Elsevier

Systems & Control Letters

Volume 106, August 2017, Pages 47-57
Systems & Control Letters

Distributed cooperative control of leader–follower multi-agent systems under packet dropouts for quadcopters

https://doi.org/10.1016/j.sysconle.2017.06.002Get rights and content

Abstract

This paper is to develop a novel distributed leader–follower algorithm for multi-agent systems in the event of stochastic communication link failure over the network. Bernoulli distribution is applied to represent the data dropout during operation while the data dropout properties of each communication links are independent from each other. Sufficient conditions for a stabilizing controller design are developed by using Lyapunov-based methodologies and Linear Matrix Inequality (LMIs) techniques. The stability condition is then decomposed into small robust stability conditions with the size of a single agent, provided that the interaction topology of the followers is an undirected graph, which leads to efficient solutions even in case that the number of agents is large and a high order system dynamics of the agent is considered. The main result is to ensure the mean square exponential stability of the overall system reaching consensus. The case of consecutive data losses in any of the communication links is also discussed in the same framework. Leader-following numerical simulations with a group of agents including quadcopters are successfully conducted to demonstrate the effectiveness of the novel consensus algorithm in this paper. The results show that the consensus achievement is incorporating the data loss probability; however a higher data loss rate may cause a longer time for agents to achieve consensus.

Introduction

In recent years, the control of multi-agent systems (MASs) has been an active area of research with formulations such as swarming, flocking, foraging, consensus and rendezvous, etc. [1], [2], [3], [4], [5]. MASs are composed of several intelligent agents which interact with each other through communication channels. Using a group of agents to achieve a common task is usually more efficient and with more operational capability than a single agent especially for tasks which are difficult or impossible for an individual agent system to complete, such as combat, surveillance, mapping, and underwater mine hunting. Among the various problems in MASs, consensus problem has received significant attention in the decentralized control of networks of dynamic agents, which can be widely applied in the cooperative control of unmanned aerial vehicles, scheduling of automated highway systems, and formation control of satellite clusters, etc. [6], [7].

In consensus problems, a group of agents is required to achieve group behaviour or reach consensus via a local distributed control protocol over a communication network. The agent dynamics, the network topology and communication constraints are three main factors influencing the control design techniques in the literature for consensus. Studies of agent dynamics have been on single/double-integrators [8], Euler-Lagrangian systems [9], general linear systems [10], linear parameter-varying (LPV) systems [11] and nonlinear oscillators [12]. The network topology may be fixed [8], [13], time-varying [14], switching and stochastic [11]. The network constraints considered are time-delays, packet losses and quantization, etc. In MASs, agents are mostly and popularly controlled through wireless networks. However, wireless networks are not always as reliable as hardwired ones due to connection strength, bandwidth constraints, which can cause packet delays and data loss. In this paper, we consider a general linear multi-agent system with applications to a set of linearized model of quadcopters under a fixed-network topology with stochastic data losses among the communication links. A leader–follower formulation is adopted. As in the literature, [15] and [16] study the leader-following consensus problem for without communication constraints, and with fixed and switching topologies respectively.

Packet losses in communication links can be modelled as stochastic processes either by Bernoulli distribution [17], [18] for MASs or Markovian process [19] for networked control systems. Most approaches are based on Lyapunov-based methods and the overall system can be ensured to have mean square stability due to the stochastic processes in the system. [20] also considers the effect of quantization combined with packet losses, in which the small l signal l stability of a networked control system subject to packet dropouts and finite-level quantization of a networked control system is investigated. In [21], in the presence of bounded packet dropouts, sparsity-promoting optimization based predictive control is proposed for good control performance and sufficient conditions for the closed-loop stability of networked control systems are derived.

In [22], a decomposition approach is used for the distributed control of identical interconnected systems which satisfy a certain structural property. In our approach, it is not straightforward to decompose the closed-loop MASs, however, the decomposition concept is used for reducing the resultant big LMI condition to a set of small LMIs with the size of a single agent. There are advantages in applying the distributed-type control design to swarm robots and especially for the applications to a fleet of quadcopters [23]–[24] with a high system order of 12 compared with the smaller size of double integrators being two.

This paper is an extension of our previous work in [13], which considers a double integrator type system under the assumption that synchronous packet losses happen in the communication links, while a large number of agents may lead to a high dimension of the LMI stability condition and intractability. In this paper, the focus is on investigating the consensus of a general linear system with stochastic communication link failures. Multi-agent systems are modelled by directed graph topology in leader-following scenarios, which means the leader is only capable of sending signals to the followers without receiving, while the communication topology among the followers is undirected. Each communication channel in the system is represented by a Bernoulli random process at every time step. The packet loss property of each channel is assumed independent from others. The case of consecutive packet losses is also considered. As in [25] and [26], the followers know the reference input of the leader while the state of the leader is sent through the topology for the consensus. Furthermore, a sufficient condition for a stabilizing controller is developed by using Lyapunov-based methodologies and Linear Matrix Inequality (LMIs) techniques. The stability condition is then decomposed into small conditions with the size of a single agent provided that the communication topology of the followers is an undirected graph, which leads to computational efficiency even in case that the number of agents is large and a high order system dynamics of the agent is considered. In both cases of a single packet loss and consecutive packet losses, the main result can ensure the mean square exponential stability of the overall system reaching consensus. In simulation results, leader-following numerical simulations with a group of five quadcopters are successfully conducted to demonstrate the effectiveness of the novel consensus algorithm in this paper. It is shown that the followers can reach consensus following the leader under data loss probability up to a critical rate. A higher data loss rate may cause a longer time for agents to achieve consensus.

The paper is organized as follows. Section 2 is the preliminary with basic concepts in graph theory and Kronecker product properties. In Section 3, the system modelled by a Bernoulli process is described, followed by the controller design and the error dynamics analysis. In Section 4, the main results are presented. Section 5 extends the main results to the case when there are consecutive data losses in the communication channel. Section 6 shows numerical simulations, and Section 7 draws conclusions.

Section snippets

Graph theory

In multi-agent systems, the information exchange through communication or by sensing is normally modelled by directed or undirected graphs. In [27], a directed graph also called digraph consists of a set of vertices V together with a set of edges E. The digraph is represented as G(V,E), where V symbolizes the agents, and E symbolizes the communication channels. An edge (i,k)E indicates that agent i receives information from agent k. An undirected graph is a graph for which the information

System description

Consider a multi-agent system with the ith agent xi(k+1)=Axi(k)+Bui(k)yi(k)=xi(k),i=1,,nwhere xiRnp and uiRmp.

Agent 1 is chosen as the leader of the team with a reference input ur(k). Note that the state of Agent 1 is not available to all followers while the reference input ur(k) is accessible to other agents, similar as in [25] and [26]. The leader is then written as x1(k+1)=Ax1(k)+Bur(k)y1(k)=x1(k).For the followers, the controller is designed as a static controller. Assuming all followers

Control gain design

In this section, the controller design for the overall system is summarized in Theorem 1. For the arbitrary packet-loss process, a sufficient condition dependent on the packet loss rate is derived by a Lyapunov approach. The condition in Theorem 1 for the overall system is then decomposed and reduced to n conditions of the size of a single follower in Theorem 2.

Theorem 1

Define the set S={SR(np(n1))×(np(n1)):S=diag(S0,,S0)}where S0Rnp×np is symmetric positive definite. Assume that

System description under consecutive packet losses

In Section 3, when there is a packet loss in a specific channel, the controller automatically uses the transmitted data sent from the previous sampling instant. In this section, we consider the consecutive packet dropout case. The modelling and control synthesis are then discussed. Denote the number of the consecutive data loss as δk and assume that δk=δ(k) is an arbitrary integer-valued stochastic variable but bounded as 1δkτM with τM being a known integer. As in Section 3, denote the

Linearized model of quadcopters

Consider a linearized model of a quadcopter given in [30]. The state vector is given as: ξ=[xẋyẏzżψψ̇θθ̇ϕϕ̇]Tand the input vector is u=FThmgτψτθτϕT.A linear model can be derived by the Taylor series linearization around the equilibrium ξ=0 with f(0,0)=0: ξ̇=Aξ+Buy=Cξwith A=01000000000000000000g0000001000000000000000000g0000001000000000000000000000000010000000000000000000000000100000000000000000000000001000000000000,B=000000000000000000001m000000001000000001000000001.Then the system is

Conclusions

In this work, a leader–follower consensus based tracking control strategy was proposed for a group of linearized quadcopters with Bernoulli distribution based group communication link failure. The consensus controller design is based on sufficient conditions from Lyapunov-based methodologies ensuring the mean square exponential stability for MASs with asynchronous packet dropouts in different communication links. Low complexity conditions independent of the network size are proposed to reduce

Acknowledgements

The research was supported by NSERC (Grant No.: 2016-04952), Canada and Alexander von Humboldt Foundation , Germany.

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