LQR Controller for Toroidal Continuously Variable Transmission in Reverse Motion

The system considered in this paper is a Toroidal Continuously Variable Transmission (TCVT) system for cars. This system is unstable in reverse motion as some mechanical parts have been removed from the original one for cost reduction, and the gear ratio has to be regulated around its nominal value for car reverse motion. The control theory used here is the Linear Quadratic Regulator (LQR) associated to a gain-scheduling technique, as the TCVT system is nonlinear according to the car speed. Moreover, as the LQR method requires the entire TCVT state vector and as the only available signal is the gear ratio, a full-order observer is designed. In order to take the other nonlinearities of the system into account, the observer is nonlinear : a diffeomorphism is then used for converting the variables provided by the nonlinear observer into the needed variables. In order to verify the effectiveness and the robustness of the controller against the car speed and the torque shift disturbance phenomenon, several experiments with a test-bed and with an actual vehicle have been performed and showed the efficiency of the proposed controller.