Discrete-Time Model of a Two-Speed PowerShift suitable for Real-Time Control and Simulation

1. Problem Statement

Vehicle powertrains are complex systems involving mechanical, electronic, and electromechanical subsystems. A critical challenge in modeling these systems is accurately simulating clutch engagement and disengagement, which relies on static and Coulomb friction.

• Limitations of Existing Models: Traditional continuous-time models often use variable-step solvers (e.g., MATLAB's ode45) and require zero-crossing detection to handle the switching dynamics of friction.
• Real-Time Constraints: These continuous-time approaches are computationally expensive and unpredictable in execution time due to adaptive step sizes. This makes them unsuitable for real-time control applications (e.g., Hardware-in-the-Loop simulation or embedded ECU testing) where fixed, deterministic execution times are mandatory.
• Simplification Issues: To avoid complexity, some models ignore detailed friction dynamics or use average models, sacrificing accuracy.

2. Methodology

The authors propose a novel discrete-time modeling approach specifically designed for a two-speed powershift transmission. The methodology consists of three main components:

A. Mathematical Formulation

The system is modeled using Euler backward discretization.

• State Variables: The model defines discrete velocity variables $\delta_a(k)$ and $\delta_b(k)$ representing the speed differences across Clutch A (Gear 1) and Clutch B (Gear 2).
• Dynamic Equations: The system dynamics are expressed in matrix form relating inertias ($J_1, J_2$), input torques ($M_1, M_2$), and friction torques ($M_{ca}, M_{cb}$).
• Friction Modeling: The maximum friction torques are modeled as saturation functions ($K$) multiplied by the sign of the relative speed, capturing the transition between slipping and locking.

B. Exact Torque Computation

A core contribution is the derivation of the exact torque required to achieve specific clutch states without relying on iterative solvers or zero-crossing detection:

1. Single Gear Engagement: The authors derive algebraic conditions to calculate the exact friction torque needed to lock a specific clutch (e.g., Clutch A) while the other is slipping. This involves solving linear equations based on the previous time step's state and current inputs.
2. Full Lock Condition: A specific condition is derived for when both clutches are engaged simultaneously (full lock), where the relative speeds are zero. The model calculates the exact torque distribution required to maintain this state.

C. Control Logic (Algorithm 1)

A deterministic control algorithm is developed to manage the clutch states at every discrete time step $k$:

• Input: Previous angular speeds, current input torques, and friction saturation limits.
• Logic Flow:
  1. Check if a "lock" condition is possible for either clutch based on the calculated required torque vs. available saturation torque.
  2. Determine the state (Locked/Slipping) for Clutch A and Clutch B.
  3. Compute the actual friction torques ($M_{ca}, M_{cb}$) based on these states.
• Output: The vector of clutch torques for the current step.

3. Key Contributions

• Discrete-Time Exactness: The paper provides a closed-form discrete-time solution for clutch engagement that eliminates the need for variable-step solvers and zero-crossing detection.
• Real-Time Suitability: By using a fixed time step ($T_s$) and deterministic algebraic calculations, the model guarantees consistent execution times, making it viable for real-time simulation and control.
• Unified Logic: The approach seamlessly handles both single-clutch shifts (gear changes) and the full-lock condition (simultaneous engagement) within a single mathematical framework.
• Generalizability: The authors note that this modeling paradigm can be extended to other powertrain components like torque converters and brakes.

4. Simulation Results

The authors validated the proposed model by comparing it against a high-precision continuous-time model (using MATLAB ode45) in a Simulink environment.

• Scenario: A two-speed powershift with initial speeds of 1000 rpm and 100 rpm, undergoing multiple gear shifts and a full-lock condition.
• Accuracy:
  • The discrete-time model (with $T_s = 20$ ms) produced results virtually identical to the continuous-time model regarding angular speeds and clutch torque profiles.
  • As the discrete step size decreased, the accuracy further converged to the continuous solution.
• Execution Frequency:
  • The continuous-time model exhibited highly variable execution frequencies (peaking at $10^5$ Hz during clutch transitions) due to step-size reduction.
  • The discrete-time model maintained a constant execution frequency, confirming its suitability for real-time hardware.
• Computation Time: Boxplot analysis showed that the average execution time per step for the discrete-time model was comparable to (and in some cases faster than) the continuous-time model, despite the latter's adaptive complexity.

5. Significance

This work bridges the gap between high-fidelity physical modeling and real-time control requirements in automotive engineering.

• Enabling HIL Testing: The model allows for accurate Hardware-in-the-Loop (HIL) testing of transmission control units (TCUs) without the computational burden of variable-step solvers.
• Control Strategy Development: It provides a robust foundation for developing advanced control strategies for gear shifting and torque management, as the exact torque requirements for engagement are explicitly computable.
• Efficiency: It demonstrates that high-precision modeling of friction and switching dynamics does not necessarily require complex, non-deterministic numerical integration, offering a more efficient alternative for embedded systems.