To assure the stability performance of the speed estimation in the speed sensorless induction motor drives, the design strategy of feedback gain matrix for an adaptive full-order observer is a necessary issue. In this paper, the major cause of instability phenomenon based on adaptive full-order observer is analyzed, and a novel design of observer's feedback gain matrix is proposed to achieve the stability over the whole operating range especially in the low speed region, including the regenerating mode. Besides, the Popov hyper-stability theory is used to prove the stability improvement based on the proposed method. Some simulation results are given to demonstrate the effectiveness of the proposed solution.