Comments on "Assessing the Stability of Linear Time-Invariant Continuous Interval Dynamic Systems

Abstract

The first part of this note identifies and analyzes two weak points of the commented paper . i) Example 1 reports an incorrect result for which we provide a complete explanation and correction. ii) The practical use of Theorem 1 reveals major drawbacks such as a severe limitation in the applicability (due to the restrictive conditions requested by the hypothesis) and a relatively low accuracy of the results (i.e., rough values for the right outer bounds of the eigenvalue ranges of interval matrices). The second part of the note presents three methods that circumvent the use of Theorem 1 and provide the following information about the eigenvalue range of an interval matrix: a right outer bound-relying on an inequality proved by J. Rohn (see References); the right end point-by solving a constrained global maximization problem; a right outer bound (for arbitrary interval matrices) and the right end point (for some classes of interval matrices)-by calculating the spectral abscissa of a constant matrix that majorizes the considered interval matrix. For illustration by numerical examples, throughout the note we use three interval matrices related to Example 1 from the commented paper.