K-BMPC: Derivative-based Koopman Bilinear Model Predictive Control for Tractor-Trailer Trajectory Tracking with Unknown Parameters

Nonlinear dynamics bring difficulties to controller design for control-affine systems such as tractor-trailer vehicles, especially when the parameters in the dynamics are unknown. To address this constraint, we propose a derivative-based lifting function construction method, show that the corresponding infinite dimensional Koopman bilinear model over the lifting function is equivalent to the original control-affine system. Further, we analyze the propagation and bounds of state prediction errors caused by the truncation in derivative order. The identified finite dimensional Koopman bilinear model would serve as predictive model in the next step. Koopman Bilinear Model Predictive control (K-BMPC) is proposed to solve the trajectory tracking problem. We linearize the bilinear model around the estimation of the lifted state and control input. Then the bilinear Model Predictive Control problem is approximated by a quadratic programming problem. Further, the estimation is updated at each iteration until the convergence is reached. Moreover, we implement our algorithm on a tractor-trailer system, taking into account the longitudinal and side slip effects. The open-loop simulation shows the proposed Koopman bilinear model captures the dynamics with unknown parameters and has good prediction performance. Closed-loop tracking results show the proposed K-BMPC exhibits elevated tracking precision with the commendable computational efficiency. The experimental results demonstrate the feasibility of K-BMPC.