Finite-Time Formation of Unmanned Aerial Vehicles with Switching Topologies and Disturbances : An Average Dwell Time Approach

This paper investigates the finite-time formation problem of unmanned aerial vehicles (UAVs) with switching topologies and external disturbances. The formation problem is first transformed into the finite-time stability and bounded problems of a switched system, respectively. In particular, the finite-time unachievable topology is concerned for unreliable information exchanges. By employing the average dwell time (ADT) method, sufficient criteria are established to deal with the switching topologies such that the desired time-varying formation can be achieved in finite time. Then, the topology-dependent controller can be designed in terms of matrix technique. Finally, an illustrative example is given to verify the effectiveness of our proposed formation strategy.


Introduction
During the past decade, the unmanned aerial vehicles (UAVs) have grown from academic researches into practical applications capable of various missions in both civilian and military areas [1][2][3].Recently, with the substantial progresses in swarm intelligence, cooperation issues of multiple UAVs and robots have been receiving increasing attention for distinguishing advantages.Based on information exchanges according to the communication topology, a group of UAVs can work collectively to meet the need for high robustness, reliability, flexibility, and efficiency [4][5][6][7][8][9][10].In particular, the formation problem is one of the most hot research topics for UAVs, which means that the UAVs can keep a certain flying configuration while accomplishing the tasks.Note that the communication among the UAVs can play a very significant role in formation since the information exchanges are conducted via the communication networks according to the communication topology [11][12][13][14][15].However, it is worth mentioning that the communication topology may change at time due to unreliable communication networks or security reason, and this may lead to unpredictable mission failure or performance degradation.As a consequence, the UAVs with switching topologies are of theoretical importance and practical background.So far, there are different remarkable approaches reported in the literature and the references therein [16][17][18].
On another research front line, the switched systems have been receiving increasing attention due to their novel ability in describing practical systems with mode switchings [19][20][21][22].In particular, as one of the most effective analysis and synthesis methods for switched systems, the average dwell time (ADT) has been widely utilized.In comparison to the dwell time method, the ADT can further reduce the conservatism for each subsystem [23][24][25].Therefore, a natural question arises from the prior discussions: can we introduce the merit of ADT of switched systems to deal with the switching topologies?The answer to this question is challenging, but the efforts can lead to encouraging results.Furthermore, in most applications, the stability is required to be achieved over a finite-time interval, which gives rises to the researches on finite-time stability, and burgeoning investigations on finite-time consensus problems of multiagent systems with switching topologies have been made [26][27][28].However, it is worth mentioning that the unachievable switching topologies are not concerned in most existing results.It is noted that there is also a need for the finite-time formation of UAVs by taking into account its realistic background.Until now, to the best of the authors' knowledge, there still exists a considerable gap in solving the formation problem of UAVs with switching topologies based on the switched system theory, let alone those with unachievable topologies.The aim of this article is to make one of the first attempts to shorten such a gap.
Inspired by the ADT method in switched systems, in this paper, the finite-time formation of UAVs with switching topologies and disturbances is studied in terms of the ADT approach.Compared with the existing results, the main contributions of our paper can be summarized as follows.(1) The finite-time formation problem of UAVs with unachievable switching topologies is investigated for the first time.More precisely, the switching topologies are categorized into two types: the achievable topologies and the unachievable topologies.The standard of category relies on whether the finitetime formation can be reached for a fixed topology.For example, it can be verified that the directed topology without a directed spanning tree is unachievable topology.This can further extend the applications of switching topologies and is more practical for UAVs.It can be found that the established results with unachievable topologies are more practical in the applications.(2) The disturbances of UAVs are also considered for formation robustness with the finitetime boundedness problem.On the basis of ADT with the multiple Lyapunov function (MLF) method, sufficient conditions are established for ensuring the finite-time formation and the desired topology-dependent controller gains are obtained accordingly.
The remainder of our paper is given as follows: Section 2 introduces the finite-time formation control problem of UAVs with some essential preliminaries.In Section 3, the main analysis and synthesis procedures are provided with the ADT method.Section 4 presents the numerical example to demonstrate the effectiveness of our proposed formation strategies.Finally, the paper is concluded in Section 5. Notation 1. R n and R m × n denote the n dimensional Euclidean space and the space of m × n real matrices, respectively.A − B ≻ 0 A − B ≺ 0 denotes that A − B is positive definite (negative definite).λ max A and λ min A denote the maximum and minimum eigenvalues of matrix A, respectively.ℒ 2 0, ∞ denotes the space of squareintegrable vector functions over 0, ∞ .A ⊗ B stands for the Kronecker product.* represents the ellipsis symmetry terms in symmetric block matrices.All matrices are compatible for algebraic operations.

Problem Formulation and Preliminaries
2.1.The UAV Dynamics.Consider a group of UAVs with the following kinematics and dynamics: where x i , y i , z i T denotes the position of the ith UAV, v i represents the ground speed, a i stands for the flight-path angle, b i is the heading angle, T i denotes the engine thrust, D i represents the drag, m i denotes the mass, g stands for the acceleration due to gravity, L i is the lift force, n gi = L i /m i g denotes the g-load value, ϕ i represents the banking angle, and d ki , k = 1,2,3 stands for the vortex disturbances.
By applying the feedback linearization method as where u xi , u yi , u zi T denotes the virtual acceleration control input and w xi , w yi , w zi T represents the disturbances, the relationships between virtual control variables and the actual control variables are given by with u xyi = u xi cos b i + u yi sin b i , a i = arcsin z i /v i , and b i = arctan y i /x i .Consequently, by denoting International Journal of Aerospace Engineering the UAV dynamics can be obtained as follows: where 2. Graph Theory.The directed graph G = V , E, A is adopted to describe the communication topology of the UAVs.A = a ij ∈ R N×N represents the weighted adjacency matrix with N and E denote the sets of nodes and edges, respectively.The L = l ij N×N denotes the Laplacian matrix with If G has a directed spanning tree, then 0 is the eigenvalue of L.More details of the algebraic graph theory can be found in [29].
The switching topologies are studied in this paper.Let G σ t σ t ∈ S = 1, 2, … , n denote all possible switching graphs, where σ t represents the index set for all switching graphs.The switching signal σ t : 0,∞ → p, and its value is the index of the graph at time t.The switching sequence is t 1 , t 2 , … , where t h , t h+1 , h = 1, 2, … , is uniformly bounded nonoverlapping [30].
Moreover, for any possible Remark 1.It is worth mentioning that when G σ t is achievable (L σ t has a directed spanning tree), M σ t is of full column rank and the eigenvalues of EM σ t are the same as that of L σ t .When G σ t is unachievable, the above conditions of M σ t will not be satisfied.Without loss of generality, suppose that there are r achievable topologies with σ t = 1, 2, … , r which means that the formation can be achieved with the L σ t and n − r unachievable topologies with r + 1 and r + 2, … , n, otherwise.All the possible communication topologies can be divided into the achievable set denoted as AT and the unachievable set denoted as UAT .

Finite-Time Formation. For the time-varying formation problem, denote h t
Consequently, the formation controller for each UAV can be designed as follows: where h i t = Ah i t + Bμ ih t and K 1σ t and K 2σ t are topology-dependent controller gains to be determined, respectively.As a result, the closed-loop dynamics of the UAVs can be obtained by In addition, note that e t = ξ t − h t ; then, it follows that Premultiplying both sides of ( 13) by E ⊗ I yields that where ε t = E ⊗ I e t denotes the disagreement of e t and ω t = E ⊗ I w t , respectively.
To this end, the following definitions are given for later use.
3 International Journal of Aerospace Engineering Definition 1 (average dwell time, ADT) [31].For any T > t ≥ 0, denote N σ T, t as the number of switching over t, T .σ t is said to have an average dwell time (ADT) τ a > 0, if there exist positive numbers N 0 such that Definition 2 (finite-time formation, FTF) [32].Given constants c 1 > 0, c 2 > 0, and T > 0, with c 1 < c 2 , matrix R > 0, and a switching signal σ t ∈ S, the finite-time formation is said to be achieved with c 1 , c 2 , T, R, σ t , if it holds that Definition 3 (uniformly finite-time formation, UFTF) [32].
Given constants c 1 > 0, c 2 > 0, d w ≥ 0, and T > 0, with c 1 < c 2 , matrix R > 0, and a switching signal σ t ∈ S, the finitetime bounded formation is said to be achieved with Definition 5 (uniformly finite-time bounded formation, UFTBF) [32].Given constants c 1 > 0, c 2 > 0, d w ≥ 0, and T > 0, with c 1 < c 2 and matrix R > 0, the uniformly finite-time bounded formation is said to be achieved with c 1 , c 2 , T, R, d w , if for any switching signal σ t ∈ S it holds that Remark 2. It should be pointed out that the concept of finitetime stability is different from Lyapunov asymptotic stability.
A system can be finite-time stable but not Lyapunov asymptotically stable and vice versa.

Main Results
In this section, the FTF and the FTBF problems of UAVs are solved with details.

Finite-Time Formation without Disturbances
Theorem 1.Consider the group of UAVs (1) with switching topologies, for giving constants α i ≥ 0, and the topology-dependent controller gains K 1i and K 2i , if there exist matrices and the total dwell time of the unachievable topologies T + satisfies Then, the FTF can be achieved with respect to c 1 , c 2 , T, R, σ t .
Furthermore, when T + can be fixed, the switching signal has an ADT satisfying International Journal of Aerospace Engineering Proof 1. Choose the multiple Lyapunov-like functions as follows: Then, the LMIs ( 20) and ( 21) mean that For t ∈ t k , t k + 1 , it can be obtained by ( 27) that Without loss of generality, denote σ t k = i and σ t − k = j with N 0 = 0.Then, for any i ≠ j, if (22) holds then Thus, for t ∈ 0, T , letting a = max i∈S a i and β = max i∈S β i yields that V t < e aT − 0,t +βT + 0,t V σ t0 0 = μ N e a T−T + 0,t +βT + 0,t V σ 0 0 , 30 where N is the switching number of σ t over 0, T and T + 0, t is the total dwell time of UAT .
On the other hand, it holds that which implies that when the FTF in Definition 2 can be achieved it can be verified that μ N e a T−T + 0,t +βT + 0,t V σ t 0 0 μ N e aT e β−a T + 0,t ε T 0 ε 0 Moreover, the ADT τ a can be obtained for a fixed T + 0, t according to Definition 1, which completes the proof.
Based on the established conditions in Theorem 1, Theorem 2 is given to calculate the topology-dependent controller gains.Theorem 2. Consider the group of UAVs (1) with switching topologies, for giving constants α i ≥ 0, β i ≥ 0, i ∈ S, and μ > 1, if there exist matrices P i ≻ 0 and W 2i , ∀ i, j ∈ S × S, i ≠ j, such that

34
and the total dwell time of the unachievable topologies T + satisfies Then, the FTF can be achieved with respect to c 1 , c 2 , T, R, σ t .Furthermore, when T + can be fixed, the switching signal has an ADT satisfying

International Journal of Aerospace Engineering
In addition, when the topology-dependent controller gain K 1i is configured, K 2i can be obtained by . By letting P i = P −1 i and W 2i = K 2i P i and performing matrix congruent transformation, the conditions then follow directly from Theorem 1.
Remark 3. It is noted that K 1i is related with the motion modes of the formation center, such that K 1i is generally separately designed in advance of K 2i .When K 1i is configured appropriately, then the desired K 2i can be obtained by solving the conditions in Theorem 2 accordingly.For the large-scale UAVs, since the formation configuration h t is prescribed, the controllers of UAVs can be designed according to u i t , where the information exchanges among the UAVs are also considered.
Note that if a common Lyapunov-like function can be found for (6), Corollary 1 can be derived.

Corollary 1.
Consider the group of UAVs (1) with switching topologies, for giving constants α i ≥ 0 and β i ≥ 0, i ∈ S, if there exist matrices P i ≻ 0 and W 2i , ∀ i, j ∈ S × S, i ≠ j, such that and the total dwell time of the unachievable topologies T + satisfies the UFTF can be achieved with respect to c 1 , c 2 , T, R .
In addition, when the topology-dependent controller gain K 1i is configured, K 2i can be obtained by Finite-Time Formation with Disturbances.When the external disturbances of UAVs are considered, the theorems of FTBF and UFTBF are presented.Theorem 3. Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants α i ≥ 0, β i 0, i ∈ S, μ > 1, κ 1i > 0, κ 2i > 0, and the topology-dependent controller gains K 1i and K 2i , if there exist matrices P i ≻ 0, ∀ i, j ∈ S × S, i ≠ j, such that where and the total dwell time of the unachievable topologies T + satisfies Then, the FTBF can be achieved with respect to c 1 , c 2 , T, R, σ t , d w .
Furthermore, when T + can be fixed, the switching signal has an ADT satisfying International Journal of Aerospace Engineering Proof 3. Select the multiple Lyapunov-like function as follows: Then, the LMIs ( 41) and ( 42) mean that Following same lines in the proof of Theorem 1, for t ∈ t k , t k+1 , it holds that

which means that
In addition, the ADT τ a can be obtained for a fixed T + 0, t according to Definition 1, which completes the proof.Theorem 4. Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants α i ≥ 0, β i ≥ 0, i ∈ S, μ > 1, κ 1i > 0, and κ 2i > 0, if there exist matrices T ln μ ln c 2 λ min P i λ min R − ln λ max R λ max P i e a T−T + 0,t +βT + 0,t c 1 + λ min R κe βT d w 46 and the total dwell time of the unachievable topologies T + satisfies Then, the FTBF can be achieved with respect to c 1 , c 2 , T, R, σ t , d w .
Furthermore, when T + can be fixed, the switching signal has an ADT satisfying In addition, when the topology-dependent controller gain K 1i is configured, K 2i can be obtained by . By applying the same matrix congruent transformation in Theorem 2, the results can be directly obtained.
Based on the above results, Algorithm 1 can be given as follows.
Corollary 2. Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants α i ≥ 0 and β i ≥ 0, i ∈ S, if there exist matrices P i ≻ 0 and W 2i , ∀ i, j ∈ S × S, i ≠ j, such that e a T−T + 0,t +βT + 0,t c 1 + λ min R κe βT d w 56 Then, the UFTBF can be achieved with respect to c 1 , c 2 , T, R, d w .
In addition, when the topology-dependent controller gain K 1i is configured, K 2i can be obtained by

Illustrative Example
In the following, a numerical example is given to verify the effectiveness of our proposed formation design.
Consider a group of four UAVs with the following switching communication topologies depicted in Figure 1.
The corresponding Laplacian matrices can be given as follows: It can be verified that the formation cannot be achieved since topology and topology 2 do not have directed spanning trees, which implies that they are unachievable.
The eigenvalues of I ⊗ A + BK are configured by (−1 ± i, −1 ± i, and −1 ± i), one has Based on the established results of Theorem 4, it can be obtained by solving the LMIs that and the total dwell time of the unachievable topologies T + is calculated as 4.04 s.

Conclusion
In this paper, the formation problem of UAVs with switching topologies and disturbance is concerned with an ADT approach.In particular, the concept of FTBF is introduced to deal with the finite-time stability problem and disturbances.Moreover, both achievable and unachievable topologies are taken into account for a more practical background of communications among the UAVs.Based on model transformation, sufficient conditions are established for the FTBF and the desired topology-dependent controller can be designed accordingly.In the end, a numerical example is given to demonstrate the effectiveness of our proposed control method.Our future study will focus on the cases with model uncertainties and communication delays.

Figure 2 :
Figure 2: State trajectories of desired time-varying formation.

Figure 3 :
Figure 3: State trajectories of the open-loop FTBF trajectory.