Formation Control for Unmanned Aerial Vehicles with Directed and Switching Topologies

Formation control problems for unmanned aerial vehicle (UAV) swarm systems with directed and switching topologies are investigated. A general formation control protocol is proposed firstly. Then, by variable transformation, the formation problem is transformed into a consensus problem, which can be solved by a novel matrix decomposition method. Sufficient conditions to achieve formation with directed and switching topologies are provided and an explicit expression of the formation reference function is given. Furthermore, an algorithm to design the gainmatrices of the protocol is presented. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.


Introduction
In the past decades, unmanned aerial vehicles (UAVs) have been widely used in civilian and military areas, such as surveillance and reconnaissance [1,2] and target search and localization [3].Since the performance of a team of UAVs working cooperatively exceeds the performance of individual UAVs, formation control of UAVs is of importance and has received a lot of attention.
The formation control of UAVs has been studied with many different methods, such as leader-follower [4], behavior [5], and virtual structure-based [6] approaches.Recently, with the development of consensus theory [7][8][9][10][11][12][13][14][15], some related methods are also used to deal with the formation control problems of UAVs.Consensus means that all agents reach a common state.The results in [16] show that consensus approaches can be used to deal with formation control problems, and leader-follower, behavior, and virtual structurebased formation control approaches are special cases of consensus-based approaches.
Based on consensus method, Abdessameud and Tayebi [17] proposed controllers for UAV swarm systems to achieve formation in the presence of communication delays.A consensus protocol together with an output feedback linearization method is presented in [18] such that the UAV swarm systems can achieve partially time-varying formation.Besides, indoor and outdoor flight experiments for quadrotor swarm systems to achieve formation by consensus approaches are carried out in [19] and [20], respectively.Based on consensus theory, we know that the achievement of formation depends on not only the individual UAV dynamics but also the structure of the networks between UAVs which can be modeled by directed and undirected graphs.However, the interaction topologies between UAVs in [19,20] are assumed to be fixed.In practical applications, the interaction topologies of UAV swarm systems may be switching due to the fact that the communication channel may fail and new channels may be created during flight.Time-varying formation control for UAV swarm systems and high-order LTI systems with switching interaction topologies are studied by Dong [20,21], but the topologies are assumed to be undirected.To the best of our knowledge, there is still work to do on formation control of UAV swarm systems with directed and switching topologies.
In this paper, we aim to solve the formation problem of UAV swarm systems with directed and switching topologies.Compared with the existing results, the assumptions of the communication topology are quite general.The remainder of this paper is organized as follows.In Section 2, some necessary concepts and useful results on graph theory are summarized and the problem formulation is given.Main theoretical results are proposed in Section 3. In Section 4, a numerical simulation is presented.Section 5 is the conclusion.

Preliminaries and Problem Description
2.1.Notations and Graph Theory.In this paper, the following notations will be used. × and  × denote the set of  ×  real and complex matrices, respectively.For  ∈ , the real part is Re().⊗ denotes the Kronecker product.  is the identity matrix of order .For a square matrix , () denotes the eigenvalues of matrix . > 0 ( ≥ 0) means that  is positive definite (positive semidefinite).max{()} (min{()}) denotes the largest (smallest) eigenvalue of the matrix .
A directed graph  = (V, E, A) contains the vertex set V = {1, 2, . . ., }, the directed edges set E ⊆ V × V, and the adjacency matrix A = [  ] × with nonnegative elements   .  = 1 if there is a directed edge from vertex  to ;   = 0, otherwise.The Laplacian matrix of the graph  is defined as  = [  ] × , where   = ∑  ̸ =   and   = −  ( ̸ = ).Zero is an eigenvalue of  with the eigenvector 1  .A directed graph is said to have a spanning tree if there is a vertex such that there is a directed path from this vertex to every other vertex.
Lemma 1 (see [8]).Zero is a simple eigenvalue of  and all the other nonzero eigenvalues have positive real parts if and only if the graph has a directed spanning tree.

Problem Description.
Consider UAV swarm systems with  UAVs.The interaction topology of the UAV swarm systems can be described by a directed graph , in which UAV  can be denoted by a vertex and the interaction channel from UAV  to UAV  can be denoted by an edge.Compared with the attitude dynamics, the trajectory dynamics of each UAV have much larger time constants, which means the attitude controller and trajectory controller can be designed separately.On the formation level, only trajectory control needs to be considered.Therefore, in this brief, the dynamics of each UAV can be described by the following double integrator [18,21,22]: where  = 1, 2, . . ., ,   () ∈   and V  () ∈   denote the position and velocity vectors of UAV , respectively, and   () ∈   are the control inputs.In the following, for simplicity of description, it is assumed that  = 1, if not otherwise specified.Therefore, UAV swarm systems (1) can be rewritten as where ]  ; then one has that if ℎ V () are not equal to zeros, the formation is time-varying.
Definition 2 (see [21]).UAV swarm systems (2) are said to achieve formation ℎ() if there exists a function where () is called a formation center function.
Before the consensus analysis of system (8), the following lemmas and definition are introduced.
Lemma 4 (see [23]).For a Laplacian matrix  of graph  and a full row rank matrix  defined as there exists a matrix  such that  = .Further, if the graph has a directed spanning tree,  is of full column rank and the eigenvalues of  are equal to the nonzero eigenvalues of .
Lemma 5 (see [24]).Suppose that the eigenvalues of  ∈  × have positive real parts; then there exists a positive definite matrix  > 0 such that Definition 6.For a switching signal () over time interval [0, ), the average dwell time of the switching signal is defined as   = /(  () + 1), where   () denotes the number of the switches.
Remark 7. In [11,25], the definition of the average dwell time of a switching signal () over time interval [0, ) can be described as follows.If there exist two positive numbers  0 and   such that   () ≤  0 + /  , where   () denotes the number of the switches,   is called the average dwell time.
It is inaccurate to give the definition by an inequality, but, according to Definition 6, it can be seen that   () ≤ /  .
Proof.Consider the following piecewise Lyapunov candidate of system (13): where  is a solution of inequality (14) and  () are feasible solutions of (12).
According to ( 23) and ( 24), one has Note that  > ln ℎ/  ; one has () → 0 as  → ∞.This means that the consensus problem of system (8) is solved.Furthermore, formation for UAV swarm systems (2) with directed and switching topologies is achieved.
Remark 10.As can be seen, the formation center is discontinuous due to the switching of the communication topology.
In addition,  1 can be used to design the motion modes of the formation center function.If  1 = 0, protocol (4) becomes a totally distributed controller. 2 has no effect on the formation center function.
Remark 11.Compared with [21,22], the interaction topologies are more common.Formation for UAV swarm systems with directed and switching topologies is solved.Furthermore, the gain matrix was designed by solving an LMI, which is simpler than solving an algebraic Riccati equation in [22].In fact, undirected topologies are just special cases of directed topologies.So the algorithms presented in this paper are applicable to those cases in [21,22].
Based on the above results, a design procedure of protocol (4) can be summarized as follows.First, choose  1 to design the motion modes of the formation center by assigning the eigenvalues of (+ 1 ).Then design  2 using the conclusion of Theorem 8.

Examples
In this section, we provide an example to illustrate the effectiveness of the above theoretical results.UAV swarm systems consisting of four agents are considered.The system matrices are defined as where  1 ,  2 ,  3 , and  4 stand for east position, east velocity, north position, and north velocity.The directed communication topologies are given in Figure 1.Clearly, each topology contains a directed spanning tree.The switching signal is shown in Figure 2.
Thus, we can obtain  min = 1 and then choose  = 0.9.Further, we can get that  1 = 3.4175,  2 = 0.2009, and ℎ = 17.0075.From Figure 2, we can get that the average dwell time is 1.25 s and then choose  = 5.
Assign the eigenvalues of ( +  1 ) at (±, ±); we get Solve LMI (14) with  = 0.9 and  = 5; a feasible solution can be obtained.Accordingly, we can get  Choose the following time-varying formation: If ℎ() is achieved, both the positions and velocities of the four UAVs locate at the vertexes of a rotating parallelogram, respectively.Choose initial states of four UAVs as Figure 3 shows the trajectories of the difference of UAV states and time-varying formation, which are denoted by solid line, dotted line, dash-dotted line, and dashed line.And the bold dotted line denotes the formation center trajectory.It is obvious that the differences achieve consensus after about  = 7 s and converge to the formation center.From Definition 2, one can obtain that the time-varying formation problem is solved.Figure 4 shows the snapshots of four UAV positions at different time.It can be seen that, after  = 6 s, the UAV swarm systems achieve a time-varying parallelogram formation.Therefore, the time-varying formation is achieved under the directed and switching topologies.

Conclusions
Formation problems for UAV swarm systems with directed and switching topologies are studied.The average dwell time of the switching topologies is introduced, based on which an LMI-based method to design the protocol is proposed.Though the UAV swarm systems can achieve the specified formation with the presented method, there are still problems in real application.As mentioned in Assumption 3, each of the switching topologies is supposed to have a spanning tree, which may not be applicable, so there is still work to do in our future work.

Figure 3 :
Figure 3: Difference of UAV state and time-varying formation.