Predefined formation‐containment control of high‐order multi‐agent systems under communication delays and switching topologies

Funding information Defense Industrial Technology Development Program, Grant/Award Number: JCKY2019601C106; Foundation Strengthening Program Technology Field Fund, Grant/Award Number: 2019-JCJQ-JJ243; Science and Technology Innovation 2030 -Key Project of "New Generation Artificial Intelligence", Grant/Award Number: 2020AAA0108200; Key Laboratory of Dependable Service Computing in Cyber Physical Society, Grant/Award Number: CPSDSC202001; Innovation Zone Project, Grant/Award Number: 18-163-00-TS-001-001-34; National Natural Science Foundation of China, Grant/Award Numbers: 61803014, 61873011, 61922008, 61973013 Abstract This paper concerns the problem of formation-containment control for general-linear multi-agent systems (MASs) with both communication delays and switching interaction topologies. On the one hand, the leaders can communicate with each other to form the desired formation and on the other, the followers need to enter the convex envelope spanned by the multiple leaders. Firstly, by using the neighbouring relative information, formation-containment protocols are designed for each leader and follower, where an edge-based state observer is incorporated into the formation-containment controller to evaluate the whole leaders’ state. Secondly, according to the linear matrix inequality technology, an algorithm is given to determine the unknown feedback matrixes in the protocol. Then, based on Lyapunov theory, the formation-containment error is proved to be convergent and formation feasibility conditions are also presented for the MASs to achieve formation-containment. Finally, a simulation on several MASs is provided to demonstrate the theoretical results.


INTRODUCTION
the neighbours, a formation problem for high-order MASs with communication delays was investigated in [29]. Note that switching topologies can affect the stability of the systems and cause the chaos of the agents. Therefore, a formation controller of MASs with high-order models under the influence of switching directed networks was studied in [30]. Moreover, a distributed formation approach under both switching topologies and communication delays was proposed and formation feasibility condition was also presented in the work [31]. In [32], a consensus-based approach was presented for the MAS in the presence of communication delays and switching interaction topologies. As a counterpart with formation, another popular research field in coordinated control is containment control, where the agents in this problem are divided into two categories, leaders and followers. The control target is to enforce the followers enter the convex combination spanned by multiple leaders. As a pioneering work in containment, a hybrid Stop-Go policy was presented to drive the followers entering the convex hull in [33]. Considering the time delay, an observer-based containment protocol was studied for high-order MASs with directed graph in [34]. To satisfy the complex environments, both continuous and discrete-time systems were taken into consideration with the influence of communication delays on the containment control in [35]. Note that the interaction topologies of the MASs can be switching due to the uncertain environments. Su et al. in [36] studied the containment problem for MASs with generic linear under switching networks. By using the combination of convex analysis and stochastic process, containment control of MASs with stochastic topologies was studied in [37].
Combined the formation control with containment control, a significant formation-containment problem arises which allows the leaders achieve the desired formation and followers can enter the formation. Note that compared with containment control, there exists also leaders and followers, but the leaders should interact with each other to form the expected formation in the formation-containment problem. As a result of the close relationship between the leaders and the followers, the formation-containment control has been applied in many engineering aeries, such as formation coordinated attack and coordinated penetration. In formation coordinated attack, the unmanned aerial vehicles are denoted as the leaders and the manned fighters are represented as the followers. Leaders are allowed to form the desired formation and the followers can enter the formation. Therefore, when the foes arise, the followers which are much more important can be protected. It should be pointed out that because of the interaction between the leaders and followers, it is difficult to isolate the Laplace matrix between them. Formation-containment control methods and feasible conditions for general-linear swarm systems were investigated in [38], which means the above control methods, no matter it is formation, containment or consensus problem, they all can be thought as a special case of formation-containment problem. The control strategy of formation-containment was applied into the UAVs signifying the effectiveness of the proposed method in the work of Dong et al. [39]. Both fixed and time-varying communication delays were taken into considera-tion for MASs to achieve formation-containment problems in [40,41]. For multiple UAVs with directed switching topologies, Han in [42] extends the formation-containment control for MASs to multiple UAVs under directed switching topologies. In the prior works mentioned above, the control strategies for formation-containment are step into a more practical field. However, it cannot be avoided that the communication delays and switching networks may exists simultaneously. Formationcontainment problem for MASs with general linear dynamic with communication delays and switching networks is challenging.
In the current paper, a consensus-based solution was put forward to realized formation-containment of generic linear MASs under communication delays and switching topologies. First, because of the different roles that the agents play, two different protocols are proposed for the leaders and follower. Then, necessary and sufficient conditions for the leaders to achieve the expected formation are provided under the constraints of communication delays and switching topologies. Moreover, utilizing the method LMI technology, an approach is developed to obtain the gain matrices. At last, based on the proposed Lyapunov function and formation feasibility condition, both the state observers and formation-containment systems are said to be convergent.
Compared with the outstanding works mentioned above, the main contributions of this paper can be listed as following.
1. Because the leaders have a huge impact on the followers, formation-containment control is more difficult in cooperative control. Compared with formation control in [14][15][16][27][28][29], the followers can enter the formation spanned by the multiple leaders. Besides, in contrast to the containment control in [19,[34][35][36], the leaders can exchange information with each other to realize the predefined formation. Therefore, the formation-containment problems have important value of engineering application. 2. Note that MASs with communication delays and switching interaction topologies were investigated in the current paper. In the work of Han et al. [41], formation-containment problem with communication delays was studied. Note that if switching topologies emerge, the convex combination of the leaders may change, and the formation-containment error cannot be convergent. Compared with formationcontainment problem in [42], the systems are divided into the two systems. It should be pointed out that only switching networks is taken into consideration and the convergence of the systems focuses on each switching system. Therefore, the methods mentioned before can not be directly used to solve the formation-containment systems with both time-varying communication delays and switching networks. 3. The control protocol was studied for general linear MASs on formation-containment. Hence, the formation-containment control mentioned in this paper are practical than the work in [41].
The paper is organized as follows. Section 2 gives some preliminaries on graph theory and the statements of the formation-containment problems. Both control protocols and some important theorems are introduced in Section 3. Then, simulation results which can illustrate the effectiveness of the proposed method are presented in Section 4, Finally, conclusions are demonstrated in Section 5.
The notations throughout this paper are shown as follows. Let ⊗ be the Kronecker product. 1 stands for a column vector with 1 as its element. I n represents the n × n identify matrix. Let diag{D 1 , D 2 , … , D N } represents a diagonal matrix. X T denotes the transpose matrix of the matrix X .

PRELIMINARIES AND PROBLEM FORMULATION
In this section, some preliminaries of graph theory are demonstrated and the descriptions on formation-containment are also proposed.

Graph theory
represents the edges set, as well as = [w i j ] ∈ ℝ n×n indicates a weighted adjacency matrix. Let e i j =(v i , v j ) denotes the edge form node v i to v i and w i j is the non-negative element with e ji . If and only if v j can The Laplacian matrix ℒ ∈ ℝ n×n associated with is defined as ℒ = − , where is the in-degree diagonal matrix of the graph. A path form v i to v j is defined as an ordered sequence of the edges The graph is said to be undirected if v i j ∈ ℰ implies v ji ∈ ℰ and w i j = w ji , as well as the connected undirected graph is defined as there exists a path for any two nodes v i and v j .
Note that the interaction among the MASs can be switching. The index set of the switching graph is represented by ℋ = {1, 2, … , s}. Let [t k , t k+1 ) denote an infinite sequence of non-overlapping time intervals, where t 0 = 0, t k − t k+1 ≥ T d > 0. The interaction topology can be switching at the instant t k and keeps fixed in the dwell time T d . (t ) ∶ [0, ∞) → {1, 2, … , s} is the switching signal. Define (t ) and ℒ (t ) as the graph and Laplacian matrix at t .
Assume that the neighbours of the leaders are only leader and that there exist M leaders and N followers. Let  E = {1, 2, … , M } and  F = {1, 2, … , N } represent, respectively, the leader set and follower set.
The Laplacian matrix ℒ of the graph at t is given as (t ) and E (t ) the interaction topology among follow-  Remark 1. To realize the formation-containment control with both communication delays and switching topologies, the Assumption 1 and Assumption 2 represented before are necessary to construct a state observer (see the illustrative example 1). It should be pointed out that the well-informed follower can be seen in the works [44][45][46], and in practical applications, the well-informed followers are denoted as the agents with powerful sensors, while the uninformed followers are those with poor sensors. Therefore, the well-informed followers can reach the information of all the leaders and the informed followers can only receive the information of the neighbouring followers.
To show that Assumption 1 and Assumption 2 are important for the MASs to construct a new control framework, by using the control protocol and gain matrices in [40], a numerical example is shown as follows: Illustrative example 1: Assume there exists a MAS consisting of five followers and four leaders. The time delay satisfies = 0.15. The switching topologies are denoted as in Figure 1, and will switch at t = 10, 30, 50 s.
The systems matrices are represented as: Then, the control protocol for the i-th follower is given as: ) .
The control input for i-th leader is given as following: Let the gain matrices in the formation-containment control The desired formation for the four leaders is specified by 7,8,9).
From Figure 2, one sees that if the topologies are switching, the convex combinations of the leaders may change, and then the containment error cannot be convergent. Therefore, the exiting formation-containment control protocol in [40][41][42] no longer meets the circumstance of both time-varying delays and switching topologies. A new control protocol should be put forward to solve the complex communication environment.

Lemma 1. [8] Assume that assumption 1 holds, 0 is the simple eigenvalue belonging to the ℒ F (t ) , and its corresponding right eigenvector is
N . The rest of the others N − 1 eigenvalues are greater that zeros.

Problem description
The agents dynamics are described by where A ∈ ℝ n×n , B ∈ ℝ n×m . x i (t ) ∈ ℝ n×n and z i (t ) ∈ ℝ n×n denote the states of the i-th follower and leaders, respectively. u iE (t )(i ∈  F ) and u iF (t )(i ∈  E ) represent, respectively, the control input of the i-th follower and leaders.
Let (1), there exists a vector-value function f (t ) ∈ ℝ n satisfying the following equation

Definition 1. If for each leader in MASs
Then the leaders are proved to achieve the formation h E (t ). (1) can realize the containment if (3) holds for each follower (2) and (3) are satisfying under any given bounded initial states, the MASs (1) can realize the formationcontainment.

Definition 3. If both
Remark 2. According to definition 1-3, it can be realized that the formation-containment problem can be seen as the integration of the formation and containment problem. Moreover, it is a much more complex condition on coordinated control. Note that if N = 0, which means there is no leaders in MASs. Then, the formation-containment can be transformed into the formation control. If h E (t ) = 0 and there is no interaction between the leaders, the formation-containment can be regarded as the containment control. Furthermore, if both N and f (t ) are equal to 0, the formation-containment problems discussed in the current paper becomes an state consensus control. Formationcontainment control problem arise after the consensus problem and can be regarded as the extension of the consensus control.
Remark 3. Note that although in formation-tracking problem, there exist leaders and followers. It should be pointed out that the agents play different roles comparing with formationcontainment control. In formation-containment problem, the leaders should form the expected formation rather than generate the reference trajectory, and the followers should enter the desired formation instead of achieved the predefined formation.
, the formation-containment protocols for leader i (i ∈  E ) is given as following: A state observer and the control input for i-th (i ∈  F ) follower are given as T denote the compact form of the leaders state, as well as K i (i = 1, 2, 3, 4, 5) represent the unknown feedback matrices. The predefined containment positive constant is denoted to i j , which for each follower i, is the communication delay that satisfy the following assumption. Since all agents have the same control chip and the systems are isomorphic, it makes sense that (t ) is the same.
In order to deal with the influence of the communication delays and switching topologies, the flowing lemmas are proposed. Lemma 2. [43] For any continuous derivative vector entry (t ) ∈ ℝ 2d , the following is satisfying where Let is a diagonal matrix which diagonal element is 0 or 1.

Lemma 3.
[32] According to Assumption 1, the matric H (t ) corresponding to the F (t ) is non-negative.

Lemma 4. [32] For any i
The major goal of this paper is to solve the following two problems for MASs(1) under both communication delays and switching topologies: First, under what conditions the formation-containment can be realized. Second, how to design the protocols (4) and state observer (5).

MAIN RESULTS
In this section, firstly, a neighbouring-based leaders controller as well as an observer-based followers protocol is put forward for MASs (1) to handle both communication delays and switching graphs. Then, the necessary and sufficient conditions for the leaders to achieve the time-varying formation is proposed. Moreover, an algorithm is given to design the protocols (4) and state observer (5). Finally, the stability of the closed loop MASs (1) is proved by the Lyapunov function and LMI technique.
Then, the closed loop of the leaders system can be written as followinġ Under protocol (5), the dynamic of the closed loop of the followers is shown aṡ Let̃i j (t ) =̂i j (t ) − z j (t ) denote the observing error between i-th observer and j -th leader. Then (8) could be rewritten as follows: According to (6), one haṡz . Let (t ) = (ū T 1 ⊗ I n )z(t ) and (t ) = ( T ⊗ I n )z(t ), then the leaders system can be divided into two parts.

Theorem 1. MASs (1) with communication delays and switching graphs can realize the desired formation h E (t ) if and only if
where which first component is 1 and others are 0. Theñ Note that  ⊗ I n is non-singular,z c (t ) as well asz̄c (t ) are linearly independent vector. Form (14) and (15), it can be obtained thatz̄c

the leaders of MASs (1) can realize the formation designed by h E (t ). This completes the proof. □
In the following, an Algorithm with five steps is proposed to determine the protocols and observer. Algorithm 1. The gain matrixes K i (i = 1, 2, 3, 4, 5) in the controller and state observer can be designed along the following steps Step 1: K 1 is chosen to specify the eigenvalue of A + BK 1 at the closed left-half complex plane. Note that the controllability of the pair (A, B) can guarantee the existence of the K 1 .
Step 2: If there exist positive symmetric matrices R E , Ω E , S E and real matrixK 2 for anȳE i (i = 1, 2), LMI (17) is feasible, the gain matric K 2 =K 2 Ω −1 ; otherwise , the Algorithm stops. where Step 3: Choose suitable K 3 to determine the real part of eigenvalues of A + BK 3 are greater than or equal to 0.
Step 4: Choose the suitable K 4 , following the equation Step 5: If there are positive symmetric matrices R F , Ω F , S F and real matrixK 5 for anȳH i (i = 1, 2), LMI (18) is met. Then K 5 =K 5 Ω −1 F ; otherwise, the algorithm stops. where According to the calculated parameters in Algorithm 1, the following theorem holds.

Theorem 2. If the formation feasibility condition (19) is fit for the all the leaders and the switching topologies satisfying the Assumptions
Then, MASs (1) can achieve the formation-containment with both communication delays and switching networks based on the protocols (4) and (5) designed by Algorithm 1.
Proof. The proof of the Theorem 2 can be divided into the following two parts.
Firstly, the leaders can achieve the desired formation. According to Algorithm 1, the leader can realize the desired formation if and only if lim t →∞ (t ) = 0. Then, one obtains that (20) and (21) are the necessary and sufficient conditions conditions for the leaders to achieve formation.
Consider the stability of the system (21). Construct the following common Lyapunov-Krasovskii candidate function where Let as it was mentioned that T ℒ E (t ) is symmetric; therefore, it is possible to get an orthogonal matrix (t ) which meeting ] T ; then taking the time derivative of V E (t ) along the trajectory of (27), one haṡ From the Assumption 3, one getṡ From Assumption 3 and Lemma 2, it can be obtained that Let (27) to (33), it can be obtained thatV where According to Schur complement lemma, Z i < 0 is equivalent to ] andΓ=diag{T , I , S }, then it can be obtained that As K 2 =K 2 Ω −1 , ∏ (̄E i ) < 0; then according to Schur complement lemma, ∏ (̄E i ) < 0 are equivalent to ∏ ( i (t ) ) < 0(i = 2, 3, … , M , (t ) = 1, 2, … , s), which equal toΓ T iΓ < 0, from (24) to (34), one has Remember (19) is the necessary and sufficient condition of (20). From Schur complement lemma and the calculated K 2 , one gets ∏ (̄E i ) < 0; then based on Lemma 4, ∏ (̄E i ) < 0 are equivalent to ∏ ( i (t ) ) < 0(i = 2, 3, … , M , (t ) = 1, 2, … , s) for each switching systems, which means the convergence of the formation error systems (35) is proved and MASs (1) under both switching graphs and communication delays realized the desired formation h E (t ).
The follower will enter the predefined convex combination by the leaders.
From (35) and (41) , MASs (1) can achieve formationcontainment control under switching topologies and timevarying delays. This completes the proof. □ Remark 4. Note that the systems (21) and (38) can be viewed as the time-delay switched system in control theory, where we can see that both time delays and switching graphs will have significant influences on the stability of this system. On the one hand, the proposed Lyapunov-Krasovskii function for just delayed systems on fixed topology can no longer be used in the switching case; on the other hand, time-delayed factor is incorporated into the Lyapunov function compared with the switched systems without time delay. Then, by putting forward the useful lemmas and assumptions, we have constructed the common Lyapunov-Krasovskii candidate function (27) to deal with both time delays and switching graphs. Finally, the time-delay switched system is proven to be asymptotic stability.
Remark 5. Based on the stated observer, a formationcontainment protocol is proposed for MASs (1), where the predefined weights i j are used to specify the desired combination. One sees that MASs (1) can realize formation-containment with both communication delays and switching topologies. Moreover, only the formations satisfying (19) can be achieved by leaders.
Remark 6. Although there exist some works that the distributed estimation can be achieved in [11], it should be pointed out that the proposed state observer is constructed without neither time-varying delays nor switching topologies, which means the proposed Lyapunov function cannot be used to prove the convergency of the time-delay switched systems, and the analysis method based on orthogonal transformation is only satisfied for fixed topology rather than switching topologies. Moreover, the state observer is put forward to estimate all the leaders' state rather than a single leader's state under both time-varying delays and switching topologies. Therefore, if b (t ) i > 0, the follower can get all the information of the leaders. Due to the physical meaning of the state observer and the coefficient b i , the follower i is denoted by the well-informed follower.
Remark 7. Note that the distributed observers are also asymptotically stable in the existing results [48][49][50][51]. Although the estimation errors cannot converge to zero in finite time, we can design appropriate gain matrices in the observer such that the estimation process is faster than the control process, and then the separation principle is applicable to the closed-loop system. Specifically, in [47], the containment problems with switching topologies are solved based on the state observer systems; an edge-based adaptive distributed observer is presented for each follower to estimate the whole states of all the leaders. The common Lyapunov theory is put forward to prove the convergence of the distributed observers under the influences of switching graphs. Moreover, in the research of the cooperative output regulation [48][49][50][51], by using the neighbours' information, the state observer is contrasted to estimate the leaders' state. Then, the control protocol is designed based on the state observer, and the state observers in [48][49][50][51] are proven to have asymptotic convergence. Therefore, the asymptotic convergence of distributed observer is appropriate and can guarantee the stability of the whole closed-loop system.
Remark 8. The works in [13] and [35] investigated, respectively, the formation or containment problem under both communication delays and switching topologies. Note that because of the influence of the formation on the followers, the mentioned protocol cannot be used directly to solve the MASs with both communication delays and switching topologies. By using the neighbours information and state observers, a novel protocol is put forward for the MASs (1) to realize formation-containment under both communication delays and switching topologies. Hence, the proposed controller is more versatile and general.

NUMERICAL SIMULATIONS
In this section, numerical simulations are presented to validate the effectiveness of the proposed approach. Assume that there exist third-order MASs with nine agents consisting of six leaders and three leaders.
The desired state containment for third-order MASs is specified by 71 =  Figure 4 shows the state snapshots of nine agents, where the states of the leaders and followers are denoted by red and blue, respectively. The imaginary line is shown to demonstrate the convex combination, and the black pentagram denotes the state of the formation reference. Figure 4 indicates that the leaders achieve the desired formation and the followers converge to the convex hull formed by the leaders. Figure 5 displays the time-varying formation error within t = 50 s. As shown in Figures 6 and 7, all the followers can acquire the leaders state and the containment errors can converge to zero. From Figures 2-5, one sees that MASs with timevarying delays and switching interaction topologies achieves the desired formation-containment.

CONCLUSION
Formation-containment problems for general linear MASs with both communication delays and switching interaction topolo-  gies were studied. The target of the leaders was to achieve the desired formation and the followers should enter the convex combination spanned by the leaders. Based on distributed state observer and neighbouring information, novel control protocols, which considered switching graphs and communication delays, were proposed for the leaders and followers, respectively. According to LMI technique, an approach to determine the feedback matrixes was put forward. Moreover, based on Lyapunov-Krasovskii stability theory and the proposed formation feasibility conditions, the stability of the formation-containment systems was proved. Future research will concentrate on extending the switching topologies from undirected to directed without well-informed follower. Another interesting topic for future study is to design the finite time state observer on the formation-containment control.