An LMIs-based back-stepping sliding mode control framework for robust velocity tracking of the automatic driving vehicle on sloped roads

Automatic driving has received a broad of attention from academia and industry since it is effective in greatly reducing the severity of potential traffic accidents and achieving the ultimate automobile safety and comfort. This article presents a back-stepping sliding mode controller (BSMC) through linear matrix inequalities (LMIs) for the highly automatic driving vehicle on sloped roads. It includes three key modules, namely, an extended Kalman filter (EKF)-based road slope estimation module, a robust BSMC-LMIs velocity-tracking controller based on the input-output feedback linearization, as well as a longitudinal inverse vehicle dynamics module. The nonlinear combined slip tire model with the transient behavior is introduced to calculate the tire forces properly, which would be further proven to offer more accurate road slope estimations even in a fierce acceleration or deceleration situation. The proposed BSMC-LMIs controller for velocity tracking can handle the lumped uncertainties which include the modeling error, the parameter perturbation, external disturbances, and noises, and guarantee the reachability of the sliding surface, meanwhile, alleviating the chattering phenomenon inherited from the sliding mode structure. Besides, a sufficient condition for the existence of the proposed BSMC is derived by using the LMIs, which ensures the t-alpha$$ {t}<^>{-\alpha } $$ asymptotical stability on the sliding surface. Finally, the robustness, feasibility, and effectiveness of the proposed BSMC-LMIs controller for velocity-tracking are verified by simulation tests in various working scenarios, which shows satisfying results when dealing with the lumped uncertainties on sloped roads. Moreover, the comparative study also shows that the proposed BSMC-LMIs controller has the best tracking performance when compared to the model predictive control, conventional sliding mode control, and the cubic proportional-integral controller.