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

• Published in: IEEE transactions on automatic control Vol. 56; no. 6; pp. 1442 - 1445
• Main Authors: Pastravanu, O, Matcovschi, M-H
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2011; Institute of Electrical and Electronics Engineers
• Subjects: Accuracy; Applied sciences; Computer science; control theory; systems; Control system analysis; Control theory. Systems; Eigenvalue range (right end point of; Eigenvalues and eigenfunctions; Estimation; Exact sciences and technology; interval systems; Linear matrix inequalities; Optimization; outer bounds of; robust stability analysis; Stability criteria; Maximization; Robust stability; Roughness; outer bounds of; Dynamical system; Interval arithmetic; Robust control; Linear time invariant system; Eigenvalue range (right end point of; interval systems; robust stability analysis; Linear stability; Interval matrix; Eigenvalue problem; Robustness
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2011.2122650

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.