A new characterization on stability bounds for singularity perturbed systems

In this short paper, we characterize the upper bound ε* for the parasitic parameter ε in a singularly perturbed systems, which ensures stability of such a system if 0 < ε < ε*. At the same time, a method is established to testify the system stability without the slow-fast decomposition required in the existing literature. It will be shown that this upper bound is just the minimum positive eigenvalue of a matrix pair, which is explicitly constructed from the system matrix. This reveals a direct relationship between the stability bound and the system matrix and may be useful in the study of robust-control problems for such systems.