Finite-time Stabilization of Switched Positive Systems with Time-varying Delays and Actuator Saturation

In this paper, the finite-time stabilization for a class of switched positive systems with time-varying delays and actuator saturation under average dwell time switching is investigated. Through multiple copositive Lyapunov-Krasovskii functionals method, the sufficient conditions for finite-time bounded of continuous-time case are given, and the appropriate switching rules and state feedback controllers are presented together. Moreover, the convex hull technique is employed to deal with actuator saturation. Finally, an illustrative example is given to show the effectiveness of the proposed method.

On the other hand, when it comes to stability, most of the research is about Lyapunov stability or asymptotic stability, which are defined over an infinite time interval. In fact, this concept is totally different with finite-time stability. In many practical applications in some real systems, however, the system variables are required to be bounded in a finite-time interval very precisely. Hence, the concept of finite-time bounded (FTB) is proposed.
The exploration of finite-time control for switched positive systems with actuator saturation is very fair [8]. In the literature [9], through multiple LCLFs methods and convex hull technique, the stabilization of SPLSs with actuator saturation under both state-dependent and time-dependent switching are investigated. In the literature [8], finite-time control for discrete-time switched singular time-delay systems subject to actuator saturation was investigated via static output feedback, and the uniformly FTB condition was given by LMIs. In fact, the switching action, coupled with the positivity constraint of state variables, together with the actuator saturation, make the behavior of SPLSs with actuator saturation very complicated. Which means the finite-time control of SPLSs with actuator saturation will be difficult and challenging.
Motivated by the above statement, this paper will focus on the study of finite-time control of SPLSs with both time-varying delays and actuator saturation. The main contributions of this paper are given in the following aspects: (1) Dealing with the FTB problem for switched positive system with both timevarying delays and actuator saturation is the first time; (2) Sufficient conditions of FTB for continuoustime switched positive systems with time-varying delays and actuator saturation are obtained by nonquadratic multiple copositive Lyapunov-krasovskii functionals method; (3) Realized the joint design of switching laws and state feedback controller under the average dwell time switching to ensure that the closed-loop system is FTB and positive.
The rest of this paper is organized as follows. Some necessary statements and definitions are given in Section 2. In section 3, sufficient conditions for state feedback controllers and switching laws of FTB of switched positive systems with time-varying delays and actuator saturation are proposed. A numerical example is given in section 4. Section 5 is the conclusion of this paper.

Problem statements and preliminaries
Given the following switched system with actuator saturation and time-varying delays  is a right continuous piecewise constant function of time, which is called switching signal, and it takes its values in the finite set S , s means the number of subsystems; Subsystem is the standard saturation function defined as . In this paper, we take the notation by using sat( )  to denote both the scalar valued and the vector valued saturation functions. In this paper, notation 0( 0) A A   means all the elements of A are positive (non-negative). Definition 1 [10]: System (1) is said to be positive if for any switching signals ( ) t  and any initial , and its trajectory ( ) 0 x t  for all 0 t  . Definition 2 [11]: A is called a Metzler matrix, if the off-diagonal entries of the matrix A are nonnegative.
Definition 3 [12]: For a switching signal ( ) t  and each 2 without loss of generality, in this paper we choose 0 0 N  . Definition 4 [13]: Given positive constants 1 c , 2 (1) is said to be FTB with respect to 1 2 There are some important notations in the following.
For any vector , we define ( ,1) For any matrix Lemma 1 [14]: Given two matrices , Then, we are going to design a state feedback controller ( ) ( ) where m n i K R   are gain matrix, i S   , such that the system (1) is not only switched positive system but also FTB.

Main results
In this part, sufficient conditions for finite-time stabilization of continuous-time , here, ... ...

Conclusions
The finite-time control for switched positive systems with actuator saturation and time-varying delays under average dwell time switching has been studied in this paper. Together with convex hull technique for solving actuator saturation, the state feedback controller gain matrices and the ADT switching rules are designed through multiple copositive Lyapunov functionals method.