This paper introduces a proposed adaptive linear predictive model designed to enhance sustainability and energy efficiency in battery electric vehicles (EVs) equipped with permanent magnet synchronous motors (PMSMs). Central to the approach is a third-order generalized integrator (TOGI) flux observer, which provides real-time adaptation of the predictive model by accurately estimating motor parameters, enabling precise flux and torque control under varying operating conditions. The control scheme employs an active-set optimization algorithm to solve the quadratic programming problem efficiently, dynamically adjusting control inputs to optimize traction torque and speed tracking while minimizing battery energy consumption. The proposed method is rigorously compared against a published conventional model predictive control (MPC) approach. Comprehensive simulation studies and experimental tests confirm that the proposed model significantly outperforms the conventional MPC in terms of control accuracy, energy efficiency, and battery longevity. Specifically, results demonstrate up to 98% reduction in current tracking error, a 19% increase in battery operational life, and more than 40% reduction in DC source current consumption across different driving scenarios. The combination of accurate TOGI-based parameter adaptation and active-set optimization offers a robust, computationally efficient, and energy-saving solution, advancing the performance and sustainability of EV powertrain control systems.