A plasmonic refractive index (RI) sensor based on metal-insulator-metal (MIM) waveguide coupled with concentric double rings resonator (CDRR) is proposed and investigated numerically. Utilizing the novel supermodes of the CDRR, the FWHM of the resonant wavelength can be modulated, and a sensitivity of 1060 nm/RIU with high figure of merit (FOM) 203.8 is realized in the near-infrared region. The unordinary modes, as well as the influence of structure parameters on the sensing performance, are also discussed. Such plasmonic sensor with simple framework and high optical resolution could be applied to on-chip sensing systems and integrated optical circuits. Besides, the special cases of bio-sensing and triple rings are also discussed.