# Consensus of Multi-Agent Systems with Switching Disconnected Topologies via Elementary-Unit-Based Dwell Time Approach

Authors: Jian Sun; Chen Guo; Qihe Shan; Yanming Wu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this study, a leader-following consensus problem is investigated for a class of multi-agent systems with switching disconnected topologies. Different from the existing results on switching disconnected topologies, the multi-agent systems considered in this study are unstable. In this situation, the disconnected agents can disperse over some periods even if there exists control protocol on them. To break through this challenge, we draw lessons from which an appropriate switching law can stabilize the switching unstable systems and propose a novel approach called elementary-unit-based dwell time (EUBDT) approach. Based on this approach, each switching interval is considered to consist of a certain number of elementary time segments. Then, by analyzing the local variation of the error states within each elementary time segment, the stabilization properties of switching behaviors are derived to compensate the divergence within the switching intervals. Based on this, the sufficient conditions for the leader-following consensus can be obtained by using a novel kind of piecewise time-varying Lyapunov functions (PTVLFs). Moreover, a time-scheduled controller is designed for such system. Finally, a numerical simulation is given to illustrate the theoretical approach.


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## Body

## 1. Introduction

In the last decade, leader-following consensus control of multi-agent systems has been extensively investigated because of their large applications in biological systems, spacecraft formation, and robot manipulators [1–6]. Because of the broad applications, leader-following consensus of multi-agent systems plays an important role in the automatic control field [7, 8]. Accordingly, a lot of efforts have been put into their analysis for leader-following consensus problem in recent years [9–11].In the field of multi-agent systems, the communication topology may not be fixed because of changes in the agent relations or failures in communication channels [12, 13]. Various results have studied the leader-following consensus problem under the switching directed topologies [13–20]. In the early works, the efforts mainly investigated the switching topologies where all the topologies are connected [13–17]. In recent years, some works have investigated the consensus problem under switching topologies in which the topology is frequently connected [18–20]. In these works, the error state decreases when the topology is connected and increases when the topology is disconnected. The overall consensus can be guaranteed by a relatively long connected topology. If we consider the severe situation that all the topologies are disconnected, this promising idea will not be applicable. How to find general methods to guarantee the consensus of such systems has aroused the interest of researchers in recent years.Without loss of generality, switching disconnected topologies are more realistic, since the disconnected topology can exist all the time [21, 22]. Some works have explored the leader-following consensus problem under switching disconnected topologies [21–24]. In [21], the sufficient conditions for the leader-following consensus control problem under switching disconnected topologies are proposed. Then, the algebraic criteria of consensus control under switching disconnected topologies are developed in [22]. Based on this, the consensus problem of multiple linear systems with switching disconnected topologies via the event-triggering control is investigated in [23]. Until now, most existing works on the switching disconnected topologies are concentrated on the critical stable (or stable) multi-agent systems (Assumption 3 in [21], Assumption 2 in [22], Assumption 3 in [23], and Assumption 1 in [24]). In these results, the connected agents are close to each other and the disconnected agents are not diverging away from each other. The global consensus can always be achieved through cooperative control under different topologies. However, in practice, if we consider the unstable multi-agent systems such as distributed voltage control of microgrid [25], multi-link manipulators driven by DC motor [26, 27], the disconnected agents will disperse though there exists the control protocol on them. Accordingly, reaching the consensus of unstable multi-agent systems under switching disconnected topologies is challenging. This study aims to overcome this challenge and reach the consensus control for such systems.In the field of switching systems, the switching unstable systems can be stabilized by designing an appropriate switching strategy [28, 29]. The main idea is that the stabilization properties at switching instants are utilized to offset the divergence of the unstable systems within the switching periods. Drawing lessons from this idea, we propose a novel approach named elementary-unit-based dwell time (EUBDT) approach to tackle the leader-following consensus problem of unstable multi-agent systems under switching disconnected topologies. Based on the EUBDT approach, we divide the switching intervals into a certain number of elementary time segments and analyze the local variation of error states within each elementary time segment. Then, the stabilization properties of switching behaviors can be obtained to offset the divergence of error states within the switching intervals. Then, by confining the dwell time constraints on each topology, the overall leader-following consensus can be reached. Finally, a simulation example is developed to illustrate the theoretical approach. To sum up, to illustrate the main contribution clearly, the following flow diagram is given.In Figure1, the main problem, challenge, approach, theoretical breakthrough, and application values have been condensed. It can be seen that the main problem and the research values of this study are illustrated clearly.Figure 1
The flow diagram of the main contribution.Notation:ℛm refers to the set of m-vectors. N∗ and N refer to the set of positive integers and natural number, respectively. The notation Q<0 represents that Q is negative definite. IN refers to the N×N identity matrix. ⊗ refers to the Kronecker product.


## 2. Preliminaries

The multi-agent systems are expressed by a directed graphG=V,E,A, where V=v1,v2,…,vN is a set of nodes, E⊆V×V is a set of directed edges, and A=aijN×N is an adjacency matrix. An edge eij=vi,vj∈E represents that an arrow from i to j in the graph, which implies that agent j can acquire information from agent i. If eij∈G,i≠j, we will have aij=1, otherwise, aij=0. A directed path in G denotes a sequence of nodes v1,v2,…,vq such that vi,vi+1∈E,i∈1,2,…,q−1. If G has a directed spanning tree, it will imply that at least a node has a directed path to all the other nodes. The Laplacian matrix of the graph is denoted as ℒ=D−A∈ℛN×N, in which the matrix D=dii∈ℛN×N with dii=∑j=1Naij and(1)lij=∑k=1,k≠iNaik;i=j,−aij;i≠j.Consider a group ofN agents, whose dynamics is expressed as(2)x˙it=Axit+Buit,i=1,2,…,N,where xit∈ℛn represents the state of agent i, uit∈ℛp represents the control input of agent i, and A and B are constant matrices.The dynamics of the leader is shown as(3)x˙0t=Ax0t,where x0t∈ℛn represents the leader’s state. The main objective is to guarantee the consensus of followers (2) and the leader (3), which can be illustrated as(4)limt⟶∞xit−x0t=0,∀i=1,2,…,N.We use a diagonal matrixℬ=diagb1,b2,…,bN to denote the access between the agents and the leader, where ℬ∈ℛN×N. If the agent i can receive the leader’s information, it will be bi=1,i=1,2,…,N; otherwise, bi=0. For convenience, a matrix ℋ=ℒ+ℬ is utilized to denote the information-exchange matrix.Define a piecewise constant functionσt:1,2,…,m as the switching signal of switching topologies, which satisfies the switching sequence Tp,p=0,1,… with T0=0 and limp⟶∞Tp=∞. Moreover, denote τp=Tp+1−Tp as the dwell time of the pth topology. The dwell time τp is constrained by minimum dwell time τmin and maximum dwell time τmax, which is denoted as τmin≤τp≤τmax. It can be observed that the topology can only be switched within the interval t∈Tp+τmin,Tp+1. For convenience, we use Dτmin,τmax to denote the set of all the switching policies in the framework of dwell time τp∈τmin,τmax. Therefore, G1,G2,…,Gm,m≥1 denotes the set of all the possible directed graphs under the switching graphs. The information-exchange matrices can be denoted as ℋ1,ℋ2,…,ℋm,m≥1. The communication topology under the switching signal σt satisfies the following assumption.Assumption 1.
The switching graphGσtis not connected. The union of all the graphsG¯=G1∪G2∪…∪Gmhas a directed spanning tree with the leader as the root, wherem∈N∗.
Based on this, we consider the controller adopting the following form:(5)uiσtt=K∑j=1Naijσtxjt−xit+biσtx0t−xit.where K is the gain matrix, aijσt represents the adjacency element of Gσt, biσt=1 when agent i can receive the leader’s information under the graph Gσt and biσt=0, otherwise.
Defineeit=xit−x0t as the tracking error for agent i. Then, the error system of agent i under the switching signal σt can be obtained as(6)e˙it=Aeit+BK∑j=1Naijσtejt−eit−biσteit.
Defineet=e1Tt,e2Tt,…,eNTtT, then the error systems for σt=r can be given by(7)e˙t=IN⊗Aet−Hr⊗BKet,t∈Tp,Tp+1,p∈N.
Therefore, the control objective is to achieve the stability of error system (7) under switching laws σt.Remark 1.
Most existing works on the consensus problem under switching disconnected topologies require that the multi-agent systems are critical stable (or stable), which is reflected in that the system matrixA in these works contains no positive real part eigenvalues (Assumption 3in [21], Assumption 2 in [22], Assumption 3 in [23], and Assumption 1 in [24]). In this case, the connected agents will close to each other and the disconnected agents will not disperse. Then, the overall consensus can be achieved by the jointly connected topology. However, this idea cannot be applied to the unstable multi-agent systems such as distributed voltage control of microgrid and multi-link manipulators driven by DC motor, since the disconnected agents will always disperse even if there exists the control protocol on them. Thus, it is challenging or even impossible to achieve consensus of all the agents. How to overcome this difficulty and achieve the seemingly impossible consensus is the main work of this study.


## 3. Main Results

In this section, the EUBDT approach and the conditions of consensus control are presented and proved.


### 3.1. The Elementary-Unit-Based Dwell Time Approach

Most results on the consensus control problem under switching disconnected topologies are concentrated on the stable or critical stable multi-agent systems [21–24]. In this situation, the connected agents will close to each other and the disconnected agents will not disperse. If we consider unstable multi-agent systems such as distributed voltage control of microgrid and multi-link manipulators driven by DC motor, the disconnected agents will disperse under the disconnected topology even if there exist the control protocol on them. Thus, how to stabilize the error states under switching disconnected topologies is challenging. In order to break through this challenge, one has to utilize the stabilization properties of switching behaviors to stabilize the error states. It is widely known that the Lyapunov functions are effective tools to describe the dynamics of multi-agent systems [19–23]. If the multiple Lyapunov functions are constructed to describe the error dynamics of unstable multi-agent systems under switching disconnected topologies, they will probably divergent within the switching intervals. Thus, we consider utilizing the “decline” characteristics at the transition instants to offset the divergence within the switching intervals. Suppose that Vσtt are multiple nonnegative functions under switching law σt:1,2,…,l. Then, we propose the following useful lemma first.Lemma 1.
Consider the multiple nonnegative functionsVσtt for σt:1,2,…,l. If there exist constants 0<μ<1 and α>0 such that(8)Vqt<eαt−TpVqTp,∀t∈Tp,Tp+1,(9)VrTp+≤μVqTp−,r≠q,(10)lnμ+ατmax<0,where τp=Tp+1−Tp, p=0,1,2,…, then global convergence of functions Vσtt can be reached.Proof: .
Define a Lyapunov function asVt=∑k=1zϵktVkt, where(11)ϵkt=1,ifσt=k,0,otherwise.
Assuming thatσt=q when t∈Tp,Tp+1, then according to (8), one has Vt<eαt−TpVTp,∀t∈Tp,Tp+1. If the functions switch from mode q to mode r when t=Tp+1, we can derive VTp+1+≤μVTp+1− from (9). Then, we can further obtain VTp+1≤μeαTp+1−TpVTp. Considering τp=Tp+1−Tp, condition (10) guarantees μeαTp+1−Tp<1. Then, one has(12)Vt<eαt−TpVTp+<μeαt−TpVTp−<μeαt−Tp−1VTp−1+<⋯<μpeαt−T0VT0.
Becausep≥t−T0/τmax, one further has(13)Vt<eϵtVT0,t∈Tp,Tp+1.where ϵ=plnμ+ατp. From (10), we have ϵ<0. Therefore, the global convergence of Vσtt is reached. The proof is completed.
From Lemma1, it can be concluded that despite the divergence of all the Lyapunov functions, we can utilize the stabilization characteristics at switching instants to guarantee the global convergence. In order to derive the stabilization properties at switching instants, the elementary-unit-based dwell time (EUBDT) approach is proposed in this study. The EUBDT approach can be divided into three steps. The first step is to divide the switching interval into a certain number of elementary time segments. The second step is to analyze the local variation of error state within each elementary time segment. The last step is to derive the stabilization properties at switching instants to offset the divergence within the switching intervals. Considering that the dwell time of the switching interval Tp,Tp+1 is uncertain, we divide Tp,Tp+1 into the certain interval Tp,Tp+τmin and uncertain interval Tp+τmin,Tp+1. It is obvious that the communication topology can only be switched within the uncertain interval. Assume that the certain interval Tp,Tp+τmin consists of M elementary time segments and each elementary time segment is denoted as tp,l,tp,l+1, where l∈1,2,…,M−1. The length of each elementary time segment is denoted as h, which satisfies h=τmin/M. Under the EUBDT method, the corresponding piecewise time-varying Lyapunov functions (PTVLFs) is constructed as(14)Vrt=eTtIN⊗Prtet,t∈Tp,Tp+1,where Prt is the piecewise time-varying matrix, which is described as follows.
For the certain intervalTp,l,Tp,l+1 with l∈1,2,…,M−1 and M∈N∗, Prt is given by(15)Prt=1−ρltPr,l+ρltPr,l+1,t∈tp,l,tp,l+1,where Pr,l>0 and ρlt=t−tp,l/h. For the uncertain interval Tp+τmin,Tp+1, Prt is expressed as(16)Prt=Pr,M.Remark 2.
The main advantage of the piecewise time-varying Lyapunov function in (14) is that it can be utilized to derive the “decline” properties at switching instants by adjusting the time-varying Lyapunov matrix Prt. Such “decline” properties can be utilized to offset the divergence made by disconnected topology and unstable systems. If the classic time-invariant Lyapunov functions in [20–24] are utilized to describe the states, the Lyapunov matrix will be fixed and the “decline” properties will be hard to derive.


### 3.2. The Consensus Conditions and the Controller Synthesis

The previous subsection has developed the EUBDT approach. Considering that the error states are always divergent within the switching intervals due to the coexistence of disconnected topology and unstable multi-agent systems, we need to utilize the cooperative control to guarantee that the exponential divergence rate within a threshold. Then, the following lemma discussing the exponential divergence rate of error system (7) is obtained.Lemma 2.
For given constantsα>0 and h>0, if there exist matrices Pr,l>0,l=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(17)Σ1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirKTBTPr,l+1−λirPr,l+1BK−αPr,l+1<0.(18)Σ2=Pr,l+1−Pr,lh+ATPr,l+Pr,lA−λirKTBTPr,l−λirPr,lBK−αPr,l<0.(19)Σ3=ATPr,M+Pr,MA−λirKTBTPr,M−λirPr,MBK−αPr,M<0.where λir,i=1,2,…,N is the ith positive eigenvalue of ℋr. Then, the exponential divergence rate of error system (7) can be limited within α for the switching interval t∈Tp,Tp+1.Proof: .
First of all, we will prove the exponential divergence rate of system (7) is limited within α for the certain interval t∈Tp,Tp+τmin. Assume σt=r for t∈Tp,Tp+τmin. Then, calculating V˙rt for t∈tp,l,tp,l+1,l=1,2,…,M−1, we have(20)V˙rt−αVrt=eTtIN⊗P.rtet+e˙TtIN⊗Prtet+eTtIN⊗Prte˙t−αeTtIN⊗Prtet.
According to (15), one has(21)P.rt=Pr,l+1−Pr,lh.
Substituting (21) into (20), one obtains(22)V˙rt−αVrt=eTtIN⊗Pr,l+1−Pr,lhet+IN⊗Aet−ℋr⊗BKetTIN⊗Prtet+eTtIN⊗PrtIN⊗Aet−ℋr⊗BKet−αeTtIN⊗Prtet=eTtIN⊗Pr,l+1−Pr,lhet+IN⊗Aet−ℋr⊗BKetT×IN⊗Pr,l+t−tp,lhPr,l+1−Pr,let+eTtIN⊗Pr,l+t−tp,lhPr,l+1−Pr,l×IN⊗A−ℋr⊗BKet.−αeTtIN⊗Pr,l+t−tp,lhPr,l+1−Pr,let
Considering thatet=e1Tt,e2Tt,…,eNTtT, we can further obtain(23)V˙rt−αVrt≤∑i=1NeiTtPr,l+1−Pr,lheit+Aeit−λirBKeitT×Pr,l+t−tp,lhPr,l+1−Pr,leit+eiTtt−tp,lhPr,l+1−Pr,lA−λirBKeit−αeiTtPr,l+t−tp,lhPr,l+1−Pr,leit.
Due toρlt=t−tp,l/h, one obtains(24)V˙rt−αVrt≤∑i=1NeiTtρltΣ1+1−ρltΣ2eit.
Then, conditions (17) and (18) guarantee V˙rt−αVrt<0,∀t∈tp,l,tp,l+1,l=1,2,…,M−1, which means that Vrt<eαt−tp,lVrtp,l,∀t∈tp,l,tp,l+1,l=1,2,…,M−1. Thus, we can further get(25)V˙rt<eαt−tp,lVrtp,l<eαt−tp,l−1Vrtp,l−1<…<eαt−TpVrTp.
Next, we will prove that the exponential divergence of system (7) is limited within α for the uncertain interval t∈Tp+τmin,Tp+1 when σt=r. Calculating V˙rt for t∈Tp+τmin,Tp+1, it yields(26)V˙rt−αVrt=e˙TtIN⊗Pr,Met+eTtIN⊗Pr,Me˙t−αeTtIN⊗Pr,Met=IN⊗Aet−ℋr⊗BKetTIN⊗Pr,Met+eTtIN⊗Pr,MIN⊗Aet−ℋr⊗BKet−αeTtIN⊗Pr,Met.
From the similar guideline of (23), it yields(27)V˙rt−αVrt≤∑i=1NAeit−λirBKeitTPr,Meit+eiTtPr,MA−λirBKeit−αeiTtPr,Meit=∑i=1NeiTtΣ3eit.
Then, condition (19) guarantees V˙rt−αVrt<0,∀t∈Tp+τmin,Tp+1, which means that Vrt<eαt−Tp−τminVrTp+τmin,∀t∈Tp+τmin,Tp+1.
To sum up, conditions (17)–(19) guarantee Vrt<eαt−TpVrTp,∀t∈Tp,Tp+1, which implies that the exponential divergence rate of error system (7) is limited within α for the switching interval t∈Tp,Tp+1. The proof is completed.
Then, based on Lemma1 and Lemma 2, the sufficient conditions for the stability of error system (7) can be obtained in Theorem 1.Theorem 1.
Suppose that Assumption1 is satisfied. For given constants α>0, 1>μ>0, and h>0, if there exist matrices Pr,l>0,l=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(28)Σ1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirKTBTPr,l+1−λirPr,l+1BK−αPr,l+1<0,(29)Σ2=ATPr,l+Pr,lA+Pr,l+1−Pr,lh−λirKTBTPr,l−λirPr,lBK−αPr,l<0,(30)Σ3=ATPr,M+Pr,MA−λirKTBTPr,M−λirPr,MBK−αPr,M<0,(31)Pj,0−μPr,M≤0,(32)lnμ+ατmax<0,where τmax denotes the maximal dwell time. Then, the stability of error system (7) can be guaranteed under the switching law σt∈Dτmin,τmax.Proof: .
From Lemma2, we can get that conditions (28)–(30) guarantee VσTpt<eαt−TpVσTpTp,∀t∈Tp,Tp+1. According to (31), we have VσTqTq+≤μVσTpTq−, which implies that VσTp+1Tp+1≤μeαTp+1−TpVσTpTp. Then, from Lemma 1, one has(33)VσTpt<eplnμ+ατptVσ00<eϵ¯tVσ00,t∈Tp,Tp+1.where ε¯=plnμ+ατmax. According to (32), one has ϵ¯<0. Therefore, the stability of error system (7) is guaranteed. The proof is completed.Remark 3.
In Theorem1, by selecting appropriate values of M and h, the corresponding maximal dwell time τmax can be computed by(34)τmax∗=maxτmax:28−32hold.Remark 4.
In this study, due to the coexistence of unstable systems and disconnected topologies, some agents inevitably diverge from each other though there exist the control protocols on them. Thus, it is hard to guarantee the exponential convergence for error system within switching intervals and we inevitably leave the error system divergent within the switching intervals. Therefore, we chooseα>0 to denote the exponential divergence rate of error states within the switching intervals.Remark 5.
Reference [28] has studied the stabilization problem of switching unstable systems. It derives the stabilization properties of the switching behaviors to offset the divergence property across the switching intervals and further achieve the overall stability. Drawing lessons from this, the EUBDT approach is proposed in this study to tackle the consensus problem of unstable multi-agent systems under switching disconnected topologies. The main idea is to utilize the stabilization properties at switching instants to offset the divergence of error states within the switching intervals. By analyzing the local variation of error states within each elementary time segment tp,l,tp,l+1, the stabilization properties at switching instants can be derived in condition (31). Then, condition (32) guarantees that the “decline” properties at switching instants are larger than the divergence of error states within the switching intervals. Based on this, the global stability of the error system can be guaranteed.
Next, we focus on the controller synthesis of error system (7) under switching disconnected topologies. A time-scheduled controller is adopted here, which is easy to be calculated. Instead of the controller (5), we will rather consider the following time-scheduled controller:(35)ui,σtt=K¯σtt∑j=1Naijσtxjt−xit+biσtx0t−xit,where K¯σtt is the time-scheduled gain to be determined. Substituting this controller into error system (7), it yields(36)e˙t=IN⊗Aet−ℋr⊗BK¯rtet,t∈Tp,Tp+1.
Then, the consensus of multi-agent systems (2) and (3) is equivalent to the stability of error system (36). Then, the following theorem can be obtained.Theorem 2.
Suppose that Assumption1 is satisfied. For given constants M∈N∗, α>0, 1>μ>0, h>0, if there exist matrices Pr,l>0,χr,ll=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(37)Σ¯1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirBTχr,l+1−λirχr,l+1TB−αPr,l+1<0,(38)Σ¯2=ATPr,l+Pr,lA+Pr,l+1−Pr,lh−λirBTχr,l−λirχr,lTB−αPr,l<0,(39)Σ¯3=ATPr,M+Pr,MA−λirχr,MTB−λirBTχr,M−αPr,M<0,(40)Pj,0−μPr,M≤0,(41)lnμ+ατmax<0,where τmax denotes the maximal dwell time. Then, consensus of multi-agent systems (2) and (3) can be achieved under the switching law σt∈Dτmin,τmax and controller gain K¯rt=χrtPr−1t, where χrt is given by(42)χrt=Λr,lt,t∈tp,l,tp,l+1,χr,M,t∈Tp+τmin,Tp+1,where Λr,lt=ρltχr,l+1+1−ρltχr,l, ρlt=t−tp,l/h and l=1,2,…,M−1.Proof: .
First of all, we consider error system (36) for t∈Tp,Tp+1+τmin. Calculating V˙rt for t∈tp,l,tp,l+1,l=1,2,…,M−1, one obtains(43)V˙rt−αVrt=eTtIN⊗P.rtet+2eTtIN⊗A−ℋr⊗Bχrtet−αeTtIN⊗Pr,ltet.
Taking the time-varying parameter (42) into (43), it yields(44)V˙r,kt−αVr,kt=eTtIN⊗Pr,l+1−Pr,lhet+2eTtIN⊗A−ℋr⊗Bχr,l+ρltχr,l+1−χr,let−αeTtIN⊗Pr,l+ρltPr,l+1−Pr,let.
From similar guideline of (23), one obtains(45)V˙rt−αVrt≤∑i=1NeiTt1−ρltPr,l+1h−Pr,lh+ATPr,l+Pr,lA−λirBTχr,l−λirχr,lTB−αPr,leit+eiTtρltPr,l+1h−Pr,lh+ATPr,l+1+Pr,l+1A−λirBTχr,l+1−λirχr,l+1TB−αPr,l+1eit=∑i=1NeiTt1−ρltΣ¯1+ρltΣ¯2eit.
Then, conditions (37) and (38) can guarantee V˙rt−αVrt<0 for t∈tp,l,tp,l+1,l=1,2,…,M−1. It means that Vrt<eαt−TpVrTp for t∈Tp,Tp+τmin
Next, we will prove the exponential divergence of system (36) for t∈Tp+τmin,Tp+1. Calculating V˙rt for t∈Tp+τmin,Tp+1, one has(46)V˙rt−αVrt≤∑i=1NeiTtATPr,l+1+Pr,l+1A−λirχr,l+1TB−λirBTχr,l+1−αPr,l+1eit=∑i=1NeiTtΣ¯3eit.
Then, condition (39) guarantees V˙rt−αVrt<0 for t∈Tp+τmin,Tp+1.
To sum up, conditions (37)–(39) guarantee Vrt<eαt−TpVrTp for t∈Tp,Tp+1. Then, by the similar guideline of (33), the stability of system (36) is reached from conditions (40) and (41). This also implies that the consensus of systems (2) and (3) is guaranteed. The proof is completed.


## 3.1. The Elementary-Unit-Based Dwell Time Approach

Most results on the consensus control problem under switching disconnected topologies are concentrated on the stable or critical stable multi-agent systems [21–24]. In this situation, the connected agents will close to each other and the disconnected agents will not disperse. If we consider unstable multi-agent systems such as distributed voltage control of microgrid and multi-link manipulators driven by DC motor, the disconnected agents will disperse under the disconnected topology even if there exist the control protocol on them. Thus, how to stabilize the error states under switching disconnected topologies is challenging. In order to break through this challenge, one has to utilize the stabilization properties of switching behaviors to stabilize the error states. It is widely known that the Lyapunov functions are effective tools to describe the dynamics of multi-agent systems [19–23]. If the multiple Lyapunov functions are constructed to describe the error dynamics of unstable multi-agent systems under switching disconnected topologies, they will probably divergent within the switching intervals. Thus, we consider utilizing the “decline” characteristics at the transition instants to offset the divergence within the switching intervals. Suppose that Vσtt are multiple nonnegative functions under switching law σt:1,2,…,l. Then, we propose the following useful lemma first.Lemma 1.
Consider the multiple nonnegative functionsVσtt for σt:1,2,…,l. If there exist constants 0<μ<1 and α>0 such that(8)Vqt<eαt−TpVqTp,∀t∈Tp,Tp+1,(9)VrTp+≤μVqTp−,r≠q,(10)lnμ+ατmax<0,where τp=Tp+1−Tp, p=0,1,2,…, then global convergence of functions Vσtt can be reached.Proof: .
Define a Lyapunov function asVt=∑k=1zϵktVkt, where(11)ϵkt=1,ifσt=k,0,otherwise.
Assuming thatσt=q when t∈Tp,Tp+1, then according to (8), one has Vt<eαt−TpVTp,∀t∈Tp,Tp+1. If the functions switch from mode q to mode r when t=Tp+1, we can derive VTp+1+≤μVTp+1− from (9). Then, we can further obtain VTp+1≤μeαTp+1−TpVTp. Considering τp=Tp+1−Tp, condition (10) guarantees μeαTp+1−Tp<1. Then, one has(12)Vt<eαt−TpVTp+<μeαt−TpVTp−<μeαt−Tp−1VTp−1+<⋯<μpeαt−T0VT0.
Becausep≥t−T0/τmax, one further has(13)Vt<eϵtVT0,t∈Tp,Tp+1.where ϵ=plnμ+ατp. From (10), we have ϵ<0. Therefore, the global convergence of Vσtt is reached. The proof is completed.
From Lemma1, it can be concluded that despite the divergence of all the Lyapunov functions, we can utilize the stabilization characteristics at switching instants to guarantee the global convergence. In order to derive the stabilization properties at switching instants, the elementary-unit-based dwell time (EUBDT) approach is proposed in this study. The EUBDT approach can be divided into three steps. The first step is to divide the switching interval into a certain number of elementary time segments. The second step is to analyze the local variation of error state within each elementary time segment. The last step is to derive the stabilization properties at switching instants to offset the divergence within the switching intervals. Considering that the dwell time of the switching interval Tp,Tp+1 is uncertain, we divide Tp,Tp+1 into the certain interval Tp,Tp+τmin and uncertain interval Tp+τmin,Tp+1. It is obvious that the communication topology can only be switched within the uncertain interval. Assume that the certain interval Tp,Tp+τmin consists of M elementary time segments and each elementary time segment is denoted as tp,l,tp,l+1, where l∈1,2,…,M−1. The length of each elementary time segment is denoted as h, which satisfies h=τmin/M. Under the EUBDT method, the corresponding piecewise time-varying Lyapunov functions (PTVLFs) is constructed as(14)Vrt=eTtIN⊗Prtet,t∈Tp,Tp+1,where Prt is the piecewise time-varying matrix, which is described as follows.
For the certain intervalTp,l,Tp,l+1 with l∈1,2,…,M−1 and M∈N∗, Prt is given by(15)Prt=1−ρltPr,l+ρltPr,l+1,t∈tp,l,tp,l+1,where Pr,l>0 and ρlt=t−tp,l/h. For the uncertain interval Tp+τmin,Tp+1, Prt is expressed as(16)Prt=Pr,M.Remark 2.
The main advantage of the piecewise time-varying Lyapunov function in (14) is that it can be utilized to derive the “decline” properties at switching instants by adjusting the time-varying Lyapunov matrix Prt. Such “decline” properties can be utilized to offset the divergence made by disconnected topology and unstable systems. If the classic time-invariant Lyapunov functions in [20–24] are utilized to describe the states, the Lyapunov matrix will be fixed and the “decline” properties will be hard to derive.


## 3.2. The Consensus Conditions and the Controller Synthesis

The previous subsection has developed the EUBDT approach. Considering that the error states are always divergent within the switching intervals due to the coexistence of disconnected topology and unstable multi-agent systems, we need to utilize the cooperative control to guarantee that the exponential divergence rate within a threshold. Then, the following lemma discussing the exponential divergence rate of error system (7) is obtained.Lemma 2.
For given constantsα>0 and h>0, if there exist matrices Pr,l>0,l=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(17)Σ1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirKTBTPr,l+1−λirPr,l+1BK−αPr,l+1<0.(18)Σ2=Pr,l+1−Pr,lh+ATPr,l+Pr,lA−λirKTBTPr,l−λirPr,lBK−αPr,l<0.(19)Σ3=ATPr,M+Pr,MA−λirKTBTPr,M−λirPr,MBK−αPr,M<0.where λir,i=1,2,…,N is the ith positive eigenvalue of ℋr. Then, the exponential divergence rate of error system (7) can be limited within α for the switching interval t∈Tp,Tp+1.Proof: .
First of all, we will prove the exponential divergence rate of system (7) is limited within α for the certain interval t∈Tp,Tp+τmin. Assume σt=r for t∈Tp,Tp+τmin. Then, calculating V˙rt for t∈tp,l,tp,l+1,l=1,2,…,M−1, we have(20)V˙rt−αVrt=eTtIN⊗P.rtet+e˙TtIN⊗Prtet+eTtIN⊗Prte˙t−αeTtIN⊗Prtet.
According to (15), one has(21)P.rt=Pr,l+1−Pr,lh.
Substituting (21) into (20), one obtains(22)V˙rt−αVrt=eTtIN⊗Pr,l+1−Pr,lhet+IN⊗Aet−ℋr⊗BKetTIN⊗Prtet+eTtIN⊗PrtIN⊗Aet−ℋr⊗BKet−αeTtIN⊗Prtet=eTtIN⊗Pr,l+1−Pr,lhet+IN⊗Aet−ℋr⊗BKetT×IN⊗Pr,l+t−tp,lhPr,l+1−Pr,let+eTtIN⊗Pr,l+t−tp,lhPr,l+1−Pr,l×IN⊗A−ℋr⊗BKet.−αeTtIN⊗Pr,l+t−tp,lhPr,l+1−Pr,let
Considering thatet=e1Tt,e2Tt,…,eNTtT, we can further obtain(23)V˙rt−αVrt≤∑i=1NeiTtPr,l+1−Pr,lheit+Aeit−λirBKeitT×Pr,l+t−tp,lhPr,l+1−Pr,leit+eiTtt−tp,lhPr,l+1−Pr,lA−λirBKeit−αeiTtPr,l+t−tp,lhPr,l+1−Pr,leit.
Due toρlt=t−tp,l/h, one obtains(24)V˙rt−αVrt≤∑i=1NeiTtρltΣ1+1−ρltΣ2eit.
Then, conditions (17) and (18) guarantee V˙rt−αVrt<0,∀t∈tp,l,tp,l+1,l=1,2,…,M−1, which means that Vrt<eαt−tp,lVrtp,l,∀t∈tp,l,tp,l+1,l=1,2,…,M−1. Thus, we can further get(25)V˙rt<eαt−tp,lVrtp,l<eαt−tp,l−1Vrtp,l−1<…<eαt−TpVrTp.
Next, we will prove that the exponential divergence of system (7) is limited within α for the uncertain interval t∈Tp+τmin,Tp+1 when σt=r. Calculating V˙rt for t∈Tp+τmin,Tp+1, it yields(26)V˙rt−αVrt=e˙TtIN⊗Pr,Met+eTtIN⊗Pr,Me˙t−αeTtIN⊗Pr,Met=IN⊗Aet−ℋr⊗BKetTIN⊗Pr,Met+eTtIN⊗Pr,MIN⊗Aet−ℋr⊗BKet−αeTtIN⊗Pr,Met.
From the similar guideline of (23), it yields(27)V˙rt−αVrt≤∑i=1NAeit−λirBKeitTPr,Meit+eiTtPr,MA−λirBKeit−αeiTtPr,Meit=∑i=1NeiTtΣ3eit.
Then, condition (19) guarantees V˙rt−αVrt<0,∀t∈Tp+τmin,Tp+1, which means that Vrt<eαt−Tp−τminVrTp+τmin,∀t∈Tp+τmin,Tp+1.
To sum up, conditions (17)–(19) guarantee Vrt<eαt−TpVrTp,∀t∈Tp,Tp+1, which implies that the exponential divergence rate of error system (7) is limited within α for the switching interval t∈Tp,Tp+1. The proof is completed.
Then, based on Lemma1 and Lemma 2, the sufficient conditions for the stability of error system (7) can be obtained in Theorem 1.Theorem 1.
Suppose that Assumption1 is satisfied. For given constants α>0, 1>μ>0, and h>0, if there exist matrices Pr,l>0,l=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(28)Σ1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirKTBTPr,l+1−λirPr,l+1BK−αPr,l+1<0,(29)Σ2=ATPr,l+Pr,lA+Pr,l+1−Pr,lh−λirKTBTPr,l−λirPr,lBK−αPr,l<0,(30)Σ3=ATPr,M+Pr,MA−λirKTBTPr,M−λirPr,MBK−αPr,M<0,(31)Pj,0−μPr,M≤0,(32)lnμ+ατmax<0,where τmax denotes the maximal dwell time. Then, the stability of error system (7) can be guaranteed under the switching law σt∈Dτmin,τmax.Proof: .
From Lemma2, we can get that conditions (28)–(30) guarantee VσTpt<eαt−TpVσTpTp,∀t∈Tp,Tp+1. According to (31), we have VσTqTq+≤μVσTpTq−, which implies that VσTp+1Tp+1≤μeαTp+1−TpVσTpTp. Then, from Lemma 1, one has(33)VσTpt<eplnμ+ατptVσ00<eϵ¯tVσ00,t∈Tp,Tp+1.where ε¯=plnμ+ατmax. According to (32), one has ϵ¯<0. Therefore, the stability of error system (7) is guaranteed. The proof is completed.Remark 3.
In Theorem1, by selecting appropriate values of M and h, the corresponding maximal dwell time τmax can be computed by(34)τmax∗=maxτmax:28−32hold.Remark 4.
In this study, due to the coexistence of unstable systems and disconnected topologies, some agents inevitably diverge from each other though there exist the control protocols on them. Thus, it is hard to guarantee the exponential convergence for error system within switching intervals and we inevitably leave the error system divergent within the switching intervals. Therefore, we chooseα>0 to denote the exponential divergence rate of error states within the switching intervals.Remark 5.
Reference [28] has studied the stabilization problem of switching unstable systems. It derives the stabilization properties of the switching behaviors to offset the divergence property across the switching intervals and further achieve the overall stability. Drawing lessons from this, the EUBDT approach is proposed in this study to tackle the consensus problem of unstable multi-agent systems under switching disconnected topologies. The main idea is to utilize the stabilization properties at switching instants to offset the divergence of error states within the switching intervals. By analyzing the local variation of error states within each elementary time segment tp,l,tp,l+1, the stabilization properties at switching instants can be derived in condition (31). Then, condition (32) guarantees that the “decline” properties at switching instants are larger than the divergence of error states within the switching intervals. Based on this, the global stability of the error system can be guaranteed.
Next, we focus on the controller synthesis of error system (7) under switching disconnected topologies. A time-scheduled controller is adopted here, which is easy to be calculated. Instead of the controller (5), we will rather consider the following time-scheduled controller:(35)ui,σtt=K¯σtt∑j=1Naijσtxjt−xit+biσtx0t−xit,where K¯σtt is the time-scheduled gain to be determined. Substituting this controller into error system (7), it yields(36)e˙t=IN⊗Aet−ℋr⊗BK¯rtet,t∈Tp,Tp+1.
Then, the consensus of multi-agent systems (2) and (3) is equivalent to the stability of error system (36). Then, the following theorem can be obtained.Theorem 2.
Suppose that Assumption1 is satisfied. For given constants M∈N∗, α>0, 1>μ>0, h>0, if there exist matrices Pr,l>0,χr,ll=0,1,…,M−1 such that for any r=1,2,…,m, the following conditions hold:(37)Σ¯1=ATPr,l+1+Pr,l+1A+Pr,l+1−Pr,lh−λirBTχr,l+1−λirχr,l+1TB−αPr,l+1<0,(38)Σ¯2=ATPr,l+Pr,lA+Pr,l+1−Pr,lh−λirBTχr,l−λirχr,lTB−αPr,l<0,(39)Σ¯3=ATPr,M+Pr,MA−λirχr,MTB−λirBTχr,M−αPr,M<0,(40)Pj,0−μPr,M≤0,(41)lnμ+ατmax<0,where τmax denotes the maximal dwell time. Then, consensus of multi-agent systems (2) and (3) can be achieved under the switching law σt∈Dτmin,τmax and controller gain K¯rt=χrtPr−1t, where χrt is given by(42)χrt=Λr,lt,t∈tp,l,tp,l+1,χr,M,t∈Tp+τmin,Tp+1,where Λr,lt=ρltχr,l+1+1−ρltχr,l, ρlt=t−tp,l/h and l=1,2,…,M−1.Proof: .
First of all, we consider error system (36) for t∈Tp,Tp+1+τmin. Calculating V˙rt for t∈tp,l,tp,l+1,l=1,2,…,M−1, one obtains(43)V˙rt−αVrt=eTtIN⊗P.rtet+2eTtIN⊗A−ℋr⊗Bχrtet−αeTtIN⊗Pr,ltet.
Taking the time-varying parameter (42) into (43), it yields(44)V˙r,kt−αVr,kt=eTtIN⊗Pr,l+1−Pr,lhet+2eTtIN⊗A−ℋr⊗Bχr,l+ρltχr,l+1−χr,let−αeTtIN⊗Pr,l+ρltPr,l+1−Pr,let.
From similar guideline of (23), one obtains(45)V˙rt−αVrt≤∑i=1NeiTt1−ρltPr,l+1h−Pr,lh+ATPr,l+Pr,lA−λirBTχr,l−λirχr,lTB−αPr,leit+eiTtρltPr,l+1h−Pr,lh+ATPr,l+1+Pr,l+1A−λirBTχr,l+1−λirχr,l+1TB−αPr,l+1eit=∑i=1NeiTt1−ρltΣ¯1+ρltΣ¯2eit.
Then, conditions (37) and (38) can guarantee V˙rt−αVrt<0 for t∈tp,l,tp,l+1,l=1,2,…,M−1. It means that Vrt<eαt−TpVrTp for t∈Tp,Tp+τmin
Next, we will prove the exponential divergence of system (36) for t∈Tp+τmin,Tp+1. Calculating V˙rt for t∈Tp+τmin,Tp+1, one has(46)V˙rt−αVrt≤∑i=1NeiTtATPr,l+1+Pr,l+1A−λirχr,l+1TB−λirBTχr,l+1−αPr,l+1eit=∑i=1NeiTtΣ¯3eit.
Then, condition (39) guarantees V˙rt−αVrt<0 for t∈Tp+τmin,Tp+1.
To sum up, conditions (37)–(39) guarantee Vrt<eαt−TpVrTp for t∈Tp,Tp+1. Then, by the similar guideline of (33), the stability of system (36) is reached from conditions (40) and (41). This also implies that the consensus of systems (2) and (3) is guaranteed. The proof is completed.


## 4. Simulation Examples

In this section, the theoretical approach is illustrated by a simulation example.Consider the multi-agent system described in (2), where the system matrices are given as(47)A=0−110.1,B=10.50.51.The eigenvalues ofA are 0.05+0.9987i and 0.05−0.9987i. It can be observed that the eigenvalues of A contain positive real parts, which implies that the multi-agent systems are unstable. The switching graphs G1 and G2 are shown in Figures 2 and 3 respectively.Figure 2
The graphG1.Figure 3
The graphG2.It is obvious thatG1 and G2 are disconnected and the union of G1 and G2 has a directed spanning tree with the leader as the root, which satisfies Assumption 1. The objective is to design controller (35) and constrain the dwell time τmin,τmax for the switching topologies such that the consensus of systems (2) and (3) can be achieved. The initial states are given as x00=52T, x10=−2−2T, x20=−25T, x30=−1−5T,x40=05T. Then, if we fixed μ=0.6, M=3, h=0.5, and α=0.4, by calculating conditions (37)–(41), the controller gain K¯rt=χrtPr−1t can be obtained as follows:(48)P1,0=2.36040.02040.02042.3583,P1,1=13.03170.05380.053813.0521,P1,2=23.99580.08720.087224.0237,P1,3=35.48340.12260.122635.5032,P2,0=15.81890.06140.061415.8264,P2,1=14.24450.06380.063814.2436,P2,2=12.39070.06060.060612.3900,P2,3=10.17040.05440.054410.1714,χ1,0=8.8749−4.4233−4.42338.8866,χ1,1=9.3661−5.0156−5.01569.9033,χ1,2=11.3905−5.9022−5.902211.8102,χ1,3=14.0194−6.8220−6.822013.4112,χ2,0=13.1102−5.7990−5.799012.5510,χ2,1=12.9337−5.0140−5.014011.8834,χ2,2=11.5902−4.9334−4.933410.9590,χ2,3=9.4147−4.2010−4.20109.1437.Moreover, the maximum dwell time can be calculated asτmax=2.24s. The minimum dwell time can be derived as τmin=Mh=1.5s. Based on this, the switching signal is shown in Figure 4, where σt=1 denotes graph G1 and σt=2 denotes graph G2.Figure 4
The switching signal for switching topologies.By imposing the obtained controller and the switching signal on the system, the state trajectories are obtained in Figures5 and 6.Figure 5
The statexi1 of the multi-agent systems.Figure 6
The statexi2 of the multi-agent systems.In Figures5 and 6, xi1t and xi2t denote the first and second state trajectories of the agent i, respectively. It is obvious that the leader-following consensus is achieved. Furthermore, we define the tracking error for agent i as ei1t=xi1t−x01t and ei2t=xi2t−x02t for i=1,2,3,4. Then, ei1t and ei2t are depicted in Figures 7 and 8 respectively.Figure 7
The state tracking errorei1.Figure 8
The state tracking errorei2.From Figures7 and 8, it is obvious that the tracking errors decay to 0, which means that the consensus of the unstable multi-agent systems can be achieved under the switching disconnected topologies.


## 5. Conclusion

In this study, the consensus control problem of unstable multi-agent systems under switching disconnected topologies was investigated. Drawing lessons from the theory of switching unstable systems, a novel elementary-unit-based dwell time (EUBDT) approach was proposed to divide the switching intervals into a certain number of elementary time segments. Then, by analyzing the local variation of error state within each elementary time segment, the divergence properties made by the coexistence of disconnected topology and unstable systems were overcome. Moreover, the consensus was proven by the utilization of a novel function named PTVLF. Lastly, the proposed results were illustrated by a simulation example. Future works conclude reducing the computation burden of the main theorem.



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