Piecewise Affine Relaxation of Discrete Value Functions in Learning Model Predictive Control With Application to Autonomous Racing

Learning Model Predictive Control (LMPC) is a data-driven approach to MPC that enhances closed-loop performance by leveraging data from successive task iterations to approximate the solution of optimal control problems. The value function in LMPC is pivotal for performance enhancement, but its discrete nature-where each point corresponds to a data point-renders the LMPC problem computationally intensive due to its mixed-integer nature. This letter introduces a novel method to construct the LMPC value function. The proposed value function is a piecewise affine approximation that interpolates the discrete data points of the original value function, resulting in a nonlinear relaxation of the mixed-integer LMPC problem. By connecting the discrete data points with piecewise affine segments, the essential characteristics of the original value function are preserved. The proposed algorithm’s effectiveness is demonstrated through numerical simulations in autonomous racing.