Abstract: The speed-sensorless vector control of induction motor based on the traditional speed adaptive law full-order observer is unstable at very low and zero speed. This paper proposes a new feedback gain matrix design method and a new speed adaptive law. The nonlinear voltage error of the inverter and the stator resistance parameter change are the fundamental reasons for the instability of the motor at very low and zero speed, which leads to the increase of excitation current error, and in turn affects the increase of the rotor flux linkage error. As a result, the traditional speed adaptive law ignoring the flux linkage error term is no longer applicable. Therefore, reducing the excitation current error as soon as possible and designing a new speed adaptive law are the key to ensuring the stability of the induction motor at very low speed and zero speed. The feedback gain matrix designed based on the convergence of the excitation current error is limited by adding the excitation current error term to zero in mathematical formula; and a new speed adaptive law without approximations and omitted parts is deduced compared to the traditional speed adaptive law, so as to achieve the full-speed stability effect of the induction motor vector control under the improved full-order flux linkage observer and the new speed adaptive law. The method is verified by using the dSPACE control experiment platform of a 3.7kW induction motor, which could operate stably at the very low and zero speed at the rated conditions. It shows that the method has good steady-state and dynamic performance.