Further Results on Reachable Set Bounding for Discrete-Time System with Time-Varying Delay and Bounded Disturbance Inputs

Wei Kang,Hao Chen,Kaibo Shi,Jun Cheng

doi:10.1155/2018/9259284

Abstract

This paper investigates the problem of reachable set bounding for discrete-time system with time-varying delay and bounded disturbance inputs. Together with a new Lyapunov-Krasovskii functional, discrete Wirtinger-based inequality, and reciprocally convex approach, sufficient conditions are derived to find an ellipsoid to bound the reachable sets of discrete-time delayed system. The main advantage of this paper lies in two aspects: first, the initial state vectors are not necessarily zero; second, the obtained criteria in this paper do not really require all the symmetric matrices involved in the employed Lyapunov-Krasovskii functional to be positive definite. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

Highlights

The reachable set of dynamic system is defined as a collection of system state vectors in presence of all admissible input disturbances
This paper investigates the problem of reachable set bounding for discrete-time system with time-varying delay and bounded disturbance inputs
Together with a new Lyapunov-Krasovskii functional, discrete Wirtinger-based inequality, and reciprocally convex approach, sufficient conditions are derived to find an ellipsoid to bound the reachable sets of discrete-time delayed system

Summary

Introduction

The reachable set of dynamic system is defined as a collection of system state vectors in presence of all admissible input disturbances. In [25], the authors studied the problem of reachable set estimation and synthesis for delayed systems by using delay-decomposition technique and reciprocally convex approach. The ellipsoidal reachable set estimation conditions of discrete-time linear systems are obtained by using discrete Wirtinger-based inequality and reciprocally convex approach. The novelty of this paper is three aspects: first, a relaxed Lyapunov-Krasovskii functional, which does not require all the involved symmetric matrices to be positive definite, is employed to solve the addressed problem; second, the initial state vectors are not necessarily zero, which is more general than the existing results [38, 39]; third, discrete Wirtinger-based inequality is taken into account in this paper to deal with the problem of reachable set bounding for delayed discrete-time systems. Throughout this paper, R > 0(R ⩾ 0, R < 0, R ⩽ 0) means that the matrix R is positive definite (positive semidefinite, negative definite, and negative semidefinite); Rm×n is the set of m × n real matrices; the superscripts −1 and T denote the inverse and transpose of a matrix, respectively; ∗ denotes the symmetric block in symmetric matrix; I denotes the identity matrix with compatible dimensions; N denotes the set of natural number; and Z denotes the set of integers

Problem Statement and Preliminaries

Main Results

Numerical Examples

11 Conflicts of Interest

Conclusions