from side-to-side of the motor vehicle  12 . Likewise, an eLSD  36  system may transfer torque from side-to-side of the motor vehicle  12 . In some examples, the eAWD  34  and/or eLSD  36  may directly alter or manage torque delivery from the ICE  22  and/or electric motors  24  and/or the eAWD  34  and eLSD  36  may act on a braking system  40  to adjust a quantity of torque delivered to each of the tires  18  of the motor vehicle  12 . 
     In further examples, the motor vehicle  12  may include a means of altering a normal force on each of the tires  18  of the motor vehicle  12  via one or more out-of-plane actuators  42  such as active aerodynamic actuators  44  and/or active suspension actuators  46 . The active aerodynamic actuators  44  may actively or passively alter an aerodynamic profile of the motor vehicle via one or more active aerodynamic elements  48  such as wings, spoilers, fans or other suction devices, actively-managed Venturi tunnels, and the like. The active suspension actuators  46  such as active dampers  50  or the like. In several aspects, the active dampers  50  may be magnetorheological dampers or other such electrically, hydraulically, or pneumatically-adjustable dampers without departing from the scope or intent of the present disclosure. For the sake of simplicity in the description that follows, ICEs  22 , electric motors  24 , eAWD  34 , eLSD  36 , the braking system  40 , aerodynamic control system, active aerodynamic elements  48 , active dampers  46 , and the like will be referred to more broadly as actuators  52 . 
     The terms “forward”, “rear”, “inner”, “inwardly”, “outer”, “outwardly”, “above”, and “below” are terms used relative to the orientation of the motor vehicle  12  as shown in the drawings of the present application. Thus, “forward” refers to a direction toward a front of a motor vehicle  12 , “rearward” refers to a direction toward a rear of a motor vehicle  12 . “Left” refers to a direction towards a left-hand side of the motor vehicle  12  relative to the front of the motor vehicle  12 . Similarly, “right” refers to a direction towards a right-hand side of the motor vehicle  12  relative to the front of the motor vehicle  12 . “Inner” and “inwardly” refers to a direction towards the interior of a motor vehicle  12 , and “outer” and “outwardly” refers to a direction towards the exterior of a motor vehicle  12 , “below” refers to a direction towards the bottom of the motor vehicle  12 , and “above” refers to a direction towards a top of the motor vehicle  12 . Further, the terms “top”, “overtop”, “bottom”, “side” and “above” are terms used relative to the orientation of the actuators  52 , and the motor vehicle  12  more broadly shown in the drawings of the present application. Thus, while the orientation of actuators  52 , or motor vehicle  12  may change with respect to a given use, these terms are intended to still apply relative to the orientation of the components of the system  10  and motor vehicle  12  components shown in the drawings. 
     The controllers  14  are non-generalized, electronic control devices having a preprogrammed digital computer or processor  54 , non-transitory computer readable medium or memory  56  used to store data such as control logic, software applications, instructions, computer code, data, lookup tables, etc., and input/output (I/O) ports  58 . Computer readable medium or memory  56  includes any type of medium capable of being accessed by a computer, such as read only memory (ROM), random access memory (RAM), a hard disk drive, a compact disc (CD), a digital video disc (DVD), or any other type of memory. A “non-transitory” computer readable memory  56  excludes wired, wireless, optical, or other communication links that transport transitory electrical or other signals. A non-transitory computer readable memory  56  includes media where data can be permanently stored and media where data can be stored and later overwritten, such as a rewritable optical disc or an erasable memory device. Computer code includes any type of program code, including source code, object code, and executable code. The processor  54  is configured to execute the code or instructions. The motor vehicle  12  may have controllers  14  including a dedicated Wi-Fi controller or an engine control module, a transmission control module, a body control module, an infotainment control module, etc. The I/O ports  58  may be configured to communicate via wired communications, wirelessly via Wi-Fi protocols under IEEE 802.11x, or the like without departing from the scope or intent of the present disclosure. 
     The controller  14  further includes one or more applications  60 . An application  60  is a software program configured to perform a specific function or set of functions. The application  60  may include one or more computer programs, software components, sets of instructions, procedures, functions, objects, classes, instances, related data, or a portion thereof adapted for implementation in a suitable computer readable program code. The applications  60  may be stored within the memory  56  or in additional or separate memory. Examples of the applications  60  include audio or video streaming services, games, browsers, social media, etc. In other examples, the applications  60  are used to manage body control system functions, suspension control system functions, aerodynamic control system functions, or the like in an exemplary motor vehicle  12 . 
     Referring now to  FIG.  2    and with continuing reference to  FIG.  1   , the system  10  utilizes one or more applications  60 , stored in memory  56 , for managing chassis and driveline actuators  52  of the motor vehicle  12 . In several aspects, the applications  60  include computer control code portions that coordinate actuators  52  to redistribute tire  18  forces in axle and/or wheel  27  level, and/or to adjust tire  18  capacity to allow increased force generation at the tire  18 /road surface contact patch  38 . The computer control code portions operate using a physics-based technique that models each actuator’s  52  functionality as well as the actuator’s  52  impact on motor vehicle motion  12  through body  62  and wheel  27  dynamics, as well as through combined tire  18  slip models. The combined tire  18  slip model computes normalized longitudinal and lateral forces in tire  18 /road surface contact patch  38  due to tire  18  deformation and characteristics. Subsequently, based on available vertical force, tire  18  forces are calculated in longitudinal and lateral directions, and interconnected to wheel  27  and body  62  dynamics to comprehend the impact of the adjusted forces on motor vehicle  12  dynamics. 
     More specifically, at block  100 , the system  10  receives a driver input  102  to a driver control interpreter (DCI)  104 . The DCI  104  reads a variety of driver inputs such as steering input, throttle input, braking input, or the like, and interprets the driver inputs before generating a desired dynamics signal  106  in the form of an actuator output. In several aspects, the DCI  104  determines boundaries for optimization and optimal coordination of actuators  52 . In complex driving scenarios at or near the limits of tire  18  adhesion, the driver’s inputs may exceed predefined actuator  52  capacities, tire  18  capacities, and the like. Accordingly, the system  10  utilizes a constrained optimization to reallocate sufficient capacity to X and Y directions in real-time and to redistribute tire  18  forces within the functionality and hardware limitations of each of the actuators  52 . Out-of-plane actuators  42  can modify normal forces and alter force generation in X and/or Y directions at a tire  18  adhesion limit. The constrained optimization examines whether the current tire  18  capacity is sufficient with actuator  52  functionality and hardware limitations to redistribute tire  18  forces to achieve desired motor vehicle  12  motion using eAWD  34  and/or eLSD  36 , or where tire  18  capacity must be increased via active aerodynamic actuators  44  if increased traction or lateral grip is requested by a driver of the motor vehicle  12 . The constrained optimization is solved in a real-time fashion to optimally harmonize control commands from different actuators  52  so that motor vehicle  12  performance capabilities are maximized and so that control interventions are minimized. That is, the motor vehicle  12  performance capabilities are increased from a first level to a second level greater than the first level so that control interventions, such as: a traction control system (TCS) input, a stability control system input, an antilock braking system (ABS) input, or the like are reduced or substantially eliminated. 
     The system  10  includes several control devices, one or more of which may be integrated into a single controller  14 , or may be integrated into distinct, separate controllers  14  in electronic communication with one another. The controllers  14  include a feed-forward controller  108  which commands the actuators  52  to achieve certain transient response characteristics, provide optimal reference control actions, and linearize control responses around an operation point. More specifically, the feed-forward controller  108  provides active aerodynamic preemptive control signals, eLSD  36  preemptive control signals, and eAWD  34  preemptive control signals. Preemptive control signals, adjust actuator  52  outputs to comport with control signal estimates from a sensor/estimation module  110 . 
     The sensor/estimation module  110  provides information to the optimization-based feed-forward controller  108  and feedback controller  112 . In several aspects, the sensor/estimation module  110  generates estimations  116  for each of the various active chassis and dynamics systems equipped to the motor vehicle  12 . In a particular example, the sensor/estimation module  110  includes an aero model estimation, an eLSD  36  model estimation, an eAWD  34  model estimation, and vehicle dynamics estimations  116 . The aero model estimation calculates downforce and maximum downforce possible given current vehicle state information. Likewise, the eLSD  36  model estimation includes clutch torque estimations and maximum clutch torque capacity estimations. The eAWD  34  model estimation produces maximum electric motor  24  torque estimations. Finally, the vehicle dynamics estimations  116  include vehicle states, road surface information, tire  18  force calculations, and road angles. 
     A dynamic constraint calculator  114  applies actual physical limitations, and tire  18  and road grip limitations to signals from the sensor/estimation module  110 . 
     Finally, the feedback controller  112  operates to achieve maximum feasible performance, stability, handling, maneuverability, steerability of the motor vehicle  12  in utilizing in-plane and out-of-plane actuators  32 ,  42 . The feedback controller  112  receives a desired motor vehicle  12  dynamics signal  106  from the DCI  104 , a reference control action  120  from the feed-forward controller  108 , control action constraints  122  from the dynamic constraint calculator  114 , and measurements  124  from the various sensors  64  equipped to the motor vehicle  12 . The feedback controller  112  then integrates the desired dynamics signal  106 , reference control action  120 , control action constraints  122 , and measurements  124  into a model which accounts for body  62  and wheel  27  dynamics and in-plane and out-of-plane actuators  32 ,  42 . The feedback controller  112  models electric motor  24  torque, eLSD  36  outputs, eAWD  34  outputs, combined tire  18  slip data, and active aerodynamic actuator  44  outputs, as well as front-to-rear and/or side-to-side interactions of various actuators  52  of the motor vehicle. 
     In addition, a model predictive control (MPC) approach is used in the feedback controller  112 . The feedback controller  112  receives a variety of motor vehicle  12  state variables from sensors  64  equipped to the motor vehicle  12 . The sensors  64  may measure and record a wide variety of motor vehicle  12  data. In several examples, the sensors  64  may include inertial measurement units (IMUs)  66 , suspension control units such as Semi Active Damping Suspension (SADS)  68 , global positioning system (GPS) sensors  70 , wheel speed sensors  72 , throttle position sensors  74 , accelerator pedal position sensors  76 , brake pedal position sensors  78 , steering position sensors  80 , tire pressure monitoring sensors  82 , aerodynamic element position sensors  84 , and the like. The IMUs  66  can measure movement, acceleration, and the like in several degrees of freedom. In a specific example, the IMUs  66  may measure position, movement, acceleration, etc. in at least three degrees of freedom. Likewise, the SADS  68  sensors may be IMUs  66  capable of measuring in three or more degrees of freedom. In some examples, the SADS  68  may be suspension hub accelerometers, or the like. Accordingly, the motor vehicle  12  state variables may include any of a wide variety of data including but not limited to: wheel  27  speed data, SADS and IMU data including attitude, acceleration and the like. 
     The MPC control logic portion or algorithm in the feedback controller  112  produces state predictions based on initial state variables measured or estimated by the sensors  64  on the motor vehicle  12 . Additional estimations  116  may also be made in order to model the impact of different factors on state variables. In cases of nonlinearity in the prediction model, a linearized model is provided for specific operational parameters using motor vehicle  12  state variable measurements and/or estimations  116  and reference control actions  120 . In order to come up with feasible control commands for the various actuators  52 , actuator  52  capacity and tire  18  capacity limits should be taken into consideration in calculations. That is, a given actuator  52  in the motor vehicle  12  may have a limited range of outputs including but not limited to a limited range of motion, limitations on velocity and/or acceleration, actuator  52  torque, and the like. Similarly, tire  18  capacity may be limited by tread depth, tire  18  wear, tire  18  pressure, tire  18  compound, tire  18  temperature, a coefficient of friction of a road surface at the contact patch  38 , and the like. Accordingly, the feedback control portion of the MPC in the feedback controller  112  includes an offline control logic portion that contains a formulation of the state variables of the motor vehicle  12  as well as a control objective design, as well as an online optimization control logic portion. 
     A prediction model control logic portion forecasts the evolution of state variables (X) and evaluates a connection between a control action sequence (U) and outputs (Y) over a finite prediction horizon. The prediction model control logic portion includes motor vehicle core dynamics such as a body  62  dynamics model including longitudinal, lateral, yaw, bounce, and pitch characteristics of the motor vehicle  12 . Likewise, the prediction model control logic includes a wheel  27  dynamics model that including angular speed and relative speed data, as well as longitudinal slip and slip ratio characteristics of each wheel  27 . The prediction model control logic also includes a tire  18  mechanics model that contains a combined slip tire model for each tire  18  of the motor vehicle  12 . Finally, the prediction model control logic includes an actuator  52  model that contains actuator  52  dynamics, constraints, and functionality. 
     Referring now to  FIG.  3   , and with continuing reference to  FIGS.  1  and  2   , a diagram of body  62  dynamics is shown. Planar body  62  dynamics of the motor vehicle can be calculated according to the following equations: 
     
       
         
           
             
               
                 
                   
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                               x 
                               2 
                             
                           
                         
                       
                     
                       
                     cos 
                       
                     
                       δ 
                       f 
                     
                       
                     − 
                     
                       
                         
                           F 
                           
                             
                               y 
                               1 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               y 
                               2 
                             
                           
                         
                       
                     
                     sin 
                     
                       δ 
                       f 
                     
                   
                 
                 / 
               
             
             
               
                 m 
                 + 
                 
                   
                     
                       
                         
                           F 
                           
                             
                               x 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               x 
                               4 
                             
                           
                         
                       
                     
                     c 
                     o 
                     s 
                       
                     
                       δ 
                       r 
                     
                     − 
                     
                       
                         
                           F 
                           
                             
                               y 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               y 
                               4 
                             
                           
                         
                       
                     
                     s 
                     i 
                     n 
                       
                     
                       δ 
                       r 
                     
                   
                 
                 / 
                 m 
                 + 
                   
                 r 
                 
                   v 
                   y 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 
                   
                     v 
                     ˙ 
                   
                   y 
                 
               
             
             
               
                 = 
                 
                   
                     
                       
                         
                           F 
                           
                             
                               x 
                               1 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               x 
                               2 
                             
                           
                         
                       
                     
                       
                     s 
                     i 
                     n 
                       
                     
                       δ 
                       f 
                     
                     + 
                     
                       
                         
                           F 
                           
                             
                               y 
                               1 
                             
                           
                         
                         
                           F 
                           
                             
                               y 
                               2 
                             
                           
                         
                       
                     
                     c 
                     o 
                     s 
                       
                     
                       δ 
                       f 
                     
                   
                 
                 / 
                 m 
                 + 
               
             
             
               
                 
                   
                     
                       
                         
                           F 
                           
                             
                               x 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               x 
                               4 
                             
                           
                         
                       
                     
                     s 
                     i 
                     n 
                       
                     
                       δ 
                       r 
                     
                       
                     + 
                       
                     
                       
                         
                           F 
                           
                             
                               y 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               y 
                               4 
                             
                           
                         
                       
                     
                     c 
                     o 
                     s 
                       
                     
                       δ 
                       r 
                     
                   
                 
                 / 
                 m 
                 − 
                 r 
                 
                   v 
                   x 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     υ 
                   
                   ¨ 
                 
                 = 
                 
                   
                     
                       L 
                       f 
                     
                     ( 
                     
                       F 
                       
                         
                           y 
                           1 
                         
                       
                     
                     + 
                     
                       F 
                       
                         
                           y 
                           2 
                         
                       
                     
                     ) 
                     + 
                     
                       L 
                       
                         
                           w 
                           f 
                         
                       
                     
                     
                       
                         
                           F 
                           
                             
                               x 
                               2 
                             
                           
                         
                         − 
                         
                           F 
                           
                             
                               x 
                               1 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     c 
                     o 
                     s 
                       
                     
                       δ 
                       f 
                     
                   
                   / 
                   
                     
                       I 
                       
                         z 
                         z 
                       
                     
                   
                 
               
             
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 + 
                 
                   
                     
                       L 
                       f 
                     
                     ( 
                     
                       F 
                       
                         
                           x 
                           1 
                         
                       
                     
                     + 
                     
                       F 
                       
                         
                           x 
                           2 
                         
                       
                     
                     ) 
                     + 
                     
                       L 
                       
                         
                           w 
                           f 
                         
                       
                     
                     
                       
                         
                           F 
                           
                             
                               y 
                               2 
                             
                           
                         
                         − 
                         
                           F 
                           
                             
                               y 
                               1 
                             
                           
                         
                       
                     
                   
                 
                   
                 
                   
                     s 
                     i 
                     n 
                       
                     
                       δ 
                       f 
                     
                   
                   / 
                   
                     
                       I 
                       
                         z 
                         z 
                       
                     
                   
                 
               
             
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 − 
                 
                   
                     
                       L 
                       r 
                     
                     
                       
                         
                           F 
                           
                             
                               y 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               y 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       L 
                       
                         
                           w 
                           r 
                         
                       
                     
                     
                       
                         
                           F 
                           
                             
                               x 
                               4 
                             
                           
                         
                         − 
                         
                           F 
                           
                             
                               x 
                               3 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     c 
                     o 
                     s 
                       
                     
                       δ 
                       f 
                     
                   
                   / 
                   
                     
                       I 
                       
                         z 
                         z 
                       
                     
                   
                 
               
             
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 − 
                 
                   
                     
                       L 
                       r 
                     
                     
                       
                         
                           F 
                           
                             
                               x 
                               3 
                             
                           
                         
                         + 
                         
                           F 
                           
                             
                               x 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       L 
                       
                         
                           w 
                           r 
                         
                       
                     
                     
                       
                         
                           F 
                           
                             
                               y 
                               4 
                             
                           
                         
                         − 
                         
                           F 
                           
                             
                               y 
                               4 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     s 
                     i 
                     n 
                       
                     
                       δ 
                       f 
                     
                   
                   / 
                   
                     
                       I 
                       
                         z 
                         z 
                       
                     
                   
                 
               
             
           
         
       
     
     Similarly, the wheel  27  dynamics model is formulated based on angular velocity or slip ratio as: 
     
       
         
           
             
               
                 ω 
                 ˙ 
               
               i 
             
             = 
             
               
                 
                   
                     
                       T 
                       i 
                     
                     − 
                     
                       T 
                       
                         
                           b 
                           i 
                         
                       
                     
                     − 
                     
                       R 
                       
                         
                           e 
                           i 
                         
                       
                     
                     
                       F 
                       
                         
                           x 
                           i 
                         
                       
                     
                   
                 
               
               / 
               
                 
                   J 
                   
                     
                       w 
                       i 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     κ 
                     ˙ 
                   
                   i 
                 
                 = 
                 
                   R 
                   
                     
                       e 
                       i 
                     
                   
                 
                 
                   
                     
                       
                         
                           T 
                           i 
                         
                         − 
                         
                           T 
                           
                             
                               b 
                               i 
                             
                           
                         
                         − 
                         
                           R 
                           
                             
                               e 
                               i 
                             
                           
                         
                         
                           F 
                           
                             
                               x 
                               i 
                             
                           
                         
                       
                     
                   
                   / 
                   
                     
                       J 
                       
                         
                           w 
                           i 
                         
                       
                     
                     
                       v 
                       
                         x 
                         , 
                         
                           w 
                           i 
                         
                       
                     
                     − 
                     
                       
                         
                           
                             v 
                             ˙ 
                           
                           
                             x 
                             , 
                             
                               w 
                               i 
                             
                           
                         
                       
                       / 
                       
                         
                           v 
                           
                             x 
                             , 
                             
                               w 
                               i 
                             
                           
                         
                         
                           
                             w 
                             + 
                             
                               κ 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 i 
                 = 
                 1 
                 , 
                 2 
                 , 
                   
                 … 
                   
                 , 
                 4. 
               
             
           
         
       
     
     The combined slip tire  18  model used herein may be based on a so-called “magic formula” (MF) based tire  18  model, however, it should be appreciated that any tire  18  model that can properly represent nonlinear and combined slip behavior of tires  18  may be used instead, or in combination with the MF tire  18  model without departing from the scope or intent of the present disclosure. The MF tire  18  model fits a wide variety of tire  18  types, constructions, and operating conditions. In the MF tire model, each tire  18  is characterized by a plurality of coefficients for each force relevant to tire  18  performance. In some examples, the plurality of coefficients relate to contact patch, lateral and longitudinal forces, self-aligning torque, and the like. These coefficients are used as a best-fit between experimentally-determined tire  18   performance data and the MF model. The coefficients then may be used to generate equations showing how much force is generated for a particular vertical load on the tire  18 , as well as camber angle, slip angle and the like. In an example, the MF based tire  18  model may be presented as follows for a longitudinal force F x  : 
     
       
         
           
             
               F 
               x 
             
             = 
             
               
                 
                   D 
                   x 
                 
                 sin 
                 
                   
                     
                       C 
                       x 
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           x 
                         
                         
                           κ 
                           x 
                         
                         − 
                         
                           E 
                           x 
                         
                         
                           
                             
                               B 
                               x 
                             
                             
                               κ 
                               x 
                             
                             − 
                             
                               
                                 tan 
                               
                               
                                 − 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               B 
                               x 
                             
                             
                               κ 
                               x 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             + 
             
               
                 
                   S 
                   
                     V 
                     x 
                   
                 
               
             
             
               G 
               
                 x 
                 a 
               
             
           
         
       
     
     In a pure slip condition, i.e. one in which the tires  18  of the motor vehicle  12  are sliding with relation to the contact patch  38 , the following formulae apply: 
     
       
         
           
             
               κ 
               x 
             
             = 
             κ 
             + 
             
               S 
               
                 H 
                 x 
               
             
           
         
       
     
     
       
         
           
             
               C 
               x 
             
             = 
             
               p 
               
                 
                   C 
                   
                     x 
                     1 
                   
                 
               
             
             
               λ 
               
                 C 
                 x 
               
             
           
         
       
     
     
       
         
           
             
               D 
               x 
             
             = 
             
               μ 
               x 
             
             
               F 
               z 
             
             
               ξ 
               1 
             
           
         
       
     
     
       
         
           
             
               μ 
               x 
             
             = 
             
               
                 
                   p 
                   
                     D 
                     x 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     x 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               
                 1 
                 + 
                 
                   p 
                   
                     p 
                     x 
                     3 
                   
                 
                 d 
                 p 
                 i 
                 + 
                 
                   p 
                   
                     p 
                     x 
                     4 
                   
                 
                 d 
                 p 
                 
                   i 
                   2 
                 
               
             
             
               
                 1 
                 − 
                 
                   p 
                   
                     D 
                     x 
                     3 
                   
                 
                 
                   γ 
                   2 
                 
               
             
             
               λ 
               
                 
                   μ 
                   x 
                 
               
             
           
         
       
     
     
       
         
           
             
               E 
               x 
             
             = 
             
               
                 
                   p 
                   
                     E 
                     x 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     E 
                     x 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
                 + 
                 
                   p 
                   
                     E 
                     x 
                     3 
                   
                 
                 d 
                 
                   f 
                   z 
                 
                 
                     
                   2 
                 
               
             
             
               
                 1 
                 − 
                 
                   p 
                   
                     E 
                     x 
                     4 
                   
                 
                 s 
                 g 
                 n 
                 
                   
                     
                       κ 
                       x 
                     
                   
                 
               
             
             
               λ 
               
                 
                   E 
                   x 
                 
               
             
           
         
       
     
     
       
         
           
             
               K 
               
                 x 
                 κ 
               
             
             = 
             
               F 
               z 
             
             
               
                 
                   p 
                   
                     K 
                     x 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     K 
                     x 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               e 
               
                 
                   p 
                   
                     K 
                     x 
                     3 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               
                 1 
                 + 
                 
                   p 
                   
                     p 
                     x 
                     1 
                   
                 
                 d 
                 p 
                 i 
                 + 
                 
                   p 
                   
                     p 
                     x 
                     2 
                   
                 
                 d 
                 p 
                 
                   i 
                   2 
                 
               
             
             
               λ 
               
                 
                   K 
                   
                     x 
                     κ 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               B 
               x 
             
             = 
             
               
                 
                   K 
                   
                     x 
                     k 
                   
                 
               
               
                 
                   C 
                   x 
                 
                 
                   D 
                   x 
                 
               
             
           
         
       
     
     
       
         
           
             
               S 
               
                 H 
                 x 
               
             
             = 
             
               
                 
                   p 
                   
                     H 
                     x 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     H 
                     x 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               λ 
               
                 
                   H 
                   x 
                 
               
             
           
         
       
     
     
       
         
           
             
               S 
               
                 V 
                 x 
               
             
             = 
             
               F 
               z 
             
             
               
                 
                   p 
                   
                     V 
                     x 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     V 
                     x 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               λ 
               
                 
                   V 
                   x 
                 
               
             
             
               λ 
               
                 
                   μ 
                   x 
                 
               
             
             
               ξ 
               1 
             
           
         
       
     
     A combined slip tire  18  model may therefore be represented as: 
     
       
         
           
             
               G 
               
                 x 
                 a 
               
             
             = 
             
               
                 c 
                 o 
                 s 
                 
                   
                     
                       C 
                       
                         x 
                         α 
                       
                     
                     t 
                     a 
                     
                       n 
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           
                             x 
                             α 
                           
                         
                         
                           α 
                           s 
                         
                         − 
                         
                           E 
                           
                             x 
                             α 
                           
                         
                         
                           
                             
                               B 
                               
                                 x 
                                 α 
                               
                             
                             
                               α 
                               s 
                             
                             − 
                             t 
                             a 
                             
                               n 
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   B 
                                   
                                     x 
                                     α 
                                   
                                 
                                 
                                   α 
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 c 
                 o 
                 s 
                 
                   
                     
                       C 
                       
                         x 
                         α 
                       
                     
                     t 
                     a 
                     
                       n 
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           
                             x 
                             α 
                           
                         
                         
                           S 
                           
                             H 
                             x 
                             α 
                           
                         
                         − 
                         
                           E 
                           
                             x 
                             α 
                           
                         
                         
                           
                             
                               B 
                               
                                 x 
                                 α 
                               
                             
                             
                               S 
                               
                                 H 
                                 x 
                                 α 
                               
                             
                             − 
                             t 
                             a 
                             
                               n 
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   B 
                                   
                                     x 
                                     α 
                                   
                                 
                                 
                                   S 
                                   
                                     H 
                                     x 
                                     α 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               α 
               s 
             
             = 
             
               α 
               F 
             
             + 
             
               S 
               
                 H 
                 x 
                 α 
               
             
           
         
       
     
     
       
         
           
             
               B 
               
                 x 
                 α 
               
             
             = 
             
               
                 
                   r 
                   
                     B 
                     x 
                     1 
                   
                 
                 + 
                 
                   r 
                   
                     B 
                     x 
                     3 
                   
                 
                 
                   γ 
                   2 
                 
               
             
             c 
             o 
             s 
             
               
                 t 
                 a 
                 
                   n 
                   
                     − 
                     1 
                   
                 
                 
                   
                     
                       r 
                       
                         B 
                         x 
                         2 
                       
                     
                     κ 
                   
                 
               
             
             
               λ 
               
                 
                   x 
                   a 
                 
               
             
           
         
       
     
     
       
         
           
             
               C 
               
                 x 
                 a 
               
             
             = 
             
               r 
               
                 C 
                 x 
                 1 
               
             
           
         
       
     
     
       
         
           
             
               E 
               
                 x 
                 a 
               
             
             = 
             
               r 
               
                 E 
                 x 
                 1 
               
             
             + 
             
               r 
               
                 E 
                 x 
                 2 
               
             
             d 
             
               f 
               z 
             
           
         
       
     
     
       
         
           
             
               S 
               
                 H 
                 x 
                 a 
               
             
             = 
             
               r 
               
                 H 
                 x 
                 1 
               
             
           
         
       
     
      However, when combined slip in longitudinal and lateral directions is not used, G xa  = 1. Similarly, when turning-slip is not used in the formulae above, ξ 1  =1. 
     The lateral force F y  may also be calculated using a MF based tire  18  model in accordance with the following simplified MF equation. 
     
       
         
           
             
               F 
               y 
             
             = 
             
               G 
               
                 y 
                 k 
               
             
             
               F 
               
                 y 
                 p 
               
             
             + 
             
               S 
               
                 V 
                 y 
                 k 
               
             
           
         
       
     
      In a pure slip condition, the following modified MF based tire  18  model equations may be used: 
     
       
         
           
             
               
                 
                   F 
                   
                     y 
                     p 
                   
                 
                 = 
                 
                   D 
                   y 
                 
                 sin 
                 
                   
                     
                       C 
                       y 
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                   
                 
                 
                   
                     
                       B 
                       y 
                     
                     
                       α 
                       y 
                     
                     − 
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           y 
                         
                         
                           
                             
                               α 
                               y 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 	 
                 
                   
                     + 
                     
                       C 
                       γ 
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           γ 
                         
                         γ 
                         − 
                         
                           E 
                           γ 
                         
                         
                           
                             
                               B 
                               γ 
                             
                             γ 
                             − 
                             
                               
                                 tan 
                               
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   B 
                                   γ 
                                 
                                 γ 
                               
                             
                           
                         
                       
                     
                   
                 
                 + 
                 
                   S 
                   
                     V 
                     y 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               α 
               y 
             
             = 
             
               α 
               F 
             
             + 
             
               S 
               
                 H 
                 y 
               
             
           
         
       
     
     
       
         
           
             
               C 
               y 
             
             + 
             = 
             
               p 
               
                 C 
                 y 
                 1 
               
             
             
               λ 
               
                 C 
                 y 
               
             
           
         
       
     
     
       
         
           
             
               C 
               γ 
             
             = 
             
               p 
               
                 C 
                 y 
                 2 
               
             
             
               λ 
               
                 C 
                 γ 
               
             
           
         
       
     
     
       
         
           
             
               D 
               y 
             
             = 
             
               μ 
               y 
             
             
               F 
               z 
             
           
         
       
     
     
       
         
           
             
               μ 
               y 
             
             = 
             
               p 
               
                 D 
                 y 
                 1 
               
             
             
               e 
               
                 p 
                 D 
                 y 
                 
                   2 
                   
                     d 
                     f 
                     z 
                   
                 
               
             
             
               
                 1 
                 + 
                 
                   p 
                   
                     p 
                     y 
                     3 
                   
                 
                 d 
                 p 
                 i 
                 + 
                 
                   p 
                   
                     p 
                     y 
                     4 
                   
                 
                 d 
                 p 
                 
                   i 
                   2 
                 
               
             
             
               λ 
               
                 μ 
                 ​ 
                 y 
               
             
             
               
                 1 
                 − 
                 
                   p 
                   
                     D 
                     y 
                     3 
                   
                 
                 
                   γ 
                   2 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 K 
                 y 
                 a 
               
             
             
               
                 
                   
                     
                       
                         = 
                         
                           p 
                           
                             K 
                             y 
                             1 
                           
                         
                         
                           F 
                           
                             z 
                             0 
                             
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         + 
                         
                           p 
                           
                             p 
                             y 
                             1 
                           
                         
                         
                           
                             d 
                             p 
                             i 
                           
                         
                         sin 
                         
                           
                             
                               p 
                               
                                 K 
                                 y 
                                 2 
                               
                             
                             
                               
                                 tan 
                               
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   
                                     
                                       F 
                                       z 
                                     
                                   
                                   
                                     
                                       
                                         
                                           p 
                                           
                                             K 
                                             y 
                                             3 
                                           
                                         
                                         + 
                                         
                                           P 
                                           
                                             K 
                                             y 
                                             4 
                                           
                                         
                                         
                                           γ 
                                           2 
                                         
                                       
                                     
                                     
                                       F 
                                       
                                         z 
                                         0 
                                       
                                     
                                     
                                       
                                         1 
                                         + 
                                         
                                           p 
                                           
                                             p 
                                             y 
                                             2 
                                           
                                         
                                         
                                           
                                             d 
                                             p 
                                             i 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         − 
                         
                           
                             
                               p 
                               
                                 K 
                                 y 
                                 5 
                               
                             
                             γ 
                           
                         
                         
                           λ 
                           
                             K 
                             y 
                             α 
                             
                               ξ 
                               3 
                             
                           
                         
                       
                     
                   
                 
                 
                   1 
                 
               
             
           
         
       
     
     
       
         
           
             
               K 
               
                 y 
                 α 
                 0 
               
             
             = 
             
               p 
               
                 K 
                 y 
                 1 
               
             
             
               F 
               
                 z 
                 0 
               
             
             
               
                 1 
                 + 
                 
                   p 
                   
                     p 
                     y 
                     1 
                   
                 
                 d 
                 p 
                 i 
               
             
             sin 
             
               
                 
                   p 
                   
                     K 
                     y 
                     2 
                   
                 
                 
                   
                     tan 
                   
                   
                     − 
                     1 
                   
                 
                 
                   
                     
                       
                         
                           F 
                           z 
                         
                       
                       
                         
                           p 
                           
                             K 
                             y 
                             3 
                           
                         
                         
                           F 
                           
                             z 
                             0 
                           
                         
                         
                           
                             1 
                             + 
                             
                               p 
                               
                                 p 
                                 y 
                                 2 
                               
                             
                             d 
                             p 
                             i 
                           
                         
                       
                     
                   
                 
               
             
             
               λ 
               
                 K 
                 y 
                 α 
               
             
           
         
       
     
     
       
         
           
             
               K 
               
                 y 
                 γ 
                 0 
               
             
             = 
             
               
                 
                   p 
                   
                     K 
                     y 
                     6 
                   
                 
                 + 
                 
                   p 
                   
                     K 
                     Y 
                     7 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               F 
               z 
             
             
               λ 
               
                 K 
                 y 
                 γ 
               
             
             
               
                 1 
                 + 
                 
                   p 
                   
                     p 
                     y 
                     5 
                   
                 
                 d 
                 p 
                 i 
               
             
           
         
       
     
     
       
         
           
             
               B 
               y 
             
             = 
             
               
                 
                   
                     
                       
                         
                           K 
                           
                             y 
                             α 
                           
                         
                       
                       
                         
                           C 
                           y 
                         
                         
                           D 
                           y 
                         
                       
                     
                     , 
                   
                 
                 
                   
                     
                       B 
                       γ 
                     
                     = 
                     
                       
                         
                           K 
                           
                             y 
                             γ 
                             0 
                           
                         
                       
                       
                         
                           C 
                           γ 
                         
                         
                           D 
                           y 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               E 
               y 
             
             = 
             
               
                 
                   p 
                   
                     E 
                     y 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     E 
                     y 
                     2 
                   
                 
                 
                   γ 
                   2 
                 
                 + 
                 
                   
                     
                       p 
                       
                         E 
                         y 
                         3 
                       
                     
                     + 
                     
                       p 
                       
                         E 
                         y 
                         4 
                       
                     
                     γ 
                   
                 
                 sgn 
                 
                   
                     
                       α 
                       y 
                     
                   
                 
               
             
             
               λ 
               
                 E 
                 y 
               
             
           
         
       
     
     
       
         
           
             
               E 
               γ 
             
             = 
             
               p 
               
                 E 
                 y 
                 5 
               
             
             
               λ 
               
                 E 
                 γ 
               
             
           
         
       
     
     
       
         
           
             
               S 
               
                 H 
                 y 
               
             
             = 
             
               
                 
                   p 
                   
                     H 
                     y 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     H 
                     y 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               λ 
               
                 H 
                 y 
               
             
               
               
           
         
       
     
     
       
         
           
             
               S 
               
                 V 
                 y 
               
             
             = 
             
               F 
               z 
             
             
               
                 
                   p 
                   
                     V 
                     y 
                     1 
                   
                 
                 + 
                 
                   p 
                   
                     V 
                     y 
                     2 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             
               λ 
               
                 V 
                 y 
               
             
             
               λ 
               
                 μ 
                 ​ 
                 y 
               
             
             
               ξ 
               2 
             
           
         
       
     
     
       
         
           
             
               S 
               
                 V 
                 y 
                 γ 
               
             
             = 
             
               F 
               z 
             
             
               
                 
                   p 
                   
                     V 
                     y 
                     3 
                   
                 
                 + 
                 
                   p 
                   
                     V 
                     y 
                     4 
                   
                 
                 d 
                 
                   f 
                   z 
                 
               
             
             γ 
             
               λ 
               
                 K 
                 y 
                 γ 
               
             
             
               λ 
               
                 μ 
                 ​ 
                 y 
               
             
             
               ξ 
               2 
             
           
         
       
     
      In a combined slip condition, the following modified MF based tire  18  model equations may be used: 
     
       
         
           
             
               
                 
                   D 
                   
                     V 
                     y 
                     k 
                   
                 
                 = 
                 
                   μ 
                   y 
                 
                 
                   F 
                   z 
                 
                 
                   
                     
                       r 
                       
                         V 
                         y 
                         1 
                       
                     
                     + 
                     
                       r 
                       
                         V 
                         y 
                         2 
                       
                     
                     d 
                     
                       f 
                       z 
                     
                     + 
                     
                       r 
                       
                         V 
                         y 
                         2 
                       
                     
                     γ 
                   
                 
                 cos 
                 
                   
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           r 
                           
                             V 
                             y 
                             4 
                           
                         
                         
                           α 
                           F 
                         
                       
                     
                   
                 
                 
                   ξ 
                   2 
                 
               
             
             
               
                 	 
                 
                   s 
                   
                     V 
                     y 
                     k 
                   
                 
                 = 
                 
                   D 
                   
                     V 
                     y 
                     k 
                   
                 
                 sin 
                 
                   
                     
                       r 
                       
                         V 
                         y 
                         2 
                       
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           r 
                           
                             V 
                             y 
                             
                               6 
                               K 
                             
                           
                         
                       
                     
                   
                 
                 
                   λ 
                   
                     V 
                     y 
                     k 
                   
                 
               
             
           
         
       
     
      Weighting function: 
     
       
         
           
             
               G 
               
                 y 
                 κ 
               
             
             = 
             
               
                 cos 
                 
                   
                     
                       C 
                       
                         y 
                         κ 
                       
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           
                             y 
                             k 
                           
                         
                         
                           κ 
                           s 
                         
                         − 
                         
                           E 
                           
                             y 
                             κ 
                           
                         
                         
                           
                             
                               B 
                               
                                 y 
                                 k 
                               
                             
                             
                               κ 
                               s 
                             
                             − 
                             
                               
                                 tan 
                               
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   B 
                                   
                                     y 
                                     k 
                                   
                                 
                                 
                                   κ 
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 cos 
                 
                   
                     
                       C 
                       
                         y 
                         κ 
                       
                     
                     
                       
                         tan 
                       
                       
                         − 
                         1 
                       
                     
                     
                       
                         
                           B 
                           
                             y 
                             k 
                           
                         
                         
                           S 
                           
                             H 
                             y 
                             κ 
                           
                         
                         − 
                         
                           E 
                           
                             y 
                             κ 
                           
                         
                         
                           
                             
                               B 
                               
                                 y 
                                 k 
                               
                             
                             
                               S 
                               
                                 H 
                                 y 
                                 κ 
                               
                             
                             − 
                             
                               
                                 tan 
                               
                               
                                 − 
                                 1 
                               
                             
                             
                               
                                 
                                   B 
                                   
                                     y 
                                     k 
                                   
                                 
                                 
                                   S 
                                   
                                     H 
                                     y 
                                     κ 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               κ 
               s 
             
             = 
             κ 
             + 
             
               S 
               
                 H 
                 y 
                 κ 
               
             
           
         
       
     
     
       
         
           
             
               B 
               
                 y 
                 k 
               
             
             = 
             
               
                 
                   r 
                   
                     B 
                     y 
                     1 
                   
                 
                 + 
                 
                   r 
                   
                     B 
                     y 
                     2 
                   
                 
                 
                   γ 
                   2 
                 
               
             
               
             cos 
             
               
                 
                   
                     tan 
                   
                   
                     − 
                     1 
                   
                 
                 
                   
                     
                       r 
                       
                         B 
                         y 
                         2 
                       
                     
                     
                       
                         
                           α 
                           F 
                         
                         − 
                         
                           r 
                           
                             B 
                             y 
                             3 
                           
                         
                       
                     
                   
                 
               
             
             
               λ 
               
                 y 
                 κ 
               
             
           
         
       
     
     
       
         
           
             
               C 
               
                 y 
                 κ 
               
             
             = 
             
               r 
               
                 C 
                 y 
                 1 
                 , 
               
             
               
             
               E 
               
                 y 
                 κ 
               
             
               
             = 
               
             
               r 
               
                 E 
                 y 
                 1 
               
             
               
             + 
               
             
               r 
               
                 E 
                 y 
                 2 
               
             
             d 
             
               f 
               
                 z 
                 , 
               
             
               
             
               S 
               
                 H 
                 y 
                 κ 
               
             
               
             = 
               
             
               r 
               
                 H 
                 y 
                 1 
               
             
               
             + 
               
             
               r 
               
                 H 
                 y 
                 2 
               
             
             d 
             
               f 
               z 
             
           
         
       
     
      When combined slip is not used: S Vyκ  = 0, G yκ  = 1. When turn slip is not used: 
     
       
         
           
             
               ξ 
               i 
             
             = 
             1 
             , 
               
               
             i 
             = 
             1 
             : 
             4. 
           
         
       
     
     Referring now to  FIG.  4   , and with continuing reference to  FIGS.  1 - 3   , torque dynamics in both front and rear axles  30 ,  28  may be characterized by: 
     
       
         
           
             
               
                 T 
                 ˙ 
               
               
                 1 
                 , 
                 2 
               
             
             = 
             − 
             
               1 
               
                 
                   τ 
                   f 
                 
               
             
             
               T 
               
                 1 
                 , 
                 2 
               
             
               
             + 
               
             
               1 
               
                 2 
                 
                   τ 
                   f 
                 
               
             
             
               T 
               
                 
                   f 
                   
                     r 
                     q 
                     s 
                     t 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 T 
                 ˙ 
               
               3 
             
             = 
               
             
               
                 − 
                 
                   T 
                   
                     
                       r 
                       
                         a 
                         c 
                         t 
                       
                     
                   
                 
                 + 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
               
                 2 
                 
                   τ 
                   r 
                 
               
             
             = 
             
               
                 − 
                 
                   
                     
                       T 
                       3 
                     
                     + 
                     
                       T 
                       4 
                     
                   
                 
                 + 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
               
                 2 
                 
                   τ 
                   r 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 T 
                 ˙ 
               
               4 
             
             = 
               
             
               
                 − 
                 
                   T 
                   
                     
                       r 
                       
                         a 
                         c 
                         t 
                       
                     
                   
                 
                 + 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
               
                 2 
                 
                   τ 
                   r 
                 
               
             
             = 
             
               
                 − 
                 
                   
                     
                       T 
                       3 
                     
                     + 
                     
                       T 
                       4 
                     
                   
                 
                 + 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
               
                 2 
                 
                   τ 
                   r 
                 
               
             
           
         
       
     
      where, τ ƒ ,τ r  are front and rear axles actuators time delay, respectively. In some aspects, a clutch-based torque front and rear axle  30 ,  28  torque distribution or dynamics solution plus eLSD  36  or torque-vectoring from wheel  27  to wheel  27  may be modeled by the following: 
     
       
         
           
             
               
                 T 
                 ˙ 
               
               
                 1 
                 , 
                 2 
               
             
             = 
             − 
             
               1 
               
                 
                   τ 
                   f 
                 
               
             
             
               T 
               
                 1 
                 , 
                 2 
               
             
             + 
             
               1 
               
                 2 
                 
                   τ 
                   r 
                 
               
             
             
               T 
               
                 
                   f 
                   
                     r 
                     q 
                     s 
                     t 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     T 
                     ˙ 
                   
                   3 
                 
                 = 
                   
                 − 
                 
                   1 
                   
                     2 
                     
                       τ 
                       r 
                     
                   
                 
                 
                   
                     
                       T 
                       3 
                     
                     + 
                     
                       T 
                       4 
                     
                   
                 
                 − 
                 
                   1 
                   
                     2 
                     
                       τ 
                       c 
                     
                   
                 
                 
                   
                     
                       T 
                       3 
                     
                     − 
                     
                       T 
                       4 
                     
                   
                 
                 + 
               
             
             
               
                 
                   1 
                   
                     2 
                     
                       τ 
                       r 
                     
                   
                 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
                 + 
                 
                   1 
                   
                     2 
                     
                       τ 
                       c 
                     
                   
                 
                 
                   p 
                   1 
                 
                   
                 t 
                 a 
                 n 
                 h 
                   
                 
                   p 
                   2 
                 
                 
                   
                     
                       Δ 
                       ω 
                     
                   
                   
                     p 
                     3 
                   
                 
                 
                   T 
                   
                     
                       c 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     T 
                     ˙ 
                   
                   4 
                 
                 = 
                   
                 − 
                 
                   1 
                   
                     2 
                     
                       τ 
                       r 
                     
                   
                 
                 
                   
                     
                       T 
                       3 
                     
                     + 
                     
                       T 
                       4 
                     
                   
                 
                 + 
                 
                   1 
                   
                     2 
                     
                       τ 
                       c 
                     
                   
                 
                 
                   
                     
                       T 
                       3 
                     
                     − 
                     
                       T 
                       4 
                     
                   
                 
                 + 
               
             
             
               
                 
                   1 
                   
                     2 
                     
                       τ 
                       r 
                     
                   
                 
                 
                   T 
                   
                     
                       r 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
                 − 
                 
                   1 
                   
                     2 
                     
                       τ 
                       c 
                     
                   
                 
                 
                   p 
                   1 
                 
                   
                 t 
                 a 
                 n 
                 h 
                   
                 
                   p 
                   2 
                 
                 
                   
                     
                       Δ 
                       ω 
                     
                   
                   
                     p 
                     3 
                   
                 
                 
                   T 
                   
                     
                       c 
                       
                         r 
                         q 
                         s 
                         t 
                       
                     
                   
                 
               
             
           
         
       
     
      where, τ ƒ ,τ r , τ c  are front and rear axles and clutch time delays, respectively. Similarly, in a motor vehicle  12  equipped with active aerodynamic actuators  44 , active aerodynamics downforce dynamics may be modeled by: 
     
       
         
           
             
               
                 ν 
                 ˙ 
               
               z 
             
             = 
             
               
                 
                   
                     
                       F 
                       
                         s 
                         p 
                       
                       f 
                     
                     + 
                     
                       F 
                       
                         A 
                         n 
                         t 
                         i 
                         D 
                         i 
                         v 
                         e 
                       
                     
                   
                 
               
             
             + 
             
               
                 
                   F 
                   
                     s 
                     p 
                   
                   r 
                 
                 + 
                 
                   F 
                   
                     A 
                     n 
                     t 
                     i 
                     S 
                     u 
                     q 
                     a 
                     t 
                   
                 
               
             
             − 
             
               F 
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               f 
             
             − 
             
               
                 
                   F 
                   
                     
                       z 
                       
                         a 
                         e 
                         r 
                         o 
                       
                     
                   
                   r 
                 
               
             
             / 
             m 
             − 
             g 
           
         
       
     
     
       
         
           
             
               θ 
               ¨ 
             
             = 
             ( 
             
               L 
               f 
             
             ( 
             
               F 
               s 
               f 
             
             + 
             
               
                 
                   F 
                   
                     A 
                     n 
                     t 
                     i 
                     D 
                     i 
                     v 
                     e 
                   
                 
                 ) 
               
               
                 − 
                 m 
                 h 
                 
                   a 
                   x 
                 
                 ) 
                 / 
                 
                   I 
                   θ 
                 
               
             
             − 
             
               L 
               r 
             
             ( 
             
               F 
               s 
               r 
             
             + 
             
               F 
               
                 A 
                 n 
                 t 
                 i 
                 S 
                 q 
                 u 
                 a 
                 t 
               
             
             ) 
             − 
             
               L 
               f 
               
                 D 
                 F 
               
             
             
               F 
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               f 
             
             + 
             
               L 
               r 
               
                 D 
                 F 
               
             
             
               F 
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               r 
             
           
         
       
     
      where, 
     
       
         
           
             
               F 
               
                 s 
                 p 
               
               f 
             
             = 
             
               F 
               
                 
                   f 
                   0 
                 
               
             
             + 
             
               K 
               f 
             
             Δ 
             
               z 
               f 
             
             + 
             
               C 
               f 
             
             Δ 
             
               
                 z 
                 ˙ 
               
               f 
             
             
               F 
               
                 s 
                 p 
               
               r 
             
             = 
             
               F 
               
                 
                   r 
                   0 
                 
               
             
             + 
             
               K 
               r 
             
               
             Δ 
             
               z 
               r 
             
               
             + 
               
             
               C 
               r 
             
             Δ 
             
               
                 
                   z 
                   ⋅ 
                 
               
               r 
             
           
         
       
     
      where F Anti Dive  and F Anti Squat  are the portion of load transfer that is transferred to wheels through anti-dive and anti-squat mechanism, and  
     
       
         
           
             
               F 
               
                 s 
                 p 
               
               f 
             
           
         
       
     
     and  
     
       
         
           
             
               F 
               
                 s 
                 p 
               
               r 
             
           
         
       
     
     are front and rear spring forces. 
     
       
         
           
             
               F 
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               f 
             
             = 
             
               1 
               2 
             
             ρ 
             
               A 
               f 
             
             
               v 
               x 
               2 
             
             
               C 
               L 
             
             
               
                 
                   θ 
                   
                     w 
                     , 
                   
                   f 
                 
                 
                   h 
                   w 
                   f 
                 
               
             
             , 
               
             
               F 
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               r 
             
             = 
             
               1 
               2 
             
             ρ 
             
               A 
               r 
             
             
               v 
               x 
               2 
             
             
               C 
               L 
             
             
               
                 
                   θ 
                   
                     w 
                     , 
                   
                   r 
                 
                 
                   h 
                   w 
                   r 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 F 
                 ˙ 
               
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               f 
             
             = 
             
               1 
               2 
             
             ρ 
             
               A 
               f 
             
             
               v 
               x 
               2 
             
             
               
                 
                   
                     ∂ 
                     
                       C 
                       L 
                     
                     
                       
                         
                           θ 
                           
                             w 
                             , 
                           
                           f 
                         
                         
                           h 
                           w 
                           f 
                         
                       
                     
                   
                   
                     ∂ 
                     
                       θ 
                       w 
                       f 
                     
                   
                 
                 
                   
                     θ 
                     ˙ 
                   
                   w 
                   
                     f 
                     , 
                     r 
                     e 
                     q 
                   
                 
                 + 
                 
                   
                     ∂ 
                     
                       C 
                       L 
                     
                     
                       
                         
                           θ 
                           
                             w 
                             , 
                           
                           f 
                         
                         
                           h 
                           w 
                           f 
                         
                       
                     
                   
                   
                     ∂ 
                     
                       h 
                       w 
                       f 
                     
                   
                 
                 
                   
                     h 
                     ˙ 
                   
                   w 
                   f 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 F 
                 ˙ 
               
               
                 
                   z 
                   
                     a 
                     e 
                     r 
                     o 
                   
                 
               
               f 
             
             = 
             
               1 
               2 
             
             ρ 
             
               A 
               r 
             
             
               v 
               x 
               2 
             
             
               
                 
                   
                     ∂ 
                     
                       C 
                       L 
                     
                     
                       
                         
                           θ 
                           
                             w 
                             , 
                           
                           f 
                         
                         
                           h 
                           w 
                           r 
                         
                       
                     
                   
                   
                     ∂ 
                     
                       θ 
                       w 
                       f 
                     
                   
                 
                 
                   
                     θ 
                     ˙ 
                   
                   w 
                   
                     f 
                     , 
                     r 
                     e 
                     q 
                   
                 
                 + 
                 
                   
                     ∂ 
                     
                       C 
                       L 
                     
                     
                       
                         
                           θ 
                           
                             w 
                             , 
                           
                           f 
                         
                         
                           h 
                           w 
                           r 
                         
                       
                     
                   
                   
                     ∂ 
                     
                       h 
                       w 
                       f 
                     
                   
                 
                 
                   
                     h 
                     ˙ 
                   
                   w 
                   r 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 θ 
                 ˙ 
               
               w 
               f 
             
             = 
             − 
             
               1 
               
                 
                   τ 
                   w 
                   f 
                 
               
             
             
               θ 
               w 
               f 
             
             + 
             
               1 
               
                 
                   τ 
                   w 
                   f 
                 
               
             
             
               θ 
               w 
               
                 f 
                 , 
                   
                 r 
                 e 
                 q 
               
             
             , 
               
               
               
               
               
               
               
               
               
               
             
               
                 θ 
                 ˙ 
               
               w 
               r 
             
             = 
             − 
             
               1 
               
                 
                   τ 
                   w 
                   f 
                 
               
             
             
               θ 
               w 
               r 
             
             + 
             
               1 
               
                 
                   τ 
                   w 
                   f 
                 
               
             
             
               θ 
               w 
               
                 r 
                 , 
                   
                 r 
                 e 
                 q 
               
             
           
         
       
     
     
       
         
           
             
               
                 h 
                 ˙ 
               
               w 
               f 
             
             = 
             
               v 
               z 
             
             + 
             
               L 
               f 
               
                 D 
                 F 
               
             
             
               θ 
               ˙ 
             
             , 
               
               
               
               
               
               
               
               
               
               
             
               
                 h 
                 ˙ 
               
               w 
               r 
             
             = 
             
               v 
               z 
             
             − 
             
               L 
               r 
               
                 D 
                 F 
               
             
             
               θ 
               ˙ 
             
           
         
       
     
      Finally, integrating the above calculations, a state-space model of the motor vehicle  12  may be shown as: 
     
       
         
           
             
               x 
               ˙ 
             
             ( 
             t 
             ) 
             = 
             f 
             
               
                 x 
                 ( 
                 t 
                 ) 
                 , 
                   
                   
                 u 
                   
                 ( 
                 t 
                 ) 
               
             
           
         
       
     
     
       
         
           
             y 
             
               t 
             
             = 
             g 
             
               
                 x 
                 
                   t 
                 
                 , 
                 u 
                 
                   t 
                 
               
             
           
         
       
     
      Likewise a linear time-varying (LTV) model that approximates above equation can be computed as follows: 
     
       
         
           
             A 
             = 
             
               
                 
                   ∂ 
                   f 
                 
                 
                   ∂ 
                   x 
                 
               
             
             , 
             B 
             = 
             
               
                 
                   ∂ 
                   f 
                 
                 
                   ∂ 
                   u 
                 
               
             
             , 
             C 
             = 
             
               
                 
                   ∂ 
                   g 
                 
                 
                   ∂ 
                   x 
                 
               
             
             , 
             D 
             = 
             
               
                 
                   ∂ 
                   g 
                 
                 
                   ∂ 
                   u 
                 
               
             
           
         
       
     
      And can be presented as: 
     
       
         
           
             
               x 
               ˙ 
             
             
               t 
             
             = 
             A 
             x 
             
               t 
             
             + 
             B 
             u 
             
               t 
             
             + 
             W 
             
               t 
             
           
         
       
     
     
       
         
           
             y 
             
               t 
             
             = 
             C 
             x 
             
               t 
             
             + 
             D 
             u 
             
               t 
             
             + 
             V 
             
               t 
             
           
         
       
     
      Where, state variables and control actions can be defined as: 
     
       
         
           
             x 
             = 
             
               
                 
                   
                     
                       v 
                       x 
                     
                       
                     
                       v 
                       y 
                     
                       
                     
                       v 
                       z 
                     
                       
                     
                       ψ 
                       ˙ 
                     
                       
                     
                       θ 
                       ˙ 
                     
                       
                     
                       T 
                       1 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       T 
                       2 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       T 
                       3 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       T 
                       4 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       F 
                       
                         
                           z 
                           
                             a 
                             e 
                             r 
                             o 
                           
                         
                       
                       f 
                     
                       
                     
                       F 
                       
                         
                           z 
                           
                             a 
                             e 
                             r 
                             o 
                           
                         
                       
                       f 
                     
                       
                     
                       θ 
                       w 
                       f 
                     
                       
                     
                       θ 
                       w 
                       f 
                     
                   
                 
               
               T 
             
           
         
       
     
     
       
         
           
             u 
             = 
             
               
                 
                   
                     
                       T 
                       f 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       T 
                       r 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       T 
                       c 
                       
                         r 
                         e 
                         q 
                       
                     
                       
                     
                       θ 
                       w 
                       
                         f 
                         , 
                         r 
                         e 
                         q 
                       
                     
                     ​ 
                     
                       θ 
                       w 
                       
                         r 
                         , 
                         r 
                         e 
                         q 
                       
                     
                   
                 
               
               T 
             
           
         
       
     
     In an example, the first component of the matrix A can be calculated as: 
     
       
         
           
             A 
             = 
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             f 
                             
                               
                                 1 
                                 
                                   ∂ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           ⋯ 
                         
                         
                           
                             ∂ 
                             f 
                             
                               
                                 1 
                                 
                                   ∂ 
                                   
                                     x 
                                     
                                       11 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             ∂ 
                             f 
                             
                               
                                 
                                   11 
                                 
                                 
                                   ∂ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           ⋯ 
                         
                         
                           
                             ∂ 
                             f 
                             
                               
                                 
                                   11 
                                 
                                 
                                   ∂ 
                                   
                                     x 
                                     
                                       11 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 11 
                 × 
                 11 
               
             
           
         
       
     
      Similar matrix calculations can be carried out for each of the components of Matrices A, B, C and D. For examples  
     
       
         
           
             ∂ 
             f 
             
               
                 1 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
           
         
       
     
     can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         + 
                         cos 
                         
                           
                             
                               δ 
                               f 
                             
                           
                         
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         x 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         , 
                                         
                                           K 
                                           i 
                                         
                                         , 
                                         
                                           F 
                                           
                                             z 
                                             i 
                                           
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       a 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         x 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         
                                           K 
                                           i 
                                         
                                         , 
                                         F 
                                         
                                           z 
                                           i 
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       K 
                                       i 
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       K 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         z 
                                         i 
                                       
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         x 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         
                                           K 
                                           i 
                                         
                                         
                                           F 
                                           
                                             z 
                                             i 
                                           
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         z 
                                         i 
                                       
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         z 
                                         i 
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         − 
                         sin 
                         
                           
                             
                               δ 
                               f 
                             
                           
                         
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         y 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         , 
                                         
                                           K 
                                           i 
                                         
                                         , 
                                         
                                           F 
                                           i 
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       a 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       x 
                                       i 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         y 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         
                                           K 
                                           i 
                                         
                                         , 
                                         
                                           F 
                                           
                                             z 
                                             i 
                                           
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       k 
                                       i 
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       K 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         y 
                                         i 
                                       
                                     
                                     
                                       
                                         
                                           a 
                                           i 
                                         
                                         , 
                                         
                                           K 
                                           i 
                                         
                                         , 
                                         
                                           F 
                                           
                                             z 
                                             i 
                                           
                                         
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         z 
                                         i 
                                       
                                     
                                   
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       F 
                                       
                                         z 
                                         i 
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             / 
             m 
           
         
       
     
     The LTV portion of the MPC algorithm is designed to solve a constrained optimization problem at each sampling time with the following objectives and cost terms: 
     
       
         
           
             J 
             
               
                 x 
                 
                   t 
                 
                 , 
                 U 
               
             
             = 
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         p 
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         y 
                                         
                                           t 
                                           + 
                                           k 
                                           , 
                                           t 
                                         
                                       
                                       − 
                                       
                                         V 
                                         x 
                                       
                                       
                                           
                                         
                                           t 
                                           + 
                                           k 
                                           , 
                                           t 
                                         
                                         
                                           Re 
                                           f 
                                         
                                       
                                     
                                     ︸ 
                                   
                                   
                                     m 
                                     o 
                                     t 
                                     i 
                                     o 
                                     n 
                                       
                                     e 
                                     r 
                                     r 
                                     o 
                                     r 
                                       
                                     t 
                                     e 
                                     r 
                                     m 
                                   
                                 
                               
                             
                           
                           
                             
                               w 
                               1 
                             
                           
                           2 
                         
                         + 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               p 
                               − 
                               1 
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             u 
                                             
                                               t 
                                               + 
                                               k 
                                               , 
                                               t 
                                             
                                           
                                           − 
                                           
                                             u 
                                             
                                               t 
                                               + 
                                               k 
                                               , 
                                               t 
                                             
                                             
                                               Re 
                                               f 
                                             
                                           
                                         
                                         ︸ 
                                       
                                       
                                         c 
                                         o 
                                         n 
                                         t 
                                         r 
                                         o 
                                         l 
                                           
                                         a 
                                         c 
                                         t 
                                         i 
                                         o 
                                         n 
                                           
                                         e 
                                         r 
                                         r 
                                         o 
                                         r 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   w 
                                   2 
                                 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     + 
                     
                       
                         
                           
                             
                               
                                 
                                   Δ 
                                   
                                     u 
                                     
                                       t 
                                       + 
                                       k 
                                       , 
                                       t 
                                     
                                   
                                   − 
                                   Δ 
                                   
                                     u 
                                     
                                       t 
                                       + 
                                       k 
                                       , 
                                       t 
                                     
                                     
                                       Re 
                                       f 
                                     
                                   
                                 
                                 ︸ 
                               
                               
                                 c 
                                 o 
                                 n 
                                 t 
                                 r 
                                 o 
                                 l 
                                   
                                 a 
                                 c 
                                 t 
                                 i 
                                 o 
                                 n 
                                   
                                 var 
                                 i 
                                 a 
                                 t 
                                 i 
                                 o 
                                 n 
                                   
                                 e 
                                 r 
                                 r 
                                 o 
                                 r 
                               
                             
                           
                         
                       
                       
                         
                           w 
                           3 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   
                     + 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     T 
                                     
                                       
                                         f 
                                         
                                           t 
                                           + 
                                           k 
                                           , 
                                           t 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     T 
                                     
                                       r 
                                       
                                         l 
                                         
                                           t 
                                           + 
                                           k 
                                           , 
                                           t 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     T 
                                     
                                       r 
                                       
                                         r 
                                         
                                           t 
                                           + 
                                           k 
                                           , 
                                           t 
                                         
                                       
                                     
                                   
                                   − 
                                   
                                     T 
                                     
                                       d 
                                       r 
                                       i 
                                       v 
                                       e 
                                       r 
                                     
                                   
                                   
                                       
                                     
                                       t 
                                       + 
                                       k 
                                       , 
                                       t 
                                     
                                     
                                       Re 
                                       f 
                                     
                                   
                                 
                                 ︸ 
                               
                               
                                 t 
                                 o 
                                 t 
                                 a 
                                 l 
                                   
                                 t 
                                 o 
                                 r 
                                 q 
                                 u 
                                 e 
                                   
                                 e 
                                 r 
                                 r 
                                 o 
                                 r 
                               
                             
                           
                         
                       
                       
                         
                           w 
                           4 
                         
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             s 
             .t 
             . 
               
             
               x 
               
                 t 
                 + 
                 k 
                 + 
                 1 
                 , 
                 t 
               
             
             = 
             A 
             
               
                 
                   x 
                   
                     t 
                     + 
                     k 
                     , 
                     t 
                   
                 
                 − 
                 
                   
                     x 
                     ¯ 
                   
                   t 
                 
               
             
             + 
             B 
             
               
                 
                   u 
                   
                     t 
                     + 
                     k 
                     , 
                     t 
                   
                 
                 − 
                 
                   
                     u 
                     ¯ 
                   
                   t 
                 
               
             
             + 
             W 
             , 
             k 
             = 
             0 
             , 
             … 
             , 
             N 
             − 
             1 
           
         
       
     
     
       
         
           
             
               x 
               0 
             
             = 
             x 
             
               t 
             
           
         
       
     
     
       
         
           
             
               
                 
                   u 
                   
                     min 
                   
                 
                 ≤ 
                 
                   u 
                   
                     t 
                     + 
                     k 
                   
                 
                 ≤ 
                 
                   u 
                   
                     max 
                   
                 
                 , 
                 k 
                 = 
                 0 
                 , 
                   
                 … 
                 , 
                 N 
                 − 
                 1 
               
             
             
               
                 
                   y 
                   
                     min 
                   
                 
                 ≤ 
                 C 
                 
                   x 
                   
                     t 
                     + 
                     k 
                   
                 
                 ≤ 
                 
                   y 
                   
                     max 
                   
                 
                 , 
                 k 
                 = 
                 1 
                 , 
                   
                 … 
                 , 
                 N 
               
             
           
         
       
     
      where y t+k,t  and  
     
       
         
           
             
               y 
               
                 t 
                 + 
                 k 
                 , 
                 t 
               
               
                 Re 
                 f 
               
             
           
         
       
     
     denote predicted and reference longitudinal velocity, respectively, and where u t+k,t  and  
     
       
         
           
             
               u 
               
                 t 
                 + 
                 k 
                 , 
                 t 
               
               
                 R 
                 e 
                 f 
               
             
           
         
       
     
     denote control actions that includes, front, rear left and rear right torque distributions, respectively. Additionally, Δu t+k,t  and  
     
       
         
           
             Δ 
             
               u 
               
                 t 
                 + 
                 k 
                 , 
                 t 
               
               
                 R 
                 e 
                 f 
               
             
           
         
       
     
     denote control action variation and its reference, respectively.  
     
       
         
           
             
               T 
               
                 
                   f 
                   
                     t 
                     + 
                     k 
                     , 
                     t 
                   
                 
               
             
             , 
             
               T 
               
                 r 
                 
                   l 
                   
                     t 
                     + 
                     k 
                     , 
                     t 
                   
                 
               
             
             , 
             
               T 
               
                 r 
                 
                   r 
                   
                     t 
                     + 
                     k 
                     , 
                     t 
                   
                 
               
             
           
         
       
     
     are front, rear left, and rear right torque distribution sequence and  
     
       
         
           
             
               T 
               
                 d 
                 r 
                 i 
                 v 
                 e 
                 r 
               
             
             
                 
               
                 t 
                 + 
                 k 
                 , 
                 t 
               
               
                 R 
                 e 
                 f 
               
             
           
         
       
     
     is driver torque request. Utilizing the above equations and variables output can be shown as: 
     
       
         
           
             y 
             = 
             
               
                 
                   
                     
                       v 
                       x 
                     
                       
                       
                     
                       v 
                       y 
                     
                       
                       
                     
                       v 
                       z 
                     
                       
                       
                     
                       θ 
                       ˙ 
                     
                       
                       
                     
                       ψ 
                       ˙ 
                     
                       
                       
                     
                       a 
                       x 
                     
                       
                       
                     
                       a 
                       y 
                     
                       
                       
                     
                       a 
                       z 
                     
                       
                       
                     
                       M 
                       θ 
                     
                       
                       
                     
                       M 
                       ψ 
                     
                       
                       
                     
                       ω 
                       1 
                     
                       
                       
                     
                       ω 
                       2 
                     
                       
                       
                     
                       ω 
                       3 
                     
                       
                       
                     
                       ω 
                       4 
                     
                   
                 
               
               T 
             
           
         
       
     
      where  
     
       
         
           
             
               
                 x 
                 ¯ 
               
               t 
             
               
             , 
               
             
               
                 u 
                 ¯ 
               
               t 
             
           
         
       
     
     represent nominal operating point for state and command. A nominal state,  
     
       
         
           
             
               
                 x 
                 ¯ 
               
               t 
             
             , 
           
         
       
     
     is calculated based on solving the nonlinear vehicle /wheel model and  
     
       
         
           
             
               
                 u 
                 ¯ 
               
               t 
             
           
         
       
     
     is calculated outside MPC, for example, by feedforward control within the feed-forward controller  108 . 
     Once the motor vehicle  12 , tire  18 , and actuator  52  modeling has been completed, and the cost function(s) and constraints described above have been determined, the controller  14  executes a control logic that functions as a quadratic problem (QP) solver that dynamically solves quadratic problems in real-time to optimize control signals to the actuators  52  of the motor vehicle  12  so as to achieve maximum grip, stability, and the like. Specifically, the above-described adaptive cost function is optimized in an online fashion to find feasible sets of control actions to actuators  52  which minimize any potential error. By substituting 
     
       
         
           
             
               x 
               k 
             
             = 
             
               A 
               k 
             
             x 
             ( 
             t 
             ) 
             + 
             
               
                 ∑ 
                 
                   i 
                   = 
                   0 
                 
                 
                   k 
                   − 
                   1 
                 
               
               
                 
                   A 
                   i 
                 
                 B 
                 
                   u 
                   
                     k 
                     − 
                     1 
                     − 
                     i 
                   
                 
               
             
           
         
       
     
     in the cost function and recasting: 
     
       
         
           
             U 
             * 
             
               
                 x 
                 
                   t 
                 
               
             
             ≜ 
               
             arg  
             
               
                 min 
               
               U 
             
             
               1 
               2 
             
             
               U 
               ′ 
             
             H 
             U 
             + 
             
               x 
               ′ 
             
             
               t 
             
             
               C 
               ′ 
             
             U 
             + 
             
               1 
               2 
             
             
               x 
               ′ 
             
             
               t 
             
             Y 
             x 
             
               t 
             
           
         
       
     
     
       
         
           
             s 
             .t 
             . 
               
             G 
             U 
             ≤ 
             W 
             + 
             S 
             x 
             
               t 
             
           
         
       
     
      where  
     
       
         
           
             U 
             * 
             
               
                 x 
                 
                   t 
                 
               
             
             = 
             
               
                 
                   
                     
                       
                         u 
                         ′ 
                       
                       0 
                       * 
                     
                     
                       
                         x 
                         
                           t 
                         
                       
                     
                       
                     … 
                       
                     
                       
                         u 
                         ′ 
                       
                       
                         N 
                         − 
                         1 
                       
                       * 
                     
                     
                       
                         x 
                         
                           t 
                         
                       
                     
                   
                 
               
               ′ 
             
           
         
       
     
     is the optimal solution, H &gt; 0 and C, Y, G, W, S are matrices of appropriate dimensions. The MPC control algorithm is based on the following iterations: at time t, measure or estimate the current state x(t), solve the QP problem to get the optimal sequence if future input moves 
     
       
         
           
             U 
             * 
             
               
                 x 
                 
                   t 
                 
               
             
             , 
           
         
       
     
     
       
         
           
             u 
             
               t 
             
             = 
               
             
               u 
               0 
               * 
             
             
               
                 x 
                 
                   t 
                 
               
             
           
         
       
     
      To the process, discard the remaining optimal moves, repeat the procedure again at time t + 1. 
     Referring now to  FIG.  5   , and with continuing reference to  FIGS.  1 - 4   , a method  200  of implementing MPC-based chassis and driveline actuator management in a motor vehicle  12  is shown. The method  200  begins at block  202 . At block  204 , the system  10  receives a driver control input  102  via the I/O ports  58  of the controller  14 , and more specifically, of the DCI  104 . The DCI  104  reads and interprets the driver control inputs  102  to generate a desired dynamics signal  106  in the form of an actuator output. At block  206 , the system  10  obtains motor vehicle  12  state information measurements via the plurality of sensors  64  equipped to the motor vehicle  12 . More specifically, at block  206 , the controller  14  receives motor vehicle  12  state information reported by IMUs  66 , SADS  68 , GPS  70  sensors, and the like. The motor vehicle state information from the sensors  64  may be obtained continuously, cyclically, intermittently, or the like without departing from the scope or intent of the present disclosure. 
     At block  208  the motor vehicle  12  state information measurements are received and processed by the sensor/estimation module  110  and the feedback controller  112 . More specifically, at block  208  the sensor/estimation module  110  executes control logic that processes the motor vehicle  12  state information measurements to generate estimations  116  for each of the various active chassis and dynamics systems equipped to the motor vehicle  12 . In a particular example, the sensor/estimation module  110  includes an aerodynamics model estimation, an eLSD  36  model estimation, an eAWD  34  model estimation, and vehicle dynamics estimations. The aero model estimation calculates downforce and maximum downforce possible given current vehicle state information. Likewise, the eLSD  36  model estimation includes clutch torque estimations and maximum clutch torque capacity estimations. The eAWD  34  model estimation produces maximum electric motor  24  torque estimations. Finally, the vehicle dynamics estimations include vehicle states, road surface information, tire  18  force calculations, and road angles. 
     At block  210 , the estimations  116  for each of the active chassis and dynamics systems from the sensor/estimation module  110  are received and processed by the DCI  104 . More specifically, at block  210  the DCI  104  generates a desired dynamics signal  106 . The desired dynamics signal  106  defines a desired output of the actuators  52  equipped to the motor vehicle  12  based on driver input and data from the sensors  64  and from the sensor/estimation module  110 . 
     At block  212 , the estimations  116  for each of the active chassis and dynamic systems from the sensor/estimation module  110  are received within the dynamic constraint calculator  114 . The dynamic constraint calculator  114  applies actual physical limitations, and tire  18  and road grip limitations to signals from the sensor/estimation module  110 . At block  214 , the dynamic constraint calculator  114  generates control action constraints  122 . 
     At block  216  the estimations  116  for each of the active chassis and dynamics systems from the sensor/estimation module  110  are received within the feed-forward controller  108  along with the control action constraints  122  from the dynamic constraint calculator  114 . The feed-forward controller  108  processes the estimations  116  and control action constraints  122  and commands the actuators  52  to achieve certain transient response characteristics, provide optimal reference control actions  120 , and linearize control responses around an operation point for each of the active chassis and dynamic systems equipped to the motor vehicle  12 . In an example, the feed-forward controller  108  provides active aerodynamic preemptive control signals, eLSD  36  preemptive control signals, and eAWD  34  preemptive control signals. Preemptive control signals, adjust actuator  52  outputs to comport with control signal estimates from a sensor/estimation module  110 . At block  218 , the feed-forward controller  108  generates the reference control action  120 . 
     At block  220 , the feedback controller  112  receives and processes the control action constraints  122 , the reference control action  120 , the desired actuator output or desired dynamics signal  106 , and the estimates and sensor measurements from the sensor/estimation module  110  and the sensors  64 . More specifically, at block  220 , the feedback controller  112  executes control logic that integrates the desired dynamics signal  106 , reference control action  120 , control action constraints  122 , and measurements  124  into a model which accounts for body  62  and wheel  27  dynamics and in-plane and out-of-plane actuators  32 ,  42 . The feedback controller  112  models electric motor  24  torque, eLSD  36  outputs, eAWD  34  outputs, combined tire  18  slip data, and active aerodynamic actuator  44  outputs, as well as front-to-rear and/or side-to-side interactions of various actuators  52  of the motor vehicle. 
     At block  222 , the feedback controller  112  generates an optimal control action signal  126  that provides the motor vehicle  12  driver with desired motor vehicle  12  performance characteristics including, but not limited to maximum feasible performance, stability, handling, maneuverability, steerability. At block  224 , the method  200  ends and returns to block  202  where the method  200  continuously operates while the motor vehicle  12  is in use. 
     The system  10  and method  200  of the present disclosure offer several advantages. These include providing a motor vehicle  12  driver or operator maximum feasible performance, stability, handling, maneuverability, steerability of the motor vehicle  12  in a wide variety of conditions, including in inclement weather, instances of tire  18  deformation, and the like. Additionally, the system  10  and method  200  may operate on a motor vehicle  12  in complex driving scenarios, including performance driving situations in which the driver may attempt a powerslide, drift, or the like, and the system  10  and method  200  will operate to provide an appropriate quantity of force generation at the tire  18 /road interface or contact patch  38 , while also providing maximum tire  18 /road interface or contact patch  38  adhesion in driving scenarios in which maximum grip is desired. These benefits may all be obtained using the system  10  and method  200  described herein while maintaining or reducing cost and complexity, reducing calibration efforts, and improving simplicity, and while also providing increased redundancy and robustness. 
     The description of the present disclosure is merely exemplary in nature and variations that do not depart from the gist of the present disclosure are intended to be within the scope of the present disclosure. Such variations are not to be regarded as a departure from the spirit and scope of the present disclosure.