Cooperative containment for second-order multi-agent systems with asynchronous setting and random link failures

Highlights

• Asynchronous containment control for discrete-time second-order multi-agent systems with random link failures is studied.

• Matrix theory and the composition of the binary relations are explored to analyze system stability.

• A necessary and sufficient condition in term of the interaction topology is established.

Abstract

The issue of cooperative containment control in second-order multi-agent systems with asynchronous setting and random link failures is examined in this paper. Asynchronous setting implies that each agent uses its neighbors’ information to update the state independently at certain discrete instants. The phenomenon of random link failure on each communication link is characterized by a Bernoulli stochastic variable. An asynchronous distributed protocol is designed by randomly measuring the position information of the previous one or two steps of the neighbors. Matrix theory and the composition of binary relation are explored to handle the containment control problem with both asynchronous setting and random link failures. Under a loose parameter selection strategy, a necessary and sufficient condition in terms of the topology structure can be established. A numerical example is finally provided to validate the theoretical result.

Introduction

Consensus of multi-agent systems (MASs) to seek certain benefits has attracted a great deal of research attention in recent decades. The literature [1] provided comprehensive overviews of the issue of consensus. In order to reach consensus, agents often need to update their states by exchanging information with the neighbors as a means of coordinating their behavior, such as achieving a common heading direction of unmanned air vehicles [2], or achieving consensus of behavior in rendezvous and flocking problem [3], [4]. According to the number of leaders, consensus problems can be divided into several sub-areas, one of which is leaderless consensus with the purpose to ensure that all agents reach a common state [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Another important sub-area that contains a leader is called leader-following consensus, which has been widely investigated in recent years, see, for instance, [16], [17], [18], [19], [20] and the references therein.

Another especially interesting topic that differs from leaderless consensus and leader-following consensus is containment control in which there are a small number of leaders with autonomous behavioral awareness and a large number of followers with no autonomous behavioral awareness, and the leaders’ goal is to design the appropriate control protocol to drive the followers gradually into the convex hull constructed by the leaders. For instance, Ji et al. [21] proposed a valuable application of containment control in the engineering field. In a dangerous environment where a group of autonomous vehicles are migrating, a small number of autonomous vehicles, which are identified as the leaders and are able to detect potential hazards in the surroundings, successfully drive the followers who lack the ability to detect potential dangers into the safe area composed of the leaders. So far, a lot of valuable research results in different investigation scenarios, such as only position information [22], sampling [23], finite-time control [24], and heterogeneous communication topology [25] have been made public. It is remarkable that these results were obtained in a synchronized scenario where all agents adjust their states synchronously. However, the acquisition of the central synchronous clock is very difficult in many engineering applications. In this case, considering the containment control under asynchronous setting is more practical. Asynchronous setting means that each agent measures its neighbors’ information to update the state independently at certain discrete instants by its clock. Recently, asynchronous containment control problems of second-order MASs were studied in [26], [27]. From the above works, the followers gradually converge into the convex hull formed by the leaders, and the followers’ final states can be described by a mathematical expression related to the directed communication topology and the leaders’ states.

In many practical engineering applications, the communication link between agents often fails due to external blocking, background noise, and multipath fading. This practical consideration has led to widespread attention in the study of the effects of communication failures on the sensor synchronization. For example, the average consensus with random link failures was studied in [28], and the state estimation issue over wireless sensor networks with random link failures was analyzed in [29]. In recent years, the random failure phenomenon of communication links has been considered in the coordinated control of MASs. The consensus issue of heterogeneous MASs with random link failures was handled in [30] and the nonlinear consensus issue over MASs with random link failures was investigated in [31].

As can be seen from the above discussions, some interesting results on the consensus with random link failures [30], [31] and asynchronous containment control [26], [27] have been published so far. However, few papers have studied the containment control problem of MASs in the context of both asynchronous setting and random link failures. Based on this consideration, the main purpose of this paper is to study the asynchronous containment control problem of MASs under the environment of random link failures. The phenomenon of random link failure on each communication link is characterized by a Bernoulli stochastic variable, and the asynchronous setting means that each agent independently uses the control input to update the state through its own clock. Different from the existing asynchronous settings [26], [27] where each agent moves at a fixed acceleration between its any two adjacent update time instants, the asynchronous setting proposed in this paper is a new state update rule, in which each agent moves at a fixed velocity between its any two adjacent update time instants. In general, resolving the containment control issue with both asynchronous setting and random link failures is more challenging than the containment control issue with synchronous setting or without random link failures. To overcome this challenge, we first design appropriate model transformations to transform the asynchronous containment control problem with random link failures into a product convergence problem of nonnegative random matrices in which some elements obey Bernoulli distribution, and then utilize algebra theory and the composition of the binary relation to solve this convergence problem.

The outline of this paper is shown as follows. Section 2 describes the model formulation of asynchronous containment control with random link failures and introduces the supporting lemma and definitions. In Section 3, a sufficient and necessary condition is provided and analyzed in detail. The effectiveness of the proposed asynchronous distributed protocol with random link failures is verified by simulation results in Section 4. Finally, some conclusive work is drawn in Section 5.

Notations. Let ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote the set containing all n-dimensional real column vectors and the set containing all n × m real matrices, respectively. A real matrix $S={\left[s_{ij}\right]}_{n\times n}$ is nonnegative if s_ij ≥ 0 for all $i,j=1,2,\dots ,n,$ and further the nonnegative matrix S is row-stochastic if $S{\mathbf{1}}_n={\mathbf{1}}_n$. For a nonnegative matrix $S=\left[s_{ij}\right]\in {\mathbb{R}}^{n\times n},$ diag{S} denotes a diagonal matrix containing sequential diagonal elements $s_{11},s_{22},\dots ,s_{nn},$ ${\Lambda }_i\left[S\right]={\sum }_{j=1}^n{s}_{ij}$ represents the ith row’s sum of matrix S and ${\parallel S\parallel }_{\infty }={\max }_i\left\{{\Lambda }_i\left[S\right]\right\}$ is the infinite norm of matrix S. Moreover, [S]_ij can also be used to denote the element s_ij of matrix S if there is no ambiguity. ${\mathbb{Z}}_{+}$ and $\mathbb{N}$ denote, respectively, the positive integer set and the natural number set. ⊗ is the Kronecker product. $I_n\in {\mathbb{R}}^{n\times n}$ is an n-order identity matrix.