Synchronization-Based Guaranteed-Performance Formation Design for Swarm Systems

Guaranteed-performance formation control for swarm systems with the second-order dynamics is investigated based on the synchronization control strategy. Firstly, a new formation protocol is presented, where the weights of connected edges are adaptively regulated and the performance constraint is imposed. -en, on the basis of the Riccati inequality, sufficient conditions for synchronization-based guaranteed-performance formation are proposed, and an explicit expression of the guaranteedperformance cost is shown, where it is fully distributed to design gain matrices of the formation protocol in the sense that it is independent of global information of swarm systems. Moreover, the whole motion of a swarm system is determined, which is associated with initial states of all agents and formation control vectors. Finally, two numerical examples are shown to demonstrate theoretical conclusions, where the static whole motion and the dynamic whole motion are considered, respectively.


Introduction
Recently, a lot of researchers paid their attention to distributed coordinated control of swarm systems since they have wide foregrounds in many fields, such as animal flocking [1,2], network synchronization [3][4][5][6][7][8], and formation of multiple intelligent agents [9][10][11][12][13]. Distributed formation originates from the animal swarm motions with local communication interactions, and it is needed that many intelligent agents construct some structure and operate with this structure.
In [14], Ren summarized and commented on the existing formation approaches containing the virtual-structure approach, the behavior-based approach, and the synchronization-based method. It was pointed out that the virtualstructure approach is centralized, and the behavior-based approach cannot be analyzed by establishing accurate models and implementing rigorous deduction although it is distributed.
e synchronization-based formation control approach is valid to realize specific structures by designing formation control vectors, where some coordinated variables in practical swarm systems are chosen and these errors between coordinated variables and formation control vectors are required to achieve synchronization. e synchronization-based formation control approach can overcome some drawbacks of the virtual-structure approach and the behavior-based approach, and it can establish formation in a distributed mode.
On the basis of the synchronization-based formation control approach, the following several important works were finished. In [15], formation stability was discussed by the Nyquist criterion and a necessary and sufficient condition was proposed. Du et al. [16] realized the multiple intelligent robot formation by a new nonlinear control protocol, where formation can be achieved in a finite time. In [17][18][19][20][21][22][23][24][25][26], the impacts of some practical factors including time delays, intermittent communications, and external disturbances on formation control and its practical applications, were well dealt with. Xiao et al. [27] constructed nonlinear control protocols to realize finite-time formation for swarm systems with each agent modeled by a first-order integrator. In [28], formation control for swarm systems with each agent modeled as a second-order integrator was addressed, where the formation experiment of multiple quadrotors was finished and the communication network has a leader-following structure. Dong and Hu [29] investigated formation tracking for swarm systems with multiple leaders and sufficient conditions for formation tracking were presented.
In [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], the formation regulation performance was not considered, which can be modeled as global suboptimal or optimal formation design problems. Inspired by the guaranteed-cost synchronization control for swarm systems addressed in [30], Wang et al. [31] investigated guaranteedperformance formation control with limited communication. Yu et al. [32] discussed guaranteed-performance formation control for swarm systems with time-varying delays and unknown disturbances and presented the formation control criteria on the basis of the linear matrix inequality, where no approach was proposed to determine the specific structure of the guaranteed-performance cost. Moreover, the associated formation control criteria in [31,32] are associated with the global information of swarm systems, such as the Laplacian matrix of the communication network or its nonzero eigenvalues; that is, those criteria are not fully distributed since gain matrices may be changed when some agents are removed or some agents are added in swarm systems. As far as we know, the synchronization-based guaranteed-performance formation design problem for swarm systems with the fully distributed control strategy is still not comprehensively studied.
is paper investigates a fully distributed strategy to realize synchronization-based guaranteed-performance formation design for swarm systems. A formation protocol is proposed on the basis of state errors and formation errors among adjacent agents, which contains the guaranteedperformance constraint and the regulation mechanism of edge weights. Furthermore, by constructing Lyapunov functions with the adaptively regulated weight terms, the translation adaptive method is proposed to eliminate the impacts of nonzero eigenvalues, and sufficient conditions for synchronization-based guaranteed-performance formation are proposed, which are fully distributed since they are not associated with global information of swarm systems. Moreover, an explicit expression of the guaranteed-performance cost is shown, and the whole motion of a swarm system is determined, which may be static and can also move in the uniform linear mode according to different initial values of velocity-similar states of all agents.
Compared with the closely associated literatures on formation control, our work has three critical characteristics. Firstly, synchronization-based guaranteed-performance formation criteria are fully distributed; that is, the gain matrix design is independent of the global information of communication networks of swarm systems. In [31,32], the formation performance among neighboring agents can only be regulated. irdly, the whole motion of a swarm system depends on the average of initial values of velocity-similar states and position-similar states and formation control vectors.
e whole motion of a swarm system was not determined in [32], and the approach in [31] is no longer valid due to adaptively regulated edge weights. e remainder of this paper is arranged as follows. In Section 2, the problem formulation is stated on the basis of the graph theory and the neighboring relationship, and a formation protocol with adaptively regulated weights and the performance constraint is shown. Section 3 proposes sufficient conditions for synchronization-based guaranteedperformance formation control according to the Riccati inequality and determines an explicit expression of the guaranteed-performance cost. Furthermore, the motion of a swarm system as a whole is determined. In Section 4, two numerical examples are given to demonstrate theoretical results. In Section 5, conclusion comments are provided.
1.1. Notations. R d and R d×d are applied to represent the dand d × d -dimensional real column vector space and real matrix space, respectively. 0 is used to stand for the zero matrix with all components zero and the zero number as a special case. e notation I N represents the N-dimensional identity matrix, and 1 N is used to denote an N-dimension column vector with all components 1. e notation ⊗ denotes the Kronecker product. e symbols Q T � Q < 0 and Q T � Q > 0 mean that the symmetric matrix Q is negative definite and positive definite, respectively, where Q T denotes the transpose of the matrix Q.

Modeling Communication Structure.
e communication structure of a swarm system can be depicted by an is the node set and

2.2.
Synchronization-Based Guaranteed-Performance Formation. Consider a swarm system with N second-order agents with the same dynamics, and each agent is modeled as is the velocity-similar state, and u i (t) ∈ R d is the control input determining by the formation protocol, respectively. e given vector ι � [ι T 1 , ι T 2 , . . . , ι T N ] T with ι i ∈ R d is used to determine specific formation structure, which is called a formation control vector. Now, we show the definition of synchronization-based formation. (2) which means that the formation control problem is transformed into the synchronization one by (2). Here, we summarize a lemma as follows.

Lemma 2. Swarm system (1) achieves synchronization-based
Swarm system (1) contains two state components; that is, a position-similar component and a velocity-similar component. On the basis of this fact, the control input u i (t) is constructed as where And K u1 and K u2 are gain matrices with compatible dimensions.
Let K u2 ]; then, it can be found by (3) that us, we propose the following guaranteed-performance formation protocol: where i � 1, 2, . . . , N, Q T � Q > 0, and K T w � K w > 0 is a gain matrix. One can find that the derivative of the communication weight w ij (t) is nonnegative, so the communication weight w ij (t) is adaptively adjusted and positive. Particularly, it may be consistently increasing until agents i and j achieve the desired formation. In this case, the upper bound of w ij (t) is set as c ij . Here, we give the definition of synchronization-based guaranteed-performance formation control. (1) is said to be synchronizationbased guaranteed-performance formation achievable by formation protocol (6)

Definition 2. Swarm system
}. e main objective of this paper is to design gain matrices K u and K w such that swarm system (1) with formation protocol (6) achieves synchronization-based guaranteedperformance formation and to determine an explicit expression of the guaranteed-performance cost J * r . Moreover, the whole motion of swarm system (1) is determined when synchronization-based guaranteed-performance formation is achieved. Remark 1. Formation protocol (6) owns two important features. e first one is that the neighbor set of all agents is time-invariant, but the communication weight w ij (t)(i ≠ j) is not, which is time-varying and can be continuously increasing if the formation error δ ij (t) (i, j ∈ 1, 2, . . . , N { }) between agents i and j is not zero. Particularly, the increasing value w ij (t) (i ≠ j) becomes more larger when the formation error δ ij (t) (i, j ∈ 1, 2, . . . , N { }) between agents i and j becomes big; that is, more control impacts are imposed on these two agents to make them achieve the desired formation more quickly. In this case, this protocol can also be called the adaptive formation protocol. e second one is that formation protocol (6) ensures the formation regulation performance among all agents by the quadratic index function J r although some agents may be unconnected, which indicates that the whole formation performance of swarm system (1) can be regulated by formation protocol (6).

Synchronization-Based Guaranteed-Performance Formation
In this section, firstly, by the time-invariant nonsingular transformation, the formation control problems are transformed into asymptotic stability ones. en, sufficient conditions for synchronization-based guaranteed-Complexity 3 performance formation are proposed on the basis of the Riccati inequality. Finally, an approach is shown to determine the whole motion of a swarm system when synchronization-based guaranteed-performance formation is achieved. (1) can be rewritten in a compact form as where which can be used to write the whole dynamics into an impact form.
Because the formation vector l is constant, one can see (6) and (7), one can obtain that where L w(t) is the time-varying Laplacian matrix of the communication network G. Due to where i � 1, 2, . . . , N, one can derive that us, one can show by (9) and (11) that From Lemma 1, one can find that L w(t) 1 N � 0. Let 0 � λ 1 < λ 2 ≤ · · · ≤ λ N denote the eigenvalues of L w(0) . Because the communication network G owns the connected property, one can find a constant nonsingular matrix where . . , N); then swarm system (12) can be transformed by (13) into two independent parts.
One can set that where the N-dimensional column vector e i (i ∈ 1, 2, . . . , N { }) owns the property that the ith element is 1 and 0 elsewhere. One can find by (16) and (17) that Because U ⊗ I 2d is invertible, it can be found that these two vectors κ 1 (t) and κ e (t) are linearly independent. Due to one has ω(t) � κ 1 (t) + κ e (t). If lim t⟶∞ κ e (t) � 0, then lim t⟶∞ (ω(t) − κ 1 (t)) � 0. From (16), lim t⟶∞ κ e (t) � 0 is equivalent to lim t⟶∞ κ e (t) � 0 since the transformation matrix U ⊗ I 2 d is nonsingular. us, by Definition 1 and Lemma 2, we can obtain the following lemma. e following theorem proposes an approach to determine gain matrices K u and K w such that swarm system (1) with formation protocol (6) achieves synchronization-based guaranteed-performance formation, which is independent of the eigenvalues of the time-varying Laplacian matrix of the communication network. Theorem 1. For any given c > 0, swarm system (1) is synchronization-based guaranteed-performance formation achievable by formation protocol (6) with K u � B T P and K w � PBB T P if there exists a symmetric and positive definite matrix P such that PA + A T P − cPBB T P + 2Q ≤ 0. In this case, an explicit expression of the guaranteed-performance cost is

Complexity
Proof. Construct the following Lyapunov functional candidate: According to P T � P > 0 and c ij ≥ w ij (t), one can find that V(t) ≥ 0. Because the communication network G is undirected, it can be found that the time-varying Laplacian matrix L w(t) is symmetric. Let K u � B T P, then, according to (15) and (20), one can deduce that Because the time-invariant transformation matrix U is orthonormal, one can see that UU T � L N , where L N stands for the Laplacian matrix of a communication network G * and all its edge weight is 1/N; that is, By (6) and (13), it can be shown that Let K w � PBB T P; then it can be derived by (21) and (24) Due to U T L w(0) U � diag λ 2 , λ 3 , . . . , λ N with λ 2 ≤ λ 3 ≤ · · · ≤ λ N being the nonzero eigenvalues of L w(0) , it can be found that One can see that PA + A T P − cPBB T P < 0 if PA + A T P − cPBB T P + 2Q ≤ 0. In this case, it can be obtained that V .
In the sequel, we discuss the impacts of the guaranteedperformance function and determine an explicit expression of the guaranteed-performance cost J * r . It can be shown that Due to UU T � L N , one can see that en, one has where h is a scalar larger than zero and 2Q is introduced in the Riccati inequality. Since λ i > 0 (i � 2, 3, . . . , N), it can be shown by (26) that Due to one can see that From (29) to (32), one can find that If PA + A T P − cPBB T P + 2Q ≤ 0, then one can find by (34) that Due to lim h⟶∞ (c ij − w ij (h)) � 0, one can obtain by (21) that us, we can find by (35) that Furthermore, one can obtain that Due to one has From (21) and (37)-(39), one can obtain that an explicit expression of the guaranteed-performance cost is Moreover, PA + A T P − cPBB T P + 2Q ≤ 0 can guarantee that swarm system (1) is synchronization-based guaranteedperformance formation achievable by formation protocol (6) with K u � B T P and K w � PBB T P. us, the conclusion of eorem 1 can be obtained. Now, we introduce a scalar μ > 0 such that P ≤ μI and one can find that PBB T P ≤ μ 2 BB T when λ max (BB T ) ≤ 1, where λ max (BB T ) stands for the maximum eigenvalue of the matrix BB T . By Schur complement lemma in [34], the following corollary can be obtained directly.  (1) is synchronization-based guaranteed-performance formation achievable by formation protocol (6) with K u � B T P and K w � PBB T P if λ max (BB T ) ≤ 1 and there exist c > 0 and P T �P ≥ μ − 1 I such that AP + PA T − cBB T + 2PQP < 0. In this case, an explicit expression of the guaranteed-performance cost is The Riccati inequality bounded by an identity matrix in Theorem 2 can be transformed into linear matrix inequality conditions by the Schur complement lemma, which can be solved by the feasp solver in LMI's toolbox.
Let x w (t) and v w (t) denote position-similar and velocity-similar components of the whole motion of swarm system (1), respectively. e following theorem determines the whole motion of swarm system (1) with formation protocol (6) when synchronization-based guaranteed-performance formation is achieved. 6 Complexity Theorem 3. If swarm system (1) with formation protocol (6) achieves synchronization-based guaranteed-performance formation, then the motion of swarm system (1) as a whole satisfies that Proof. If swarm system (1) achieves synchronization-based guaranteed-performance formation, then one can find that Due to one can obtain by (14) that From (44) to (46), it can be shown that us, the conclusion of eorem 3 can be obtained. □ Remark 2. In [31,32], sufficient conditions for formation control depend on the global information of swarm systems, such as the Laplacian matrix and its nonzero eigenvalues, etc. In those cases, formation control strategies are not fully distributed. It should be pointed out that the dilation adaptive method can realize fully distributed control as shown in [35], but it is no longer valid when the guaranteed performance is considered and becomes an important constraint for formation design. In the proof of eorem 1, we introduce a translation scalar to eliminate the impacts of nonzero eigenvalues on guaranteed-performance formation control and realize the fully distributed control in the sense that gain matrices of the formation protocol are not associated with the global information of swarm systems. Particularly, the translation adaptive method is closely associated with the undirected property of the communication network, which introduces the Laplacian matrix of a complete graph.
Remark 3. eorem 3 shows an approach to determine the motion of swarm system (1) with formation protocol (6) as a whole, where x w (0) and v w (0) are the average of initial values of position-similar states and velocity-similar states, respectively, and ι w is the average value of all formation control vectors. Two important aspects should be emphasized.
e first one is that the formation control vectors not only determine formation structures, but also impact the whole motion.
e second one is that if the average of initial values of velocity-similar states is equal to zero, then the whole formation is static as the synchronization-based guaranteed-performance formation is achieved. In this case, the center point of the whole formation is jointly determined by the average of initial values of position-similar states and the formation control vectors. However, if the average of initial values of velocity-similar states is nonzero, then the whole formation moves in the uniform linear mode when the synchronization-based guaranteed-performance formation is achieved.

Numerical Simulations
In this section, two numerical examples are provided to demonstrate the theoretical conclusions, where the static whole motion and the dynamic whole motion are dealt with, respectively.
Example 1 (static whole motion). Consider the hexagon formation of six agents in the two-dimensional space. e dynamics of each agent in swarm system (1) is given as follows: Set Q � 0.8I 4 and c � 5; then one can obtain by eorem 1 that And J * r � 320.42. e communication network is presented in Figure 1, where the initial weights of all connected edges are one. e two-dimensional formation control vector ι � [ι T 1 , ι T 2 , ι T 3 , ι T 4 , ι T 5 , ι T 6 ] T is given in Table 1, where initial values of position-similar states and velocity-similar states are also shown. It can be found that the average value of initial states of velocity-similar states is equivalent to zero. Figure 2 shows the state snapshots of six agents, and the whole motion is given at J r (t)s, J r (t)s, J r (t)s, and J r (t)s, where six agents are represented by asterisks, plusses, circles, x-marks, pentagrams, and hexagrams, respectively, and the center point of the whole motion is denoted by red circles, which is determined by eorem 3. It can be found that all agents achieve a time-invariant hexagon formation and the center point of the whole motion is static. Figure 3 shows that the practical performance function J r (t) tends to a finite  Serial number Center 8 Complexity and stable value less than J * r , which is determined by eorem 1. It can be found that the desired structure of this swarm system is achieved and the expected performance is guaranteed; that is, this swarm system achieves synchronization-based guaranteed-performance formation.
Example 2 (dynamic whole motion). Consider the pentagram formation structure of five agents in the three-dimensional space. In this case, each agent is modeled as Let Q � I 4 and c � 3; then gain matrices can be determined from eorem 1 as And J * r � 1214.31. e communication network with five agents is shown in Figure 4, where the initial weights of all connected edges are set as 1, which is regulated according to formation protocol (5). Table 2 shows the three-dimensional formation control vector ι � [ι T 1 , ι T 2 , ι T 3 , ι T 4 , ι T 5 ] T and initial values of position-similar states and velocity-similar states, where the average of initial values of velocity-similar states is nonzero. Center     In Figure 5, the state snapshots of five agents and the whole motion are revealed at J r (t)s, J r (t)s, J r (t)s, and J r (t)s, where five agents are denoted by asterisks, plusses, circles, x-marks, and hexagrams, respectively, and the center point of the whole motion is denoted by red circles. One can see that all agents achieve a pentagram formation, where the pentagram is fixed, but the center point of the whole motion is time-varying and is consistent with eorem 3. Figure 6 reveals the practical performance function J r (t) and the guaranteed-performance cost J * r , which is determined by eorem 1. One can see that this swarm system achieves the desired synchronization-based guaranteed-performance formation.

Conclusions
Synchronization-based guaranteed-performance formation design problems for second-order swarm systems were addressed. A guaranteed-performance formation protocol was proposed, where the connected-edge weights are adaptively regulated on the basis of state errors and formation errors among neighboring agents. Furthermore, on the basis of the Riccati inequality, synchronization-based guaranteed-performance formation criteria were proposed, and an explicit expression of the guaranteed-performance cost was given.
ose criteria are fully distributed in the sense that the gain matrix design of the formation protocol is independent of global information of swarm systems. Moreover, the whole motion of a swarm system was determined, where the impacts of position-similar states, velocity-similar states, and formation control vectors are shown, respectively. Particularly, it was shown that when the synchronizationbased guaranteed-performance formation is achieved, the formation center of the whole swarm system can be static and can also move in the uniform linear mode. Moreover, it is a very interesting future work to establish a unified    Complexity criterion, which can compare similar methods associated with the guaranteed-performance formation control.

Data Availability
e simulation data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.

Authors' Contributions
Hongtao Dang and Yichen Du conceptualized the study; Lingyun Kong was responsible for methodology; Hui Yao validated the study; Jianxiang Xi contributed to formal analysis; Hongtao Dang investigated the study; Yichen Du was responsible for writing and preparation of original draft; Hui Yao and Lingyun Kong were responsible for writing-review and editing ; Jianxiang Xi supervised the study; Hongtao Dang was responsible for project administration; and Hongtao Dang acquired funding. All authors have read and agreed to the published version of the manuscript.