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

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.


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][2][3][4][5][6]. Because of the broad applications, leader-following consensus of multi-agent systems plays an important role in the automatic control eld [7,8]. Accordingly, a lot of e orts have been put into their analysis for leader-following consensus problem in recent years [9][10][11].
In the eld of multi-agent systems, the communication topology may not be xed 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][14][15][16][17][18][19][20]. In the early works, the e orts mainly investigated the switching topologies where all the topologies are connected [13][14][15][16][17]. In recent years, some works have investigated the consensus problem under switching topologies in which the topology is frequently connected [18][19][20]. In these works, the error state decreases when the topology is connected and increases when the topology is disconnected. e 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 nd 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][22][23][24]. In [21], the su cient conditions for the leader-following consensus control problem under switching disconnected topologies are proposed. en, 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) multiagent 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. e 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], multilink 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. is 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]. e 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. en, the stabilization properties of switching behaviors can be obtained to offset the divergence of error states within the switching intervals. en, 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 Figure 1, 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.
Notation: R m refers to the set of m-vectors. N * and N refer to the set of positive integers and natural number, respectively. e notation Q < 0 represents that Q is negative definite. I N refers to the N × N identity matrix. ⊗ refers to the Kronecker product.

Preliminaries
e multi-agent systems are expressed by a directed graph N×N is an adjacency matrix. An edge e ij � (v i , v j ) ∈ E represents that an arrow from i to j in the graph, which implies that agent j can acquire information from agent i. If e ij ∈ G, i ≠ j, we will have a ij � 1, otherwise, a ij � 0. A directed path in G denotes a sequence of nodes v 1 , v 2 , . . . , v q such that (v i , v i+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. e Laplacian matrix of the graph is denoted as Consider a group of N agents, whose dynamics is expressed as where x i (t) ∈ R n represents the state of agent i, u i (t) ∈ R p represents the control input of agent i, and A and B are constant matrices. e dynamics of the leader is shown as where x 0 (t) ∈ R n represents the leader's state. e main objective is to guarantee the consensus of followers (2) and the leader (3), which can be illustrated as We use a diagonal matrix B � diag b 1 , b 2 , . . . , b N to denote the access between the agents and the leader, where B ∈ R N×N . If the agent i can receive the leader's information, it will be b i � 1, i � 1, 2, . . . , N; otherwise, b i � 0. For convenience, a matrix H � L + B is utilized to denote the information-exchange matrix.
Define a piecewise constant function σ(t): [1, 2, . . . , m) as the switching signal of switching topologies, which satisfies the switching sequence T p , p � 0, 1, . . . with T 0 � 0 and lim p⟶∞ T p � ∞. Moreover, denote τ p � T p+1 − T p as the dwell time of the pth topology. e dwell time τ p is constrained by minimum dwell time τ min and maximum dwell time τ max , which is denoted as τ min ≤ τ p ≤ τ max . It can be observed that the topology can only be switched within the interval t ∈ [T p + τ min , T p+1 ). For convenience, we use D [τ min ,τ max ] to denote the set of all the switching policies in the framework of dwell time τ p ∈ [τ min , τ max ].
erefore, G 1 , G 2 , . . . , G m , m ≥ 1 denotes the set of all the possible directed graphs under the switching graphs. e information-exchange matrices can be denoted as H 1 , H 2 , . . . , H m }, m ≥ 1. e communication topology under the switching signal σ(t) satisfies the following assumption.
e switching graphG σ(t) is not connected. e union of all the graphsG � G 1 ⋃ G 2 ⋃ . . . ⋃ G m has a directed spanning tree with the leader as the root, wherem ∈ N * .
Based on this, we consider the controller adopting the following form: 2 Discrete Dynamics in Nature and Society Define e i (t) � x i (t) − x 0 (t) as the tracking error for agent i. en, the error system of agent i under the switching signal σ(t) can be obtained as Define e(t) � [e T 1 (t), e T 2 (t), . . . , e T N (t)] T , then the error systems for σ(t) � r can be given by erefore, the control objective is to achieve the stability of error system (7) under switching laws σ(t). 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 matrix A 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. en, 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. us, 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.

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

e 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][22][23][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. us, 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][20][21][22][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.
us, we consider utilizing the "decline" characteristics at the transition instants to offset the divergence within the switching intervals. Suppose that V σ(t) (t) are multiple nonnegative functions under switching law σ(t): [1, 2, . . . , l). en, we propose the following useful lemma first.   Discrete Dynamics in Nature and Society

Lemma 1. Consider the multiple nonnegative functions
Assuming that σ(t) � q when t ∈ [T p , T p+1 ), then according to (8) Because p ≥ (t − T 0 )/τ max , one further has where ϵ � p(lnμ + ατ p ). From (10), we have ϵ < 0. erefore, the global convergence of V σ(t) (t) is reached. e proof is completed. From Lemma 1, 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 elementaryunit-based dwell time (EUBDT) approach is proposed in this study. e EUBDT approach can be divided into three steps. e first step is to divide the switching interval into a certain number of elementary time segments. e second step is to analyze the local variation of error state within each elementary time segment. e 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 [T p , T p+1 ) is uncertain, we divide [T p , T p+1 ) into the certain interval [T p , T p + τ min ) and uncertain interval [T p + τ min , T p+1 where P r (t) is the piecewise time-varying matrix, which is described as follows.
For the certain interval [T p,l , T p,l+1 ) with l ∈ 1, 2, . . . , { M − 1} and M ∈ N * , P r (t) is given by where P r,l > 0 and ρ l (t) � (t − t p,l )/h. For the uncertain interval [T p + τ min , T p+1 ), P r (t) is expressed as □ Remark 2. e main advantage of the piecewise timevarying 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 P r (t). 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][21][22][23][24] are utilized to describe the states, the Lyapunov matrix will be fixed and the "decline" properties will be hard to derive.

e Consensus Conditions and the Controller Synthesis.
e 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. en, the following lemma discussing the exponential divergence rate of error system (7) is obtained.
4 Discrete Dynamics in Nature and Society where λ r i , i � 1, 2, . . . , N is the ith positive eigenvalue of H r . en, the exponential divergence rate of error system (7) can be limited within α for the switching interval t ∈ [T p , T p+1 ).
Proof: First of all, we will prove the exponential divergence rate of system (7) is limited within α for the certain interval t ∈ [T p , T p + τ min ). Assume σ(t) � r for t ∈ [T p , T p + τ min ).
e proof is completed. en, based on Lemma 1 and Lemma 2, the sufficient conditions for the stability of error system (7) can be obtained in eorem 1.

□ Theorem 1. Suppose that Assumption 1 is satisfied. For given constants α
such that for any r � 1, 2, . . . , m, the following conditions hold: where τ max denotes the maximal dwell time. en, the stability of error system (7) can be guaranteed under the switching law σ(t) ∈ D [τ min ,τ max ] .
Proof: From Lemma 2, we can get that conditions (28) en, from Lemma 1, one has 6 Discrete Dynamics in Nature and Society where ε � p(ln μ + ατ max ). According to (32), one has ϵ < 0. erefore, the stability of error system (7) is guaranteed. e proof is completed. □ Remark 3. In eorem 1, by selecting appropriate values of M and h, the corresponding maximal dwell time τ max can be computed by 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. us, 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. erefore, we choose α > 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. e 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 [t p,l , t p,l+1 ), the stabilization properties at switching instants can be derived in condition (31). en, 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 timescheduled controller is adopted here, which is easy to be calculated. Instead of the controller (5), we will rather consider the following time-scheduled controller: where K σ(t) (t) is the time-scheduled gain to be determined. Substituting this controller into error system (7), it yields _ e(t) � I N ⊗ A e(t) − H r ⊗ BK r (t) e(t), t ∈ T p , T p+1 .
(36) en, the consensus of multi-agent systems (2) and (3) is equivalent to the stability of error system (36). en, the following theorem can be obtained.
Taking the time-varying parameter (42) into (43), it yields Discrete Dynamics in Nature and Society − αe T (t) I N ⊗ P r,l + ρ l (t) P r,l+1 − P r,l e(t).
To sum up, conditions (37)-(39) guarantee en, by the similar guideline of (33), the stability of system (36) is reached from conditions (40) and (41). is also implies that the consensus of systems (2) and (3) is guaranteed. e proof is completed.

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 e eigenvalues of A 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 multiagent systems are unstable. e switching graphs G 1 and G 2 are shown in Figures 2 and 3 respectively.
It is obvious that G 1 and G 2 are disconnected and the union of G 1 and G 2 has a directed spanning tree with the leader as the root, which satisfies Assumption 1. e objective is to design controller (35) and constrain the dwell time τ min , τ max for the switching topologies such that the consensus of systems (2) and (3) can be achieved. e initial states are given as T . en, if we fixed μ � 0.6, M � 3, h � 0.5, and α � 0.4, by calculating conditions (37)-(41), the controller gain K r (t) � χ r (t)P − 1 r (t) can be obtained as follows:    x 01 x 11 x 21 x 31 x 41 Moreover, the maximum dwell time can be calculated as τ max � 2.24s. e minimum dwell time can be derived as τ min � Mh � 1.5s. Based on this, the switching signal is shown in Figure 4, where σ(t) � 1 denotes graph G 1 and σ(t) � 2 denotes graph G 2 .
By imposing the obtained controller and the switching signal on the system, the state trajectories are obtained in Figures 5 and 6.
In Figures 5 and 6, x i1 (t) and x i2 (t) 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 e i1 (t) � x i1 (t) − x 01 (t) and e i2 (t) � x i2 (t) − x 02 (t) for i � 1, 2, 3, 4.
en, e i1 (t) and e i2 (t) are depicted in Figures 7 and 8 respectively. x 02 x 12 x 22 x 32 x 42 Figure 6: e state x i2 of the multi-agent systems.  From Figures 7 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.

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.
en, 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.

Data Availability
All of the parameters of the numerical simulation are included in the study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this study.