# Intermittent Time-Varying Formation Control for High-Order Networked Agents Subject to Discontinuous Communications

Authors: Lixin Wang; Zhe Luo; Xiaoqiang Li; Xinsan Li; Xiaogang Yang
Journal: Complexity
 (2021)

---

## Abstract

This paper investigates the leaderless and leader-follower time-varying formation design and analysis problems for a group of networked agents subject to discontinuous communications. Firstly, a leaderless time-varying formation control protocol is proposed via the intermittent control strategy, where the control input of each agent is constructed by the distributed local state information and formation instructions in the communication time unit, but it is zero in the noncommunication time unit. Then, an explicit formulation of the formation center function is determined to describe the formation movement trajectory of the whole networked agents. Leaderless time-varying formation design and analysis with discontinuous communications are given in the form of linear matrix inequalities. Moreover, the main results of the leaderless cases are extended to the leader-follower cases. Finally, two numerical examples are provided to illustrate the theoretical results of leaderless and leader-follower cases, respectively.


---

## Body

## 1. Introduction

Distributed cooperative control has received more attention from scholars in the last two decades, which can be applied in many circumstances, including flocking [1], consensus [2–6], formation control [7–10], distributed computation [11–13], and multisource data analysis [14, 15]. Distributed formation control indicates that a group of networked intelligent agents form the desired geometrical shape via the distributed control protocol, which is constructed by the local information among neighboring agents. It was shown in [16] that the consensus-based formation control is distributed and can be achieved by utilizing the consensus algorithms and tools. Recently, the consensus theory developed fast, and many interesting research results emerged, as shown in [17–22]. As a result, the distributed formation control has aroused many researches, which can be divided into time-invariant formation and time-varying formation according to the time-dependent characteristics of the formation shape.For the time-invariant formation, the relative position among agents remains unchanged after forming the formation structure, which means that the geometrical shape of the whole networked agents is time-invariant. From the graph theory perspective, the time-invariant formation control conditions were derived in the form of the Nyquist criterion in [23]. Jafarian et al. [24] investigated the time-invariant formation keeping problems for nonholonomic wheeled robots, where the disturbance rejection is achieved. Finite-time time-invariant formation control was achieved in [25], where a class of nonlinear control protocol was utilized. For the time-varying formation, the formation shape can be time-varying, as shown in [26, 27], which is more flexible than the time-invariant formation, and can be utilized in many practical applications. Dong et al. [28] provided a time-varying formation tracking scheme for second-order networked agents and applied it to the formation flying of a team of quadrotors. Wang et al. [29] proposed a robust time-varying formation control protocol with a distributed extend state observer, which can compensate the external disturbances actively. For a group of agents with multiple leaders, a leader-follower time-varying formation control method was shown in [30], where necessary and sufficient conditions with the formation tracking feasibility conditions were given. Generally speaking, the time derivative of the time-varying formation is not zero and cannot be ignored in formation design and analysis, so the time-varying cases are more challenging than the time-invariant ones.Due to the temporary interrupt of communication links, the sensing device failures, and the silent period of communications, the networked agents may suffer discontinuous communications. On the one hand, the communication topologies may be switched since the links of the network are changed. Some interesting works regarding switching topologies can be found in [31, 32], where it was shown that the communication topologies switch at some moments by the switching signal. On the other hand, the discontinuous communications are intermittent in the sequence of the time units; that is, the communication time units and the noncommunication time units appear alternately. Wang et al. [33] investigated the limited-budget consensus problems with intermittent interactions, where the consensus protocol was codesigned by the limited budget and the performance index and can guarantee the weighting optimization between the consensus performance and the energy consumption. Sun and Wang [34] proposed a new sampling-based time unit method to solve the consensus problems for nonlinear networked agents with intermittent interactions. The time-invariant formation control with intermittent interactions was studied in [35]. However, to the best of our knowledge, the time-varying formation control problems for networked agents with discontinuous communication in terms of both the switching topologies and intermittent interactions are still open.In this paper, we investigate the leaderless and leader-follower time-varying formation design and analysis for high-order networked homogeneous agents with discontinuous communications caused by switching topologies and intermittent interactions. Firstly, a new time-varying formation control protocol is proposed via the intermittent control strategy, which only adopts the local intermittent information and formation instructions among neighboring agent. Secondly, by the nonsingular transformation and the orthonormal transformation, the closed-loop dynamics of the whole network for both leaderless cases and leader-follower cases are decomposed into two subdynamics, which, respectively, describe the formation movement trajectory of the networked agents as a whole and the relative movement among agents. Thirdly, leaderless and leader-follower time-varying formation design and analysis criteria are derived under the condition of discontinuous communications, where the convergency of the Lyapunov function is analyzed in the communication time units and noncommunication time units, and it can be guaranteed by satisfying the formation feasibility condition, the discontinuous communication condition, and the linear matrix inequality condition, simultaneously.Compared with the related work regarding the time-varying formation control, the contribution of this paper is twofold. Firstly, different from the works in [26–30], this paper considers the discontinuous communications of both switching topologies and intermittent communications. In this case, the right-hand side of the closed-loop system is piecewise continuous. To solve this problem, a new intermittent time-varying formation control method is proposed. However, the analysis and design method in [26–30] cannot be adopted in this paper. Secondly, this paper determines the formation movement trajectory of the networked agents for both leaderless and leader-follower cases. For the leaderless case, the formation movement trajectory is determined by the formation center function, and it is determined by the zero-input response of the leader in the leader-follower case. Besides, it is shown that the intermittent communication and the switching topology do not affect the formation movement trajectory. In contrast, the authors in [26–30] did not determine the formation movement trajectory in the situation of the discontinuous communications.The main body of this paper is arranged as follows. The model of discontinuous communications and the dynamics of the agents are established in Section2. Leaderless time-varying formation design and analysis criteria with discontinuous communications are given in Section 3, where the explicit formulation of the formation center function is also determined. Section 4 extends the main results of the leaderless time-varying formation design and analysis to the leader-follower cases. Two numerical simulation examples are provided in Section 5, and Section 6 concludes the whole paper.Throughout this paper,ℜn×d stands for the n×d-dimensional real matrix space. ℵ is the set of natural numbers. N is utilized to denote the number of the agents, and 1N represents the N-dimensional column vector with all components 1. The number, vector, and matrix of zero value are collectively called as 0. QT=Q>0 means that matrix Q is symmetric and positive definite.


## 2. Problem Formulation and Preliminaries

### 2.1. Communication Constraint Modeling

In this paper, we consider the discontinuous communication among agents. On the one hand, the agent cannot communicate with each other in some noncommunication time units. On the other hand, the communication topologies of the networks are switched in some communication time units. To show the abovementioned discontinuous communication type from the time-domain perspective, it is supposed for∀s∈ℵ that there exists a nonoverlapping time unit sequence Ts,Ts+1=Ts,T˜s∪T˜s,Ts+1, where Ts=Ts1<Ts2<⋯<Tsrs=T˜s<T˜s+εs=Ts+1 with rs and εs being positive integers. Notice that Ts,T˜s and T˜s,Ts+1 represent the communication time unit and the noncommunication time unit, respectively. Ts1,Ts2,…,Tsrs… denotes the switching time over which the communication topologies are switched. Without loss of generality, the initial time is assumed to be T0=0. The length of time unit Ts,Ts+1 satisfies that 0<Tmin∗≤Ts∗=Ts+1−Ts≤Tmax∗. It should be noted that 0<Tdwell≤Tsrs−Tsrs−1≤T¯dwell, where Tdwell is the minimum dwell time. The noncommunication rate is defined as σs=Ts+1−T˜s/Ts+1−Ts, where 0<σs≤σmax<1 and σmax is called the maximum noncommunication rate. Note that the discontinuous communication is aperiodic since the length of each time unit Ts,T˜ss∈ℵ can be unequal.The switching topologies are modeled asG=G1,G2,…,Gk with the switching signal ϖt:0,+∞⟶1,2,…,k, where G is the digraph. For each digraph G, the vertex set is denoted by V=v1,v2,…,vH, and the edge set is represented by E=vl,vm:vl,vm∈V with the edge weight blmϖt. Note that if there exists an edge vl,vm,vl,vm∈V, from the vertex vm to vl, then the edge weight blmϖt>0. Otherwise, blmϖt=0. The neighboring set of the vertex vl is defined as Nlϖt=vm∈V:vm,vl∈E, and Lϖt=llmϖtH×H stands for the Laplacian matrix of the topologies with lllϖt=∑m∈Nlϖtblmϖt and llmϖt=−blmϖtl≠m. More details about the graph theory can be found in [36]. It should be pointed out that the switching topology does not affect the requirement of the noncommunication rate. In this paper, both leaderless and leader-follower communication topologies are considered, which satisfy the following assumption.Assumption 1.
It is assumed that the leaderless communication topology is represented by connected undirected graph, and the leader-follower communication topology is denoted by digraph containing a spanning tree with the leader locating at the root of the spanning tree.Lemma 1.
For the connected undirected graph, the Laplacian matrixL is symmetric and positive semidefinite, and zero is the simple eigenvalue of L.


### 2.2. Network Dynamics Modeling

The dynamics of the networked agents with leaderless structures are described as follows:(1)x˙lt=Axlt+Bult,where l=1,2,…,N, A∈ℜn×n, B∈ℜn×d, xlt are the states of agent l, and ult is the control protocol.Definition 1.
The expected formation shape of the network is described by a vector-valued functionzt=z1Tt,z2Tt,…,zNTtT, where zltl=1,2,…,N is the piecewise continuous differentiable and is called the formation instruction.
According to the formation instruction, we propose a time-varying formation control protocol via the intermittent control strategy as follows:(2)ult=K∑m∈Nlϖtblmϖtxmt−zmt−xlt+zlt,t∈Ts,T˜s,0,t∈T˜s,Ts+1,where l=1,2,…,N, K∈ℜd×n is the gain matrix. Then, the definitions of the leaderless time-varying formation design and analysis are given as follows.Definition 2.
(leaderless time-varying formation analysis). For any given gain matrix K∈ℜd×n and bounded initial states xl0−zl0, l=1,2,…,N, if there exists a function ht∈ℜn such that limt⟶+∞xlt−zlt−ht=0, l=1,2,…,N, then it is said that network (1) with protocol (2) reaches leaderless time-varying formation, where ht is said to be the formation center function.Definition 3.
(leaderless time-varying formation design). If there exists a gain matrix K such that network (1) with protocol (2) reaches leaderless time-varying formation, then it is said to be leaderless time-varying formation reachable.
In this paper, we mainly focus on designing the gain matrixK such that network (1) subject to discontinuous communications reaches leaderless time-varying formation with protocol (2). Then, the main results on the leaderless time-varying formation are extended to the leader-follower cases.Remark 1.
It should be noticed that protocol (2) is constructed via the intermittent control strategy; that is, the control input is intermittent over the nonoverlapping time unit sequence Ts,Ts+1, ∀s∈ℵ. In the communication time unit, the control input is established according to the local state information among neighboring agents and the formation instructions. However, in the noncommunication time unit, the control input is set to be zero since it is missing. This kind of intermittent control strategy will lead to the piecewise continuous right-hand side of the closed-loop networks, which is challenging to be dealt with in the stability analysis of the networked agents.Remark 2.
Note that, for the consensus, it requires that all the agents reach an agreement of states, where the formation structure is not needed. For the time-invariant formation, the formation instruction is invariable, whose time derivate is zero; i.e.,z˙l≡0l=1,2,…,N. Compared with the consensus and the time-invariant formation, the main difficulty in designing the time-varying formation protocol is that the time derivate of the formation instruction zltl=1,2,…,N affects the analysis of the convergency of the formation. In this case, the formation feasibility condition is introduced to overcome the challenging problems in designing the gain matrix of the time-varying formation protocol.


## 2.1. Communication Constraint Modeling

In this paper, we consider the discontinuous communication among agents. On the one hand, the agent cannot communicate with each other in some noncommunication time units. On the other hand, the communication topologies of the networks are switched in some communication time units. To show the abovementioned discontinuous communication type from the time-domain perspective, it is supposed for∀s∈ℵ that there exists a nonoverlapping time unit sequence Ts,Ts+1=Ts,T˜s∪T˜s,Ts+1, where Ts=Ts1<Ts2<⋯<Tsrs=T˜s<T˜s+εs=Ts+1 with rs and εs being positive integers. Notice that Ts,T˜s and T˜s,Ts+1 represent the communication time unit and the noncommunication time unit, respectively. Ts1,Ts2,…,Tsrs… denotes the switching time over which the communication topologies are switched. Without loss of generality, the initial time is assumed to be T0=0. The length of time unit Ts,Ts+1 satisfies that 0<Tmin∗≤Ts∗=Ts+1−Ts≤Tmax∗. It should be noted that 0<Tdwell≤Tsrs−Tsrs−1≤T¯dwell, where Tdwell is the minimum dwell time. The noncommunication rate is defined as σs=Ts+1−T˜s/Ts+1−Ts, where 0<σs≤σmax<1 and σmax is called the maximum noncommunication rate. Note that the discontinuous communication is aperiodic since the length of each time unit Ts,T˜ss∈ℵ can be unequal.The switching topologies are modeled asG=G1,G2,…,Gk with the switching signal ϖt:0,+∞⟶1,2,…,k, where G is the digraph. For each digraph G, the vertex set is denoted by V=v1,v2,…,vH, and the edge set is represented by E=vl,vm:vl,vm∈V with the edge weight blmϖt. Note that if there exists an edge vl,vm,vl,vm∈V, from the vertex vm to vl, then the edge weight blmϖt>0. Otherwise, blmϖt=0. The neighboring set of the vertex vl is defined as Nlϖt=vm∈V:vm,vl∈E, and Lϖt=llmϖtH×H stands for the Laplacian matrix of the topologies with lllϖt=∑m∈Nlϖtblmϖt and llmϖt=−blmϖtl≠m. More details about the graph theory can be found in [36]. It should be pointed out that the switching topology does not affect the requirement of the noncommunication rate. In this paper, both leaderless and leader-follower communication topologies are considered, which satisfy the following assumption.Assumption 1.
It is assumed that the leaderless communication topology is represented by connected undirected graph, and the leader-follower communication topology is denoted by digraph containing a spanning tree with the leader locating at the root of the spanning tree.Lemma 1.
For the connected undirected graph, the Laplacian matrixL is symmetric and positive semidefinite, and zero is the simple eigenvalue of L.


## 2.2. Network Dynamics Modeling

The dynamics of the networked agents with leaderless structures are described as follows:(1)x˙lt=Axlt+Bult,where l=1,2,…,N, A∈ℜn×n, B∈ℜn×d, xlt are the states of agent l, and ult is the control protocol.Definition 1.
The expected formation shape of the network is described by a vector-valued functionzt=z1Tt,z2Tt,…,zNTtT, where zltl=1,2,…,N is the piecewise continuous differentiable and is called the formation instruction.
According to the formation instruction, we propose a time-varying formation control protocol via the intermittent control strategy as follows:(2)ult=K∑m∈Nlϖtblmϖtxmt−zmt−xlt+zlt,t∈Ts,T˜s,0,t∈T˜s,Ts+1,where l=1,2,…,N, K∈ℜd×n is the gain matrix. Then, the definitions of the leaderless time-varying formation design and analysis are given as follows.Definition 2.
(leaderless time-varying formation analysis). For any given gain matrix K∈ℜd×n and bounded initial states xl0−zl0, l=1,2,…,N, if there exists a function ht∈ℜn such that limt⟶+∞xlt−zlt−ht=0, l=1,2,…,N, then it is said that network (1) with protocol (2) reaches leaderless time-varying formation, where ht is said to be the formation center function.Definition 3.
(leaderless time-varying formation design). If there exists a gain matrix K such that network (1) with protocol (2) reaches leaderless time-varying formation, then it is said to be leaderless time-varying formation reachable.
In this paper, we mainly focus on designing the gain matrixK such that network (1) subject to discontinuous communications reaches leaderless time-varying formation with protocol (2). Then, the main results on the leaderless time-varying formation are extended to the leader-follower cases.Remark 1.
It should be noticed that protocol (2) is constructed via the intermittent control strategy; that is, the control input is intermittent over the nonoverlapping time unit sequence Ts,Ts+1, ∀s∈ℵ. In the communication time unit, the control input is established according to the local state information among neighboring agents and the formation instructions. However, in the noncommunication time unit, the control input is set to be zero since it is missing. This kind of intermittent control strategy will lead to the piecewise continuous right-hand side of the closed-loop networks, which is challenging to be dealt with in the stability analysis of the networked agents.Remark 2.
Note that, for the consensus, it requires that all the agents reach an agreement of states, where the formation structure is not needed. For the time-invariant formation, the formation instruction is invariable, whose time derivate is zero; i.e.,z˙l≡0l=1,2,…,N. Compared with the consensus and the time-invariant formation, the main difficulty in designing the time-varying formation protocol is that the time derivate of the formation instruction zltl=1,2,…,N affects the analysis of the convergency of the formation. In this case, the formation feasibility condition is introduced to overcome the challenging problems in designing the gain matrix of the time-varying formation protocol.


## 3. Leaderless Time-Varying Formation Design and Analysis

In this section, we give leaderless time-varying formation design and analysis criteria under the condition of discontinuous communications, and then, we determine an explicit formulation of the formation center function.Forl=1,2,…,N, let φlt=xlt−zlt, substituting (2) into (1) gives that(3)φ˙lt=Aφlt+zlt+BK∑m∈Nlϖtblmϖtφmt−φlt−z˙it,t∈Ts,T˜s,Aφlt+zlt−z˙it,t∈T˜s,Ts+1.Denoteφt=φ1Tt,φ2Tt,…,φNTtT, then we can rewrite equation (3) as(4)φ˙t=IN⊗Aφt+zt−Lϖt⊗BKφt−IN⊗Inz˙t,t∈Ts,T˜s,IN⊗Aφt+zt−IN⊗Inz˙t,t∈T˜s,Ts+1.Since Assumption1 holds, we can find from Lemma 1 that there exists an orthonormal matrix Wϖt=1N/N,W˜ϖt such that(5)WϖtTLϖtWϖt=diagλ1ϖt,λ2ϖt,…,λNϖt,where 0=λ1ϖt<λ¯≤λ2ϖt≤λ3ϖt≤⋯≤λNϖt≤λ¯ are the eigenvalues of the Laplacian matrix. Denote Ωϖt=diagλ2ϖt,λ3ϖt,…,λNϖt and υt=WϖtT⊗Inxt=υ1Tt,κTtT, where κTt=υ2Tt,υ3Tt,…,υNTtT, and then, we can transform network (4) into the following form:(6)υ˙1t=Aυ1t+1HTH⊗Azt−1HTH⊗Inz˙t,t∈Ts,Ts+1,(7)κ˙t=IH−1⊗A−Ωϖt⊗BKκt+W˜ϖtT⊗Azt−W˜ϖtT⊗Inz˙t,t∈Ts,T˜s,IH−1⊗Aκt+W˜ϖtT⊗Azt−W˜ϖtT⊗Inz˙t,t∈T˜s,Ts+1.From the above orthonormal transformation, network (4) is decomposed into two subnetworks; i.e., subnetworks (6) and (7). In the following, we will provide a theorem to describe the explicit formulation of the formation center function according to the dynamics of subnetwork (6).Theorem 1.
If network (1) with protocol (2) reaches the time-varying formation zt, then the formation center function satisfies that(8)limt⟶+∞ht−hct+hzt=0,where(9)hct=eAt1NTN⊗Inx0,hzt=1NTN⊗Inzt.Proof.
Define the following functions:(10)Γct≜Wϖte1⊗υ1t=1H1H⊗υ1t,(11)Γzt≜∑l=2HWϖtel⊗υlt,where ell∈1,2,…,N is a N-dimensional unit column vector, whose l th element is 1. According to equation (10), we have(12)Γct=Wϖt⊗Inυ1Tt,0T.
Since(13)∑l=2NWϖtel⊗υlt=0,κTtT,we can derive that(14)Γzt=Wϖt⊗In0,κTtT.
BecauseWϖt⊗In is a nonsingular matrix, we can see from the above analysis that Γct and Γzt are linearly independent. Hence, we can use υ1t/N to determine the explicit formulation of the formation center function.
According to Definition2 and equation (12), we have(15)limt⟶+∞xlt−zlt−1Nυ1t=0.
From equation (16), we have(16)υ10=1N1NT⊗Inφ0.
Then, we can derive that(17)∫0teAt−τ1N1NT⊗Inz˙τdτ=1N1NT⊗Inzt−eAt1N1NT⊗Inz0+∫0teAt−τ1N1NT⊗Azτdτ.
From equations (6), (16), and (17), we can obtain the conclusion of Theorem 1.Remark 3.
The explicit formulation of the formation center function in Theorem1 describes the formation movement trajectory of the networked agents as a whole. Note that when the time-varying formation is reached, all the networked agents will keep a formation shape and move along with the trajectory determined by the formation center function, which contains two parts. The first part hct is associated with the dynamics and the initial states of each agent, which describe the influence mechanism of the consensus states on the formation movement. The second part hzt is related to the time-varying formation instruction. From the formulation of ht, we can find that the discontinuous communication does not impact the formation movement trajectory of the whole networked agents.
According to the analysis from (10) to (14), we can conclude that network (1) reaches time-varying formation if and only if limt⟶+∞κt=0. Based on this fact, we give the time-varying formation design criterion in the following theorem.Theorem 2.
Network (1) is leaderless time-varying formation reachable by protocol (2) with K=0.5λ¯−1ϑBTQ˜−1 if the following conditions hold simultaneously:(i)
The formation feasibility conditionz˙lt=Azltl=1,2,…,N holds.(ii)
The discontinuous communication conditionα1−σmax>μσmax is satisfied, where α and μ are positive constants given previously.(iii)
There existϑ>0 and Q˜=Q˜T>0 such that(18)AQ˜+Q˜AT−μQ˜<0,AQ˜+Q˜AT+αQ˜−ϑBBT<0.Proof.
Construct the Lyapunov function as follows:(19)Vt=κTtIN−1⊗Q˜−1κt.
Fort∈Ts,T˜s with ∀s∈ℵ, we can obtain the time derivative of Vt according to the dynamics of subnetwork (7) that(20)V˙t=κTtIN−1⊗Q˜−1A+ATQ˜−1−Ωϖt⊗Q˜−1BK+KTBTQ˜−1κt+2κTtW˜ϖtT⊗Q˜−1Azt−2κTtW˜ϖtT⊗Q˜−1z˙t.
Since the formation feasibility conditionz˙lt=Azltl=1,2,…,N holds, we can deduce that W˜ϖtT⊗Q˜−1z˙t=W˜ϖtT⊗Q˜−1Azt. Then, we can see from (20) that(21)V˙t=κTtIN−1⊗Q˜−1A+ATQ˜−1−Ωϖt⊗Q˜−1BK+KTBTQ˜−1κt.
SubstitutingK=0.5λ¯ϑBTQ˜−1 into equation (21) yields(22)V˙t+αVt=∑l=2NκlTtQ˜−1A+ATQ˜−1+αQ˜−1−λlϖtλ¯−1ϑQ˜−1BBTQ˜−1κlt.
By pre- and postmultiplyingAQ˜+Q˜AT+αQ˜−ϑBBT<0 with Q˜−1 and −λlϖtλ¯−1≤1l=2,3,…,N, we can obtain from (22) that(23)V˙t<−αVt.
Fort∈T˜s,Ts+1 with ∀s∈ℵ, we can obtain the time derivative of Vt along the dynamics of subnetwork (7) that(24)V˙t=κTtIN−1⊗Q˜−1A+ATQ˜−1κt.
Then, it follows that(25)V˙t−μVt=∑l=2NκlTtQ˜−1A+ATQ˜−1−μQ˜−1κlt.
According toAQ˜+Q˜AT−μQ˜<0, we have(26)V˙t<μVt.
In the sequel, we discuss the convergency ofVt along the time unit sequence Ts,Ts+1, ∀s∈ℵ. Firstly, for t∈T0,T1, we can show that(27)VT1<eμT1−T˜0VT˜0<eμT1−T˜0e−αT˜0−T0VT0=e−β0V0,where β0=α−α+μσ0T0∗. By the discontinuous communication condition α1−σmax>μσmax, we can see that β0>0. Then, we have for any positive integer s that(28)VTs+1<V0e−∑q=0sβq,where βq=α−α+μσqTq∗, q=1,2,…,s. For any t>0, we can see that an integer i≥1 exists such that Ti<t≤Ti+1. Hence, we have(29)Vt≤eμTmax∗VTi≤eμTmax∗V0e−∑j=0i−1βj≤eμTmax∗V0e−iα−α+μσmaxTmin∗≤eμTmax∗V0e−α−α+μσmaxTmin∗/Tmax∗t.
According to equation (29), we can conclude that limt⟶+∞κt=0, which means that network (1) reaches time-varying formation exponentially. The proof of Theorem 2 is finished.
The time-varying formation design criterion in Theorem2 provides an approach to design the gain matrix of protocol (2). However, if the gain matrix is given previously, then it is interesting to check that the given gain matrix is feasible or not. Let Q=Q˜−1, then the following corollary gives the time-varying formation analysis criterion.Corollary 1.
For any given gain matrixK, network (1) with protocol (2) reaches leaderless time-varying formation if the following conditions hold simultaneously:(i)
The formation feasibility conditionz˙lt=Azltl=1,2,…,N holds.(ii)
The discontinuous communication conditionα1−σmax>μσmax is satisfied, where α>0 and μ>0.(iii)
There existQ=QT>0 such that(30)QA+ATQ−μQ<0,QA+ATQ+αQ−λ¯QBK+KTBTQ<0,QA+ATQ+αQ−λ¯QBK+KTBTQ<0.Remark 4.
The formation feasibility condition in Theorem2 and Corollary 1 indicates that not all the desired formation instructions can be reached by the networked agents. We can find intuitively that the formation structure to be formed is restrained by the dynamics of each agent. For example, due to the constraint of the maneuvering characteristics, a team of unmanned aerial vehicles cannot perform some specific actions and thus cannot form the corresponding formation shape. For the time-varying formation, some formation feasibility conditions similar to the condition z˙lt=Azltl=1,2,…,N can be found in [28–30]. Note that if the derivative of the time-varying formation instruction is zero; i.e., z˙lt≡0, l=1,2,…,N, then the formation feasibility condition becomes Azlt=0. In this case, the formation instruction is time-invariant.Remark 5.
Since the intermittent control strategy leads to the piecewise continuous right-hand side of the closed-loop systems, the stability property of the closed-loop systems should be analyzed in the communication time units and the noncommunication time units. It should be pointed out that the states of the closed-loop systems may be divergent in the noncommunication time units. To ensure the whole convergency of the Lyapunov function, the discontinuous communication conditionα1−σmax>μσmax is provided, which can establish the relationship between the convergency factor α and the divergent factor μ via the maximum noncommunication rate σmax. As the result, the Lyapunov function Vt can be convergent with the rate faster than α−α+μσmaxTmin∗/Tmax∗ in virtue of inequality (29).


## 4. Extensions to Leader-Follower Cases

This section extends the main results of the leaderless time-varying formation design and analysis with discontinuous communications to the leader-follower cases.The dynamics of networked agents with leader-follower structures are modeled as(31)x˙Nt=AxNt,x˙lt=Axlt+Bult,where l=1,2,…,N−1 are the labels of N−1 followers, and the leader is labeled by the subscript N.Assumption 2.
It is assumed that the communication topology among followers is represented by the connected undirected graph.
For leader-follower structures, we propose the following time-varying formation control protocol:(32)ult=Klf∑m∈Nlϖtblmϖtxmt−zmt−xlt+zlt,+KlfblNϖtxNt−xlt+zlt,t∈Ts,T˜s,0,t∈T˜s,Ts+1,where l=1,2,…,N−1, Klf∈ℜd×n is the gain matrix. Then, the definitions of the leader-follower time-varying formation design and analysis are given as follows.Definition 4.
(leader-follower time-varying formation analysis). For any given gain matrix Klf∈Rd×n and bounded initial states xl0−zl0, l=1,2,…,N−1, if limt⟶+∞xlt−zlt−xNt=0, l=1,2,…,N−1, then it is said that network (1) with protocol (2) reaches leader-follower time-varying formation.Definition 5.
(leader-follower time-varying formation design). If there exists a gain matrix Klf such that network (1) with protocol (2) reaches leader-follower time-varying formation, then it is said to be leader-follower time-varying formation reachable.
Form (31) and (32), we can obtain that(33)φ˙t=IN⊗Aφt+zt−Llfϖt⊗BKlfφt−IN⊗Inz˙t,t∈Ts,T˜s,IN⊗Aφt+zt−IN⊗Inz˙t,t∈T˜s,Ts+1,where Llfϖt satisfies that(34)Llfϖt=Lfϖt+Δlϖt−llϖt00,Δlϖt=diagb1Nϖt,b2Nϖt,…,bN−1Nϖt,llϖt=b1Nϖt,b2Nϖt,…,bN−1NϖtT.
Lfϖt is the Laplacian matrix of the topology among followers.
Construct the following nonsingular matrix:(35)Yϖt=IN−11N−101.
Letφ˜lt=φlt−xNtl=1,2,…,N−1, then we have(36)Yϖt−1⊗Inφt=φ˜2Tt,…,φ˜NTt,x1TtT.
Then, it follows fromΔlϖt1N−1=llϖt that(37)Yϖt−1LϖtYϖt=Lfϖt+Δlϖt000.
Since Assumptions1 and 2 hold, we can find that Lfϖt+Δlϖt is positive definite and symmetric. Hence, there exists an orthonormal matrix Y˜ϖt∈ℜN−1×N−1 such that(38)Y˜ϖtTLfϖt+ΔlϖtY˜ϖt=Ωfϖt=diagλ˜1ϖt,λ˜2ϖt,…,λ˜N−1ϖt,where 0<λ˜1ϖt≤λ˜2ϖt≤⋯≤λ˜N−1ϖt are the eigenvalues of Lfϖt. Let φ˜t=φ˜1Tt,φ˜2Tt,…,φ˜N−1TtT and Yϖt−1⊗Inφ˜t=ηt=η1Tt,η2Tt,…,ηN−1TtT, then network (33) is converted to the following two subnetworks:(39)x˙Nt=AxNt,η˙t=IN−1⊗A−Ωfϖt⊗BKlfηt+Y˜ϖtT⊗Azt−Y˜ϖtT⊗Inz˙t,t∈Ts,T˜s,IN−1⊗Aηt+Y˜ϖtT⊗Azt−Y˜ϖtT⊗Inz˙t,t∈T˜s,Ts+1.
BecauseYϖt and Y˜ϖt are nonsingular, we can find that network (31) reaches leader-follower time-varying formation if and only if limt⟶+∞ηt=0. Let λ¯lf=minλ˜1i:∀i∈1,2,…,k and λ¯lf=maxλ˜1i:∀i∈1,2,…,k, then we give the following theorem to show the sufficient conditions of the leader-follower time-varying formation design.Theorem 3.
Network (31) is leader-follower time-varying formation reachable by protocol (32) with Klf=0.5λ¯lf−1ξBTR˜−1 if the following conditions hold simultaneously:(i)
The formation feasibility conditionz˙lt=Azltl=1,2,…,N holds.(ii)
The discontinuous communication conditionα1−σmax>μσmax is satisfied, where α>0 and μ>0.(iii)
There existξ>0 and R˜=R˜T>0 such that(40)AR˜+R˜AT−μR˜<0,AR˜+R˜AT+αR˜−ξBBT<0.Corollary 2.
For any given gain matrixKlf, network (31) with protocol (32) reaches leader-follower time-varying formation if the following conditions hold simultaneously:(i)
The formation feasibility conditionz˙lt=Azltl=1,2,…,N holds.(ii)
The discontinuous communication conditionα1−σmax>μσmax is satisfied, where α>0 and μ>0.(iii)
There existsR=RT>0 such that(41)RA+ATR−μR<0,RA+ATR+αR−λ¯lfRBKlf+KlfTBTR<0,RA+ATR+αR−λ¯lfRBKlf+KlfTBTR<0.Remark 6.
The leader-follower time-varying formation can be regarded as an extension of the leaderless time-varying formation since their main conclusions are similar. In this case, the control gains of these two cases can be designed via solving the linear matrix inequalities in the similar form. However, the topology structures of these two cases are different, which can be reflected on the difference of the eigenvalues of the Laplacian matrices. Moreover, the formation trajectories of the leaderless time-varying formation and the leader-follower time-varying formation are different. For the leaderless case, the formation trajectory is determined by the states of all agents and the formation instruction, which is described by the formation center function. For the leader-follower case, the formation trajectory is determined by the leader.


## 5. Numerical Simulations

In this section, we provide two numerical simulation examples to demonstrate the effectiveness of the proposed theorems regarding the leaderless and leader-following time-varying formation design and analysis with discontinuous communications.Example 1.
(leaderless topologies). Consider a group of networked agents labeled by 1–6, which are of third order as follows:(42)A=0100010.5−10.5,B=122.
The initial states of the agents are set as(43)x10=3.5,−5.1,−2.8T,x20=−2.3,−7.5,1.6T,x30=2.8,−4.6,1.3T,x40=2.7,−2.1,2.1T,x50=3.7,−1.4,−5.9T,x60=7.2,−2.9,4.3T.
The switching topologies are given in Figure1, where the dwell time is set as 0.3 s.
The communication time unit and noncommunication time unit are set to bet∈s,s+0.8s and t∈s+0.8,s+1s, respectively, which are period time units for better operability of the simulation. In this case, the maximum noncommunication rate σmax=0.2. Choose α=1.6 and μ=6, and then, we can find that the discontinuous communication condition α1−σmax>μσmax is satisfied. The time-varying formation instruction is chosen as follows:(44)zlt=sint+l−1π3cost+l−1π3−sint+l−1π3,l=1,2,…,6.
Based on Theorem2, the matrix variable and the gain matrix are calculated as(45)β=8.5573,Q˜=4.1772−2.24754.2758−2.24757.37661.45734.27581.457320.4770,K=8.2490,6.9729,−0.6591.
Figure2 shows the trajectories of φll=1,2,…,6 for the networked agents, where the full curves with different colors denote the φll=1,2,…,6 for the six agents and the sequence of blue circles represents the formation center function. We can see from Figure 2 that the curve of each agent converges to that of the formation center function; that is, they reach the consensus with the states of the formation center functions.
Figure3 depicts the state snapshots of six agents at t=0s, t=8s, t=9s, and t=10s, where six agents are depicted by blue circles, pink plusses, red hexagrams, black squares, green hexagrams, and indigo x-marks. We can find from Figure 3 that the six agents reach a time-varying regular hexagon around the formation center at different times, which can keep rotating. The above simulation results indicate that network (1) with protocol (2) can reach the leaderless time-varying formation with discontinuous communications.Figure 1
Leaderless switching topologies.Figure 2
Trajectories ofφll=1,2,…,6.
(a)(b)(c)Figure 3
State snapshots of six agents at different times: (a)t=0s; (b) t=8s; (c) t=9s; (d) t=10s.
(a)(b)(c)(d)Example 2.
(leader-follower topologies). Consider a leader labeled by 6 and five followers labeled by 1–5, whose dynamics and initial states are the same as the leaderless cases. The switching topologies are given in Figure 4, where the dwell time is set as 0.3 s. The communication time unit and noncommunication time unit are t∈s,s+0.8s and t∈s+0.8,s+1s, respectively, and the corresponding parameters are chosen as the same as the leaderless cases.
The time-varying formation instruction is chosen as(46)zlt=sint+2l−1π5cost+2l−1π5−sint+2l−1π5,l=1,2,…,5.
According to Theorem3, the matrix variable and the gain matrix are calculated as(47)ξ=8.5573,R˜=4.1772−2.24754.2758−2.24757.37661.45734.27581.457320.4770,Klf=27.2831,23.0624,−2.1799.
Figure5 shows the trajectories of φll=1,2,…,5 for five followers and that of the leader, where the full curves with different colors stand for the φll=1,2,…,5 for the five followers and the sequence of red imaginary curves denotes that of the leader. It can be found from Figure 5 that the curves of all followers converge to that of the leader; that is, they can track the trajectory of the leader.
Figure6 depicts the state snapshots of five followers and the leader at t=0s, t=8s, t=9s, and t=10s, where they are depicted by blue circles, pink plusses, red hexagrams, black squares, green hexagrams, and indigo x-marks. We can find from Figure 3 that the six agents reach a time-varying regular pentagon and keep rotating around the leader, which means that network (26) with protocol (27) can reach the leader-follower time-varying formation with discontinuous communications.Figure 4
Leader-follower switching topologies.Figure 5
Trajectories ofφll=1,2,…,5 and that of the leader.
(a)(b)(c)Figure 6
State snapshots of five followers and the leader at different times: (a)t=0s; (b) t=8s; (c) t=9s; (d) t=10s.
(a)(b)(c)(d)


## 6. Conclusions

The leaderless and leader-follower time-varying formation design and analysis for networked agents with discontinuous communications were studied. The leaderless time-varying formation control protocol was proposed via the intermittent control strategy, which contains both the communication time unit and the noncommunication time unit. An explicit formulation of the formation center function was determined, which can describe the formation movement trajectory of the networked agents as a whole. Leaderless time-varying formation design and analysis with discontinuous communications were given, where the formation feasibility conditions and discontinuous communication conditions were constructed to ensure the stability of the closed-loop subnetworks. Moreover, the main results of the leaderless cases were extended to the leader-follower cases, where the trajectory of the formation movement was determined by the state of the leader.



---

*Source: 6694587-2021-03-03.xml*