Stability of Linear Systems with the Generalized Lipschitz Property

WSEAS Transactions on Systems

Print ISSN: 1109-2777
E-ISSN: 2224-2678

Volume 18, 2019

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 18, 2019

Stability of Linear Systems with the Generalized Lipschitz Property

AUTHORS: Michael Gil

Download as PDF

ABSTRACT: We consider non-autonomous multivariable linear systems governed by the equation u˙ = A(t)u with the matrix A(t) satisfying the generalized Lipschitz condition kA(t) − A(τ )k ≤ a(|t − τ |) (t, τ ≥ 0), where a(t) is a positive function. Explicit sharp stability conditions are derived. In the appropriate situations our results generalize and improve the traditional freezing method. An illustrative example is presented.

KEYWORDS: linear systems; stability; generalized Lipschitz conditions.

REFERENCES:

[1] Bylov, B. F., Grobman, B. M., Nemyckii V. V. and Vinograd R. E. The Theory of Lyapunov Exponents, Nauka, Moscow, 1966. In Russian.

[2] Daleckii, Yu L. and Krein, M. G. Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. 1 974.

[3] Desoer, C. A. Slowly varying systems x˙ = A(t)x, IEEE Transactions on Automatic Control, 14, (1969) 780-781.

[4] Gil’, M. I. Stability of linear nonautonomous multivariable systems with differentiable matrices, Systems & Control Letters, 81, (2015) 31-33.

[5] Gil’, M. I. Operator Functions and Operator Equations World Scientific, New Jersey, 2017.

[6] Ilchmann A., Owens, D. H, and Pratcel-Wolters, D. Sufficient conditions for stability of linear time-varying systems, Systems & Control Letters, 9, (1987) 157-163.

[7] Jetto, L. and Orsini, V. Relaxed Conditions for the Exponential Stability of a Class of Linear Time-Varying Systems IEEE Transac. on Automat. Control, 54, no.7, (2009) 1580–1585.

[8] Kamen, H. W., Khargonekar, P. P. and Tannembaum, A. Control of slowly varying linear systems, IEEE Trans. Automat. Control, 34, no. 12, (1989) 1283-1285.

[9] Mullhaupt, P., Buccieri, D. and Bonvin, D. A numerical sufficiency test for the asymptotic stability of linear time-varying systems,Automatica, vol. 43, (2007) 631-638, .

[10] Rugh, W.J. Linear System Theory. Prentice Hall, Upper Saddle River, New Jersey, 1996.

[11] Solo, V. On the stability of slowly time-varying linear systems, Mathemat. Control, Signals, Syst., vol. 7, (1994) 331-350, .

[12] Vinograd, R. E. An improved estimate in the method of freezing, Proc. Amer. Soc. 89 (1), (1983) 125-129.

WSEAS Transactions on Systems, ISSN / E-ISSN: 1109-2777 / 2224-2678, Volume 18, 2019, Art. #3, pp. 25-28

Copyright Β© 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

Quick Links

Login

Other Articles by Author(s)

Author(s) and WSEAS

WSEAS Transactions on Systems

Bulletin Board