PURPOSE: To improve transient responsiveness ever so better, by making a high- speed type supercharger operate in a low-speed range to some extent and, in turn, fully operating the high-speed supercharger at once by means of a back- pressure selector with shifting to the high-speed range, in the case where double supercharging takes place with both low- and high-speed superchargers.

CONSTITUTION: In time of low-speed operation on an engine 1, since back pressure in the exhaust gas is small enough, a waste gate valve 8 of a back-pressure selector 10 being set up in an exhaust passage is closed. In consequence, the whole lot of exhaust gases out of the engine 1 flows in a turbine 5b of a low- speed type supercharger 5 set up in a suction passage 3 and performs its scavenging action. At this time, the exhaust gas after driving the supercharger 5 flows in a turbine 4b of a high-speed type supercharger 4 through an exhaust passage 7 and gets ready for supercharging. And, when the engine 1 reaches the specified engine speed and the above valve 8 is opened by back pressure in the exhaust gas, a part of the exhaust gas directly flows in the turbine 4b by way of a bypass passage 9 and exhaust passage 7. With this constitution, the high-speed supercharger 4 performs its operation at once.