A system for controlling the behaviour of a vehicle relative to the ground is described wherein the linear ground speed of a vehicle that has at least two wheels (1) each equipped with a sensor suitable for supplying a measurement that is a function of the rotation of said wheel is determined at each instant. The system comprises a circumferential speed indicator (101, 102) for each wheel (1) based on the measurement from the corresponding sensor (11) and a system for diagnosing characteristic parameters of the instantaneous condition of said wheels to validate the use of the circumferential speed Vr measured for each of these wheels to produce an estimation of the instantaneous speed Vv of the vehicle relative to the ground. The diagnostic system notably comprises a means (109) of diagnosing loss of road grip of a wheel as a function of its instantaneous acceleration, a means (107) of diagnosing the validity of the wheel speed indication based on a measurement of the road grip coefficient of the wheel and a means (113) of diagnosing the maintaining or restoration of road grip of the wheel as a function of a determination of its slip. A vehicle speed indicator (119) produces an estimation of the vehicle speed relative to the ground as a function of indications that are not rejected on completion of the diagnoses carried out. The invention also targets the application of this speed estimation system to the determination of the instantaneous slip when rolling of the wheels of a vehicle, each coupled to an individual electric machine and to the control of the torque applied to each of these wheels as a function of this slip to improve the behaviour of the vehicle.