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The problem of input-output finite-time control of positive switched systems with time-varying and distributed delays is considered in this paper. Firstly, the definition of input-output finite-time stability is extended to positive switched systems with time-varying and distributed delays, and the proof of the positivity of such systems is also given. Then, by constructing multiple linear copositive Lyapunov functions and using the mode-dependent average dwell time (MDADT) approach, a state feedback controller is designed, and sufficient conditions are derived to guarantee that the corresponding closed-loop system is input-output finite-time stable (IO-FTS). Such conditions can be easily solved by linear programming. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Positive switched systems are a class of dynamics whose state and output are nonnegative whenever the initial conditions and inputs are nonnegative. In the last decades, the research of positive switched systems is a hot topic due to their many practical applications in communication networks [

Contrary to asymptotic stability, some researchers focus on finite-time stability (FTS) of positive switched systems: that is, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval. Some works dealing with analysis and controller design of FTS control problems have been published [

Motivated by the aforementioned factors, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is considered. The main contributions of this paper are as follows: (

Consider the following positive switched systems with time-varying and distributed delays:

Next, we will present some definitions and lemmas for the positive switched system (

System (

For any switching signal

A matrix

Let

System (

Then, suppose that

In the same way, if

Finally, assume that there exists an element

From the above, system (

The proof is completed.

Next, we will give the definitions of input-output finite-time stability for the positive switched system (

Consider zero initial condition

Consider zero initial condition

[

The aim of this paper is to design a state feedback controller

In this subsection, we will focus on the problem of IO-FTS of positive switched system (

Consider system (

Construct the multiple copositive type Lyapunov-Krasovskii functional for system (

Along the trajectory of system (

In this subsection, the state feedback controller

Consider the positive switched system (

According to Lemma

The proof is completed.

Next, an algorithm is presented to obtain the feedback gain matrices

By adjusting the parameters

Substituting

The gain

Consider the positive switched system (

State trajectories of closed-loop system (

Switching signal of system (

The evolution of

In this paper, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is investigated. The definition of IO-FTS of continuous positive switched systems is proposed. Then, by constructing multiple linear copositive Lyapunov functions, a state feedback controller is designed. Based on the MDADT approach, some sufficient conditions are obtained to guarantee that the closed-loop system is IO-FTS. Such sufficient conditions can be solved by linear programming. Our further work will focus on the IO-FTS of positive switched nonlinear systems with multiple delays.

The authors declare that they have no conflicts of interest.

The authors are grateful for the support of the National Natural Science Foundation of China under Grants U1404610, 61473115, 61374077, and 61203047.

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