By relying on a two-dimensional (2-D) face test, Ref [1,2] obtained a necessary and sufficient condition for the robust Hurwitz and Schur stability of interval matrices.Ref [1,2] revealed that it is impossible that there are some isolated unstable points in the parameter space of the matrix family, so the stability of exposed 2-D faces of an interval matrix guarantees stability of the matrix family.
Ref [1,2] provides the examples to demonstrate the applicability of the robust stability test of interval matrices. Remarks:
- The 2-D face of an interval matrix is Hurwitz stable, if and only if the maximum real part of the eigenvalues of the 2-D face of the interval matrix is smaller than 0 [1].
- An interval matrix is Hurwitz stable, if and only if all the 2-D faces of the interval matrix is Hurwitz stable.
- The 2-D face of an interval matrix is Schur stable, if and only if the maximum absolute of the eigenvalues of all the 2-D faces of the interval matrix is smaller than 1 [1].
- An interval matrix is Schur stable, if and only if all the 2-D face of the interval matrix is Schur stable.
- To determine the stability of interval matrix, needs to test all the 2-D faces of matrices.
Refs:
[1] Yang Xiao; Unbehauen, R., Robust Hurwitz and Schur stability test for interval matrices, Proceedings of the 39th IEEE Conference on Decision and Control, 2000. Volume 5, Page(s):4209 – 4214. The paper [1] can be downloaded from Web site of IEEE Explore.
[2] XIAO Yang, Stability Analysis of Multidimensional Systems, Shanghai Science and Technology Press, Shanghai, 2003.
Now, I hope that someone can discuss or comment the results of [1].
