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This paper investigates the problems of stability and

Positive systems mean that their states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative [

The delta operator, a novel method with good finite word length performance under fast sampling rates, has drawn considerable interest in the past three decades. As we know, the standard shift operator was mostly adopted in the study of control theories for discrete-time systems. However, the dynamic response of a discrete system does not converge smoothly to its continuous counterpart when the sampling period tends to zero; namely, data are taken at high sampling rates. Until Goodwin et al. proposed a delta operator method in [

In real engineering, time delays are involved in many fields, such as mechanics, medicine, chemistry, biology, physics, economics, engineering, and control theory [

In addition, exogenous disturbances are commonly unavoidable in practical process, and the output will be inevitably affected by the disturbance in a system. Because of the peculiar nonnegative property of positive systems, the

In this paper, we focus our attention on investigating the stability and

The remainder of the paper is as follows. The problem formulation and some necessary lemmas are provided in Section

Consider the following switched delta operator system with time-varying delays:

To illustrate the main advantage of delta operator systems directly, we consider a typical continuous system without time delays as follows:

Since a delta operator system can be regarded as a quasicontinuous system when

System (

Definition

System (

From the definition of delta operator

When

In the light of Lemma

System (

For any switching signal

Without loss of generality, one chooses

For

system (

under zero initial condition, that is,

In Definition

The purposes of this paper are

This section will focus on the problems of stability analysis and

First, we consider the following switched positive delta operator system:

Sufficient conditions of exponential stability of system (

Given a positive constant

Furthermore, the state decay of system (

Choose the following piecewise copositive type Lyapunov functional for the

For simplicity,

The Lyapunov function in delta domain has the following form:

According to (

From (

Let

Therefore, according to Definition

This completes the proof.

When

When

Given a positive constant

When the sampling period

We can obtain sufficient conditions of exponential stability of system (

Given a positive constant

Let

Given a positive constant

The following theorem establishes sufficient conditions of exponential stability with

For given positive constants

By Theorem

For any

Combining (

This completes the proof.

When

In this section, we are interested in designing a state feedback controller

Considering system (

Denote

This completes the proof.

Consider the controller design of the following positive switched delta operator system without time delay:

Considering system (

Based on Theorem

Consider the following.

Input the matrices

Choose the parameters

By the equation

Check condition (

Construct the feedback controller

Consider positive switched delta operator system (

Subsystem 1:

Subsystem 2:

Obviously, condition (

According to (

Switching signal.

State responses of the closed-loop system.

In this paper, the stability and

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120.

_{1}-gain and control synthesis for positive switched systems with time-varying delay

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