We consider the problem of controlling inductions motors driven through AC/DC rectifiers and DC/AC inverters. The control objectives are threefold: (i) forcing the motor speed to track a reference signal, (ii) regulating the DC Link voltage, (iii) assuring a satisfactory power factor correction (PFC) with respect to the power supply net. First, a nonlinear model of the whole controlled system is developed in the Park-coordinates. Then, a nonlinear multi-loop controller is synthesized using the backstepping design technique. A formal analysis based on Lyapunov stability and average theory is developed to describe the control system performances. In addition to closed-loop global asymptotic stability, it is proved that all control objectives (motor speed tracking, DC link voltage regulation and unitary power factor) are asymptotically achieved up to unavoidable but small harmonic errors (ripples).