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This paper is concerned with finite-time stabilization (FTS) analysis for a class of uncertain switched positive linear systems with time-varying delays. First, a new definition of finite-time boundedness (FTB) is introduced for switched positive system. This definition can simplify FTS analysis. Taking interval and polytopic uncertainties into account, a robust state feedback controller is built such that the switched positive linear system is finite-time bounded. Finally, an example is employed to illustrate the validities of obtained results.

As a special kind of hybrid dynamical systems, the switched system consists of finite subsystems and a switching law. The switching law orchestrates the switches between subsystems [

In control theory, the stability and stabilization are basic problems for switched system. Many valuable results on these problems have been obtained, such as stability [

Since switched system contains several dynamic subsystems and a switching law, it is difficult to analyze FTB and FTS. Recently, these problems have been extensively studied. Based on the average dwell-time approach, Lin et al. investigated FTB for switched systems with fixed delays [

About the mentioned literatures, we need to pay attention to the following facts. First, switched positive system is an important kind of switched system. However, there are few works on its FTB and FTS. Second, most of the obtained results are based on a common assumption that the norm of external disturbance is bounded over infinite time interval. But the external disturbance usually does not satisfy this strict assumption. In this case, the obtained results may be unsuitable for application. Thus, it is very significant to reduce the restriction on external disturbance. Besides, the definition of FTB contains a product term of gain matrix and state vector; therefore complex mathematical calculations are unavoidable. Finally, the state of switched positive system is nonnegative. Then, whether a more concise definition of FTB could be proposed for switched positive system to simplify the analysis of FTS is a problem that naturally arises.

Motivated by the above considerations, we investigate FTS for uncertain switched positive linear systems. The main object of this paper is to design a robust state feedback controller such that the system is finite-time bounded. The main contributions and novelties of this paper are summarized as follows:

The remainder of the paper is organized as follows. In Section

Consider a switched linear system described as

The state feedback controller is given as follows:

The main task is to choose appropriate

Define

Applying (

Assume that system (

The external disturbance

In most of the existing literatures, the external disturbance should satisfy

The time-varying delay satisfies

The state trajectory is continuous everywhere. Furthermore, the switching number of

Next, Lemma

System (

Definition

Forgiven positive constants

According to the definition of FTB in most of other literatures, system (

The essence of FTB lies in that the system state must remain within the prescribed bound over fixed interval. Since both (

For

Assume that the uncertainties in system (

Assume that

Due to the existence of interval uncertainties, system matrices

Since (

Construct multiply copositive Lyapunov-Krasovskii function for system (

Taking the derivatives of

From (

It is derived from (

Applying (

Since

Let

Noting (

It is obtained via (

Assume that the switching number of

According to the definition of

Substituting (

Obviously,

It is derived from (

On the other hand, (

Consequently,

According to Definition

Note (

Compared with other literatures, such as [

Next, the attention is focused on designing a state feedback controller such that the system (

Assume that

By the property of convex hull,

Construct multiply copositive Lyapunov-Krasovskii functions which have the same structure of (

It is derived from (

It follows that

Note that

Similarly, we get

Applying (

The remainder of the proof is similar to (

Inequalities (

A numerical example is given to illustrate the validities of the obtained results in this section. Consider the following system with interval uncertainties:

Obviously,

Then, it follows that

Noting that

Consequently,

On the other hand, from (

According to Theorem

Finally, the simulation of system (

The figure of switching law.

The figure of system state.

The figure of

In addition, though system (

This paper focuses on the FTS analysis for a class of uncertain switched positive linear systems. Based on multiply copositive Lyapunov-Krasovskii functions and average dwell-time method, a robust state feedback controller is built such that the uncertain system is finite-time bounded. The innovation points lie in that a concise definition of FTB is proposed for switched positive system. Since the new definition is adopted, the subsequent mathematical derivation is greatly simplified. Finally, a numerical example illustrates the validities of the obtained results and the difference between LAS and FTB.

However, multiply copositive Lyapunov-Krasovskii functions in this paper are traditional. In the literature [

The authors (Tianjian Yu, Yanke Zhong, Tefang Chen, and Chunyang Chen) declare that there is not any conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 61273158.

_{1}control for positive switched linear systems with time-varying delay