Global stabilization of a chain of two integrators by a feedback in the form of nested sigmoids

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The problem of stabilizing a chain of two integrators by a feedback in the form of nested sigmoids is considered. Such a feedback allows one to easily take into account boundedness of the control resource and ensure the fulfillment of desired characteristics of the transient process, such as a given exponential rate of the deviation decrease near the equilibrium state and the constraint on the maximum velocity. Global stability of the closed-loop system is proved by constructing its Lyapunov function.

About the authors

Yu. V. Morozov

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences

Author for correspondence.
Email: tot1983@inbox.ru
Russian Federation, Moscow

A. V. Pesterev

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences

Email: alexanderpesterev.ap@gmail.com
Russian Federation, Moscow

References

  1. Kurzhanski A.B., Varaiya P. Solution Examples on Ellipsoidal Methods: Computation in High Dimensions. Cham, Switzerland: Springer, 2014.
  2. Teel A.R. Global Stabilization and Restricted Tracking for Multiple Integrators with Bounded Controls // Sys. & Cont. Lett. 1992. V. 18. № 3. P. 165–171.
  3. Teel A.R. A Nonlinear Small Gain Theorem for the Analysis of Control Systems with Saturation // Trans. Autom. Contr., IEEE, 1996. V. 41. № 9. P. 1256–1270.
  4. Morozov Yu.V., Pesterev A.V. Глобальная устойчивость гибридной аффинной системы 4-го порядка // Изв. РАН. ТиСУ. 2023. № 5. С. 3–15.
  5. Hua M.-D., Samson C. Time Sub-optimal Nonlinear Pi and Pid Controllers applied to Longitudinal Headway Car Control // Int. J. Control. 2011. V. 84. P. 1717–1728.
  6. Olfati-Saber R. Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles, Ph.D. Dissertation, Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science, 2001.
  7. Pesterev A.V., Morozov Yu.V., Matrosov I.V. On Optimal Selection of Coefficients of a Controller in the Point Stabilization Problem for a Robot-wheel // Communicat. Comput. Inform. Sci. (CCIS). 2020. V. 1340. P. 236–249.
  8. Pesterev A.V., Morozov Yu.V. Optimizing Coefficients of a Controller in the Point Stabilization Problem for a Robot-wheel // Lect. Notes Comput. Sci. 2021. V. 13078. P. 191–202.
  9. Pesterev A.V., Morozov Yu.V. The Best Ellipsoidal Estimates of Invariant Sets for a Third-Order Switched Affine System // Lect. Notes Comput. Sci. 2022. V. 13781. P. 66–78.
  10. Пестерев А.В. Глобальная устойчивость аффинной системы второго порядка с переключениями // АиТ. 2023. № 9. С. 95–105.
  11. Барбашин Е.А. Введение в теорию устойчивости. Серия: Физико-математическая библиотека инженера. М.: Наука, 1967.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences