Abstract
Time varying formation control problem for a group of quadrotor unmanned aerial vehicles (UAVs) under Markovian switching topologies is investigated through a modified dynamic event-triggered control protocol. The formation shape is specified by a time varying vector, which prescribes the relative positions and bearings among the whole agents. Instead of the general stochastic topology, the graph is governed by a set of Markov chains to the edges, which can recover the traditional Markovian switching topologies in line with the practical communication network. The stability proof for the state space origin of the overall closed-loop system is derived from the singular perturbation method and Lyapunov stability theory. An event-triggered formation control protocol in terms of a dynamically varying threshold parameter is delicately carried out, while acquiring satisfactory resource efficiency, and Zeno behavior of triggering time sequences is excluded. Finally, simulations on six quadrotor UAVs are given to verify the effectiveness of the theoretical results.
1. Introduction
Along with the increasing applications in various areas, such as aerial photography, express delivery, and disaster relief, formation control of multiagent systems has attracted considerable attention from many researchers [1, 2]. In particular, as a typical class of physical systems with practical interest, the quadrotor UAVs is widely used in the military and civilian fields [3, 4]. Actually, due to the strong nonlinear coupling and limited communication resources [5], the control problem of multiple quadrotor UAVs will be very challenging and difficult [6]. Therefore, how to design the formation control protocol for multiple quadrotor UAVs subject to limited communication resources becomes a significant research focus.
A defining feature of formation control problem is that multiple agents work together to accomplish a collaborative formation task [7]. Several classic formation control strategies, including leader-follower, virtual structure, and behavior based methods, were applied in the scientific community [8, 9]. For example, formation control of multiple quadrotor UAVs, based on position estimation [10], backstepping design technique [11], and finite time algorithms [12], respectively, was investigated so as to make a construct and keep the formation shape during flying. It should be pointed out that time varying formation tracking problems arise in some scenarios, such as source seeking and target enclosing. For example, time varying formation analysis and design problems for multiagent systems with switching topologies were solved in [13, 14]. Based on the fact that multiagent systems subject to random abrupt variations could be modeled as the switching systems, then some results have been obtained on it [14]. Compared with the previous works, time varying formation control results for multiagent systems with switching topologies were provided in [15, 16]. Besides, due to random link failures, variation meeting the need and sudden environmental disturbances [17], some dynamical systems could be modeled as Markovian switching systems, which were governed by a set of Markov chains [18, 19]. By considering the complex network as Markovian switching topologies, it plays a crucial role in the field of networked control system [20].
In practice, under a limited bandwidth, it is necessary and important to consider the issues of energy waste and competition [21]. Therefore, event-triggered communication mechanism was born at the right moment [22, 23]. As a popular research topic, some latest event-triggered control results were provided in [24–28]. In particular, compared with the general event-triggered controller with a fixed threshold parameter, the authors in [24, 29] developed the dynamic/adaptive event-triggered control protocol of multiagent systems for acquiring satisfactory resource efficiency, respectively. Meanwhile, take the strong nonlinear coupling and underactuated of the quadrotor UAVs into account, time scaling based control method has also been recognized as a powerful tool in the analysis and design of controllers, which is with crucial importance in applications to the mobile inverted pendulum [30], the ball-beam system [31], and the quadrotor UAVs [32]. Therefore, it is of great importance to extend the event-triggered formation results to multiple quadrotor UAVs under Markovian switching topologies. In addition, it is difficult to obtain all the transition rates under the realistic communication environment [33]. So that randomly occurring control strategy is more realistic and meaningful to accomplish attitude stabilization and formation missions under the limited communication resources.
Motivated by these observations, the contributions of this paper are proposing a novel time varying formation control strategy and an event-triggered communication scheme to solve the formation problem of multiple quadrotor UAVs with Markovian switching topologies. The main highlights of this paper are summarized as follows. First, a modified graph of entire system is governed by a set of Markov chains to the edges, and the traditional Markovian switching topologies can be recovered through adjusting the modes of edges and the transition rates. Second, the dynamic event-triggered controller is derived from a time scaling based control strategy, which consists of two parts: the closed-loop system stability analysis based on the framework of singularly perturbed theory and the event-triggered control scheme in terms of a new dynamically varying threshold parameter to guarantee time varying formation shape. Third, Markovian switching topologies involve partly unknown transition rates, which are of great importance to be considered and thus closer to the realistic communication environment. In addition, Zeno behavior can be excluded during the whole running process. Finally, several simulations can illustrate the theoretical results.
The rest of this paper is organized as follows. The system dynamics and some preliminaries on graph theory are introduced in Section 2; Section 3 provides main results on event-triggered formation control for multiple quadrotor UAVs. In Section 4, simulation results are given and this paper is concluded in Section 5.
Notations 1. Throughout this paper, denotes -vector norm and stands for the spectral radius for matrices. The notation means the Kronecker product of matrices and , and and represent its maximum and minimum eigenvalues.
2. Preliminaries and System Formulation
2.1. Graph Theory
Define a time varying random undirected graph with a nonempty finite vertex set and an edge set . Different with the general ones, it consists of a time sequence of random graphs in which the edge set varies with . Namely, each edge evolves according to a homogeneous Markov process , which takes values in with the transition rate asAssume that do not change infinitely fast; thus, if . It means that the total number of system modes is and the total transition rate is given by
The weighted adjacency matrix is associated with . Here if and otherwise. Assumed that there is no self-loop in the graph, which implies that . In this paper, the set of neighbors with respect to the agent is . A graph is connected, if there is a path between any two vertices; otherwise, it is disconnected. A diagonal matrix with being the th row sum of . Then, the Laplacian of the graph is defined as . Consider the formation with a leader-follower structure by introducing a diagonal matrix , which evolves according to the Markov process with a finite mode set and a time interval , where if is a leader and otherwise. Hence, the interaction matrix is given by .
Assumption 2. The undirected graph is connected.
Remark 3. For simplicity, we just consider the undirected graph in this paper, that is, . Note that, if the graph is a general directed one, there will be some small differences. A possible approach to consider a directed graph is to introduce two Markov chains in each two agents; an alternative is to extend the state space set , which could be defined according to the weight and direction of the graph. Both approaches will be addressed in our future work.
Remark 4. With a limited bandwidth, the switching of is caused by the sensing/detecting failure and communication failure, which is passive. In fact, it is difficult to obtain all the elements of the transition rate matrix, or some of the elements are not necessary to guarantee the system stability. Since that, the transition rate matrices are assumed to be partly accessed; even some of them are unknown completely, which could be descried as follows:where “” represents the unknown transition rate.
2.2. Problem Formulation
Consider a group of quadrotor UAVs as shown in Figure 1; the dynamics of agent are given as the following form [12]:where are positions and denote the three Euler angles of rotation, representing pitch, roll, and yaw, respectively. are the aerodynamic friction coefficients, and are the coefficients of the translation drag forces. are the quadrotor moments of inertias, denotes the mass of the quadrotor, is half length of the helicopter, is thrust to moment gain, and is gravitational acceleration.
Each system dynamics consist of six degrees of freedom model, which can be separated into position dynamic and attitude dynamic. With the choice of state variables , ; , ; , ; , ; , ; , . Then, it can be represented by
As is well known to us, the quadrotor is underactuated and differentially flat. Accordingly, choose four variables and specify the desired trajectory as .
2.3. Formation Definition
The formation shape can be described by a vector , where is the continuously differentiable formation vector, such thatDefine the desired position of agent as . For , there exists . Hence, there are such shape vectors satisfying the following properties:
Remark 5. The formation reference vector is not available to all agents, but the desired interdistances of agent and its neighbors are known. Note that the time varying formation vector is not unique, and the relative position between the reference vector and the formation can be adjusted.
Lemma 6 (see [34]). For any and , it holds that .
3. Main Results
In this section, time varying formation control problem for multiple quadrotor UAVs is solved through an event-triggered control scheme. The system stability analysis and exclusion of Zeno behavior are also provided.
3.1. Singularly Perturbed System
Define the error vectors aswhere is positive constant denoted as perturbing parameter, which satisfies . The overall system can be written as
System (9)–(11) has the standard form of a singularly perturbed system with a two-time scale; that is, represents the attitude states with fast time scale and represents the position states with slow time scale. Roughly speaking, in order to maintain a predefined formation shape for multiple quadrotor UAVs, the attitude stabilization should be guaranteed firstly.
3.2. Controller Design
Dynamic event-triggered control scheme is introduced to the multiple quadrotor UAVs. Define the following measurement errors:LetThen, the auxiliary control variables are given byTherefore, the choice of control inputswhere the reference angles and are addressed as
Remark 7. It should be stressed that (17) and (18) are needed to be nonsingular; in other words, the denominator of (17) or (18) cannot be zero. In practice, the quadrotor has to take a certain thrust to overcome gravity in order to maintain hovering; otherwise, it would sink vertically. It means that the denominator of (17) or (18) is approximate to , and during the flight, it will also be greater than zero based on the defined domain of dynamics , .
3.3. Dynamic Event-Triggered Communication
Figure 2 depicts the control law running in the th quadrotor. Generally speaking, each agent updates its controller whenever the designed trigger condition is reached, called the triggered event. Based on local information, it decides when to broadcast its current state over the network. In other words, the key problem is to find a triggering rule that determines when agent has to broadcast the new state information to its neighbors.
In contrast to most of the existing works, a new dynamic event-triggered communication mechanism is developed to schedule interagent communication. The threshold parameter in the proposed event triggering condition will not be fixed permanently but vary with time by following a dynamic rule. The detailed dynamic rule is provided in the following section. The event-triggered control strategy works as follows: define a trigger function for each agent, which depends on local information only; an event is triggered as soon as the trigger condition is fulfilled, while each agent recomputes its control law in accordance with the measurement error, such that all the agents could reach and keep the predesigned formation shape. It will be shown numerically that the dynamic event-triggered communication mechanism can achieve a better tradeoff between reducing data transmissions and preserving favorable formation performance.
3.4. System Stability via Two-Time Scale
The stability analysis for each subsystem will be done by starting from the faster one to the slower one. The main results of this paper are presented next with the help of the following theorems.
Theorem 8. Consider the th quadrotor dynamics, which is also a singularly perturbed system. Then, there always exists such that the state space origin of the closed-loop system (9)–(11) is exponentially stable with .
Proof. The proof is set down in the following five items.
System (9)–(11) has a unique equilibrium point at .
The quasi-steady-state solution of closed-loop system (9)-(10) is . Hence, by substituting into (9)-(10), the isolated root is given by . It is noteworthy that the isolated root evaluated in (16)–(18) gives the specific values , , and .
System (9)–(11) and the isolated root have bounded partial derivatives in compact sets.
By considering and using the isolated root into (11), the following slow dynamics system is obtained:The local stability of the slow dynamics (11) can be derived by using event-triggered control method. For the sake of simplicity, it will be given in Theorem 12 later.
To obtain the boundary layer system, by deriving with respect to scaled time and setting , the boundary layer system is obtained:Thus, for the and subsystemsIt is obvious that and are Hurwitz, where system (22) is exponentially stable. Meanwhile, the stability result of subsystem will be established in Theorem 10 as follows.
According to [32], there are sufficient conditions to claim that there exists such that guarantees that system (9)–(11) achieves the following limit .
Remark 9. Based on the system stability analysis by the singular perturbation method, instead of using the feedback angles , , , the control input is related to the reference angles , , , and the coupling between the attitude subsystem and the position subsystem is diminished.
Theorem 10. Given and , then define and . Consider the yaw subsystem (21); the system stability could be obtained through the distributed control law (14) and (15) and the following event-triggered communication condition:where is the dynamic threshold parameter with , , , and the initial value , which ensures that always holds.
Proof. Obviously, is a monotone nonincreasing function;thus, always holds. Particularly, if the threshold parameter is preset as a constant before dynamic event-triggered controller is implemented, this is called the static event-triggered controller. Define , , and . Based on (13), (21) becomesHence, it follows from [35] that there exist positive constants , , and such thatwhere , , andLet and . Consider the following Lyapunov function:Then, its derivative isBased on Lemma 6, the following inequality holds:Based on (23) and (25), (28) can be written asHence, , and exponentially.
Next, Zeno behavior is excluded. For any , it follows from (23) thatwhere . Hence,Thus,The next event will not be triggered before . Thus, a lower bound on the interevent intervals is given by that solves the following equation:Therefore, for all the solutions are greater than or equal to given bywhich is strictly positive, so Zeno behavior is excluded. This completes the proof.
Remark 11. It should be pointed out that the vectors and are obtained inside of the controller; as shown in Figure 2, the pitch and roll attitude dynamics are not event-triggered. In other words, only the yaw angle in the attitude subsystem is controlled through an event-triggered strategy. In addition, compared with some existing works, in which assuming in the position control loop, the yaw angle is not always zero in practice. Since that the control method is more suitable in this paper.
After the attitude angles reach quasi-steady states, hierarchically, , , , . Substituting (14), (16)–(18) into (19), it is possible to show that the closed-loop position subsystem can be written asWithout loss of generality, we only provide the proof along -subsystem in the following.
Theorem 12. Given and , then let and . Considering the position subsystem (19), the system stability could be obtained through the distributed control law (16)–(18) and the following event-triggered communication condition:where is the dynamic threshold parameter with , , and the initial value .
Proof. Let ; the -subsystem could be rewritten asSimilarly, there exist positive constants , , and such thatwhere . Note that , and let and . Design the following Lyapunov function:Then, its derivative isBased on Lemma 6, the following inequality holds:Based on (37) and (39), (41) can be written asHence, , and exponentially.
Next, Zeno behavior is excluded. For any , one haswhere . Hence,Thus,The next event will not be triggered before ; it meansBy the same graphical argument as in the proof of Theorem 10, it can be concluded that a lower bounded on the interevent times is given by , so Zeno behavior is excluded.
This completes the proof.
Remark 13. Compared with the existing works, the threshold parameter of event triggering condition will not be fixed permanently but vary with time by following the measurement errors. Since that it can achieve a better tradeoff between reducing event times and preserving favorable formation performance.
4. Simulation Results
In this section, simulations are given to demonstrate the effectiveness of the theoretical results. Consider a group of agents, which consists of six quadrotor UAVs, and the dynamics can be written as (4). The system parameters are shown in Table 1. Furthermore, the initial mode of undirected topology is given in Figure 3. A special case is considered, where takes the switch value in with an equal probability and the time interval . Meanwhile, assume that each Markov chain takes values in a finite state space , and the time interval . Otherwise, each undirected edge has a Markov chain, and the six agents have 15 chains totally. The transition rate matrices are considered as Table 2.
As an example, the time evolutions of some Markov chains are depicted in Figure 4.
Let the initial states of six quadrotor UAVs beThe desired trajectory is given by , and the formation shape specified by the desired interagent distances is described aswhich means that if the quadrotor system achieves the desired time varying formation, then the six quadrotor UAVs will follow a circle while keeping a phase separation of . In order to verify the effectiveness and advantage of the developed formation control protocol, the following two cases are evaluated:
Case 1. The distributed event-triggered function (23), (37) and the controller are implemented with the parameters as follows:Simulation results are shown in Figures 5 and 6. The response curves of formation trajectory and attitude are given in Figure 5, which reveal that positions of six agents successfully achieve and maintain a predesigned formation shape. As depicted in Figure 6, it demonstrates the snapshots of position trajectories of the six robots at , , , and , respectively.
(a) Evolution of the position states
(b) Evolution of the attitude states
(c) Evolution of the velocity states
(d) Evolution of the angular velocity
Case 2. The same formation task is carried out by implementing the event-triggered controller with a constant threshold parameter . The aforementioned results derived from theorems indicate that the formation control problem is feasible under the general event-triggered controller. Figure 7 shows the trajectories of positions and attitude angles within 50 s and Figure 8 shows the snapshots of position trajectories at , , , and , respectively.
The actual event release instants for all agents are shown in Figures 9 and 10. A quantitative comparison of how many events on each agent are actually transmitted is provided in Table 3, and Table 4 shows the mean and standard deviation for each error signal in .
(a) Evolution of the position states
(b) Evolution of the attitude states
Remark 14. Some conclusions could be summarized as follows:
Simulation results in both Cases 1 and 2 show that the formation shape could be guaranteed under Markovian switching topologies with partially unknown transition rates.
In Figure 5, the convergence time of and (i.e., 0.80 s) is shorter than that of , , , , , and (i.e., 7.20 s); these plots have different time horizons; it confirms the two-time-scale structure controller derived from the perturbing parameter . Particularly, the reason why the convergence time of and in Figure 5 is as long as that of the position states is that and are tracking the reference and rather than zero.
As shown in Figures 6 and 8, at , , and , the real shape of six agents is always uniform. When , there is a little difference because the graph is randomly governed by Markov chains.
Compared with Case 2, there are more events for all agents in Case 1, which means that it has more potential to reduce the occupancy of limited communication resources in Case 2. That is because the threshold parameter is fixed as all the time, while in Case 1, is monotonically nonincreasing (shown in Figure 11). Generally, the larger the value of the threshold parameter is, the less events are derived.
Based on most of the computed performance indexes in Table 4, the controller in Case 1 can achieve better formation performance than that in Case 2. Although less events are derived by using a fixed threshold parameter, the formation performance is compromised.
5. Conclusion
In this paper, formation control problem of multiple quadrotor UAVs under Markovian switching topologies is addressed through an event-triggered control strategy. Markovian switching topologies are redesigned through utilizing the Markov chains to the edge set, which could recover the traditional ones by adjusting the modes of edges and the transition rates. Then, a predesigned formation shape can be kept along with a distributed formation controller constructed with a reasonably event-triggered updated rule, even subject to the unknown transition rates. Rigorous analyses of the convergence results are obtained based on singular perturbations theory and Lyapunov stability theory. In addition, Zeno behavior is excluded for the triggered time sequences. Simulation results have been given to illustrate the effectiveness of the proposed control strategy. In some of the cases, the velocity states are unmeasurable. Since that, event-triggered formation control problem for multiple quadrotor UAVs without collision subject to output feedback will be considered in our future work.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by NSFC under Grant 61473248 and Natural Science Foundation of Hebei Province under Grant F2016203496.