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- Open Access
Mean square containment control problems of multi-agent systems under Markov switching topologies
- Guoying Miao^{1}Email author and
- Tao Li^{1}
https://doi.org/10.1186/s13662-015-0437-3
© Miao and Li; licensee Springer. 2015
- Received: 23 October 2014
- Accepted: 3 March 2015
- Published: 16 May 2015
Abstract
The paper investigates containment control for multi-agent systems under Markov switching topologies. By using graph theory and the tools of stochastic analysis, sufficient conditions of mean square containment control problems are derived for the second-order multi-agent systems. Then the obtained results are further extended to high-order multi-agent systems.
Keywords
- multi-agent systems
- containment control
- mean square consensus
1 Introduction
Recently, cooperative control for multi-agent systems has attracted increasing attention, due to its many applications in different fields. As is well known, consensus is the basic problem of cooperative control for multi-agent systems, which means that every agent tends to the same value in a team. Moreover, the topic of consensus problems with multiple leaders is interesting, which is called containment control for multi-agent systems.
Containment control for multi-agent systems means that all followers are driven into the convex hull generated by the leaders. For example, when some robots are used to carry poisonous materials, since they do not pollute other places, a group of robots are needed to drive them in the designed route, which is called leaders. At present, many results about containment control have been obtained [1–12]. In [1], dynamic containment algorithms based on observers for the high-order continuous-time multi-agent systems were proposed under the fixed topology. Then the results of continuous-time multi-agent systems were extended to that of the discrete-time multi-agent systems. In [2], the containment control problem of double-integrator dynamics with multiple leaders was discussed, where velocities of leaders are unavailable. In [3], the output feedback algorithm and state feedback algorithm were proposed for solving the containment control problem of the discrete-time multi-agent systems, in which time delay is considered in communication networks of multi-agent systems. By using the Lyapunov functional method and the tools of the linear matrix inequality, containment control problems of second-order multi-agent with time delays were investigated, where the stationary leaders and dynamic leaders were considered in [4], respectively. In [5], containment algorithms based on the sampled data for the second-order multi-agent systems were given, where the necessary and sufficient conditions of multi-agent systems were derived.
There are deterministic topologies in the above literature. In fact, communication topologies of networks usually change randomly. By using the graph theory and knowledge of stochastic analysis, mean square consensus of the discrete-time multi-agent systems was discussed under Markov switching topologies in [6]. Then the authors in [7] extended the results in [6] to leader-following consensus of discrete-time multi-agent systems under Markov switching topologies. Under randomly switching topologies, consensus conditions of continuous-time and discrete-time high-order multi-agent systems were given, respectively, where the random link failures between agents were discussed in [8]. In [9], the convergence speed of the first-order discrete-time multi-agent systems was studied. Then the authors in [10] extended the results in [9] to that of the second-order and high-order multi-agent system, respectively. Moreover, under random switching topologies, target containment control for the second-order multi-agent systems was discussed in [11], where switching topologies were driven by a Markov process. In addition, mean square containment control problems of the first-order and second-order multi-agent systems with communication noises was investigated in [12].
Inspired by the results in [1–12], this paper further investigates the containment control for multi-agent systems under Markov switching topologies. In this paper, containment algorithms for continuous-time and discrete-time multi-agent systems are given, respectively. By using the graph theory and theory of stochastic analysis, sufficient conditions of mean square containment control for multi-agent systems are derived. Then we extend the results of the second-order multi-agent systems to high-order multi-agent systems.
2 Mean square containment control for discrete-time multi-agent systems
Before we give the main results, basic graph theory is introduced. Suppose that there are N agents in the topology. \(\mathcal{G}=\{\mathcal{V},\mathcal{A},\mathcal{E}\}\) denotes the graph corresponding to the communication topology, where \(\mathcal {V}=\{1,\ldots,N\}\) is the set of nodes, \(\mathcal {A}=[a_{ij}]_{N\times N}\) is the adjacency matrix. If \((i,j)\in \mathcal{E}\) holds, then \(a_{ij}=1\) and otherwise \(a_{ij}=0\). The Laplacian matrix is defined as \(L=[l_{ij}]_{N\times N}\), where \(l_{ij}=-a_{ij}\) with \(i\neq j\) and \(l_{ii}=\sum_{j=1}^{N}a_{ij}\) with \(i=j\).
Definition
Under Markov switching topologies, the mean square containment control problem of the multi-agent system is solved for any initial distribution, if the followers are driven into the convex hull generated by the leader’s states, \(\lim_{k\rightarrow \infty}E(x_{i}(k)-x_{i}^{\ast})^{2}=0\) for \(i=1,\ldots, M\), where \(x_{i}^{\ast}\in \mathit{co}_{L}=\{\sum_{j=M+1}^{N}\alpha_{j}x_{j}(k),j=M+1,\ldots,N, \alpha_{j}\geq0,\sum_{j=M+1}^{N}\alpha_{j}=1\}\).
Assumption 1
Assume that there exists a path from the leaders to each follower.
Theorem 1
Suppose that Assumption 1 holds. In the fixed directed topology, under the algorithm (3), the containment control problem for system (1) can be solved, that is, follower (1) is driven into the leaders’ sets (2).
Proof
Assumption 2
Under Markov switching topologies, for each follower, there exists at least one path form one leader to that follower in the union of the topologies \(\{\mathcal{G}_{1},\ldots, \mathcal{G}_{s}\}\).
Lemma 1
Under Assumption 2, all the eigenvalues of \(I_{M}-D_{1}\) have positive real parts.
Theorem 2
Assume that Assumption 2 holds. Under the containment algorithm (3), system (1) can solve the containment control problem in the mean square sense.
Proof
Remark 1
We extend the results in [1] to the case under Markov switching topologies. The mean square consensus of multi-agent systems was investigated in [6], while the mean square containment control problems are discussed in this paper.
Remark 2
We can extend the results of Theorem 2 to the mean square containment control problem of high-order discrete-time multi-agent systems. In order to save space, we omit it here.
3 Mean square containment control for continuous-time multi-agent systems
Lemma 2
Under Assumption 2, all eigenvalues of \(L_{1}\) have positive real parts, and \(L= \bigl[{\scriptsize\begin{matrix} L_{1} &L_{2}\cr 0 & 0 \end{matrix}} \bigr]\) is the Laplacian matrix corresponding to the union set of \(\{\mathcal{G}_{1},\ldots,\mathcal{G}_{s}\}\).
Proof
From Lemma 1 in [1], we obtain the results. □
Theorem 3
Suppose Assumption 2 is satisfied. Under Markov switching topologies and the containment algorithm (17), system (15) can solve the mean square containment control problem.
Proof
Remark 3
In [11], containment control for second-order multi-agent systems was discussed under random switching topologies. However, in this paper, we have given another method to solve mean square containment control problem, which is different from the one in [11].
Theorem 4
Assume that \((A, B)\) is stabilizable and Assumption 2 holds. Under the containment control protocol (31) with \(K=\theta B^{T}P\), system (29) can be achieved by the mean square containment control under the Markov switching topologies, where \(\theta\geq\frac{1}{\min\{\bar{\pi}_{i}\}\epsilon}\), ϵ is the minimum eigenvalue of the matrix \(L_{1}+L_{1}^{T}\), P is defined in (35).
Proof
Remark 4
In [8], the mean square consensus problem of the high-order multi-agent systems was investigated, while the mean square containment control of high-order multi-agent systems has been discussed in this paper.
4 Conclusions
In the paper, we have investigated the mean square containment control for discrete- and continuous-time second-order multi-agent systems under Markov switching topologies, respectively. By using graph theory and the tools of stochastic analysis, sufficient conditions for mean square containment control are derived. In addition, we extend the results of the second-order multi-agent systems to that of the high-order multi-agent system. Moreover, the topic of containment control problems of multi-agent systems with communication noise is interesting, which is our future work.
Declarations
Acknowledgements
This work was supported by Natural Science Fundamental Research Project of Jiangsu Colleges and Universities under Grants 14KJB120006, the Natural Science Foundation of Jiangsu Province under Grant BK20140045, the Scientific Research Foundation of Nanjing University of Information Science and Technology under Grant 2013X047, C-MEIC in School of Information and Control of Nanjing University of Information Science and Technology, Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology of Nanjing University of Information Science and Technology.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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