Systems & Control Letters 65 (2014) 13–22 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Quantized consensus over directed networks with switching topologies Dequan Li a,∗ , Qiupeng Liu b , Xiaofan Wang b , Zhixiang Yin a a School of Science, Anhui University of Science and Technology, Huainan 232001, Anhui Province, PR China b Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, PR China article info Article history: Received 7 November 2012 Received in revised form 16 November 2013 Accepted 25 November 2013 Keywords: Consensus Multi-agent systems Uniform quantization Digraph Switching networks abstract This paper studies the quantized consensus problem for a group of agents over directed networks with switching topologies. We propose an effective distributed protocol with an adaptive finite-level uniform quantized strategy, under which consensus among agents is guaranteed with weaker communication conditions. In particular, we analytically prove that each agent sending 5-level quantized information to each of its neighbors, together with 3-level quantized information to itself at each time step, which suffices for attaining consensus with an exponential convergence rate as long as the duration of all link failures in the directed network is bounded. By dropping the typical common left eigenvector requirement for the existence of common quadratic Lyapunov function, we conduct the convergence analysis based on the notion of input-to-output stability. The proposed quantized protocol has favorable merits of requiring little communication overhead and increasing robustness to link unreliability, and it fits well into the digital network framework. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Recently, the problem of distributed consensus in networked multi-agent systems has received significant attention due to its important applications. Roughly speaking, the purpose of the con- sensus problem is to design a distributed protocol in the presence of limited information communication and dynamically switch- ing network topologies, such that a group of agents achieve some agreement between the states. If the final consensus value is the weighted linear combination of the initial states of all agents, then the weighted average consensus problem is solved. In particular, if the final agreed value is the exact average of the initial values of all agents, then average consensus is achieved, which is of par- ticular interest in the applications of load balancing [1,2] and task assignment [3]. Early efforts on related distributed consensus problems focused primarily on the assumption that communication channels of net- works have unlimited capacity. However, this assumption may not be true in practice, because digital channels are subject to band- width constraints and only finite number of bits of information can be transmitted along each channel. As such, information trans- mitted among agents has to be quantized prior to being sent. As ∗ Corresponding author. Tel.: +86 554 6668892. E-mail address: leedqseu@gmail.com (D. Li). a result, quantized consensus or consensus with quantized com- munication has attracted wide interest over the past few years [2,4–19]. Applications of quantized consensus can be found in [3,20], where a framework denoted as discrete consensus was pro- posed, in which quantized average consensus algorithms were performed via network gossiping. Serving as a generalization of quantized consensus, this framework is particularly suitable for applications in load balancing and task assignment, and thus sheds light on applications over directed networks. While in the existing works about quantized consensus, it is commonly assumed that all weighted adjacency matrices (or Laplace matrices) associated to the directed networks having a common left eigenvector. With this basic assumption, state av- erage or weighted average invariance of the networks is pre- served, and thus the final consensus value can be specified [21,22]. More technically, the squared norm or weighted squared norm of the disagreement vector can be chosen as the common quadratic Lyapunov function to carry out the consensus convergence analy- sis [21,22]. For undirected or directed networks with fixed topolo- gies, the above assumption is easily satisfied. While for general directed networks with dynamically switching topologies, the left eigenvectors are also time-dependent, except the case where the directed switching networks are always balanced (i.e. networks in which the in-degree and out-degree of each node are the same), or equivalently, the corresponding weighted adjacency matrices are double stochastic [21], otherwise the final achieved consensus value is time-varying and there does not exist a common quadratic 0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.11.013