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Issues of global stabilization of a centrifugal compressor with spool dynamics are presented. Control schemes are designed for systems with and without system uncertainty. While activating only the close-coupled valve or the throttle as the single control input, backstepping tools are first used to achieve global stability of the working equilibrium for systems without uncertainty. The uncertainties in the spool dynamics and the compressor characteristic are known to be likely to happen in practical applications. Under such circumstances, the compressor torque and the close-coupled valve are then employed to achieve robust global stability of the working equilibrium via the Lyapunov redesign method. Numerical simulations are given to demonstrate the main results.