# Stability Test of 2-D Face of an Interval Matrix 1.0

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Created : Aug 23, 2007

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## The program can test the stability of 2-D face of an ...

The program can test the stability of 2-D face of an interval matrix.

Copyright (C) Yang XIAO, Beijing Jiaotong University, Aug. 2, 2007, E-Mail: yxiao@bjtu. edu. cn.

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 stable 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. This program provides the examples to demonstrate the applicability of the robust stability_test of interval matrices in Ref [1, 2].

Remarks:

(1) 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].

(2) An interval matrix is Hurwitz stable, if and only if all the 2-D faces of the interval matrix is Hurwitz stable.

(3) 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].

(4) An interval matrix is Schur stable, if and only if all the 2-D face of the interval matrix is Schur stable.

(5) To determine the stability of interval matrix, needs to test all the 2-D faces of matrices.

[2] XIAO Yang, Stability Analysis of Multidimensional Systems, Shanghai Science and Technology Press, Shanghai, 2003.

The paper [1] can be downloaded from Web site of IEEE Explore.

Copyright (C) Yang XIAO, Beijing Jiaotong University, Aug. 2, 2007, E-Mail: yxiao@bjtu. edu. cn.

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 stable 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. This program provides the examples to demonstrate the applicability of the robust stability_test of interval matrices in Ref [1, 2].

Remarks:

(1) 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].

(2) An interval matrix is Hurwitz stable, if and only if all the 2-D faces of the interval matrix is Hurwitz stable.

(3) 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].

(4) An interval matrix is Schur stable, if and only if all the 2-D face of the interval matrix is Schur stable.

(5) To determine the stability of interval matrix, needs to test all the 2-D faces of matrices.

**Ref:**

[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[2] XIAO Yang, Stability Analysis of Multidimensional Systems, Shanghai Science and Technology Press, Shanghai, 2003.

The paper [1] can be downloaded from Web site of IEEE Explore.

**• MATLAB Release: R13**

**Demands:****Stability Test of 2-D Face of an Interval Matrix 1.0 scripting tags:**interval, matlab mathematics, test, schur, stability test, hurwitz, interval matrix, face, stable.

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