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

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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.

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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

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Supplementary files

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2. Fig. 1. Examples of saturation functions: sat(x) (1); tanh(x) (2); 2arctg(x)/p (3); x/(1 + | x |) (4).

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3. Fig. 2. Graphs of deviation x1(t) (1), velocity x2(t) (2) and control U(t) (3).

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4. Fig. 3. Phase trajectories of the system.

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