Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4063184, 11 pages

https://doi.org/10.1155/2017/4063184

## Consensus Control for a Multiagent System with Time Delays

^{1}Department of Mechanical Engineering, Graduate School of Science and Engineering, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan^{2}Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, Netherlands

Correspondence should be addressed to Yiran Cao

Received 2 September 2016; Revised 14 December 2016; Accepted 26 December 2016; Published 28 February 2017

Academic Editor: Olfa Boubaker

Copyright © 2017 Yiran Cao 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

In this paper, we consider the consensus control problem for a multiagent system (MAS) consisting of integrator dynamics with input and output time delays. First, we investigate a consensus condition for the MAS with a linear controller and without any delay compensation. We then propose a consensus controller with a state predictor to compensate the effect of time delay. The consensus condition for this controller is derived and investigated. Finally, we present an example of solving the consensus control problem for two-wheel mobile robots with feedback loops that pass through a computer network with time delays. To demonstrate the validity of the predictor-based controller, we conduct experiments with two-wheel mobile robots and present the results.

#### 1. Introduction

Achieving cooperative control of robotic systems is of increasing interest and has attracted a great attention in recent years. There are many potential applications for multirobot systems, including unmanned aerial vehicles, satellite clusters, automated highways, and search and rescue operations. Control tasks for robotic systems include consensus [1, 2], flocking [3–5], formation control [6, 7], and tracking [8–10]. Of these, consensus constitutes a fundamental problem for the coordination control of distributed systems. Since cooperative multirobot systems rely on communication between robots in order to collaborate, time delays due to communication through networks and computations are a problem that cannot be neglected. Time delays in general can degrade system performance or even destroy stability. When each robot is considered to be an individual agent, multirobot systems can be considered multiagent systems (MASs). Here, we consider the consensus problem of MASs with time delay.

For nonlinear systems with input delay, Oguchi and Nijmeijer [11] proposed a delay compensation method with a state predictor based on anticipating synchronization. Several studies [1, 12–16] have focused on the consensus problem in a MAS with time delay. The papers [12, 13] showed the upper bound of allowable input time delay under which consensus could be achieved. We attempt to consider both input and output time delay due to both communication and computation. In this study, we first introduce a linear time-delay control protocol with a corresponding consensus condition, similar to that in [12]. The consensus condition is used to give the consensus region of the allowable time delay corresponding to coupling strength. To compensate the time-delay effect, we focus on using a prediction control scheme for MASs with input and output time delays to allow the system to achieve consensus. A previously proposed state predictor-based controller for nonlinear systems with time delay is based on anticipating synchronization [11]. Kojima et al. used it for a tracking-control problem with time delay [17], as did Alvarez-Aguirre et al. [18]. In our previous work [2], we used a controller based on this predictor [11] to solve the consensus problem of MASs in undirected graph networks with time delay and derived the corresponding consensus condition. In this paper, by extending the results, we show that the system achieves the average consensus, and the MASs with directed communication graph can also achieve consensus under the consensus condition. To show the validity of the obtained results, we make the experiments in a multirobot system with the proposed controllers to solve the consensus problem of the coordinates of mobile robots.

This article is organized as follows. In Section 2, we introduce consensus problems for networks of dynamic agents with input and output time delays and show the necessary and sufficient conditions with linear coupling without the predictor to achieve consensus. We then propose a predictor-based consensus controller, for which we derive the necessary and sufficient conditions. Simulation results are presented in Section 3. Section 4 shows experimental results of nonholonomic mobile robots with input and output time delays to show the effectiveness of the proposed control scheme. Finally, Section 5 contains our conclusions.

#### 2. Problem Formulation

Consider a network that consists of identical integrator agents with invariant input and output time delays given as the following dynamics: Here , , and denote the state, output, and input vectors of the th agent, respectively. and separately denote the input and output delays corresponding to agent .

For this system, the consensus problem is formulated as follows.

*Definition 1 (consensus problem). *For multiagent system (1) with input and output time delays, the consensus problem is to find a control protocol to make the states of all agents reach agreement such that for all as .

Following the consensus control protocol proposed by [12], we assume that these agents are interconnected by the following controller:for , where denotes the coupling strength between agents and . denotes the set of agents adjacent to agent , which means these agents are connected to agent in the network topology. We now introduce some definitions about a graph . is the Laplacian matrix of a graph corresponding to the network topology constructed by the interconnection of the agents. If the information communication between agent and is bidirectional, the graph is undirected, and the corresponding Laplacian has the following entries:It is well known that the bidirectional graph Laplacian is diagonalized and has a zero eigenvalue, and positive real eigenvalues such as corresponding to the agents system.

Assuming that , each round-trip time delay is given by for . All coupling strengths are identical and denoted as . Controller (2) is simplified as where denotes the Kronecker product of two matrices and and denote the output vector and the input vector, respectively.

The dynamics of the total system can then be derived aswhere denotes the state vector. The initial condition of the states is given as , where .

Therefore, from the stability of system (5), the following consensus condition is proven following the results of Olfati-Saber and Murray [12].

Theorem 2. *Assume that each system (1) is interconnected by (2) with a coupling strength and constant input and output time delay . The constructed network topology is fixed, undirected, and connected. If the pair satisfiesthe delayed system achieves consensus. Here is the maximum eigenvalue of for an -agents system.*

Rewriting as , pairs of satisfying can stabilise the delayed system. Thus, has a maximum value boundary corresponding to each value of . In general, if the number of agents increases, corresponding to the network structure also tends to increase. Based on , for a fixed time delay , decreases as increases, and this slows the convergence rate. Therefore, this condition means that the convergence rate gets slower as the number of agents and the allowable delay increases. To overcome this problem, in the next section, we propose a state predictor-based controller that can counteract the effect of .

Moreover, if the graph is undirected and connected, [12] shows that MAS without time delays achieve the consensus, and the consensus solution is given as the average of the states of all agents; that is, . In [19], the necessary and sufficient condition for an average consensus problem for MAS with nonuniform and asymmetric time delay is given. From these results, we know that the MAS in Theorem 2 achieves the average consensus and that the consensus value is given aswhere .

#### 3. Main Results

Based on the MAS (1) with controller (2), we propose a controller with a state predictor based on anticipating synchronization for the consensus control of agents. We present the consensus controller and discuss its stability problem in this section.

##### 3.1. State Predictor-Based Controller

Anticipating synchronization is a kind of master-slave synchronization. The predictor is constituted by the given system dynamics and coupling of the difference of the system output and delayed predictor states. The dynamics of this predictor can be stated as follows:where is the prediction gain. Meanwhile, , where denotes the predicted outputs. The initial condition of the predicted states is given as , where . Then, using the output of the predictor instead of the output of the actual system, the main controller is given as

Controller (9) with state predictor (8) compensates the effect of time delays at input and output. If the predictor has prior knowledge of the initial states of the system, the prediction error always remains , and the predictor can predict the exact future value of the states of the system. Thereafter, the total system is shown as

##### 3.2. Consensus Condition

As we use a predictor to predict the states, it is important to prove that the prediction error converges to . The prediction error is defined as

When the prediction error converges to zero, this means that estimates the exact future value of , which is . The time-delay is totally compensated at this time.

With the use of (8) and (9), the dynamics of prediction error can be obtained as

To derive the necessary and sufficient conditions such that the whole system converges to consensus, we consider the coordinate transformation as follows:whereand denotes the synchronization error. Substituting (8), (9), and (11) for the derivative of (13), we obtain the following dynamics:The prediction error dynamics (12) and the dynamics (15) can be rewritten in a matrix form as

From this equation, the consensus condition is given in the following theorem.

Theorem 3. *Assume that each agent (1) is controlled by predictor (8) and controller (9) with gain , constant input and output time delay and prediction gain . The constructed network topology is fixed, undirected, and connected. Then, if the pair and satisfythe MAS achieves consensus.*

*Proof. *The proof is given for the stability of the total synchronization error dynamics (16). After the Laplace transformation, the characteristic equation of (16) can be derived asTo make the equation hold, one of the above determinates should be equal to . As can be seen, the first determinant of (18) represents the synchronization error, and the latter represents the prediction error. To make both errors converge to , all solutions satisfying the following equations must have negative real parts:Since for a fixed, undirected, and connected graph is a symmetric real matrix and for the eigenvalues it holds that , the term of the first equation in (19) satisfies the following results:where is the matrix that transforms into the diagonalized form. Thus, the first equation can be written as . To make have negative real part, we get the condition from the first equation.

For the second equation, we consider the smallest value of , such that , which has a zero real part on the imaginary axis. Then we haveAssuming , we can get and . Since the delay-free system is described by (which is exponentially stable for ), and the continuity of eigenvalues for LTI systems holds, the roots of the dynamics of the second equation of (19) lie on the open left half-plane. Therefore, from the second equation, we get .

The first condition corresponds to the consensus condition for the system without delay, and the second comes from the stability of the prediction error. This discussion means that the synchronization-based predictor is an extension of the full-state observer, and a counterpart of the separation principle holds for the stability of the system with the synchronization-based predictor.

Compared with the consensus condition (6) of Theorem 2, the ranges of both and are extended. The coupling strength is independent of time delay in Theorem 3 and holds for any constant . Moreover, if the prediction error is not zero, both and affect the convergence rate. If the predictor has prior knowledge of the initial condition, is the only influence factor for the convergence rate. We can choose a larger value of in order to make the system converge to consensus faster.

*Remark 4. *It is known that a directed graph contains a directed spanning tree, if and only if the corresponding graph Laplacian always has one zero eigenvalue and eigenvalues that have positive real parts [20]. In this network topology, the MAS satisfying inequalities (17) can also reach a consensus.

Concerning the average value of agent states, we have the following results.

Theorem 5. *Consider that the system with agent (1) satisfies Theorem 3, so that the MAS achieves consensus. This MAS achieves average consensus for any initial states and the consensus solution is given as*

*Proof. *The total system can be summarised asFollowing the method shown in [19], we consider the following functional vector :where represents the solution on the time interval such that , . Since this graph Laplacian is a symmetric zero column-sum matrix, holds. Thus, the integral term is vanished in (24). The time-derivative of (24) along the solution of (23) is given asTherefore, the functional vector is time-invariant, and the value always equals the average of the initial state given as

##### 3.3. Simulation Results

By using controller (9) and predictor (8), we can obtain the simulation results for the system (10) of three and four agents, respectively, connected by networks in Figure 1.