Finite-Time Asynchronous Event-Triggered Formation of UAVs with Semi-Markov-Type Topologies

In this paper, the finite-time formation problem of UAVs is investigated with consideration of semi-Markov-type switching topologies. More precisely, finite-time passivity performance is adopted to overcome the dynamical effect of disturbances. Furthermore, an asynchronous event-triggered communication scheme is proposed for more efficient information exchanges. The mode-dependent formation controllers are designed in terms of the Lyapunov–Krasovskii method, such that the configuration formation can be accomplished. Finally, simulation results are given to demonstrate the validity of the proposed formation approach.


Introduction
With the rapid development of control and network technology, unmanned aerial vehicles (UAV) have been a prominent research topic in both academic and engineering fields in recent years. A variety of application scenarios have been employed by UAVs, such as search and rescue [1][2][3], express transportation [4,5], and remote sensing [6]. Furthermore, it is worth mentioning that multiple UAVs can accomplish complex tasks with more robustness and efficient working ability than a single UAV. As one of the interesting control issues, the formation problem for UAVs has gained significant research attention [7][8][9][10]. Specifically, the communication topology for UAVs has played a key role since all the information exchanges are achieved via the communication network. It should be pointed out that it is always difficult or expensive to hold fixed communication topology within different task assignments or in an unstructured environment with disturbances. Furthermore, the varying topology may exhibit certain random features in some conditions. Hence, it is natural and important to investigate the formation protocol with random topologies [11][12][13]. On the other hand, lots of studies have shown that the Markov process could be adopted to describe these stochastic jumping features between different modes accurately. As a result, some initial efforts have been made toward Markovian jumping communication topologies. However, notice that the transition probabilities of a Markovian jumping system are conformed to be a fixed exponential distribution, which would lead to certain restrictions in practical applications. In fact, time-varying transition probabilities are more general in implementations, and the semi-Markov process is able to depict these time-varying transition probabilities [14][15][16][17]. As far as the authors' knowledge extends, there is still huge room for studies on the integration of semi-Markovian jumping topology and UAV formation problems.
Another active research field is networked control systems, where the event-triggered schemes have aroused great concern recently [18 -20]. As we well know, computational and network resources are very valuable for UAVs. Under this context, the event-triggered communication schemes could considerably improve communication efficiency and optimize data transmission compared to traditional time-triggered ones [21][22][23]. Moreover, when taking into account the topology jumpings, the vast majority of event-triggered schemes do not utilize mode information well, and thus, the acquired results are conservative in certain senses. It is noteworthy that control strategies for Markov/semi-Markov jumping systems could utilize the involved information on active modes to improve control effects substantially. In short, the mode-dependent event-triggered strategies are more applicable in practical applications [14,24]. However, it is necessary to point out the fact that constructing an available mode-dependent triggering scheme by events is difficult and challenging. To date, to the best of our knowledge, there are few results on the formation of UAVs with semi-Markovian jumping topology. This motivates us to shorten this gap.
In light of the aforementioned discussions, our aim is to explore the mode-dependent event-triggered strategy for UAVs formation problems with semi-Markovian jumping topology within a finite-time framework. More precisely, the concept of finite-time passivity is employed for disturbance attenuation during UAV formation while the transient performance of formation dynamics is achieved. Compared with most reporting works, the novelties of our paper are threefold. First, the finite-time formation model of UAVs is formulated with sufficient consideration of semi-Markovian jumping topologies. Secondly, an original asynchronous mode-dependent event-triggering strategy is developed for UAVs in order to improve the communication efficiency through reducing data transmissions. Thirdly, according to model transformation and the Lyapunov-Krasovskii functional, a distributed formation protocol has been developed to satisfy the finite-time passivity performance in the mean-square sense.
The rest of our work is as listed following. In Section 2, we construct the formation model of UAVs with regard to semi-Markovian jumping topologies and design the asynchronous mode-dependent event-triggering formation protocol. Then, sufficient conditions satisfying finite-time passivity performance would be verified and given with reliable details in Section 3. After that, the simulation results are illustrated with numerical simulations in Section 4 to verify the correctness of our derived designs. Finally, the conclusion and expectation of this paper are given in Section 5.
The notations in this paper are given in Table 1.

Problem Formulation
Given a probability space (O, F, P) and denote {δ(t), t ≥ 0} as a continuous-time discrete-state semi-Markov process, which takes values in a finite set S = {1, . . . , N }. The transition probability matrix Π := (π ij (h)), h > 0, ∀i, j ∈ S is defined as follows Accordingly, we introduce a directed graph G δ(t) = {V δ(t) , E δ(t) , A δ(t) } to depict the communication topology among the UAVs. V δ(t) = {v 1δ(t) , · · · , v δ(t) } and E δ(t) represent for the sets of nodes and edges, A δ(t) = [a ijδ(t) ] N×N ∈ R N×N represents the weighted adjacency matrix by respectively. Moreover, the Laplacian matrix of By employing the graph theory, one can verify for any possible L δ(t) and a full row rank matrix E ∈ R (N−1)×N [25] For the formation of UAVs, a typical outer/inner-loop design structure is utilized [26]. Then, consider the group of N quadrotors with double integrator dynamics described as follows: where p m ∈ R n , v m ∈ R n and u m ∈ R n represent the position, velocity and formation control input vector, respectively.

Remark 1.
The notable outer/inner-loop formation control configuration for quadrotors has been widely studied, where the outer-loop control is devoted to the desired position and velocity of UAVs. Under this context, the double integrator point-mass model can be effectively adopted to describe the formation dynamics of UAVs.
By taking into account the external disturbances, the following state-space model of each UAV can be rewritten by: where x m = p T m , v T m T , u m (t) denotes the formation control input, w(t) represents the external disturbances and Suppose that all UAVs share a unified communication network with semi-Markovian jumping topologies. Accordingly, we assume that a mode-dependent sampling period h δ(t) , h δ(t) ≤h, is adopted, such that local information exchanges are accomplished between neighboring UAVs. Furthermore, the following asynchronous mode-dependent event triggering communication strategy is applied and t m k represents the latest kth updating instant of the mth UAV, 0 < κ δ(t) < 1 represents the triggering threshold, W 1δ(t) > 0 and W 2δ(t) > 0 represents the weighting matrices. It can be found that based on the proposed asynchronous event-triggered strategy, the information updates of the UAVs only require the neighboring information exchanges instead of the overall states of UAVs, which is more applicable for the distributed local communications of the UAVs. In conclusion, detailed notations of these parameters are given in the following Table 2.

Symbol Implication
Triggering weighting matrices

Local neighboring UAVs state changes
In the sequel, the formation configuration definition is given as follows: where d mn represents the relative formation configuration distance.
Denote d mn = d m − d n , and design the formation control input by where K δ(t) represents the mode-dependent gains to be determined. As a result, by letting Based on the above results, one can further obtain thaṫ ). Note that the resulting closed-loop dynamics can be rewritten as follows: where For simplicity of description, denote δ(t) = i and one haṡ Recalling that L δ(t) = M δ(t) E , it can be obtained thaṫ where , Before proceeding, the following definitions are given for the transient performance of formation dynamics [27,28].

Remark 2.
It should be pointed out that based on model transformation, the finite-time formation control problem is converted to the finite-time stability (passivity performance) control problem. The concept of finite-time stability (passivity performance) of formation errors is then investigated, where the asymptotic behavior over a finite-time interval is concerned instead of infinite-time stability, which implies that the prescribed upper bound can be satisfied during the prescribed time intervals. The mean-square finite-time passivity performance not only focuses on the dynamical features within a finite time interval but also deals with the disturbances from an energy point of view. It should be pointed out that mean-square finite-time passivity may not be Lyapunov mean-square passivity and vice versa.
Hence, our main aim is to design the mode-dependent formation control gains K i to guarantee finite-time passivity formation. The following matrix lemma can be provided to derive further results [29]. where

Main Results
This section gives the detailed formation controller design procedure with the aid of convex optimization and the matrix technique. and Proof. Construct mode-dependent Lyapunov-Krasovskii functionals as follows: where Moreover, we defined the weak infinitesimal operator for V(i, t) by where Γ i (h) represents the cumulative distribution function of the sojourn time, and q ij denotes the probability intensity. By employing the input-delay strategy, one haṡ As a result, it can be derived that Similarly, it can be obtained that By employing Lemma 1 to LV 3 (i, t), one has Meanwhile, the event-triggered function implies that Then, we can derive the following matrix inequality Hence, we can verify that when Ξ i (h) < 0 in Theorem 1 holds, it satisfies that Furthermore, it can be derived that Consequently, we have that This means that η(t) can be finite-time boundedness according to Definition 2, and we thus finish this proof. with Proof. Following similar proofs in Theorem 1, it can be derived that such thatΘ i (h) < 0 can guarantee that the above matrix inequality holds, wherẽ Consequently, it yields that Subsequently, by performing matrix congruent transformation toΘ i < 0 with I ⊗ P −1 i and letting P −1 Meanwhile, by and φ 3 < 2 χ 2 . Then, we have Finally, by adopting the method for the time-varying dwell time π ij (h), π ij (h) = ∑ L 1 λ ι π ij,ι , ∑ L 1 λ ι = 1, λ ι ≥ 0, the remainder of the proof can follow conveniently from Theorem 1, which ensures that the finite-time passivity formation can be satisfied according to Definition 3.

Remark 3.
The computation complexity for the above linear matrix inequality (LMI) convex optimization conditions is related to the convex combination representation of time-varying dwell time π ij (h) with ι ∈ L. As such, it is a trade-off to setting an appropriate ι for better describing the time-varying dwell time and computation complexity of LMIs.

Simulation Results
This section demonstrates the effectiveness of theoretical results via the numerical simulation results.
As depicted in Figure 1, consider a group of four UAVs and the corresponding Laplacian matrices of communication topology are given by  10 14 , 100, 2.0612, 5), the values of mode-dependent formation controller gains and triggering matrices are achieved by solving the convex optimization problem as follows: With initial conditions and setting t F = 5, Figures 2 and 3 show the resulting formation trajectories of UAVs. More precisely, it can be observed that the UAVs can achieve converging motion within the prescribed finite time interval. Moreover, one can verify from Figure 3 that the finite-time boundedness can be ensured with c 1 and c 2 by Definition 2. Figures 4-7 reveal the data broadcasting instants with release intervals of UAVs, from which one can find that the signal transmissions can be effectively decreased compared with the traditional time-triggered approaches, such that the local communication efficiency among the UAVs can be increased under the distributed formation control framework. Furthermore, it can be seen from Figure 8 that the finite-time passivity condition can be satisfied according to Definition 3 with

Conclusions
This paper studies the formation problem of UAVs with semi-Markov jump topologies under the finite-time passivity performance. By proposing an asynchronous eventtriggered data transmission strategy, communication efficiency can be improved considerably. With the aid of choosing the mode-dependent Lyapunov-Krasovskii function, sufficient formation criterion is established, and corresponding formation controller gains are calculated by LMIs, such that desired finite-time passivity performance is satisfied for configuration formation. A simulation example with four UAV verifications is performed to validate the effectiveness of the derived theoretical algorithm. Future research of interest can be focused on the cases with more complex network environments, i.e., limited bandwidth or channel fading of communication network.