Complexity

Volume 2018 (2018), Article ID 8124861, 15 pages

https://doi.org/10.1155/2018/8124861

## Event-Based Time Varying Formation Control for Multiple Quadrotor UAVs with Markovian Switching Topologies

Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao 066004, China

Correspondence should be addressed to Zhen Zhou; moc.361@7160nehzuohz

Received 22 December 2017; Accepted 20 February 2018; Published 23 April 2018

Academic Editor: Victor M. Becerra

Copyright © 2018 Zhen Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.