Stability Switches of a Type of Delay Systems with Delay Dependent Parameter

For a special class of characteristic equation with delay dependent parameters, at the critical points, by introducing two suitable auxiliary functions which only depends on the coefficients of the system, a new criterion is established to check the crossing directions of the characteristic roots crossing the imaginary axis as delay varies. Compare with the result in the literature, the criterion no longer requires that the phase Angle be differentiable with respect to the delay. Also, example is given to illustrate the effectiveness of the criterion.