Fuzzy Game-Theoretic Tube Model Predictive Control for Integrated Vehicle Stability System

This paper develops a fuzzy game-theoretic tube model predictive control (MPC) framework for coordinated vehicle lateral motion control using active front steering (AFS) and direct yaw moment control (DYC). The vehicle dynamics are represented by a discrete-time Takagi-Sugeno fuzzy model to capture operating-condition dependence and parametric uncertainty, while a common-feedback tube MPC structure is employed to guarantee robust constraint satisfaction through an offline-designed invariant tube and terminal set. On this basis, the nominal control problem is formulated as a two-player finite-horizon Nash game, allowing AFS and DYC to optimize individual performance objectives under shared state dynamics and constraints. To enable real-time implementation, two fixedcomplexity online Nash solvers are considered: a best-response (BR) iteration scheme and a variational inequality (VI) formulation solved by an extragradient method. The closed-loop analysis establishes recursive feasibility under bounded disturbances and finite-iteration online equilibrium computation. In addition, a practical input-to-state stability result is derived, in which the effect of inexact online Nash solutions is explicitly captured through a practical-descent framework. Compared with the BR solver, the VI-based solver provides a more direct residual-based interpretation of equilibrium approximation accuracy and its relation to closed-loop stability margins. Hardware-in-the-loop experiments under multiple driving maneuvers verify that the proposed framework is computationally tractable and effective in real time, while achieving robust tracking performance, constraint satisfaction, and coordinated actuator usage.