The article proposes a novel solution to the control problem of centrifugal gas compressors which are driven by three-phase induction motors (IMs) and three-phase permanent magnet synchronous motors (PMSMs), through a novel flatness-based control scheme which is implemented in successive loops. By re-arranging state variables and by splitting suitably the state vector of the IM-driven gas-compressor and of the PMSM-driven gas compressor into subsystems one arrives at writing the associated state-space models in the triangular (strict feedback) form. For the latter state-space description it is possible to solve the control and stabilization problem using chained control loops. The state-space model of the IM-driven gas-compressor and of the PMSM-driven gas-compressor is decomposed into cascading subsystems which satisfy differential flatness properties. For these subsystems virtual control inputs are computed, capable of inverting their dynamics and of eliminating the associated tracking error. The control inputs which are actually applied to the complete nonlinear form of the IM-actuated gas-compressor and of the PMSM-actuated gas compressor are computed from the last subsystem of the chained state-space description. These control inputs incorporate in a recursive manner all virtual control inputs which were computed from the individual subsystems included in the initial state-space equation. The control inputs that should be applied to the nonlinear system so as to assure that all state vector elements will converge to the desirable setpoints are obtained at each iteration of the control algorithm by tracing backwards the subsystems of the chained state-space model. The method is of proven global stability and ensures fast and accurate tracking of the reference setpoints by the state variables of the IM-driven gas compressor and of the PMSM-driven gas compressor.