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Article

Keywords:
diagonal stability; Cauchy indices
Summary:
We give detailed discussion of a procedure for determining the robust $D$-stability of a $4\times 4$ real matrix. The procedure begins from the Hurwitz stability criterion. The procedure is applied to two numerical examples.
References:
[1] Abed E. H.: Decomposition and stability of multiparameter singular perturbation problems. IEEE Trans. Automat. Control, AC-31 (1986), 925–934. Zbl 0601.93004
[2] Barker G. P., Berman A., Plemmons R. J.: Positive diagonal solutions to the Lyapunov equations. Linear and Multilinear Algebra 5 (1978), 249–256. MR 0469939 | Zbl 0385.15006
[3] Berman A., Shasha D.: Inertia-preserving matrices. SIAM J. Matrix Anal. Appl. 12-2 (1991), 209–212. MR 1089156 | Zbl 0733.15004
[4] Cain B.: Real $3\times 3$ $D$-stable matrices. J. Res. of the National Bureau of Standards, Sec. B 80 (1976), 75–77. MR 0419478
[5] Cain B.: Inside the $D$-stable matrices. Linear Algebra Appl. 56 (1984), 237–243. MR 0724564 | Zbl 0525.15008
[6] Cross G. W.: Three types of matrix stability. Linear Algebra Appl. 20 (1978), 253–263. MR 0480586 | Zbl 0376.15007
[7] Chen J., Fan M. K. H., Yu C.: On $D$-stability and structured singular values. Systems Control Lett. 24 (1995), 19–24. MR 1307123 | Zbl 0877.93079
[8] Carlson D., Datta B. N., Johnson C. R.: A semi-definite Lyapunov theorem and the characterization of tridiagonal $D$-stable matrices. SIAM J. Algebraic Discrete Methods 3 (1982), 293–304. MR 0666854 | Zbl 0541.15008
[9] Gantmacher F. R.: The theory of matrices II. Chelsea Publishing Company, 1959. MR 0107649
[10] Hartfiel D. J.: Concerning the interior of $D$-stable matrices. Linear Algebra Appl. 30 (1980), 201–207. MR 0568792
[11] Hershkowitz D.: Recent direction in matrix stability. Linear Algebra Appl. 171 (1992), 161–186. MR 1165452
[12] Horn R. A., Johnson C. R.: Topics in matrix analysis. Cambridge University Press, 1991. MR 1091716 | Zbl 0729.15001
[13] Johnson C. R.: Second, third, and fourth order $D$-stability. J. Res. of the National Bureau of Standards, Sec. B 78 (1974), 11–13. MR 0340287 | Zbl 0283.15005
[14] Johnson R., Tesi A.: On the $D$-stability problem for real matrices. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2, No. 2 (1999), 299–314. MR 1706600 | Zbl 0936.15014
[15] Kanovei G. V., Logofet D. O.: $D$-stability of $4\times 4$ matrices. J. Comput. Math. Math. Phys. 38 (1998), 1369–1374. MR 1669150
[16] Seidenberg A.: A new decision procedure for elementary algebra. Ann. Math. (2) 60 (1954), 365–374. MR 0063994
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