Dynamical modeling and optimal state estimation using Kalman-Bucy filter for a seamless two-speed transmission for electric vehicles

A seamless two-speed transmission that incorporates a two-stage planetary gear set with common sun and common ring gears and two braking mechanisms to control the flow of power is introduced. For an electric vehicle equipped with such a transmission, a detailed dynamical model of the driveline including the half shaft stiffness and longitudinal vehicle dynamics is derived by exploiting the torque balance and virtual work principle. A deterministic Luenberger observer and a stochastic Kalman-Bucy filter are designed to estimate the unmeasured states. These observers estimate the speed of the sun and ring gears and the input and output torques of the transmission based on the measured speeds of the electric motor and the vehicle. Due to nonlinearities in the system such as the longitudinal vehicle dynamics, nonlinear observer methods generally apply for the observer design. However, the nonlinearities are only function of measurable states. Hence, using linear output injection to design an observer results in linear error dynamics. Therefore, the nonlinear observer design problem is transformed into the design of an observer for a linear system. The simulation and experimental results are presented to verify and compare the performance of the deterministic Luenberger estimator with stochastic Kalman-Bucy filter when the system encounters noise.