ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2009. Vol. 1

On stability of a subclass of hybrid systems

S. N. Vassilyev

Hybrid systems with switching non linear vector fields of ordinary differential equations (ODE) of subsystems in given instants are considered. The analysis of the dynamical properties of stability and attractivity in those models are reduced to a problem of analysis of analogous dynamics in usual ODE, but with discontinuous solutions. A reduction method with trajectory homomorphism from initial model to auxiliary one is used. Some examples are considered.

hybrid systems, switched systems, stability, homomorphism, vector-Lyapunov function

1 DeCarlo R. A. Perspectives and results on the stability and stabilizability of hybrid systems / R. A. DeCarlo, M. S. Branicky, S. Petterson // Proc. of the IEEE. — 2000. — Vol. 88, № 7. — P. 1069–1082.

2 Матросов В.М.Метод сравнения в математической теории систем / В.М.Матросов, Л. Ю. Анапольский, С. Н. Васильев. — Новосибирск: Наука, 1980.

3 Васильев С. Н. Метод редукции и качественный анализ динамических систем, I,II/ С. Н. Васильев// ИзвестияРАН.Сер.Теорияи системыуправления. — 2006. —№1. —С.21–29 №2. —С.5–17.

4 Персидский К. П. Ко второй методе Ляпунова / К. П. Персидский// Известия АН КазССР. Математика и механика. — 1947.— Т. 42, № 1. — С. 48–55.

5 Corduneanu C. Применение дифференциальных неравенств в теории устойчивости/CorduneanuC.// An.Sti.Univ. “Al.I.Cuza”,IasiSect.I aMat. — 1960. — №6. —P.47–58(нарус.яз.).

6. Vassilyev S. N. Homomorphisms of Impulsive Differential Equations with Impulses at Unfixed Times and Comparison Method / S.N.Vassilyev // Intern. J.ofHybrid Systems. — 2002. — Vol. 2, № 3. — P. 289–296.

7. Branicky M. Studies in hybrid systems : Modeling, analysis, and control : PhD thesis/ M. Branicky. — M.I.T., Cambridge, MA, 1995.

8. Sontag E.D.Non linear regulation: The piece wise linear approach / E.D.Sontag // IEEE Transactions on Automatic Control. —1981. —26(2). —P.346–358.

9. Matveev A. Qualitative theory of hybrid dynamical systems / A. Matveev, A. Savkin. — Birkhauser, 2000.

10. Wicks M. A. Construction of piecewise Lyapunov functions for stability switched systems / M. A. Wicks, P. Peleties, R. A. DeCarlo // Proc. of the 33rd CDC. — 1994. — P. 3492–3497.

Full text (russian)