Abstract
This paper investigates the finitetime formation problem of unmanned aerial vehicles (UAVs) with switching topologies and external disturbances. The formation problem is first transformed into the finitetime stability and bounded problems of a switched system, respectively. In particular, the finitetime 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 timevarying formation can be achieved in finite time. Then, the topologydependent controller can be designed in terms of matrix technique. Finally, an illustrative example is given to verify the effectiveness of our proposed formation strategy.
1. 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 [1–3]. 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 [4–10]. 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 [11–15]. 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 [16–18].
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 [19–22]. 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 [23–25]. 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 finitetime interval, which gives rises to the researches on finitetime stability, and burgeoning investigations on finitetime consensus problems of multiagent systems with switching topologies have been made [26–28]. 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 finitetime 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 finitetime 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 finitetime 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 finitetime formation and the desired topologydependent controller gains are obtained accordingly.
The remainder of our paper is given as follows: Section 2 introduces the finitetime 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. and denote the dimensional Euclidean space and the space of real matrices, respectively. denotes that is positive definite (negative definite). and denote the maximum and minimum eigenvalues of matrix , respectively. denotes the space of squareintegrable vector functions over . stands for the Kronecker product. represents the ellipsis symmetry terms in symmetric block matrices. All matrices are compatible for algebraic operations.
2. Problem Formulation and Preliminaries
2.1. The UAV Dynamics
Consider a group of UAVs with the following kinematics and dynamics: where denotes the position of the th UAV, represents the ground speed, stands for the flightpath angle, is the heading angle, denotes the engine thrust, represents the drag, denotes the mass, stands for the acceleration due to gravity, is the lift force, denotes the load value, represents the banking angle, and stands for the vortex disturbances.
By applying the feedback linearization method as where denotes the virtual acceleration control input and represents the disturbances, the relationships between virtual control variables and the actual control variables are given by with , , and .
Consequently, by denoting the UAV dynamics can be obtained as follows: where
2.2. Graph Theory
The directed graph is adopted to describe the communication topology of the UAVs. represents the weighted adjacency matrix with
and denote the sets of nodes and edges, respectively. The denotes the Laplacian matrix with
If has a directed spanning tree, then 0 is the eigenvalue of . More details of the algebraic graph theory can be found in [29].
The switching topologies are studied in this paper. Let denote all possible switching graphs, where represents the index set for all switching graphs. The switching signal , and its value is the index of the graph at time . The switching sequence is , where , , is uniformly bounded nonoverlapping [30].
Moreover, for any possible of graph and a full row rank matrix defined as there exists a matrix such that .
Remark 1. It is worth mentioning that when is achievable ( has a directed spanning tree), is of full column rank and the eigenvalues of are the same as that of . When is unachievable, the above conditions of will not be satisfied.
Without loss of generality, suppose that there are achievable topologies with which means that the formation can be achieved with the and unachievable topologies with and , otherwise. All the possible communication topologies can be divided into the achievable set denoted as and the unachievable set denoted as .
2.3. FiniteTime Formation
For the timevarying formation problem, denote , where is a piecewise continuously differentiable, and , where .
Consequently, the formation controller for each UAV can be designed as follows: where and and are topologydependent controller gains to be determined, respectively.
As a result, the closedloop dynamics of the UAVs can be obtained by where .
In addition, note that ; then, it follows that
Premultiplying both sides of (13) by yields that where denotes the disagreement of and , respectively.
To this end, the following definitions are given for later use.
Definition 1 (average dwell time, ADT) [31]. For any , denote as the number of switching over . is said to have an average dwell time (ADT) , if there exist positive numbers such that
Definition 2 (finitetime formation, FTF) [32]. Given constants , , and , with , matrix , and a switching signal , the finitetime formation is said to be achieved with , if it holds that
Definition 3 (uniformly finitetime formation, UFTF) [32]. Given constants , , and , with and matrix , the uniformly finitetime formation is said to be achieved with , if it holds that
Definition 4 (finitetime bounded formation, FTBF) [32]. Given constants , , , and , with , matrix , and a switching signal , the finitetime bounded formation is said to be achieved with , if it holds that
Definition 5 (uniformly finitetime bounded formation, UFTBF) [32]. Given constants , , , and , with and matrix , the uniformly finitetime bounded formation is said to be achieved with , if for any switching signal 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 finitetime stable but not Lyapunov asymptotically stable and vice versa.
3. Main Results
In this section, the FTF and the FTBF problems of UAVs are solved with details.
3.1. FiniteTime Formation without Disturbances
Theorem 1. Consider the group of UAVs (1) with switching topologies, for giving constants , , , , and the topologydependent controller gains and , if there exist matrices , , , such that
where
and the total dwell time of the unachievable topologies satisfies
Then, the FTF can be achieved with respect to .
Furthermore, when can be fixed, the switching signal has an ADT satisfying
Proof 1. Choose the multiple Lyapunovlike functions as follows:
Then, the LMIs (20) and (21) mean that
For , it can be obtained by (27) that
Without loss of generality, denote and with . Then, for any , if (22) holds then
Thus, for t , letting and yields that
where is the switching number of over and is the total dwell time of .
On the other hand, it holds that
which implies that when the FTF in Definition 2 can be achieved it can be verified that
Moreover, the ADT can be obtained for a fixed according to Definition 1, which completes the proof.
Based on the established conditions in Theorem 1, Theorem 2 is given to calculate the topologydependent controller gains.
Theorem 2. Consider the group of UAVs (1) with switching topologies, for giving constants , , , and , if there exist matrices and , , , such that
where
and the total dwell time of the unachievable topologies satisfies
Then, the FTF can be achieved with respect to .
Furthermore, when can be fixed, the switching signal has an ADT satisfying
In addition, when the topologydependent controller gain is configured, can be obtained by
Proof 2. By letting and and performing matrix congruent transformation, the conditions then follow directly from Theorem 1.
Remark 3. It is noted that is related with the motion modes of the formation center, such that is generally separately designed in advance of . When is configured appropriately, then the desired can be obtained by solving the conditions in Theorem 2 accordingly. For the largescale UAVs, since the formation configuration is prescribed, the controllers of UAVs can be designed according to , where the information exchanges among the UAVs are also considered.
Note that if a common Lyapunovlike 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 and , , if there exist matrices and , , , such that
and the total dwell time of the unachievable topologies satisfies
Then, the UFTF can be achieved with respect to .
In addition, when the topologydependent controller gain is configured, can be obtained by
3.2. FiniteTime 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 , , , , , , and the topologydependent controller gains and , if there exist matrices , , , such that
where
and the total dwell time of the unachievable topologies satisfies
Then, the FTBF can be achieved with respect to .
Furthermore, when can be fixed, the switching signal has an ADT satisfying
Proof 3. Select the multiple Lyapunovlike function as follows:
Then, the LMIs (41) and (42) mean that
Following same lines in the proof of Theorem 1, for , it holds that
By letting , , and and , , and , one has
where , , and .
Consequently, it can be obtained that
which means that
In addition, the ADT can be obtained for a fixed according to Definition 1, which completes the proof.
Theorem 4. Consider the group of UAVs (1) with switching topologies and disturbances, for giving constants , , , , , and , if there exist matrices and , , , such that
where
and the total dwell time of the unachievable topologies satisfies
Then, the FTBF can be achieved with respect to .
Furthermore, when can be fixed, the switching signal has an ADT satisfying
In addition, when the topologydependent controller gain is configured, can be obtained by
Proof 4. 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 and , , if there exist matrices and , , , such that
and the total dwell time of the unachievable topologies satisfies
Then, the UFTBF can be achieved with respect to .
In addition, when the topologydependent controller gain is configured, can be obtained by
4. 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.
(a) Topology 1
(b) Topology 2
(c) Topology 3
The corresponding Laplacian matrices can be given as follows:
It can be verified that the formation cannot be achieved since topology 1 and topology 2 do not have directed spanning trees, which implies that they are unachievable.
Accordingly, it can be obtained that
The timevarying formation configuration is given as , , , , and , and the external disturbances are set by .
The design parameters are chosen as , , , , , and . For the FTBF, the parameter is chosen as 0.5, the parameter is set as 50, is given by 10 s, and is supposed to be 0.0025.
The eigenvalues of are configured by (, , and −), 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 is calculated as 4.04 s.
In the simulation, the initial conditions of the UAVs are given as and is set as 2.5 s. Figures 2–4 show the desired timevarying formation trajectory, the openloop FTBF trajectory, and the closedloop FTBF trajectory, respectively. It can be seen that the designed formation controllers for each UAV can achieve the prescribed formation with unachievable topologies and external disturbances. Figure 5 depicts the formation errors of the UAVs with the finitetime interval, and Figure 6 indicates the corresponding FTBF conditions. It can be found that the formation errors are finitetime bounded with switching topologies according to Definition 4 with and , which supports our theoretical analysis.
5. 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 finitetime 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 topologydependent 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.
Data Availability
All data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 61703038, 61627808, and 9164820, the Fundamental Research Funds for the Central Universities under Grant FRFTP18034A2, and the National Key Research and Development Program of China under Grant SQ2017YFB130092. This work is also supported by the Strategic Priority Research Program of the CAS (Grant XDB02080003).