LMI characterization of structural and robust stability. (English) Zbl 0949.93064

Authors’ abstract: This paper introduces several stability conditions for a given class of matrices expressed in terms of Linear Matrix Inequalities (LMI), being thus simply and efficiently computable. Diagonal and simultaneous stability, both characterized by polytopes of matrices, are addressed. Using this approach a method particularly attractive to test a given matrix for D-stability is proposed. Lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions. The key idea is to relate Hurwitz stability with a positive realness condition.


93D09 Robust stability
15A39 Linear inequalities of matrices
15A42 Inequalities involving eigenvalues and eigenvectors
Full Text: DOI


[1] Boyd, S. P.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia · Zbl 0816.93004
[2] Colaneri, P.; Geromel, J. C.; Locatelli, A., Control Theory and Design: An \(RH_2\) and \(RH_∞\) Viewpoint (1997), Academic Press: Academic Press New York
[3] Feron, E.; Apkarian, P.; Gahinet, P., Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Trans. Aut. Contr., 41, 7, 1041-1046 (1996) · Zbl 0857.93088
[4] Geromel, J. C., On the determination of a diagonal solution of the Lyapunov equation, IEEE Trans. Aut. Contr., 30, 4, 404-406 (1985) · Zbl 0589.65033
[5] Geromel, J. C.; Peres, P. L.D.; Bernussou, J., On a convex parameter space method for linear control design of uncertain systems, SIAM Journal on Control and Optimization, 29, 2, 381-402 (1991) · Zbl 0741.93020
[6] Hershkowitz, D., Recent directions in matrix stability, Linear Algebra Appl., 171, 161-186 (1992) · Zbl 0759.15010
[7] Kaszkurewicz, E.; Bhaya, A., Robust stability and diagonal Lyapunov functions, SIAM J. Matrix Anal. Appl., 14, 2, 508-520 (1993) · Zbl 0774.93062
[8] Sun, W.; Khargonekar, P. P.; Shim, D., Solution to the positive real control problem for linear time invariant systems, IEEE Trans. Aut. Contr., 39, 10, 2034-2046 (1994) · Zbl 0815.93032
[9] Persidskii, S. K., Problem on absolute stability, Automat. Remote Control, 12, 1889-1895 (1969) · Zbl 0218.34050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.