Global exponential stabilization of nonlinear systems via width time-dependent periodically intermittent smooth control

Jitai Liang,Wanjun Xia

doi:10.1186/s13662-020-02969-3

Jitai Liang, Wanjun Xia

Open Access

https://doi.org/10.1186/s13662-020-02969-3

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Abstract

In this paper, the global exponential stability and stabilization problems for a class of nonlinear systems are investigated. Some sufficient conditions to guarantee global exponential stable and estimate the minimum admissible value of the control width are presented in virtue of time-dependent width Lyapunov functions. Furthermore, a periodically intermittent smooth controller with variant control width is introduced and theoretical analysis is provided. The smooth index function of periodically intermittent smooth control inputs is defined and the supremum (or least upper bound) of smooth index function set can be solved. On the basis of the analysis, the designed periodically intermittent smooth controller not only can globally exponentially stabilize the nonlinear systems, but also can control the exponential convergence rate of the nonlinear systems. Finally, numerical simulations are given to verify the obtained theoretical results.

Highlights

Stabilization problem of nonlinear dynamic systems has been a topic of focus in recent years
A numerical example will be presented to characterize the relationship between the exponential convergence rate θ, the minimal work width τmin and the control input smooth index function
Instead of adopting a constant gain matrix, we propose to select among the state feedback gain matrices the one satisfying K – K(t) < ε(t) where K is a constant matrix and a real-valued function ε(t) named smooth index function to be determined

Summary

Introduction

Stabilization problem of nonlinear dynamic systems has been a topic of focus in recent years. Unlike the previous results (see [2, 3, 6, 13, 15, 18] and the references therein), another interesting intermittent control strategy which is more adaptable to the practice application and can be found in Fig. 1 is introduced and studied in this paper. Motivated by the aforementioned observations, width time-dependent periodically intermittent smooth controller design for a class of nonlinear systems is presented in this paper. The proposed periodically intermittent control scheme has time dependence and variant control width. 2, we formulate the problem of periodically intermittent control law with variant control width of a class of nonlinear systems, and introduce some necessary preliminaries. 3, some sufficient criteria of global exponential stabilization for a class of nonlinear systems by means of time-dependent width Lyapunov functions are given. · is used to refer to the Euclidean vector norm or the spectral norm for matrices. λmax(·) and λmin(·) denote the maximum and minimum eigenvalues, respectively

Preliminaries and problem formulation

Conclusion