Dynamic average consensus on synchronous communication networks Minghui Zhu and Sonia Mart´ ınez Abstract— We propose a class of dynamic average consensus algorithms that allow a group of agents to track the average of their measured signals. The algorithms are implemented in discrete time and require a synchronous communication schedule. The convergence results rely on the input-to-output stability properties of consensus algorithms and require that the union of communication graphs over a bounded period of time be strongly connected. The only requirement on the set of signals is that the difference of the n th -order derivatives of any two signals be bounded for some n ≥ 0. I. I NTRODUCTION We consider the problem in which a set of autonomous agents aims to track the average of individually measured time-varying signals by local communication with neighbors. This problem is referred to as dynamic average consensus in opposition to the more studied static consensus. The dynamic average consensus problem arises in different contexts, such as formation control [7], sensor fusion [17], [18], [25] distributed estimation [12] and distributed tracking [20], [31]. These tasks require that all agents agree on the average of time-varying signals and thus the consensus on a static aver- age value, e.g., the initial states of the agents, is insufficient. Literature review: The distributed static consensus prob- lem was introduced in the literature of parallel processors in [27] and has attracted significant attention in the con- trols community. A necessarily incomplete list of references includes [6], [16] for continuous-time consensus, [2], [9], [15], [24] for discrete-time consensus, [1], [13] discuss asynchronous consensus, and [10], [3], [26] treat quantized consensus, randomized consensus and consensus over ran- dom graphs, respectively. The convergence rate of consensus algorithms is e.g., discussed in [22], [28], consensus prop- agation is considered in [14], and conditions on consensus algorithms to achieve different consensus values is discussed in [4]. Consensus algorithms find application in a variety of areas such as load balancing [5], [30], formation control [6], [7], and, as we have mentioned, sensor fusion [12], [17], [18], [25], distributed tracking [20], [31] and consensus- based belief propagation in Bayesian networks [19]. The dynamic average consensus problem in continuous- time is studied in [8], [17], [23], [25]. By using standard frequency-domain techniques, the authors in [25] showed that their algorithm was able to track the average of ramp inputs with zero steady-state error. In the context of input-to- state stability, [8] showed that proportional dynamic average consensus algorithm could track with bounded steady-state This work was supported in part by NSF Career Award CMS-0643673 and NSF IIS-0712746. The authors are with Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Dr, La Jolla CA, 92093, {mizhu,soniamd}@ucsd.edu error the average of bounded inputs with bounded derivatives. On the other hand, they showed that proportional-integral dynamic average consensus algorithm could track the av- erage of constant inputs with sufficiently small steady-state error. The authors in [17] proposed a dynamic consensus algorithm and applied it to the design of consensus filters. The algorithm in [17] can track with some bounded steady- state error the average of a common input with a bounded derivative. The problem studied in [23] is similar to that in [17], and consensus of agents is over a common time-varying reference signal. However, the algorithm in [23] assumes that agents know the nonlinear model which generates the time-varying reference function. The problem studied in the present paper is close to those in [8] and [25] and includes those in [17] and [23] as special cases. Statement of contributions. In this paper, we propose a class of discrete-time dynamic average consensus algo- rithms and analyze their convergence properties. This paper contributes to the problem of dynamic average consensus in the following aspects: The continuous-time communi- cation assumption for dynamic average consensus in [8] and [25] is relaxed, and we consider more realistic discrete- time synchronous communication models. This allows us to obtain a direct relation between the frequency of inter- agent communication and the difference of input signals. Our dynamic average consensus algorithms are able to track the average of a larger class of time-varying inputs than [8] and [25] with zero or sufficiently small steady-state error. This includes polynomials, logarithmic-type functions, periodic functions and other functions whose n th -order dif- ferences are bounded, for n ≥ 0. We can also handle the case where the difference of the common part, that appears in all the individual inputs, explodes. Our analysis for the dynamic average consensus algorithms relies upon the input-to-output stability property of discrete-time static consensus algorithms in the presence of external disturbances. This result is the counterpart of continuous-time static consensus algorithms in [11] but more general in that we allow for unbounded disturbances. Organization of the paper. We now outline the reminder of the paper. In Section II, we introduce general notation and the statement of the problem we study. In Section III, we focus on a first-order algorithm for dynamic average consensus. Section IV generalizes this to a class of n th - order algorithms for dynamic average consensus and analyze their convergence properties. In Section V, we present some re- marks on the extension of the results in Section III and IV. In Section VI, an example and its simulation results are given. Finally, Section VII includes some concluding remarks. 2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008 FrB03.4 978-1-4244-2079-7/08/$25.00 ©2008 AACC. 4382