Investigating duality on stability conditions. (English) Zbl 1157.93457

De Oliveira, Maurício C.

Syst. Control Lett. 52, No. 1, 1-6 (2004).

Summary: This paper is devoted to investigate the role played by duality in stability analysis of linear time-invariant systems. We seek for a dual statement of a recently developed method for generating stability conditions, which combines Lyapunov stability theory with Finsler’s Lemma. This method, developed in the time domain, is able to generate a set of (primal) equivalent stability tests involving extra multipliers. The resulting tests have very attractive properties. Stability is characterized via linear matrix inequalities and we use optimization theory to obtain the duals. The dual problems are given a frequency domain interpretation.

MSC:

93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34D20	Stability of solutions to ordinary differential equations
90C90	Applications of mathematical programming
93C05	Linear systems in control theory

Keywords:

Stability; Linear systems; Duality; Linear matrix inequalities; Optimization

References:

[1]	A. Ben-Tal, L. El Ghaoui, A. Nemirovski, Robustness, in: H. Wolkowicz, R. Saigal, L. Vandenberghe (Eds.), Handbook of Semidefinite Programming: Theory, Algorithms and Applications, Kluwer Academic Press, Boston MA, 2000, pp. 139-162.
[2]	El Gahoui, L.; Niculesco, S.-L., Advances in linear matrix inequality methods in control, (2000), SIAM Philadelphia, PA
[3]	Finsler, P., Über das vorkommen definiter und semidefiniter formen in scharen quadratischer formen, Comment. math. helv., 9, 188-192, (1937) · JFM 63.0054.02
[4]	Henrion, D.; Bachelier, O.; Šebek, M., \(D\)-stability of polynomial matrices, Internat. J. control, 74, 8, 845-856, (2001) · Zbl 1011.93083
[5]	Henrion, D.; Meinsma, G., Rank one LMIs and Lyapunov’s inequality, IEEE trans. automat. control, 46, 8, 1285-1288, (2001) · Zbl 1016.93029
[6]	Henrion, D.; Tarbouriech, S.; Šebek, M., Rank-one LMI approach to simultaneous stabilization of linear systems, Systems control lett., 38, 79-89, (1999) · Zbl 1043.93545
[7]	M.C. de Oliveira, R.E. Skelton, Stability tests for constrained linear systems, in: S.O. Reza Moheimani (Ed.), Perspectives in Robust Control, Lecture Notes in Control and Information Sciences, Springer, Berlin, 2001, pp. 241-257. · Zbl 0997.93086
[8]	M.C. de Oliveira, R.E. Skelton, On stability tests for linear systems, in: Proceedings of the 15th IFAC World Congress, Barcelona, Spain, 2002, pp. 3021-3026.
[9]	A. Shapiro, Duality, optimality conditions, and perturbation analysis, in: H. Wolkowicz, R. Saigal, L. Vandenberghe (Eds.), Handbook of Semidefinite Programming: Theory, Algorithms and Applications, Kluwer Academic Press, Boston, MA, 2000, pp. 68-92.
[10]	do Valle Costa, O.L.; Fragoso, M.D., Stability results for discrete-time linear-systems with Markovian jumping parameters, J. math. anal. appl., 179, 1, 154-178, (1993) · Zbl 0790.93108
[11]	Vandenberghe, L.; Boyd, S.P., Semidefinite programming, SIAM rev., 38, 49-95, (1996) · Zbl 0845.65023