Robust flocking for non-identical second-order nonlinear multi-agent systems

This paper investigates the robust flocking problem for second-order nonlinear systems with a leader and external disturbances. In contrast with most of second-order systems in the literature, the intrinsic dynamics here are nonlinear and non-identical that depend not only on the velocity but also on the position, which is more realistic. Moreover, the interaction topology is undirected and switching. Provided that the leader’s velocity may be constant or time-varying, two distributed flocking control laws have been proposed for two cases to make the differences of the velocities between all followers and the leader approach to zero asymptotically. The proposed distributed flocking control laws are both model-independent which results in the effectiveness of the controllers to cope with the different intrinsic dynamics of the followers and the leader under some assumptions on boundedness of several states. An example is given to illustrate the validity of the theoretical results.


Introduction
Multi-agent systems have attracted enormous attention from a large variety of fields including social science, animal behavior, smart grids, physics, biology, intelligent transportation and some areas in engineer, due to its vastly useful application in practice. A multi-agent system consists of some agents communicating with each other locally by some kind of link, aiming to complete various controller goals via local interaction of designated agents by designing some suitable controllers. In control theory, multi-agent systems have been focused on a lot in several directions, such as flocking [1][2][3][4][5][6], sensor networks, spacecraft formation flying [7][8][9][10], consensus [11][12][13][14][15][16][17], rendezvous [18], axial alignment, cooperative surveillance, etc.
Among these different directions, flocking is a form of collective behavior of numerous interacting agents with the purpose of reaching a common group mission. This behavior is ubiquitous in nature like a self-organized *Correspondence: houshengsu@gmail.com 3 School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China Full list of author information is available at the end of the article group with the ability to coordinate group behavior, such as flocking of birds, fish, and bees, etc. It has a tremendous number of potential applications in reality. For instance, by employing flocking of cooperative unmanned aerial vehicles (UAVs), one nation could perform necessary military missions such as reconnaissance, etc. Therefore, it is of huge significance for researchers to concentrate on the study of flocking. As early as in 1986, three flocking rules were brought forward by Reynolds based on the animals' behavior as follows [19]. the simple second-order dynamics have been studied. And both the free-space and the obstacle-space have been taken into account in several cases. In [20] it has been shown that the flocking algorithm in [1] still drives all agents to the same velocity even if only a part of agents could receive the information of the leader's velocity. A flocking algorithm with multi-target tracking for multiagent systems has been provided in [21], in which the authors proposed a method to implement the multi-target missions under the introduction of two types of potential functions. By using virtual force and pseudo-leader mechanism, the authors in [22] have proposed an approach to determine the pseudo-leaders among all agents. A unified optimal control framework has been proposed for flocking of second-order multi-agent systems in [23] and the authors utilized a sort of new inverse optical control strategy to procure the optimal control law along with the occurrence of a challenging optimal problem. The second-order systems with nonlinear intrinsic dynamics have been discussed for the flocking problem with a general switching graph in [24] by using a leader-following strategy and pinning control. The more general secondorder systems with nonlinear intrinsic dynamics have been considered for flocking problem with a virtual leader in [2], where all agents have the same intrinsic nonlinear dynamics as well as the virtual leader governed by locally Lipschitz nonlinearity. Apart from the second-order point models aforementioned, nonlinear Euler-Lagrange systems have been investigated for the flocking problem in [25] and the flocking is reached by designing an adaptive controller. For more details on flocking problem, we refer readers to the survey [26,27]. Based on these aforesaid, this paper investigates the robust flocking problem for second-order systems with a leader on undirected switching networks. Two distributed flocking control protocols are presented to achieve flocking asymptotically for two cases according to if the leader's velocity is constant or time-varying. Compared with the existing flocking problem in the literature, the contributions of this paper are threefold. First, the intrinsic dynamics are non-identical, that is, the intrinsic dynamics of all followers are all different from each other. Second, the external disturbances are considered in the dynamic equations of followers. Third, the intrinsic dynamics are nonlinear, which rely not only on the velocity but also on the position. Additionally, it should be noted that the proposed distributed flocking control laws are modelindependent which results in the effectiveness of the controller to tackle the different intrinsic dynamics of the followers and the leader under some assumptions on boundedness of several states by virtue of Lyapunov theory.
The remainder of this paper is organized as follows. Section 2 gives some notations and the background of graph theory and nonsmooth analysis. The problem statement is presented in Section 3. And main results are provided in Section 4 which are divided into two cases: one is the case with constant leader's velocity, and the other is the case with time-varying leader's velocity. An illustrative example is given to demonstrate the effectiveness of the theoretical results in Sections 5 and 6 draws the conclusion.

Notation
Let R n and R n×m be the sets of n-dimensional vectors and n × m matrices, respectively. R + denotes the set of nonnegative real numbers and diag{a 1 , a 2 , . . . , a n } is the diagonal n × n matrix with diagonal entries equal to a 1 , a 2 , . . . , a n . 0 is a vector or matrix with compatible dimension with all entries 0 and 1 is denoted analogously. For a vector x ∈ R n , sgn(x) denotes the signum function for every element of x. And ||x|| 1 := |x 1 | + |x 2 | · · · + |x n | is the 1-norm of a vector. Given a matrix M, λ min (M) and λ max (M) are, respectively, the smallest and largest eigenvalues of M. ||x|| and ||A|| stand for the standard Euclidean norm of a vector x ∈ R n and the induced norm of a matrix A ∈ R n×n , respectively. Subscript T represents the transpose of a vector or matrix.

Graph theory
A graph with N nodes is defined by G N = (V N , E N ) consisting of a set of vertices V N = {v 1 , v 2 , . . . , v N } and a set of edges E N ⊆ V N × V N . An edge (v i , v j ) ∈ E N means node j could receive information from node i, in which node i is called a neighbor of node j. A graph is called undirected if any (v i , v j ) ∈ E N implies (v j , v i ) ∈ E N , and directed otherwise. A directed path in a directed graph is a sequence of edges of the form (i 1 , i 2 ), (i 2 , i 3 ), . . ., and an undirected path in undirected graph is defined analogously. An undirected graph is called connected if there is an undirected path between every pair of distinct nodes. The neighbor set of node i is denoted by Combined with the leader, the followers graph is expanded to be a new graph In this paper, the results are based on the following assumption on the communication graph.

Assumption 1
The graph G N is undirected and the leader in G N+1 (t) has directed paths to all followers at the initial time t = 0.
It is worth noting that Assumption 1 is a weak condition, which is only postulated at the initial time, instead of the whole time duration which has been widely used in many existing works. Compared with the whole duration, this assumption is easily satisfied at the initial time by contrived deployment, which has also been employed in such as [2,28].
Regarding the flocking problem, the following lemma will be used later.

Lemma 1 ([29])
If the graph G N is undirected and the leader in G N+1 has directed paths to all followers, then M is symmetric positive-definite.

Nonsmooth analysis
Given the differential equation ( [30]) where f : R n × R → R n is measurable and essentially locally bounded. A vector function x(·) is called a Filippov solution of (1) if x(·) is absolutely continuous anḋ denotes the smallest convex closed set containing all the limit values of the vector-valued function f (x i , t) for x i → x and a constant t. For a locally Lipschitz function V : The following nonsmooth version of LaSalle's theorem is useful later.

Problem statement
Consider a multi-agent system consists of N followers labeled as 1 to N. In the meantime, the dynamics of each follower i satisfy the following non-identical second-order nonlinear differential equations ( [17,29,31]) where x i ∈ R n , v i ∈ R n , and u i ∈ R n are the position, velocity, and control input of follower i, respectively. Furthermore, w i ∈ R n is the continuous external disturbance of follower i. And f i : R n × R n → R n is continuously differentiable nonlinear function.
Meanwhile, there exists a leader labeled as agent 0 besides those N followers, whose state is represented by position x 0 and velocity v 0 with the derivative of v 0 being bounded, i.e., ||v 0 || ≤ H 1 , where H 1 > 0 is a constant. Note that v 0 could be either constant or time-varying. For flocking of multi-agent systems, the objective is to achieve the following properties by designing distributed cooperative control law, which are also known as three flocking rules of Reynolds [19]: The agents attempt to stay close to nearby flockmates.

Collisions are avoided among all agents.
For the purpose of reaching flocking for multi-agent systems, the following assumption is utilized necessarily.
Note that w i needs to be bounded for the sake of robust flocking since its exact value could not be obtained by followers that have only local interactions. In addition, the trajectory of the leader is not necessary to be exactly known as long as its state is bounded, which is a sort of generic assumption. Therefore, this scenario is really a generation of a variety of results in the literature. For instance, the leader has the same intrinsic dynamic as the followers in [22]. In reality, it is natural to require the states of all agents to be bounded due to the consideration of safety and restricted field, etc. However, the bound could be any large what we need in practice.
For robust flocking, the connectivity-preserving mechanism (see [2,18]) for undirected switching graph, roughly speaking, guarantees that one follower or the leader wills till be a neighbor of another follower for any time t > 0 as long as it is the case at the initial time t = 0. The mechanism is described as follows.
i) Initial edges are generated by ii) Let φ i (j)(t) ∈ {0, 1} represent whether or not follower j is a neighbor of follower i, which is defined as follows.
Note that in this paper is chosen to equal the sensing radius of followers, i.e., = R, which prevents new edges from being added to the initial graph at initial time 0.
The following definition states the notion of potential function.

Definition 1
The potential function V ij is a continuously differentiable, nonnegative function of ||x i − x j || satisfying the following conditions.

V ij achieves its unique minimum when
For example, V ij could be given as Similar to Lemma 3.1 in [32], we have the following lemma.

Lemma 3 Let V ij be defined in Definition 1. The following equality holds:
For simplicity, let n = 1 in this paper. Note that it is straightforward to extend the results to the case of general dimension by means of the Kronecker product ⊗.

Main results
This section focuses mainly on the robust flocking problem for multi-agent system (2) with the velocity of the leader being time-varying. In this case, the distributed flocking algorithm is proposed as where k 1 , k 2 , k 3 are positive control gains to be determined later, andv i0 is the estimate of the leader's velocity by follower i withv 00 = v 0 , satisfying the following adaptation law: Physically speaking, the first term on the right-hand side of (7) is used to preserve the initial connectivity among agents at all times, the second term is designed to deal with the nonlinearity and disturbance in the agent's dynamics, the third term is introduced to counteract the impact of time-varying v 0 , and the last term is employed to align all agents' velocities.
Under input (7), system (2) can be rewritten aṡ means the differential inclusion defined in last section, and a.e. stands for "almost everywhere". From Assumption 2, it is obvious to see that ||L 1 || is bounded and in view of the mean value theorem, there exists a positive nondecreasing function α 0 such that ||N 1 || ≤ α 0 (||y||)||y|| (see [29]), where y = (x T ,ṽ T ) T . After denoting α(z) := α 0 (z)z, z ≥ 0, which is a positive nondecreasing function as well, one has ||N 1 || ≤ α(||y||). As a result, it follows from Assumption 1 that there exists a positive constant H 2 such that ||y|| ≤ H 2 and hence ||N 1 || ≤ α(H 2 ). We are now ready to given the flocking result. Under Assumptions 1 and 2, the distributed flocking algorithm (7) for multi-agent system (2) can ensure that velocity differences between all followers and the leader ultimately converge to zero, i.e., lim t→∞ ||v i (t) −

Theorem 1
Proof Let t 0 = 0, t 1 , t 2 , . . . be the switching time slots of interaction networks for system (2). Constructing the Lyapunov function candidate as Then the generalized derivative of V is calculated aṡ where we have used Lemma 3 and the factẋ 0 = 0. (12) directly leads to that . As a consequence, V is bounded in time [ t 0 , t 1 ). Moreover, in view of the definition of potential function, one follower is always a neighbor of another follower or the leader once it is the case at the initial time t = 0, since the potential function and thereby V will become infinite as ||x j − x i || → R. Hence, no existing edge will be lost in any time t ∈[ t 0 , t 1 ]. Additionally, from the definition of connectivity-preserving mechanism, no new edges are added to the interaction structure of initial graph at time t = 0. Consequently, inequality (12) is still correct for any time t ∈[ t 0 , t 1 ]. Similarly, it can be concluded that the network structure in any time interval [ t k , t k+1 ), k ≥ 1 is always the same as that of initial graph at time t = 0. That is, Assumption 1 holds for all time t ≥ 0.
To proceed, consider the estimator (8). Using the notationv 0 and nonsmooth analysis, (8) can be rewritten asv Constructing the Lyapunov function candidate as whose generalized derivative is derived aṡ which is negative if k 3 > √ NH 1 . As previously shown, Assumption 1 holds for all time t ≥ 0 and thus (15) is correct with fixed M for all time t ≥ 0. And M is positive definite by Lemma 1. With reference to Theorem 3.1 in [30], one has lim t→∞v 0 = 0. Moreover, by inequality (15), it can be obtaineḋ which follows that Therefore, vanishing the right-hand side yields that v i0 → v 0 in finite time T 0 , that is,v i0 = v 0 , ∀t ≥ T 0 , i = 1, 2, . . . , N, where T 0 is defined by As for T 0 in (17), there must exist a constant l ∈ N such that T 0 ∈[ t l , t l+1 ). At this stage, partitioning the time into two parts: t ∈[ 0, t l ) and t ∈[ t l , ∞). When t ∈[ 0, t l ), (12) has shown that The generalized derivative of V in (11) can be evaluated aṡ which is negative if k 2 ≥ ||L 2 || + α(H 2 ). Notice that {y(T)| ||y(T)|| ≤ α −1 (k 2 − ||L 1 ||)} is a compact set anḋ V = 0 implies thatṽ = 0, i.e.,ṽ i = v 0 . Invoking Lemma 2, one concludes that lim t→∞ ||v i −v 0 || = 0, and the collision is avoided since V will become infinite as ||x i − x j || → 0 which contradicts the boundedness of V. This completes the proof. [2,[20][21][22][23][24], this paper is the first to address the flocking problem for multiple agent systems with non-identical intrinsic dynamics and external disturbances. The main difficulties lie in the controller design and its convergence analysis for the studied problem.

Remark 1 In comparison with existing results, such as
As a special case, when the velocity of the leader is constant, i.e.,v 0 = 0, a simpler distributed flocking algorithm for (2) can designed as where notations are the same as exploited in (7). Besides, the adaptation law (8) The controller (19) has the same physical meaning as that of (7), and the main difference is that the third term on the right-hand side of (7), which are introduced to offset the time dependence of v 0 , is unnecessary here since v 0 is constant.
Similarly, as argued earlier in the proof of Theorem 1, one can obtain that all the estimatorsv i0 will converge to v 0 , i = 1, 2, . . . , N in finite time T 1 := λ max (M)(v 0 ) T Mv 0 /λ min (M), and the similar result to Theorem 1 is provided as follows.
Theorem 2 Under Assumptions 1 and 2, the distributed flocking algorithm (19) for systems (2) can ensure that velocity differences between all followers and the leader will ultimately converge to zero, i.e., lim t→∞ ||v i (t)−v 0 || = 0, i = 1, 2, . . . , N, and the collision of all agents is avoided if the following inequality holds:

An illustrative example
In this section we mainly present an example to illustrate the effectiveness of the theoretical results. In the example, the flocking problem consists of 8 followers labeled as i, i = 1, 2, . . . , 8 and a leader labeled as 0, in which each agent has the following intrinsic dynamics that is adapted from the example in [2]: (22) where , v 03 ) T with the initial position x 0 (0) = (4, 5, 3.5) T and the initial velocity v 0 (0) = (2, 2.5, 3) T . Furthermore, the external disturbances w i is given by This example is related to the following potential function Meanwhile, the initial interaction topology is given by For the estimatev i0 of the leader velocity v 0 , the initial value is given byv i0 (0) = (3, 1.5, 0.6) T , i = 1, 2 . . . , 8. For some parameters, let k 1 = 2, k 2 = 12, k 3 = 15, and consider the directed graph by making the sensing radius R = 10. Figure 1 shows that the estimates of leader velocity definitely converge to the actual leader velocity in finite time, and Fig. 2 shows the initial position and velocity structure, from which it is obvious to see that all followers are moving in different directions. And Fig. 3 presents that all followers match the velocity of the leader, which reachs the velocity matching as a flocking rule, and at the same time the collision avoidance and flocking centering rules are achieved as shown in Fig. 3. Furthermore, Fig. 4 precisely depicts the velocity matching rule, which means that the velocities of all followers will be equal to the velocity of the leader.

Conclusion
The robust flocking problem for second-order systems is investigated with external disturbances in this paper, in which the intrinsic dynamics are all distinct from each other for all agents. Moreover, the intrinsic dynamics, which are not like most of the second-order systems discussed in the literature, are nonlinear that depend not only on the velocity but also on the position which is more practical. Two distributed flocking control laws have been proposed to make the differences of the velocity between all followers and the leader approach to zero asymptotically based on that the leader's velocity may be constant or time-varying. The proposed distributed flocking control laws are both model-independent which results in the effectiveness of the controllers to cope with the different intrinsic dynamics of the followers and the leader under some assumptions on boundedness of several states. Finally, an example is presented to illustrate the efficiency of the theoretical results. Future directions include the investigation of finite-time robust flocking by using continuous-time controllers, and the scenario with stochastic and possibly unbounded disturbances.