Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Stability and Stabilization of Nonlinear Continuous and Discrete Uncertain Systems with Modal Approach

Author(s):

Irina Efremovna Zuber

Eng. Sc. D.,
Leading Researcher
St. Petersburg State University ,
199034 St.Petersburg, Universitetskaya nab.7/9

Zuber.Yanikum@gmail.com

Arkadii Khaimovich Gelig

Doctor of Engineering Science in Physics and Mathematics,
professor.
St. Petersburg State University ,
199034 St.Petersburg, Universitetskaya nab.7/9

agelig@yandex.ru

Abstract:

In this paper we consider the system of differential equations for which the elements of object matrix are functions of the system state and time. These functions are perturbed by uncertain functionals. Using the spectral decomposition of the object matrix and the Lyapunov quadratic function with unit matrix we obtain the sufficient conditions of global exponential stability of the system considered. The similar system is studied in the case of scalar control, on the assumption that the distribution vector depends on the state. Using the modal approach in the condition of the uniform controllability we perform the synthesis of scalar control which provides the global exponential stability of closed loop system. Analogous results are obtained for discrete nonlinear uncertain systems with the same structure.

Keywords

• continuous nonlinear uncertain differential systems
• discrete nonlinear uncertain systems
• global exponential stability
• stabilization

References:

1. Isidori A. Nonlinear Control Systems, London: Springer. 1993
2. Zak S. H. Systems and Control, Oxford Univ. Press. 2002
3. Yakubovich V. A., Leonov G. A., Gelig A. Kh. Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, World Scientific. London. 2004
4. Yakubovich V. A., Starzhinski V. M. Linear Differential Equations with Periodic Coefficients, Vol 1, 2, 839 pp. 1975. New York-Toronto. Jerasalem-London

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