# Exponential Stabilization for a Class of Nonlinear Switched Systems with Mixed Delays under Asynchronous Switching

Authors: Yongzhao Wang
Journal: Complexity
 (2018)

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## Abstract

This paper deals with the exponential stabilization problem for a class of nonlinear switched systems with mixed delays under asynchronous switching. The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. By constructing the parameter-dependent Lyapunov-Krasovskii functional and the average dwell time approach, some sufficient conditions in forms of linear matrix inequalities are presented to ensure the exponential stability of the switched nonlinear system under arbitrary switching signals. In addition, through the special deformation of the matrix and Schur complement, the controllers with asynchronous switching are designed. Finally, a numerical example and a practical example of river pollution control are provided to show the validity and potential of the developed results.


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## Body

## 1. Introduction

The switched system is a relatively complex and typical dynamic system. It is composed of a group of discrete or continuous dynamic subsystems and a switching rule that coordinates the operation of each subsystem. In recent years, the research achievements of a switched system are widely used in aerospace, artificial intelligence, and biochemical and industrial manufacturing [1–3]. Moreover, the research of switched systems has attracted more and more domestic and foreign scientific research workers’ attention [4, 5]. Thus, switched systems are of significant interest not only for their applicability in practice but also for their interesting theoretical properties. This is motivated by the need for systematic approach to investigate switched systems.It is well known that time delay, perturbation, and stochastic term are inevitable in some practical control systems, which are often the main cause for instability or undesirable system performance of a control system [6–8]. In the past few years, the subject of switched systems with time-varying delays has attracted considerable attention due to a strong engineering background. For instance, Bingi et al. [9] investigate the phenomenon of time-dependent transmittance of evanescent Bloch modes (EBM) in ZnS random photonic crystal (RPC) which forms the basis for photonic delay switching. Exponential stability and L1-gain analysis for positive time delay Markovian jump systems with switching transition rates subject to average dwell time have been investigated in [10]. Among these studies, stability is a crucial and fundamental problem for a switched system. Therefore, the research on stability of system has attracted a large number of domestic and foreign scientific researchers’ attention [11, 12]. However, the problems of stability for switched nonlinear systems are a great challenge and few results have been reported for a switched nonlinear system with time delay. Hence, many researchers have paid more attention to studying nonlinear switched systems in the past few years and some stability results related to switched nonlinear systems have been reported in the literature [13, 14]. Moreover, Sun et al. and Daafouz et al. [15, 16] obtain sufficient conditions guaranteeing the exponential stability by a common Lyapunov functional (CLF). However, a common Lyapunov functional approach might become too conservative when stability is assessed. To address this issue, scholars investigate the systems by using multiple Lyapunov functional (MLF) [17, 18] and average dwell time (ADT) method [19, 20] in recent years. In particular, average dwell time technique plays an important role in switched system analysis and control synthesis. In this paper, our main goal is to provide a novel multiple Lyapunov-Krasovskii functional to study the exponential stabilization of switched nonlinear uncertain systems with mixed delays by an average dwell time method under asynchronous switching.On the other hand, it is noted that the majority of the results mentioned above were based on an ideal assumption that the switching between the controller and the system is synchronous. In fact, since it is inevitable that some time is needed to identify the system mode and apply the matching controller, there is an asynchronous phenomenon between the system mode switching and the controller switching. Therefore, it is significant to study the problem of asynchronous switching and such system has gradually become a hot research field. With the deepening of research, some valuable achievements of asynchronous switching have emerged. Xiang and Wang [21] investigate robust control for uncertain switched systems with time delay under asynchronous switching, and some new delay-dependent exponential stabilization criteria for the system were established. Wang et al. [22] study the influence of random interference on the switching system by average dwell time technique under asynchronous switching, and H∞ control issue for a class of switching system is considered in [23]. However, based on the above discussion, the problem of exponential stabilization for a class of nonlinear switched systems with mixed delays under asynchronous switching has not been well reported.The core of this paper adds to the further development of switched nonlinear systems under asynchronous switching. Compared with the existing results on switched systems, the results of this paper have four contributions. Firstly, the switching signal of the Lyapunov-Krasovskii functional constructed in the paper is dependent on the controller switching signal, which is convenient for the analysis of the proposed issue. Secondly, the problem of the asynchronous switching between the subsystems and the candidate controllers is considered. We derive some sufficient conditions for exponential stability of the switched nonlinear systems with mixed delays by an average dwell time approach. Thirdly, based on matrix deformation technique and Schur complement, the state feedback controllers of switched nonlinear systems are designed under asynchronous switching, while on the existing work, the controller design problem was not considered. Finally, the results of this paper are extended to a practical example of river pollution control.The rest of the paper is organized as follows. The problem description and preliminary knowledge are presented in Section2. In Section 3, a novel multi-Lyapunov-Krasovskii functions related to parameters are constructed and a sufficient condition for exponential stabilization of a class of nonlinear switched systems with mixed delays under asynchronous switching is given by using average dwell time and matrix inequality. Moreover, the controllers of the switching system are designed through a special matrix deformation method, which is the important conclusion of this paper. Section 4 gives a numerical example and a practical example of river pollution control to show the validity and potential of the developed results. A conclusion is shown in Section 5.


## 2. Problem Description and Preliminaries

This paper studies a class of nonlinear switching systems as follows:(1)ẋt=A1σtxt+A2σtxt−τt+Bσtut+ft,xt−τt+gt,xt−ht,xt=φt,t∈max−τ,h,0,where ut∈Rm is control input, xt∈Rn denotes a system state, and φs∈Rn is the initial condition. A1i, A2i, and Bi, i∈L, are constant matrices, and σt:0,∞→L=1,2,…,n is the switching signal, which is a piecewise continuous function, n is the number of subsystems. Specially, switching sequence of the system is expressed as t0,σt0,t1,σt1,…,tk,σtk, where t0 and tk represent the initial and the kth switching time, respectively. τt and ht represent the mixed delays of the switched system and satisfy the following cases:
(2)0≤τt≤τ,τ̇t≤d<1,0≤ht≤h,ḣt≤υ<1.ft,xt−τt and gt,xt−ht are nonlinear perturbation functions, which satisfy the following condition:
(3)ft,xt−τt≤εxt−τt,gt,xt−ht≤ρxt−ht,where ε>0 and ρ>0 are known constants, noting that the nonlinear perturbations are widely applicable in practice and considered by many researchers.When the controller synchronizes with the switching subsystem, the state feedback controller is often designed as(4)ut=Kσtxt,where Ki, i∈L, denote the feedback gain matrix. However, in actual operation, the controller switching time lags behind the subsystem switching time. In other words, the controller and the switching subsystem are asynchronous. At this point, we assume that the delay of the switching signal of the controller is τd, where the lag time τd<c and c is a known constant. Then, the state feedback controller is designed as
(5)ut=Kσt−τdxt.Remark 1.
This article studies switched systems under asynchronous switching. We suppose that thejth subsystem is activated at the switching instant tk−1 and the ith subsystem is activated at the switching instant tk, then the corresponding switching controllers are activated at the switching tk−1+τd and tk+τd, respectively. Therefore, the closed-loop system of system (1) in the interval tk,tk+1 can be represented as follows:
(6)ẋt=A1i+BiKjxt+A2ixt−τt+ft,xt−ht+gt,xt−τt,∀t∈tk,tk+τdmismatchedperiods,ẋt=A1i+BiKixt+A2ixt−τt+ft,xt−ht+gt,xt−τt,∀t∈tk+τd,tk+1matchedperiods.
To facilitate the calculation, defineA¯1ij=A1i+CiKj and A¯1i=A1i+CiKi. Then, the closed-loop system is abbreviated as follows:
(7)ẋt=A¯1ijxt+A2ixt−τt+ft,xt−τt+gt,xt−ht,∀t∈tk,tk+τdmismatchedperiods,ẋt=A¯1ixt+A2ixt−τt+ft,xt−τt+gt,xt−ht,∀t∈tk+τd,tk+1matchedperiods.In order to prove the next statements, the following definitions and lemmas are introduced.Definition 1 [24].
The equilibriumx∗=0 of the closed-loop system (7) is called to be exponential stabilization under the switching signal σ and feedback control (5), if the solution of the closed-loop system (7) satisfies the following inequality.
(8)xt≤ωsup−maxτ,h≤θ≤0xt0+θe−λt−t0,∀t≥t0,ω≥1,λ>0.Definition 2 [25].
For givenN0≥0, τa≥0, and any t2>t1≥0, let Nσt1,t2 be the number of switching signals σt in the interval t1,t2, if the following equation
(9)Nσt1,t2≤N0+t2−t1τa,holds, then τa is said to be average dwell time. The authors choose N0=0 in this paper.Lemma 1 [25].
S1, S2, and S3 are symmetric matrices of appropriate dimensions with S1=S1T<0, S3=S3T>0, then S1+S2S3−1S2T<0 if and only if
(10)S1S2S2T−S3<0.


## 3. Main Results

In this section, we derive some sufficient conditions of the exponential stabilization for switched nonlinear systems by an average dwell time approach under asynchronous switching. Moreover, the state feedback controllers of switched nonlinear systems are designed.Proposition 1.
For given positive constantsα, β, τ, d, h, υ, and μ≥1, if there is Pi, Qi, and Ri, which are symmetric and positive definite matrices, such that
(11)Pi≤μPj,Qi≤μQj,Ri≤μRj,∀i,j∈L,(12)Πi=φ11iφ12i0PiPi∗φ22i000∗∗φ33i00∗∗∗−I0∗∗∗∗−I<0,(13)Ξi=ω11iφ12i0PiPi∗ω22i000∗∗ω33i00∗∗∗−I0∗∗∗∗−I<0,where
(14)φ11i=PiA¯1i+A¯1iTPi+Qi+Ri+αPi,φ22i=ε2−1−de−ατQi,φ12i=PiA2i,φ33i=ρ2−1−υe−αhRi,ω22i=ε2−1−dQi,ω33i=ρ2−1−υRi,ω11i=PiA¯1ij+A¯1ijTPi+Qi+Ri+αPi.
Then, the closed-loop system (7) is exponentially stable under the feedback controller (5) for arbitrary switching signal with the average dwell time satisfying
(15)τa>τa∗=lnμ+α+βτdα.Proof 1.
There exist two periods during the whole running time of stability for switched systems (7): matched and mismatched periods.
Firstly, whent∈tk+τd,tk+1, σtk=i∈L, the closed-loop system (7) is active within the ith subsystem and the corresponding ith switching controller is also activated. The Lyapunov-Krasovskii functional is constructed as follows:
(16)V1σtt=xTtPσtxt+∫t−τtteαs−txTsQσtxsds+∫t−htteαs−txTsRσtxsds.
We can easily obtain the following inequalities:(17)V̇1i=2xTtPiẋt−α∫t−τtteαs−txTsQixsds−α∫t−htteαs−txTsRixsds−1−τ̇te−ατtxTt−τtQixt−τt−1−ḣte−αhtxTt−htRixt−ht+xTtRixt+xTtQixt≤xTtPiA¯1i+A¯1iTPi+Qi+Rixt−1−de−ατxTt−τtQixt−τt+xTtPiA2ixt−τt+xTtPift,xt−τt+xTt−τtA2iTPixt+fTt,xt−τtPixt−1−υe−αhxTt−htRixt−ht+xTtPigt,xt−ht+gTt,xt−htPixt−α∫t−τtteαs−txTsQixsds−α∫t−htteαs−txTsRixsds.
The inequality (3) can be rearranged as follows:
(18)ε2xTt−τtxt−τt−fTt,xt−τtft,xt−τt≥0,ρ2xTt−htxt−ht−gTt,xt−htgt,xt−ht≥0.
Combining (17) with (18), we can get
(19)V̇1i+αV1i≤xTtPiA¯1i+A¯1iTPi+Qi+Ri+αPixt−gTt,xt−htgt,xt−ht+xTtPift,xt−τt+xTt−τtA2iTPixt+fTt,xt−τtPixt+xTt−htρ2I−1−υe−αhRixt−ht+xTtPigt,xt−ht+xTtPiA2ixt−τt+xTt−τtε2I−1−de−ατQixt−τt+gTt,xt−htPixt−fTt,xt−τtft,xt−τt.
Let(20)ζt=xTtxTt−τtxTt−htfTt,xt−τtgTt,xt−htT.
Then,(21)V̇1i+αV1i≤ζTtΠiζt<0,where Πi is given by (12). Therefore, the following formula is established.
(22)V̇1i<−αV1i.
Secondly, whent∈tk,tk+τd, the closed-loop system (7) is active within the ith subsystem and the corresponding jth switching controller is also activated. We choose the Lyapunov-Krasovskii functional candidate as follows:
(23)V2σtt=xTtPσtxt+∫t−τtteβt−sxTsQσtxsds+∫t−htteβt−sxTsRσtxsds.
We can get the time derivative ofV2σt as follows:
(24)V̇2i=2xTtPiẋt+xTtQixt+β∫t−τtteβt−sxTsQixsds−1−τ̇teβτtxTt−τtQixt−τt+β∫t−htteβt−sxTsRixsds−1−ḣteβhtxTt−htRixt−ht+xTtRixt≤xTtPiA¯1ij+A¯1ijTPi+Qi+Rixt−1−dxTt−τtQixt−τt+xTtPiA2ixt−τt+xTtPift,xt−τt+fTt,xt−τtPixt−1−υxTt−htRixt−ht+xTtPigt,xt−ht+gTt,xt−htPixt+xTt−τtA2iTPixt+β∫t−htteβs−txTsRixsds+β∫t−τtteβs−txTsQixsds.
Recalling (18), it follows that
(25)V̇2i−βV2i≤xTtPiA¯1ij+A¯1ijTPi+Qi+Ri−βPixt+xTt−τtε2I−1−dQixt−τt+fTt,xt−τtPixt+xTtPift,xt−τt+xTt−τtA2iTPixt+xTt−htρ2I−1−υRixt−ht+xTtPigt,xt−ht+gTt,xt−htPixt−gTt,xt−htgt,xt−ht+xTtPiA2ixt−τt−fTt,xt−τtft,xt−τt,that is,
(26)V̇2i−βV2i≤ζTtΞiζt<0.where Ξi is given by (13). Therefore, the following formula is established.
(27)V̇2i<βV2i.
By recalling (2), we have
(28)∫t−τtteαs−txTsQixsds+∫t−htteαs−txTsRixsds≤∫t−τttxTsQixsds+∫t−htteαs−txTsRixsds≤∫t−τtteβt−sxTsQσxsds+∫t−htteβt−sxTsRσxsds.
Thus,(29)V1it≤V2it.
In the entire intervalt0,t, the Lyapunov-Krasovskii function is expressed as
(30)Vt=Vt,∀t∈tk+τd,tk+1,V2σt,∀t∈tk,tk+τd,where k=0,1,2,…,n.
Whent∈tk+τd,tk+1, integrating both sides of (22) from tk+τd to t, we get
(31)Vt≤e−αt−tk+τdV1itk+τd+≤e−αt−tk+τdV2itk+τd−≤e−αt−tk+τdeβτdV2itk+≤μe−αt−tk+τdeβτdV1itk+≤⋯≤μkek+1βτde−αt−t0−k+1τdVt0≤eα+βτdelnu+α+βτd/τa−αt−t0Vt0.
Whent∈tk,tk+τd, integrating both sides of (27) from tk to t, we get
(32)Vt≤eβt−tkV2itk+≤μeβt−tkV1itk−≤μe−αt−tk−1−τdeβτdV1itk−1+τd−≤⋯≤μkek+1βτde−αt−t0−k+1τdVt0≤eα+βτdelnu+α+βτd/τa−αt−t0Vt0.
Using the equality (16) and (23), we obtain
(33)Vt≥axt2,Vt0≤bsup−maxτ,h≤θ≤0xt0+θ2,where
(34)a=mini∈LλminPi,b=maxi∈LλmaxPi+τeβτmaxi∈LλmaxQi+heβhmaxi∈LλmaxRi.
So,(35)xt≤e1/2α+βτdbasup−maxτ,h≤θ≤0xt0+θeι,where ι=−1/2α−lnu+α+βτd/τat−t0.
According to Definition1, it is easily to prove that the closed-loop system (7) is exponential stabilization.Remark 2.
In [17, 21], stabilization of nonlinear switched systems with delay was investigated under asynchronous switching and some criteria of stability for nonlinear switched systems were obtained. However, the nonlinear term in [17, 21] did not consider time delay. Liu et al. [8] focus on the problem of stability for a class of switched nonlinear systems with time-varying delay, but they [8] do not contain asynchronous switching. In this paper, we consider nonlinear switched systems and the nonlinear term in our system contains time-varying delay. In addition, some exponential stabilization criteria are obtained under asynchronous switching. Compared with [8, 17, 21], we have overcome the situation where the controller switching instant lags behind the subsystem switching instant and we have a greater advantage when dealing with complex systems in practice.Remark 3.
It is worth noting that the parameter-dependent Lyapunov-Krasovskii functional constructed in the form of (30) has three main features. On the one hand, the Lyapunov-Krasovskii functional depends on the switching signal of the controller, which is convenient for the analysis of the controller design under asynchronous switching. On the other hand, the Lyapunov-Krasovskii functional is incremental both at switching instants and during the mismatched periods, but it is decreasing as a whole and the stability of the system is guaranteed. Finally, the subsystems are allowed to be unstable during mismatched periods resulted from asynchronous switching.Proposition 2.
For givenα, β, τ, d, h, υ, and μ≥1, if there is Xi, Gi, Oi, and Yi, which are symmetric and positive definite matrices, such that
(36)Xj≤μXi,Gj≤μGi,Oj≤μOi,∀i,j∈L,i≠j,(37)φ¯11iφ¯12i0IIεXi0XiXi∗φ¯22i0000ρXi00∗∗φ¯33i000000∗∗∗−I00000∗∗∗∗−I0000∗∗∗∗∗−I000∗∗∗∗∗∗−I00∗∗∗∗∗∗∗−Oi0∗∗∗∗∗∗∗∗−Gi<0,(38)ω¯11iφ¯12i0IIεXi0XiXi∗ω¯22i0000ρXi00∗∗ω¯33i000000∗∗∗−I00000∗∗∗∗−I0000∗∗∗∗∗−I000∗∗∗∗∗∗−I00∗∗∗∗∗∗∗−Oi0∗∗∗∗∗∗∗∗−Gi<0,where
(39)φ¯11i=A1iXi+BiKiXi+A1iXi+BiKiXiT+αXi,φ¯22i=1−de−ατGi−2Xi,φ¯12i=A2iXi,φ¯33i=1−υe−αhOi−2Xi,ω¯22i=1−dGi−2Xi,ω¯11i=A1iXi+BiKjXi+A1iXi+BiKjXiT−βXi,ω¯33i=1−υOi−2Xi.
Then, system (1) is exponentially stable if average dwell time τa>τa∗=lnμ+α+βτd/α holds. Furthermore, the controller can be designed by the following formula:
(40)Ki=YiXi−1.Proof 2.
FromGi>0 and Oi>0, we can get
(41)Gi−XiTGi−1Gi−Xi≥0,Oi−XiTOi−1Oi−Xi≥0.
Then,(42)Gi−2Xi≥−XiGi−1Xi,Oi−2Xi≥−XiOi−1Xi.
Both sides of (37) multiplies simultaneously Xi−1,Xi−1,Xi−1,I,I,I,I,I,I, and we have the following inequality:
(43)φ~11iφ~12i0Xi−1Xi−1ε0II∗φ~22i0000ρ00∗∗φ~33i000000∗∗∗−I00000∗∗∗∗−I0000∗∗∗∗∗−I000∗∗∗∗∗∗−I00∗∗∗∗∗∗∗−Oi0∗∗∗∗∗∗∗∗−Gi<0,where
(44)φ~11i=Xi−1A1i+Xi−1BiYiXi−1+Xi−1A1i+Xi−1BiYiXi−1T+αXi−1,φ~12i=Xi−1A2i,φ~22i=−1−de−ατGi−1,φ~33i=−1−υe−αhOi−1.
Let(45)Yi=KiXi,Xi−1=Pi,Gi−1=Qi,Oi−1=Ri.Using Schur complement in (43), we can get that (12) holds and the same method can be used to prove that formula (13) holds. From (45), the controller gains are given by (40). The proof is completed.


## 4. Simulation Examples

In this section, a numerical example and a practical example are given to illustrate the effectiveness and applicability of the proposed approach.Example 1.
We consider nonlinear switched systems with mixed delays, which is consisted of two subsystems with the following parameters:(46)A11=−0.500−0.6,A21=−0.400−0.6,A12=−0.800−0.6,A22=−0.600−0.7,B1=0.30.5,B2=0.20.6.
We chooseα=1.05, β=0.4, h=0.9, τ=0.8, d=0.4, υ=0.3, τd=0.3, μ=1.9, ε=0.3, ρ=0.4,
(47)ft,xt−τt=0.1cosx1t,0.2sinx2t−τt−0.1,gt,xt−ht=0.1sinx1t−ht,0.2cosx2t−0.1.Through calculation, we get the average dwell time:
(48)τa>τa∗=lnμ+α+βτdα=1.0256.
According to (38), (40), and (42), we can obtain
(49)X1=0.27770.12020.12020.0866,X2=0.26860.11790.11790.0841,G1=0.41630.02700.02700.3832,G2=0.41090.02450.02450.3827,O1=0.42400.03400.03400.3813,O2=0.41770.03150.03150.3809,Y1=0.93550.9169,Y2=0.41030.8936.
Controller gains of the system can be calculated by (40) and be expressed as
(50)K1=−2.681913.9815,K2=−8.157222.0690.Figure1 shows the switching signals of the system and controller, while Figure 2 shows the state response with an initial state x0=−11T of the considered switched nonlinear system under the designed switching signal depicted in Figure 1. It is easy to find that the system is exponentially stabilizable under asynchronous switching.Example 2.
We will illustrate the effectiveness of our approach through river pollution control issues.
In a reach of a polluted river, the concentrations per unit volume of biochemical oxygen demand and dissolved oxygen are denoted aszt and qt, respectively. Let z∗ and q∗ corresponding to some measure of water quality standards denote the desired steady values of zt and qt, respectively. Define x1t=zt−z∗, x2t=qt−q∗, and xt=x1Ttx2TtT. Then, the dynamic equation for xt can be written as [18, 19]:
(51)ẋt=Axt+A~xt−τt+But+ωt,where
(52)A=−k10−η1−η20−k30−k20−η1−η2,A~=η200η2,B=η11.where ut=u1Ttu2TtT is the control variables of river pollution, ki0i=1,2,3, η1 and η2 are known constants, and ωt is the disturbance input of the system. The physical meaning of these parameters can be found in [26, 27].
In accordance with the actual situation, we assumed that the system actuators are subject to good performance or failure in this paper. Therefore, the model is divided into two subsystems for discussion. Then, system (51) can be described as the following switched system:
(53)ẋt=A11xt+A21xt−τt+B1ut+ft,xt−τt+gt,xt−htnofailuresoccur,A12xt+A22xt−τt+B2ut+ft,xt−τt+gt,xt−htfailuresoccur.
For the simulation of our purposes, we choosek10=1.6, k20=1, k30=1.6, η1=0.3, and η2=0.7 and we can get
(54)A11=−2.60−1.6−2,A21=0.7000.7,B1=0.31.
Letτt=0.3sint, ht=0.3sint, ωt=ω1t+ω2t,
(55)ω1t=ft,xt−τt=0.1sinx1t0.2sinx2t−τt,ω2t=gt,xt−ht=0.2cosx1t0.1cosx2t−ht.
Then, we will use the above parameters to design a set of switching sequences to stabilize the above system (53). At the same time, we choose
(56)A12=−2.901.2−1.8,A22=0.5000.5,B2=0.20.8.
α=0.3, β=0.5, μ=1.7, h=0.8, τ=0.9, τd=0.4, d=0.3, υ=0.3, τt=0.3sint, ht=0.3cost, we get the average dwell time:
(57)τa>τa∗=lnμ+α+βτdα=2.8354.
By solving (38), (40), and (42), we have
(58)X1=0.41610.06900.06900.4290,X2=0.22930.22410.22410.4260,G1=1.30800.02440.02441.2977,G2=1.32630.64890.64891.3533,O1=0.73890.07990.07990.7482,O2=0.55900.21370.21370.7364,Y1=0.95010.2311,Y2=0.60680.4860.
Then, the controller gains constructed by (40) are
(59)K1=2.25410.1760,K2=3.150−0.5166.Figure 1
The switching law.Figure 2
State response of the system.Figure3 shows the switching signals of the system and controller, while Figure 4 shows the state response with an initial state x0=−1.5,1.5T of the considered switched nonlinear system under the designed switching signal depicted in Figure 3. Therefore, the effectiveness of our approach is verified by its application in the control of the river pollution process.Remark 4.
Zheng et al. [27] have investigated the pollution problem of a single reach river modelled by the dynamics of water quality subject to uncertainty in system parameters; an adaptive controller is developed based on linear matrix inequality technique, and it is shown that the controller can guarantee the closed-loop system to converge, globally and exponentially. However, the performance of the actuator for the river pollution model may be deviated in practical applications. This article has solved this situation by changing the original system model to a switched system for processing. Compared with [26, 27], we have stronger application value in practice.Figure 3
The switching law.Figure 4
State response of the system.


## 5. Conclusion

In this paper, the problem of exponential stabilization for a class of a nonlinear switched system with mixed time delays has been studied. The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. Based on a novel parameter-dependent Lyapunov-Krasovskii functional, some sufficient conditions for the exponential stability of the switched system under asynchronous switching are obtained by the average dwell time approach. Moreover, the controllers of the switched system are designed through a special matrix transformation method. Finally, a numerical example and a practical example of river pollution control are provided to show the validity and potential of the developed results.Through the research of this paper, we learned that different piecewise Lyapunov functionals may lead to different conservatism. It deserves further study to choose an improved piecewise Lyapunov functional so as to reduce the conservativeness. In order to better study the asynchronous switching problem in multiple aspects, we will further optimize Lyapunov functionals and the dual controller design for better performance. In this article, we did not consider stochastic term. It is well known that stochastic term is inevitable in some practical control systems [28], which is often the main cause for instability or undesirable system performance of a control system. Specifically, the stabilization of stochastic switched nonlinear systems with Markov jumps will be taken as a main direction of our future research.



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