On local asymptotic stabilization of the nonlinear systems with time-varying perturbations by state-feedback control

ABSTRACT

In this paper, we are interested in the relation between the solutions of the control system $\dot{x}=f(x,u)$ and the solutions of its (potentially unknown) perturbation $\dot{x}=f(x,u)+w(x,t).$ Under the assumption that the linear part of the unperturbed system at the point $(0,0)$ is controllable and that disturbance $w(x,t)$ is asymptotically sufficiently small, there exists a state-feedback controller of the form u=−Kx such that the perturbed system preserves the local asymptotic stability of the zero solution of unperturbed system. The main result of this paper gives the sufficient conditions, more specifically, the relations between the important parameters of the system, to ensure this property and at the same time provides the method for calculating the lower bound of region of attraction. Moreover, we obtain a nontrivial extension of the classical result of H. K. Khalil regarding the behavior of the (uncontrolled) perturbed systems whose nominal part is exponentially asymptotically stable at the origin x=0.

KEYWORDS:

• Nonlinear systems
• perturbations
• asymptotic stabilization
• state-feedback

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

R. Vrabel http://orcid.org/0000-0002-2640-595X

Additional information

Funding

This work was supported by the Vedecká Grantová Agentúra MŠVVaŠ SR a SAV (VEGA) of Slovakia under Grant “Holistic approach of knowledge discovery from production data in compliance with Industry 4.0 concept” [grant number 1/0272/18] and the Research and Development Operational Programme funded by the European Regional Development Fund under Grant “University Scientific Park: Campus MTF STU - CAMBO” [grant number 26220220179].

Notes on contributors

Robert Vrabel

Robert Vrabel received the Masters degree in Applied Mathematical Analysis in 1990 and Ph.D. in 1998 at the Slovak Academy of Sciences. At the present he is Associate Professor with the Institute of Applied Informatics, Automation and Mechatronics of the Faculty of Materials Science and Technology in Trnava, Slovak University of Technology in Bratislava, Slovakia. His current research interests include control of linear and nonlinear systems, singular perturbations in dynamical systems and qualitative theory of ordinary differential equations generally.