Consensus of second-order and high-order discrete-time multi-agent systems with random networks

doi:10.1016/j.nonrwa.2011.12.009

Nonlinear Analysis: Real World Applications

Volume 13, Issue 5, October 2012, Pages 1979-1990

https://doi.org/10.1016/j.nonrwa.2011.12.009 Get rights and content

Abstract

This paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks and arbitrary weights. Random networks mean that the existence of any edge is probabilistic and independent of any other edge. By introducing the agreement set, velocity control gain and high-order state control gain, some consensus protocols are provided for the discrete-time random networks. Moreover, the per-step and asymptotic convergence factors are proposed to measure the convergence and convergence speed. Some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results.

Introduction

The consensus problem of multi-agent systems has attracted great attention due to its wide application in many fields, such as biology, physics, robotics and control engineering. Roughly speaking, the consensus on control generally means to design a network protocol such that all the agents asymptotically reach an agreement on their states. A growing number of literature have been presented on the consensus problem of multi-agent dynamic systems [1], [2], [3], [4], [5], [6], [7]. Most of these papers assume that the network topology is fixed or evolves deterministically with time. However, practical networks are usually in uncertain communication environments [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Li and Zhang [5] studied the average-consensus for the first-order integrator networks under fixed and directed topologies. In this case, each agent can only obtain the information from its local state or the states of its neighbors. However, the information flow between any pair of agents may be subject to failure with a certain probability in the real world. Therefore, it is important to study the system with random network, the existence of whose edges is probabilistic. That is each link in the network, which represents the information flow between any pair of agents, could be subject to failure with a certain probability. Hence the topology of the network varies randomly over time. Practically, random networks are very common [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. The consensus problem over a difference equation has wide applications in random network theory. Porfiri and Stilwell [7] presented new generalizations on consensus seeking for arbitrary weighted directed random graphs. Song et al. [9] found that the consensus problem in three different convergence modes (almost surely, in probability, and in $L_{1}$ ) is equivalent. They gave the necessary and sufficient condition on the consensus over a random network, which is deduced by independent identically distributed (i.i.d.) stochastic matrices, through the stability in a projected subspace. Abaid and Porfiri [10] studied the discrete-time consensus problem for a group of agents that communicate through a stochastic directed network with fixed out-degree. They found the necessary and sufficient conditions for mean square consentability of the averaging protocol and derived a closed form expression for the asymptotic convergence factor. Hatano and Mesbahi [11] studied the consensus problem over random information networks. Tahbaz-Salehi and Jadbabaie [16] investigated the consensus problem for stochastic discrete-time linear dynamical systems.

Though numerous studies on consensus have been presented in various communication environments, such as fixed topology, varying topology or random switching topology, most of the relevant studies focus on the first-order integrator agent systems. However, when the driving force (acceleration) is considered as control input in practical systems, each agent should be modeled as a double-integrator or even multi-integrator. Thus it is necessary and significative to study the consensus for the second-order and high-order multi-agent systems. Different from the first-order integrator agent system, the consensus of the double-integrator and multi-integrator agent systems depend on not only the topology of the network but also the coupling strength of relative positions and the high-order states between the neighboring agents. The consensus problem of double-integrator agent system has been studied extensively [17], [18], [19], [20], [21], [22]. Unlike the first-order consensus, Ren and Atkins [17] showed that the second-order consensus may not be achieved even if the interaction topology has a directed spanning tree. Therefore, the second-order consensus problem is more complicated and challenging. Cao and Ren [18] studied the convergence of two consensus algorithms for the double-integrator dynamics with intermittent interaction in a sampled-data setting. Hu and Hong [21] showed that if the leader is always reachable and the maximum delay is sufficiently small, the followers can track the leader successfully under proper scaling factors. Lin et al. [22] studied the effect of constant input delays on convergence of second-order consensus protocols and provided the delay-dependent consensus conditions.

As described above, the study of the second-order and high-order multi-agent systems is important and significant. In addition, the information flow between any pair of agents, could be subject to failure with a certain probability in the real world. Because of these problems, this paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks, the existence of whose edges is probabilistic. The convergence analysis of any distributed algorithm on such a network is difficult, because the behavior depends on the actual probability distribution of the network topology. Unlike [9], [10], [11], which studied the consensus over random network with determined weights, we study the consensus for the second-order and high-order multi-agent systems over random network with arbitrary weights.

The rest of this paper is organized as follows. The necessary concepts and knowledge of graph theory are given in Section 2. The problems are formulated in Section 3, and the main results are provided in Sections 4 Consensus of the second-order system with random networks, 5 Consensus of the high-order system with random networks. Some examples and simulations are presented in Section 6 to illustrate the effectiveness of the obtained theoretical results. The conclusions and discussions are given in Section 7.

Section snippets

Preliminaries

Throughout this paper, $R^{n}$ denotes the set of $n$ -dimensional real column vectors and $I_{n}$ the $n$ -dimensional identity matrix; The symbol $\otimes$ denotes the Kronecker product; $1 = {[1, 1, \dots, 1]}^{T}$ is a column vector with appropriate dimension, and $0$ denotes a zero vector or a zero matrix with an appropriate dimension.

Let $G = {V, E, A}$ be a weighted graph with the set of nodes $V = {v_{1}, v_{2}, \dots, v_{n}}$ , where $i \in Υ$ represents the $i$ th agent, and $Υ = {1, 2, \dots, n}$ is the node index of $G$ belonging to a finite index set. $E \subseteq V \times V$ is the set of

Problem formulation and convergence analysis

Suppose that the multi-agent system under consideration consists of $n$ agents. Each agent is regarded as a node in the directed communication graph $G = {V, E, A}$ . Let $x_{i} (k)$ denote the state of agent $i$ at time instant $k$ , and $x (k) = {(x_{1} (k), x_{2} (k), \dots, x_{n} (k))}^{T}, v (k) = {(v_{1} (k), v_{2} (k), \dots, v_{n} (k))}^{T}$ denote the position vector and the velocity vector of the agents, respectively. Moreover, each agent updates its current state based upon the information received from its neighbors according to a control law, which is

Consensus of the second-order system with random networks

In this section, the per-step convergence factor for the second-order discrete-time multi-agent system with random networks are presented. Before giving the main results, some important lemmas are given first for the further analysis.

Lemma 4.1

The evolution of the error vector for the discrete-time consensus algorithm (3.2) is given by the following recursive equation $ξ (k + 1) = H^{T} (I_{n} \otimes E - L (k) \otimes F) H ξ (k) .$

Proof

By (3.5), (3.6), $ξ (k + 1) = H^{T} ϕ (k + 1) = H^{T} (I_{n} \otimes E - L (k) \otimes F) ϕ (k) = H^{T} [I_{n} \otimes E - L (k) \otimes F] [H ξ (k) + \frac{1}{n} β β^{T} ϕ (k)] = H^{T} [I_{n} \otimes E - L (k) \otimes F] H ξ (k) + \frac{1}{n} H^{T}$

Consensus of the high-order system with random networks

This part studies the convergence and convergence speed for the high-order discrete-time multi-agent systems with random networks.

Consider the following high-order multi-agent system ${\begin{matrix} x_{i}^{(0)} (k + 1) = x_{i}^{(0)} (k) + x_{i}^{(1)} (k), \\ x_{i}^{(1)} (k + 1) = x_{i}^{(1)} (k) + x_{i}^{(2)} (k), \\ \dots \\ x_{i}^{(N - 1)} (k + 1) = x_{i}^{(N - 1)} (k) + u_{i}, \end{matrix}$ where $x_{i}^{(l)} (k) \in R$ is the $l$ -order state of the $i$ th agent at time instant $k, l = 0, 1, 2, \dots, N - 1, i \in Υ$ , and $x_{i}^{(0)} (k) = x_{i} (k)$ .

Design the consensus protocol as $u_{i} (k) = d_{0} \sum_{v_{j} \in N_{i} (k)} a_{i j} (x_{j} (k) - x_{i} (k)) - \sum_{l = 1}^{N - 1} d_{l} x_{i}^{(l)} (k),$ where $u_{i} (k) \in R$ is the control

Example and simulations

In this section, some numerical simulations are presented to illustrate the correctness of the obtained theoretical results.

Example

Consider the multi-agent system with six agents (denoted by node $i, i = 1, 2, 3, 4, 5, 6$ ) over directed random networks. Both the probability matrix $P_{k}$ and the weight matrix $W_{k}, k = 1, 2, 3$ , are generated randomly. $P_{1} = [\begin{matrix} 0 & 0.8663 & 0.7463 & 0.4386 & 0.9667 & 0.0225 \\ 0.0881 & 0 & 0.0046 & 0.6997 & 0.9091 & 0.0953 \\ 0.6318 & 0.2611 & 0 & 0.0976 & 0.5901 & 0.4625 \\ 0.4614 & 0.7348 & 0.1587 & 0 & 0.4263 & 0.2807 \\ 0.3779 & 0.9385 & 0.7076 & 0.2588 & 0 & 0.3887 \\ 0.5348 & 0.8801 \end{matrix}$

Conclusions

This paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks. Consensus protocols are proposed by introducing the agreement set and the control gains of high-order status. The per-step convergence factor and asymptotic convergence factor are proposed to measure the convergence and convergence speed of the multi-agent systems. Some simulation results are given to illustrate the effectiveness of the

Acknowledgments

The authors thank the editors and the reviewers very much for their valuable comments and constructive suggestions, which helped to significantly improve the quality of this paper.

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^☆: This work was partially supported by the National Natural Science Foundation of China under Grants 60973012, 61073025, 61073026, 61073065 and 61100076, and the Doctoral Foundation of Ministry of Education of China under Grant 20090142110039.

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Consensus of second-order and high-order discrete-time multi-agent systems with random networks☆

Abstract

Introduction

Section snippets

Preliminaries

Problem formulation and convergence analysis

Consensus of the second-order system with random networks

Consensus of the high-order system with random networks

Example and simulations

Conclusions

Acknowledgments

Physica A

Automatica

Automatica

Nonlinear Analysis. Real World Applications

Automatica

Nonlinear Analysis. Real World Applications

Physica A

Nonlinear Analysis. Real World Applications

Automatica

Automatica

Physica A

Systems and Control Letters