The stabilization of a pendulum system with position feedback and delayed position feedback controller is studied. Considering the fact that the controller itself has delay, a second-order differential equation with  two delays is established. We present the characteristic equation, obtain the relation between the system parameters and stability, and give the results for delay-dependent and delay-independent stability. By the technique for finding roots of algebra equation, the multiplicity of characteristic roots is analyzed, the parameter condition that the multiplicity is at most 4 is obtained, and it is proved that there are at most two triple characteristic roots. When the modulus of the characteristic root tends to infinity, we prove that the real part must tend to negative infinity. By using the asymptotic analysis technique, the asymptotic expression of the characteristic root is calculated. The validity of the conclusion is verified by Matlab numerical simulations. This method can be extended to $n$-order differential equations with multiple time-delays.