A multi-stage compression system includes a plurality of centrifugal compression stages. Each stage includes an impeller and a variable speed motor coupled to the impeller. Each variable speed motor is operable at a speed between a first speed and a second speed. The multi-stage compression system also includes a control system that is connected to each of the variable speed motors and is operable to vary the speed of each motor. The speed of each motor is varied simultaneously such that a ratio of the speed of any two variable speed motors remains constant.