This paper proposes a coordinated switching power system stabilizer (SPSS) to enhance the stability of multimachine power systems. The SPSS switches between a bang-bang power system stabilizer (BPSS) and a conventional power system stabilizer (CPSS) based on a state-dependent switching strategy. The BPSS is designed as a bang-bang constant funnel controller (BCFC). It is able to provide fast damping of rotor speed oscillations in a bang-bang manner. The closed-loop stability of the power system controlled by the SPSSs and the CPSSs is analyzed. To verify the control performance of the SPSS, simulation studies are carried out in a 4-generator 11-bus power system and the IEEE 16-generator 68-bus power system. The damping ability of the SPSS is evaluated in aspects of small-signal oscillation damping and transient stability enhancement, respectively. Meanwhile, the coordination between different SPSSs and the coordination between the SPSS and the CPSS are investigated therein.