Leader-following consensus of data-sampled multi-agent systems with stochastic switching topologies
Introduction
Leader-following consensus has received significant attention in the control field in recent years. This is mainly due to its wide applications in engineering, such as unmanned air vehicles, mobile robotic swarms, wireless sensor networks and cooperative surveillance [1], [2], [3], [4], [5]. The main idea of leader-following consensus is that the leader sends its state information to the followers directly or indirectly such that the tracking errors between leader and all followers are as small as possible. There are many publications on the topic of the leader-following consensus problem [6], [7], [8]. In [6], the multi-agent system considered was with measurement noises and directed interconnection topology. A sufficient condition for mean-square stability of the closed loop tracking control system was obtained by designing distributed estimators. In [7], both leaderless and leader-following consensus problems were studied. The stability or boundedness conditions were presented based on Lyapunov theorems and Nyquist stability criterion. By using the sampled-data control approach, the leader-following consensus for multi-agent systems was studied in [8]. The topology considered is deterministic.
Data-sampled approach is frequently used to discretize the continuous-time system in control community. In recent years this method is also used to study the multi-agent systems [8], [9], [10], [11], [12]. Two sampled-data coordination algorithms for double-integrator dynamics were studied in [10] where the interaction topology is fixed undirected/directed. In [11], the consensus problem of double-integrator multi-agent systems with both fixed and switching topologies was studied. The switching signal is arbitrary and only a sufficient condition is derived to solve a consensus problem in this case. In [12], the authors researched the stochastic bounded consensus tracking problems of multi-agent systems, where the sampling delay induced by the sampling process was considered.
The topologies in the above literature are all deterministic or switching in a deterministic framework. However, the system models are sometimes switching stochastically due to the internal or/and external disturbance. Similar to some other control systems, the Markovian switching model has been used to describe the interaction topology among the agents in very recent years [13], [16]. In [13], the static stabilization problem of a decentralized discrete-time single-integrator network with Markovian switching topologies was studied. In [14], the authors considered the consensus for a network of single-integrator agents with Markovian switching topologies. In [15], the authors studied the mean-square consentability problem for a network of double-integrator agents with Markovian switching topologies. In [16], the authors studied the distributed discrete-time coordinated tracking problem for multi-agent systems with Markovian switching topologies in case of the transition probabilities are equal.
Motivated by the former considerations, we will extend the leader-following consensus problem in [8] to the case of Markovian switching topologies in this paper. In this case, the leader-following consensus problem will become more challenging. Both time-invariant and time-varying leader are considered. Based on algebra graph theory and Markovian jump system theory, we present the necessary and sufficient conditions for the convergence of the tracking error systems. An optimization algorithm will be given to derive the allowable control gains and sampling period.
Notation: Let and represent, respectively, the real number set and the non-negative integer set. Denote the spectral radius of the matrix M by . Suppose that . Let (respectively, ) denote that is symmetric positive semi-definite (respectively, symmetric positive definite). Denote the determinant of the matrix A by . Given , define , where is the mathematical expectation. “⊗” represents the Kronecker product of matrices. In denotes the identity matrix. Let and denote, respectively, the column vector with all components equal to 1 and zero matrix.
Section snippets
Graph theory notions
Denote the directed graph by , where and represent, respectively, the node set and the edge set. Suppose that there exist n followers and one leader label as agents 1 to n, and agent r, respectively. Suppose with order n be the interaction topology among the n followers. is the adjacency matrix associated with . Here if agent i can obtain information from agent j and otherwise. We assume that . Let be a directed graph of order used
Main results
In this section, we will analyze the convergence of the tracking error systems (12), (15). Before giving the main results, the following lemmas are necessary [21]. Lemma 1 Let . Let . Then, , where denotes the determinant of a matrix, if and D commute pairwise. Lemma 2 For given , denote , then the real parts of all eigenvalues of H are positive if and only if the leader has directed paths to all followers in . Here
Simulation results
In this section, we give two examples to show the effectiveness of the presented results. For simplicity, we let if agent i can obtain the information from agent j, , if the ith follower can obtain the information from the leader. Example 1 Suppose the multi-agent system considered be composed of one leader and four followers. We assume that the Markov chain has two modes. The transition probability matrix is assumed to be . The corresponding interaction topologies
Conclusion
In this paper, we have investigated the leader-following consensus problem for multi-agent systems with Markovian switching topologies in a sampled-data setting. Both time-varying and time-invariant reference states have been considered. Based on algebraic graph theory and Markovian jump linear system theory, the necessary and sufficient conditions for the boundedness of the tracking errors with a time-varying reference state and the mean-square stability of the tracking error system with a
Acknowledgements
The work was supported by the National Natural Science Foundation of China under Grants no. 61203056, and Qing Lan Project.
Huanyu Zhao received his B.S. degree in mathematics and applied mathematics from Hubei University in 2005, and Ph.D. degree in control science and engineering from Nanjing University of Science and Technology in 2011. From May, 2010 to November, 2010, he was an exchange Ph. D. student supported by Nanjing University of Science and Technology with Department of Electrical and Computer Engineering, Utah State University. From February 2013 to January 2014, he was a postdoctoral candidate of the
References (22)
Multi-vehicle consensus with a time-varying reference state
Syst Control Lett.
(2007)- et al.
Consentability for high-order multi-agent systems under noise environment and time delays
J. Frank. Inst.
(2013) - et al.
Distributed tracking control of leader-following multi-agent systems under noisy measurement
Automatica
(2010) - et al.
Stochastic bounded consensus tracking of leader-follower multi-agent systems with measurement noises based on sampled-data with small sampling delay
Physica A
(2013) - et al.
Static decentralized control of a single-integrator network with Markovian sensing topology
Automatica
(2005) - et al.
Consentability and protocol design of multi-agent systems with stochastic switching topology
Automatica
(2009) - et al.
Distribute discrete-time coordinated tracking with Markovian switching topologies
Syst. Control Lett.
(2012) - et al.
Tracking control for multi-agent consensus with an active leader and variable topology
Automatica
(2006) Consensus tracking under directed interaction topologiesalgorithms and experiments
IEEE Trans. Control Syst. Technol.
(2010)- et al.
Distributed coordinated tracking with a dynamic leader for multiple Euler–Lagrange systems
IEEE Trans. Autom. Control
(2011)
Distributed leader-following consensus for second-order multi-agent systems with nonlinear inherent dynamics
Int. J. Syst. Sci.
Cited by (18)
Mean square consensus of double-integrator multi-agent systems under intermittent control: A stochastic time scale approach
2019, Journal of the Franklin InstituteLeader-following consensus of second-order nonlinear multi-agent systems with unmodeled dynamics
2018, NeurocomputingCitation Excerpt :Distinguished by the number of leaders, consensus (or synchronization) of MASs includes leaderless consensus, leader-following consensus, and containment control [23–26]. The leader-following consensus problem means designing appropriate protocols to steer all followers to keep pace with the leader [27]. To date, much work has been done on leader-following consensus of MASs, especially on linear MASs (such as [28–32]).
Group tracking control of second-order multi-agent systems with fixed and Markovian switching topologies
2018, NeurocomputingCitation Excerpt :Xie and Liang [13] studied the group consensus of MASs with time-delays by Lyapunov first method directly. On the other hand, the consensus of MASs with a reference signal has drawn much attention and many results have been obtained in [14–16]. In [14], Hu and Hong gave a necessary and sufficient condition to guarantee the positive stability of the matrix H, which played a key role in the convergence analysis of tracking error system.
Discrete-time consensus strategy for a class of high-order linear multiagent systems under stochastic communication topologies
2017, Journal of the Franklin InstituteCitation Excerpt :The consensus problem over second-order MASs in the presence of switching topologies and random communication delays was investigated in [31]. In [32], mean square stability of tracking errors in the leader-follower consensus problem in second-order MASs under stochastic networks was analyzed, and in [33], an output-feedback consensus protocol for second-order stochastic networks was proposed. Considering all the above-mentioned issues, on the one hand, it is necessary to study consensus control of high-order MASs under stochastic networks; on the other hand, dependency of consensus protocols design on the information of the set of feasible switching topologies was the main drawback of the existing protocols for consensus control of high-order stochastic networks.
Distributed cooperative control for a class of stochastic upper-triangular nonlinear multi-agent systems
2016, NeurocomputingCitation Excerpt :Yu et al. [5] study some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Zhao [6] studies the leader-following consensus problem for multi-agent systems with Markovian switching topologies in a sampled-data setting. Ma and Zhang [7] investigate the joint impact of the agent dynamic structure and the communication topology on consensus ability for linear MASs.
Leader-following consensus of multi-agent system with a smart leader
2016, NeurocomputingCitation Excerpt :In addition, Wang and Cheng [19] study the containment problem for the first-order and second-order multi-agent system with communication noises, assuming the existence of multiple leaders. For more details, the readers can refer to [20–24]. Notably, much of the previous works adopt the assumption that the leader is a special agent who is independent of all the other agents.
Huanyu Zhao received his B.S. degree in mathematics and applied mathematics from Hubei University in 2005, and Ph.D. degree in control science and engineering from Nanjing University of Science and Technology in 2011. From May, 2010 to November, 2010, he was an exchange Ph. D. student supported by Nanjing University of Science and Technology with Department of Electrical and Computer Engineering, Utah State University. From February 2013 to January 2014, he was a postdoctoral candidate of the Department of Electrical Engineering, Yeungnam University, Republic of Korea. He is currently an Associate Professor of Faculty of Electronic and Electrical Engineering, Huaiyin Institute of Technology. His research interests include Markov switching systems, cooperative control of multi-agent systems, and game theory.