Complete stability analysis of linear neutral time-delay systems with multiple commensurate delays

The aim of this study is to provide a practical paradigm to ascertain the complete characterization of stability intervals for neutral time-delay systems with multiple commensurate delays. Firstly, the matrix pencil test is used to determine the complete set of imaginary characteristic roots. Only at these frequencies points may the stability switching occur. Then compute the principle delay value tau for each zero-crossing frequency omega within [0, 2pi] and their infinite extension with the period 2pi/omega. Sort all critical delays by size, then the exact intervals can be obtained exhaustively. Lastly, a boundary criterion is used to check the stability for any given delay point value in an interval. According to the continuity of stability exponent, the interval has identical stability with any delay point therein. Therefore, the stability property of all intervals is obtained. Numerical examples are given to illustrate the theoretical results.