Volume 18 Issue 2
Jun.  2020
Article Contents

Ali Alouache, Qing-He Wu. Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State[J]. Journal of Electronic Science and Technology, 2020, 18(2): 190-200. doi: 10.1016/j.jnlest.2020.100053
Citation: Ali Alouache, Qing-He Wu. Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State[J]. Journal of Electronic Science and Technology, 2020, 18(2): 190-200. doi: 10.1016/j.jnlest.2020.100053

Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State

doi: 10.1016/j.jnlest.2020.100053
Funds:  This work was supported by the National Natural Science Foundation of China under Grant No. 61321002
More Information
  • Author Bio:

    Ali Alouache received the B.Eng. degree from the Institute of Electronic and Electrical Engineering IGEE (ex INELEC), Boumerdes University, Boumerdes in 2012. He received his M.S. degree from École Militaire Polytechnique (EMP), Algiers in 2015. Currently, he is pursuing his Ph.D. degree with the Automation School, Beijing Institute of Technology (BIT), Beijing. His research interests include tracking control of robotics, consensus of multi-agent systems, formation control of multi robots, computer vision, and image processing

    Qing-He Wu received his B.Eng. degree in electrical engineering from Huazhong University of Science and Technology, Wuhan in 1982. He obtained the M.S. and Ph.D. degrees from the Swiss Federal Institute of Technology (ETH), Zurich in 1984 and 1990, respectively. From 1986 to 1994, he was an assistant and oberassistent with the Institute of Automatic Control, ETH. Since 1995, he has been working with BIT, where he has been a professor since 1997. He was a visiting research fellow from July 2002 to June 2003 with Akita Prefectural University, Akita. His research interests include robust control theory, H-infinity control, and tracking control of multi-agent systems

  • Authors’ information: A. Alouache and Q.-H. Wu are with the School of Automation, Beijing Institute of Technology, Beijing 100081 (e-mail: alouache15@yahoo.fr; qinghew@bit.edu.cn).
  • Received Date: 2017-08-11
  • Rev Recd Date: 2019-06-27
  • Available Online: 2020-07-08
  • Publish Date: 2020-06-01
  • This paper discusses consensus tracking of single-integrator multi-agent systems with a time varying reference state based on the least squares estimation theory to deal with the case of communications disturbance. Since none of the followers can communicate with the leader within the accepted time due to communications disturbance, least squares estimation is applied for online estimation of the time varying reference state and reducing the tracking errors occurred due to communications disturbance. A theoretical proof is provided under some assumptions while the necessary and sufficient conditions are demonstrated for which consensus is reached on the time-varying reference state in case of communications disturbance. Finally, a Matlab example is given to validate the effectiveness of the proposed approach.
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    [25] C. Yang, W. Hong, L. Huang, et al., “Fast consensus tracking of multiagent systems with diverse communication delays and input delays,” Mathematical Problems in Engineering, vol. 2014, no. 5, pp. 1-10, Jan. 2014.
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Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State

doi: 10.1016/j.jnlest.2020.100053
Funds:  This work was supported by the National Natural Science Foundation of China under Grant No. 61321002
  • Author Bio:

  • Corresponding author: A. Alouache and Q.-H. Wu are with the School of Automation, Beijing Institute of Technology, Beijing 100081 (e-mail: alouache15@yahoo.fr; qinghew@bit.edu.cn).

Abstract: This paper discusses consensus tracking of single-integrator multi-agent systems with a time varying reference state based on the least squares estimation theory to deal with the case of communications disturbance. Since none of the followers can communicate with the leader within the accepted time due to communications disturbance, least squares estimation is applied for online estimation of the time varying reference state and reducing the tracking errors occurred due to communications disturbance. A theoretical proof is provided under some assumptions while the necessary and sufficient conditions are demonstrated for which consensus is reached on the time-varying reference state in case of communications disturbance. Finally, a Matlab example is given to validate the effectiveness of the proposed approach.

Ali Alouache, Qing-He Wu. Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State[J]. Journal of Electronic Science and Technology, 2020, 18(2): 190-200. doi: 10.1016/j.jnlest.2020.100053
Citation: Ali Alouache, Qing-He Wu. Consensus Based Least Squares Estimation for Single-Integrator Multi-Agent Systems with a Time-Varying Reference State[J]. Journal of Electronic Science and Technology, 2020, 18(2): 190-200. doi: 10.1016/j.jnlest.2020.100053
  • In recent years, multi-agent systems have gained a lot of interest due to their advantages for accomplishing complex tasks compared with a single agent. Consensus tracking of multi-agent systems is the most studied issue in the literature. Consensus tracking means to reach an agreement regarding a certain quantity of interest that depends on the state of all the agents[1]-[3].

    The applications of consensus tracking algorithms are numerous in different domains, such as rendezvous control of multi nonholonomic agents[4]-[6], formation control[7],[8], and flocking attitude alignment[9],[10]. The special cases are consensus tracking with a dynamical leader based on the virtual structure approach[11] and consensus of the second-order multi-agent system with multi leaders[12].

    The basic theoretical framework of consensus problems for networked dynamical systems was introduced by the studies of Saber and Murray[13],[14], which were inspired by the earlier work of Fax and Murray[15],[16]. Jadbabaie et al.[17] demonstrated that all the agents could eventually move in the same direction with a simple neighbor rule despite the absence of the centralized coordination and each agent’s set of neighbors changing with time as the system evolved under a joint connection topology. So as a special consensus problem, tracking has also been explored by many scholars, consensus tracking control was proposed in [18] for double integrator dynamics, when there was a coupling between the neighbors’ information state derivatives. Consensus tracking for multi-vehicle control with a constant or time-varying reference state was proposed by Ren[19],[20]. Later consensus tracking control was also studied in other different conditions, such as consensus with noise measurement[21], with time delay[22], and with uncertain dynamics[23].

  • Most of the previously cited works operate under the hypothesis that the leader always keeps connected with a partial set of followers. However in the physical applications, the communications between the followers and the leader may fail due to many reasons like noise and the errors of the sensors, therefore the followers can not receive the information of the leader, which may cause the instability of the multi-agent system. Therefore many works have investigated the problems of the communications failure between the agents of the systems. Leader-following consensus tracking for multi-agent systems with multiplicative noise was analyzed by Djaidja and Wu[24]. A fast consensus tracking approach for discrete time multi-agent systems with input and communications delay was studied by [25]. The formation control problem for single-integrator multi-agent systems with diverse communications and input delay was investigated by [26]. In [27], particle swarm optimization was used to tackle communications failure for tracking control of multiple mobile robots formation. Formation control of multi-agent systems in realistic fading environments was proposed in [28] and [29]. Song et al.[30] used tools from stochastic differential delay equation (SDDE), martingale theory, and stochastic inequality for establishing sufficient conditions which led mean square consensus with multiplicative noise and time delay under directed fixed topologies. Jin and Yuan[31] proposed a robust adaptive sliding mode approach to the asymptotic consensus problem for a class of multi-agent systems with time-varying additive actuator faults and communications perturbation, however the design is complex and the performance of the multi-agent system is not further analyzed for the timing of communications disturbance.

  • The contribution of this paper is to propose least squares estimation for consensus tracking of single-integrator multi-agent systems with a time varying reference state to overcome the case of communications disturbance.

    The proposed approach is primarily based on Ren’s fundamental algorithm presented in [19] for multi-vehicle consensus with a time varying reference state. If none of the followers can communicate with the leader within the accepted time due to communications disturbance, therefore the proposed approach is adapted to improving the stability and robustness of the multi-agent system for achieving consensus even communications fail among the agents. The main idea behind this work is that the followers try to predict the coming information from the leader based on the information which had been received before and take the corresponding action in case of communications disturbance by estimating the time varying reference state according to online collected data for reducing the tracking errors occurred due to communications failure.

    A theoretical proof is provided for the proposed approach under some assumptions while the necessary and sufficient conditions are also demonstrated, for which consensus is reached on the time-varying reference state in case of communications disturbance. At the end, a Matlab example is given to validate the effectiveness of the proposed approach by supposing the case of communications disturbance and comparing the results with the fundamental algorithm of Ren[19].

    The rest of this paper is structured as follows: Section 2 gives some background about the graph theory and least squares estimation technique. Section 3 formulates the consensus algorithm for single-integrator multi-agent systems with a time-varying reference state and the problem statement. Section 4 describes in details the proposed approach and provides a theoretical proof under some assumptions with the necessary and sufficient conditions. The comparative results of Matlab simulations are given in Section 5. Finally, conclusions and further investigations are given in Section 6.

  • The graph theory is used to represent the allowed information flow between the agents of the system. Each node of the graph represents an agent, which is connected to its neighbors through an adjacency matrix, where each node has some effects on its neighbors for sharing communications information. This section gives a brief tutorial on the algebraic graph theory and some basic concepts of the least squares estimation theory.

  • The information exchange among vehicles is modelled by directed or undirected graphs[13]. A directed graph is a pair (Vp, εp), where the set ${V_p} \!=\! \left\{ {1,{\rm{ }}2,{\rm{ }} \cdots,{\rm{ }}p} \right\}$ is a finite nonempty node set and εp$\subseteq $Vp×Vp is an edge set of ordered pairs of nodes, called edges. The edge (i, j ) in the edge set of a directed graph denotes that the vehicle j can receive information from the vehicle i but not necessarily vice versa.

    An undirected graph can be viewed as a special case of a directed graph, where an edge (i, j ) in the undirected graph corresponds to edges (i, j ) and (j, i ) in the directed graph. A directed tree is a directed graph in which every node has exactly one parent expect for one node, called the root, which has no parent and has a directed path to every other node. In undirected graphs, a tree is a graph in which every pair of nodes are connected by exactly one undirected path.

    The adjacency matrix ${{\text{A}}_p} \!=\! [{a_{i,j}}] \!\in\! {\mathbb{R}_{p \times p}}$ of a directed graph (Vp, εp) is defined such that ${a_{i,j}}$ is a positive weight if $(j,i)\!\in$εp, and ${a_{i,j}}$=0 if (j, i)$\notin$εp. Self-edges are not allowed unless otherwise indicated. The adjacency matrix of an undirected graph is defined analogously except that ${a_{i,j}}\!=\!{a_{j,i}}$ for all i≠j because $(j,i)\!\in$εp implies$(i,\,j)\!\in$εp.

    Define the matrix ${{\text{L}}_p} \!=\! \left[ {{l_{i,j}}} \right] \!\in\! {\mathbb{R}_{p \times p}}$ as ${l_{i,i}} = \sum\limits_{j = 1,j \ne i}^n {{a_{i,j}}} $, ${l_{i,j}} \!=\! - {a_{i,j}},i\!\ne\! j.$ Note that if (j, i )$\notin$εp, then ${l_{i,j}}\! =\! - {a_{i,j}} \!=\! 0$.

    The matrix Lp satisfies ${l_{i,j}} \!\le\! 0$, $\sum\limits_{j = 1,j\,\!\ne\!\, i}^n {{l_{i,j}}} \!=\! 0$, $i \!=\! 1,{\rm{ }}2 , {\rm{ }} \cdots , {\rm{ }}p$. For an undirected graph, Lp is symmetrical and it is called the Laplacian matrix.

  • Least squares estimation is a mathematical method that can realize the best fitting in terms of the minimum square error according to the given data.

    Given the data of the input vector ${{\text{x}}_i} = {\left[ {{x_{1 , i}} , {\rm{ }}{x_{2 , i}} , {\rm{ }} \cdots ,{\rm{ }}{x_{n , i}}} \right]^{\rm{T}}}$ and the output vector ${{\text{Y}}_i} = {\left[ {{y_1} , {\rm{ }}{y_2} , {\rm{ }} \cdots , {\rm{ }}{y_n}} \right]^{\rm{T}}}$, where n denotes the number of the variables that are related to Y. Therefore we may define the mathematical model for the given data as

    where ${\hat{{{\text{θ}}}}}$ is the regression vector to be solved ${{\hat {{\text{θ}}}}}=\left[ {θ _1} , {θ _2} , \cdots , {θ_n}\right]^{\rm{T}}$, ${{\text{Φ}}}={\left[ {{x_1} , {x_2} ,\cdots ,{x_m}} \right]^{\rm{T}}}$ is the input vector, m denotes the number of sampling points, and ${{\text{ε}}}{\rm{ = }}{\left[ {{{{{ε}}} _1} , {\rm{ }}{{{{ε}}} _2} , {\rm{ }} \cdots , {\rm{ }}{{{{ε}}} _m}} \right]^{\rm{T}}}$ is the residual vector.

    The least squares estimation problem aims to search a regression coefficient ${\hat{{{\text{θ}}}}}$ that minimizes the following value $J = \sum\limits_{i = 1}^m {{{ε}} _i^2} = {{\text{ε}}^{\rm{T}}}{\text{ε}}.$ The regression vector ${{\hat {{\text{θ}}}}}{\rm{ = }}{\left( {{{{\text{Φ}}}^{\rm{T}}}{{\text{Φ}}}} \right)^{ - 1}}{{{\text{Φ}}}^{\rm{T}}}{\text{Y}}$ is obtained, such that the matrix ${{\text{Φ}}^{\rm{T}}}{\text{Φ}} $ is reversible.

    In order to accelerate the estimation process for the vehicle and to perform regularly prediction with new data, then the recursive least squares estimation algorithm with an attenuation factor is adopted.

  • Consider an interconnected system composed of n agents with single-integrator dynamics:

    Or in a matrix form, (1) may be rewritten as follows

    where ${{\text{ξ}}} \triangleq {\left[ {{{\text{ξ}}}_1^{\rm{T}} , {\rm{ }}{{\text{ξ}}}_{\rm{2}}^{\rm{T}} , {\rm{ }} \cdots , {\rm{ }}{{\text{ξ}}}_n^{\rm{T}}} \right]^{\rm{T}}}.$ ${{{\text{ξ}}}_i} \in {\mathbb{R}^m}$ and ${{\text{u}}_i} \in {\mathbb{R}^m}$ are respectively the information state and the information control input of the ith vehicle. ${{\text{L}}_n}(t) \in {\mathbb{R}^n}$ is the non-symmetrical Laplacian matrix at time t and $\otimes $ denotes the Kronecker product. Im denotes the m dimensional identity matrix.

    Suppose a multi-agent system which consists of n identical vehicles as described above with an additional vehicle labelled n+1, which acts as the unique leader of the team. Therefore the (n+1)th vehicle is named as the leader and the vehicles $1 , {\rm{ }}2 , {\rm{ }} \cdots , {\rm{ }}n$ are named as followers.

    The vehicle n+1 has the information state as following ${{\text{ξ}}_{n + 1}} \triangleq {{\text{ξ}}^r} \in {\mathbb{R}^m}$, where ${{{\text{ξ}}}^r}$ denotes the time varying reference state.

    The consensus reference state satisfies

    where (·,·) is bounded, piecewise continuous in t, and locally Lipschitz in ${{{\text{ξ}}}^r}$.

    The consensus problem with a time varying reference state is solved if and only if ${{{\text{ξ}}}_i}(t) \to {{{\text{ξ}}}^r}(t) ,$ where ${i} = 1 , {\rm{ }}2 , {\rm{ }} \cdots , {n}$ as t→∞.

    Assume the team of the vehicles as an interaction topology with n+1 nonempty nodes. Therefore according to the graph theory tutorial given in the subsection 2.1, we can get the corresponding directed graph ${{\mathcal{G}}_{n + 1}}$ for the multi-agent system, i.e., ${{\mathcal{G}}_{n + 1}} \!\triangleq\! ({V_n}_{ + 1},{{{{ε}}} _n}_{ + 1})$, the adjacency matrix An+1, and the matrix Ln+1 which is the non-symmetrical Laplacian matrix associated with the graph ${{\mathcal{G}}_{n + 1}}$.

    In general, there are two primary methods to solve the consensus tracking problem for the multi-agent system. A consensus tracking algorithm was proposed in [19] as

    where ai is the positive constant scalar and ${a_{i,j}}$ is the (i, j ) entry of the adjacency matrix An+1 with the additional assumption that ai,n+1=1, if $\left( {n + 1,{\rm{ }}i\,} \right) \in {{{ε}}_{n + 1}}$, and ai,n+1=0 otherwise.

    When An+1 is constant and ai,n+1=1, $i = 1,\,2,\,\cdots,\,n$, the consensus tracking problem with a time-varying reference state is solved with (1). However, the topology condition is too rigorous, hence another control law was proposed in [19] as

    where ${\rm{ }}i \!=\! 1,{\rm{ }}2,{\rm{ }} \cdots,{\rm{ }}n$, $\;j \!= \!1,{\rm{ }}2,{\rm{ }} \cdots,{\rm{ }}n + 1$, ai(t) is the (i, j ) entry of the adjacency matrix An+1(t) at time t, γ is a positive constant scalar, and ${{{{η}}} _i}(t) \triangleq \sum\limits_{j = 1}^{n + 1} {{a_{i,j}}(t)} $.

    The consensus problem is solved if and only if the corresponding directed graph ${{\mathcal{G}}_{n + 1}}$ has a directed spanning tree. Equation (5) solves consensus tracking of the time varying reference state for the multi-agent system if the communications are available among the agents as demonstrated in [19], however in case of communications disturbance[27], this paper proposes a solution for consensus tracking based on the least squares estimation theory as demonstrated in the following section.

  • The cooperative control approach proposed in this paper is developed based on (5), thus it is valid when the communications between the leader and the followers are available, specifically the followers which can communicate with the leader directly. Moreover, the proposed algorithm is adapted for estimating the value of the time varying reference state ${{{\text{ξ}}}^r}$ using least squares estimation. When the communications between the leader and the followers are failed, the reference state ${{{\text{ξ}}}^r}$ is substituted by the estimated state, i.e., ${\tilde{{{\text{ξ}}}}}_{}^r.$

    Before getting into the development of the proposed approach, the following assumptions are presented as sufficient conditions for the application of the proposed algorithm.

    Assumption A. We assume that the reference state ${{\text{ξ}}^r}$ has N derivatives in the time interval of the process, i.e., [t0, T ], where N is a positive integer number. T is the duration of the process, and T=t0+td+tr as indicated in Fig. 1.

    Figure 1.  Timing of communications disturbance.

    As shown in Fig. 1, we consider that the instant t0 denotes the moment when the reference state ${{\text{ξ}}^r}$ is available to a portion of the followers. td denotes the moment when the reference information is not available to the followers due to communications disturbance. tr denotes the moment when the reference information becomes available again.

    Assumption B. We assume that the information packet about the reference state that the follower received from the leader is proportional to the time td. Here, it is supposed that the sampling frequency is 1, thus td means the number of the information packets that the follower receives during the period [t0, td]. For reducing the influence of the old data and paying more attention to the new states, an attenuation factor is added at least squares estimation.

    Assumption C. We assume that the multi-agent system is stable and also can achieve consensus tracking after the moment T (i.e., T=t0+td+tr).

    The reference state ${{{\text{ξ}}}^r}$ is assumed to be available again if and only if

    where $i = 1,\,2 ,\, \cdots ,\, n $ and ${δ}$ denotes the critical deviation. Notice that if the cumulative state deviation of some followers is more than ${δ}$, the whole system will be instable and it will never reach consensus tracking even if the reference state ${{\text{ξ}}^r}$ is available again.

    Assumption D. Since communications disturbance is more likely to occur in the tracking process when the states of followers are very close to the reference state ${{\text{ξ}}^r}$, then the estimated value ${\tilde{{{\text{ξ}}}}}_{}^r$ is also very close to the reference state ${{{\text{ξ}}}^r}$. Hence we assume that the time, which the followers spend on tracking ${\tilde{{{\text{ξ}}}}}_{}^r$ and achieving ${{{\text{ξ}}}_i} = {\tilde{{{\text{ξ}}}}}_{}^r$, is very short and can be ignored compared with the time tr.

    Theorem 1. If the time-varying reference state ${{{\text{ξ}}}^r}$ has N derivatives in the time interval of the process [t0, T ], therefore an estimation function ${\tilde{{{\text{ξ}}}}}_{}^r$ can be obtained according to the received information (td) states, which satisfies the following inequality $\displaystyle\int_{{t_0}{\rm{+}}{t_d}}^T {{{\left| {{\tilde{{{\text{ξ}}}}}_{}^r - {{\text{ξ}}}_{}^r} \right|}^T}\left| {{\tilde{{{\text{ξ}}}}}_{}^r - {{\text{ξ}}}_{}^r} \right|{\rm{d}}t} < {δ}_1$, where ${δ}_1$ is affected by td, N, and tr.

    Proof.

    We consider that ${\tilde{{{\text{ξ}}}}}_J^r$ is the estimation for the Jth state of the reference state ${{{\text{ξ}}}^r}.$

    Firstly, ${\tilde{{{\text{ξ}}}}}_J^r$ is expressed as

    where $k \!=\! {\rm{min}} (N, {t_d} \!-\! 1)\,\, {\rm {and}} \, J \!=\! 1,{\rm{ }}2,{\rm{ }} \cdots ,{\rm{ }}M$ with M denoting the dimension of the state.

    Secondly, ${\tilde{{{\text{ξ}}}}}_J^r$ is transformed into a matrix form as follows

    with ${{\text{Φ}}} = {[{{{\text{φ}}} _{J,1}},{\rm{ }}{{{\text{φ}}} _{J,2}},{\rm{ }} \cdots ,{\rm{ }}{{{\text{φ}}} _{J,m}}]^{\rm{T}}},{\rm{ }}{{{\text{φ}}} _{J,i}} = [1,{\rm{ }}t,{\rm{ }} \cdots ,{\rm{ }}{t_k}]$, and ${{\text{θ}}} = [{a_{J,0}},{\rm{ }}{a_{J,1}},{\rm{ }} \cdots ,{\rm{ }}{a_{J,k}}]^{\rm T}$, where m=td, therefore

    with ${{\text{ε}}} = {[{{{{ε}}}_{J,1}},{\rm{ }}{{{{{ε}}}}_{J,2}},{\rm{ }} \cdots ,{\rm{ }}{{{{{ε}}}}_{J,m}}]^{\rm{T}}}$ denoting the residual vector.

    Then, according to the least squares estimation theory, there exists a vector θ which can minimize the value of εTε.

    Hence an estimation function $\tilde {{\text{ξ}}} _J^r$ is obtained and it is the best fitting to the state ${{\text{ξ}}}_J^r$ in the time interval [t0, t0+td].

    Since ${{\text{ξ}}}_J^r$ has N derivatives, so in the time interval [t0+td, T ], the state ${{\text{ξ}}}_J^r$ can be expressed in the form of the Taylor expansion as follows: ${{\text{ξ}}}_J^r = {\tilde{{{\text{ξ}}}}}_J^r + R(\text{x}),$ where R(x) is a higher order infinitesimal.

    Therefore in the time interval [t0+td, T ], the following inequality $\displaystyle\int_{{t_0}{\rm{+}}{t_d}}^T {{{\left| {{\tilde{{{\text{ξ}}}}}_J^r - {{\text{ξ}}}_J^r} \right|}^2}{\rm{d}}t} < {{{δ}_1}/M}$ is reasonable, and obviously the following $\displaystyle\int_{{t_0}{\rm{+}}{t_d}}^T {{{\left| {{\tilde{{{\text{ξ}}}}}_J^r - {{\text{ξ}}}_J^r} \right|}^T}\left| {{\tilde{{{\text{ξ}}}}}_J^r - {{\text{ξ}}}_J^r} \right|{\rm{d}}t} < {{δ}_1}$ is reasonable, too. Meanwhile, $k\! =\! {\rm{min}} (N,\,{t_d} \!-\! 1)$, thus ${δ}_1$ is inversely proportional to td or N, also it is proportional to tr. Finally, Theorem 1 is proved.

    Theorem 2. If the time-varying reference state ${{{\text{ξ}}}^r}$ has N derivatives in the time interval [t0, T ] and N>μ1, td1, and tr2. The values μ1 and μ2 are the critical values which make ${δ}_1$<${δ}$, and the directed graph ${{\mathcal{G}}_{n + 1}}$ has a directed spanning tree. Therefore the consensus tracking problem is solved if the estimated value ${\tilde{{{\text{ξ}}}}}_{}^r$ is adopted at the time interval [t0+td, T ].

    Proof.

    The proof of this theorem can be given by the following two steps.

    Step 1: Firstly, when the reference state is available, it is necessary to prove that the consensus tracking problem is solved if the graph ${\tilde{{{\text{ξ}}}}}_{}^r$ has a directed spanning tree.

    Since ${\tilde{{{\text{ξ}}}}}_{}^r$ has a directed spanning tree, only all entries of the last row of An+1 are zero and ${{{{η}}} _i} \!= \!\sum\limits_{j \!=\! 1}^{n + 1} {{a_{i,j}} \!\ne\! 0} ,$ $i \!=\! 1,{\rm{ }}2,{\rm{ }} \cdots ,{\rm{ }}n.$

    According to (1), (5) can be rewritten as

    Equation (9) can be rewritten in a matrix form as follows

    where ${{\text{U}}} \triangleq {\left[{{\text{u}}}_1^{\rm{T}},{{\text{u}}}_2^{\rm{T}}, {\cdots} ,{\text{u}}_n^{\rm{T}}, {{\dot{\text{ξ}}}^r}^{\rm{T}} \right]^{\rm{T}}}$, ${{\text{Ξ}}} \triangleq {\left[ {{{\text{ξ}}}_1^{\rm{T}},{\rm{ }}{{\text{ξ}}}_2^{\rm{T}},{\rm{ }} \cdots ,{\rm{ }}{{\text{ξ}}}_n^{\rm{T}}, {{{\text{ξ}}}^r}^{\rm{T}}} \right]^{\rm{T}}}$, and ${\text{L}}_{n,n + 1}$ is the first n rows of the Laplacian matrix ${{\text{L}}_{n + 1}}$.

    Since the directed graph of ${{\text{L}}_{n + 1}}$ has a directed spanning tree, we can induce that rank$({{\text{L}}_{n + 1}})\!=\!n$. This implies that rank$({{\text{L}}_{n,n + 1}})\!=\!n$.

    ${{\text{L}}_{n,n + 1}}$ can be rewritten as $\left[ {\left. {\text{B}} \right|{\text{b}}} \right]$, where B constitutes the first n columns of ${{\text{L}}_{n,n + 1}}$, and b is the last column of ${{\text{L}}_{n,n + 1}}$. Notice that the summation of each row of ${{\text{L}}_{n,n + 1}}$ is zero, hence b=−B1n. Therefore we can get that rank(B)=n.

    When we substitute $\left[ {\left. {\text{B}} \right|{\text{b}}}\right] $ fo ${{\text{L}}_{n,n\! +\! 1}}$ in (9), we get

    where ${\text{u}} \!=\! {\left[ {{\text{u}}_{\rm{1}}^{\rm{T}},{\rm{ }}{\text{u}}_{\rm{2}}^{\rm{T}},{\rm{ }} \cdots ,{\rm{ }}{\text{u}}_n^{\rm{T}}} \right]^{\rm{T}}}$ and ${{\text{ξ}}}\! \triangleq \! {\left[ {{{\text{ξ}}}_{\rm{1}}^{\rm{T}},{\rm{ }}{{\text{ξ}}}_2^{\rm{T}},{\rm{ }} \cdots ,{\rm{ }}{{\text{ξ}}}_n^{\rm{T}}} \right]^{\rm{T}}}$.

    Since B is full rank, it is invertible. Then (9) can be rewritten as

    Moreover, (12) can be rewritten as

    Notice that ${\dot{{{\text{ξ}}}}}{\rm{ = }}{\text{u}}$, hence (13) can be written as

    From (14), we can get that ${{\text{ξ}}}\!\left( t \right) \to {\text{1}_n} \otimes {{{\text{ξ}}}^r}\!\left( t \right)$ as t→∞. It follows that ${{{\text{ξ}}}_i}(t) \to {{{\text{ξ}}}^r}\!(t)$ where i = 1, 2,···, n as t→∞, then the consensus tracking is solved.

    Step 2: Secondly, we need to prove that the consensus tracking problem is solved when the estimated state ${\tilde{{{\text{ξ}}}}}_{}^r$ is adopted. At time interval [t0+td, T ], the cumulative state deviation of followers at that time interval can be written as

    with ${\rm{ }}i \!=\! 1,{\rm{ }}2,{\rm{ }} \cdots ,{\rm{ }}n$.

    According to Assumption D, (15) can be rewritten as

    From Theorem 1, we can induce the following as

    where ${{δ}_1}\!$<${δ}$. Equation (17) implies that after the moment of disturbance T and when ${{{\text{ξ}}}^r}$ is available again, the consensus problem still can be solved.

    Theorem 3. If the time-varying reference state ${{{\text{ξ}}}^r}$ has N derivatives in the time interval [t0, T ] and N>μ1, td >μ1, and tr<μ2, where μ1 and μ2 are critical values, then ${{δ}_1}$<${δ}$, and the directed graph ${{\mathcal{G}}_{n + 1}}$ has a directed spanning tree, therefore

    where td denotes the last state that the follower can get before ${{{\text{ξ}}}^r}$ becomes unavailable.

    Proof.

    According to the definition of μ2, we can get

    According to Theorem 1, we can write the following as

    From (18) and (19), the following inequality can be deduced

    The inequality (20) is given by Theorem 3, which has been proved. Through the above analysis, it can be concluded that the proposed approach predicts the coming reference state by analysing the states that have been received before. Therefore in case of communications disturbance, the least squares estimation algorithm is applied to estimate the time-varying reference state for the consensus tracking of the multi-agent system. Moreover, the robustness and stability of the multi-agent system are also enhanced as demonstrated by Theorem 3.

  • This section presents a simulation example for verifying the effectiveness of the proposed approach comparing with the fundamental algorithm given by (5).

    The simulation is performed on a group of four follower robots and one leader robot in one dimensional environment (M=1), where each robot is modelled by (1).

    Assume that the communications topology among the robots as indicated in Fig. 2 with the time varying reference state ${{{\text{ξ}}}^r}\!(t)$ is available only to the leader robot numbered 1.

    Figure 2.  Communications topology for the multi-agent system.

    Assume that the function of the reference state $ {{{\text{ξ}}}^r}\!(t) $ is sin(t). Suppose that the reference state ${{{\text{ξ}}}^r}\!(t)$ is not available during the time interval [18 s, 21 s], hence the disturbance instant is td=18 s and the reference is available again at the instant tr=21 s. Figs. 3 and 4 illustrate the results obtained by applying the control law of (5) to the multi-agent system. Notice that the followers keep the reference state of the moment td=18 s during the time interval [18 s, 21 s] and the multi-agent system fails to reach consensus as shown in Fig. 3 and large tracking errors are resulted as illustrated in Fig. 4.

    Figure 3.  Consensus tracking with a time-varying reference state using (5) under communications disturbance.

    Figure 4.  Tracking errors using (5) under communications disturbance.

    By applying (5) for the multi-agent system under communications disturbance, a large cumulative state deviation and large tracking errors are obtained as illustrated in Figs. 3 and 4, respectively. In the practical applications, such a large deviation will cause the system to be instable and even the system cannot achieve consensus tracking after the reference state is available again due to many reasons, such as the limited communications distance.

    Now the proposed least squares estimation algorithm is adopted for the multi-agent system, and the obtained results of consensus tracking can be seen in Figs. 5 and 6.

    Figure 5.  Consensus tracking with a time-varying reference state under communications disturbance using the proposed least squares estimation algorithm.

    Figure 6.  Tracking errors under communications disturbance using the proposed least squares estimation algorithm.

    The results of Fig. 5 show a less cumulative state deviation compared with the results of Fig. 3 and the system successfully achieves consensus even with communications disturbance. The results of Fig. 6 show that the tracking errors are reduced, compared with the results of Fig. 4.

    The obtained results demonstrate that the proposed leader follower cooperative approach provides stability and robustness for the multi-agent system to achieve consensus under communications disturbance compared with the fundamental algorithm of Ren given by (5).

  • This paper described the development of a leader-follower approach for consensus tracking of single-integrator multi-agent systems with a time varying reference state based on least squares estimation to deal with the case of communications disturbance. The estimation of the reference state was done according to online collected data for reducing the tracking errors due to communications failure.

    Firstly, the significance of the consensus tracking algorithm for the multi-agent system was illustrated. Secondly, the assumptions as well as the necessary and sufficient conditions were given for the application of the algorithm. These assumptions can be satisfied in many practical applications. After that the theoretical proof was provided to show the validity of the proposed algorithm.

    A simulation example was carried out to compare the performance of the developed least squares estimation approach with the fundamental algorithm proposed by Ren[19] for consensus tracking of single-integrator multi-agent systems and supposing the case of communications disturbance. The comparative results demonstrated the effectiveness and the advantages of the proposed approach to deal with the communications disturbance for consensus tracking of single-integrator multi-agent systems compared with the fundamental algorithm of Ren.

    In the future work, we attempt to investigate the double integrator dynamics and nodes with nonlinear dynamics.

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