Input-to-State Stability and Stabilization of Nonlinear Impulsive Positive Systems

Yiqing Xue,Ping Zhao

doi:10.3390/math9141663

Abstract

This paper focuses on the problems of input-to-state stability (ISS) and stabilization for nonlinear impulsive positive systems (NIPS). Using the max-separable ISS Lyapunov function method, a sufficient condition on ISS is given for general NIPS. On that basis, the ISS criteria for linear impulsive positive systems (LIPS) and affine nonlinear impulsive positive systems (ANIPS) are given. Through them, ISS properties can be directly judged from the algebraic and differential characteristics of the systems. Then, utilizing the ISS criteria, state-feedback and impulsive controllers are designed for LIPS and ANIPS, respectively, which make the systems input-to-state stabilizable. Lastly, some numerical examples are given to verify the effectiveness of our results.

Highlights

A positive system is a special kind of dynamical system whose state and output variables are non-negative whenever and wherever the initial state and the input variables are non-negative [1]
The input-to-state stability (ISS) problem was investigated in [25,26,27], with some sufficient conditions provided on the basis of the Lyapunov function method
For linear impulsive positive systems (LIPS) (13), the following proposition can be obtained by the max-separable ISS Lyapunov function method

Summary

Introduction

A positive system is a special kind of dynamical system whose state and output variables are non-negative whenever and wherever the initial state and the input variables are non-negative [1]. The ISS problem was investigated in [25,26,27], with some sufficient conditions provided on the basis of the Lyapunov function method. The ISS problem of NIPS is studied for the first time, with some criteria on ISS and input-to-state stabilization being provided. On the basis of the ISS criteria for LIPS and ANIPS, state-feedback and impulsive controllers were designed, respectively, which make the systems input-to-state stabilizable.

Preliminaries

ISS of Nonlinear Impulsive Positive Systems

Input-to-State Stabilization

Conclusions