Algorithms to Estimate the Distance between a Stable Matrix to the Nearest Unstable One

In this paper we give effective estimates for the distance of a stable matrix to the set of unstable matrices for both the continuous- and discrete-time cases. The first estimate is based on preliminary reduction of the system matrix into Schur form and further use of the Gershgorin theorem. In this case we treat the continuous- and discrete-time systems in an unified manner. In the second estimate we use bounds for the matrix exponential. Here we obtain even more for continuous-time systems, estimating the distance to the nearest time varying matrix, corresponding to an unstable system.