Accelerated linear iterations for distributed averaging q Ji Liu, A. Stephen Morse ⇑ Department of Electrical Engineering, Yale University, New Haven, Connecticut, United States article info Article history: Received 1 August 2011 Accepted 17 September 2011 This paper is dedicated to David Q. Mayne on the occasion of his 80th birthday. Keywords: Consensus Distributed algorithms Linear iterations Convergence rates Matrix analysis abstract Distributed averaging deals with a network of n > 1 agents and the constraint that each agent is able to communicate only with its neighbors. The purpose of the distributed averaging problem is to devise a protocol which will enable all n agents to asymptotically determine in a decentralized manner, the aver- age of the initial values of their scalar agreement variables. Most distributed averaging protocols involve a linear iteration which depends only on the current estimates of the average. Building on the idea pro- posed in Muthukrishnan, Ghosh, and Schultz (1998), this paper investigates an augmented linear itera- tion for fast distributed averaging in which local memory is exploited. A thorough characterization of the behavior of the augmented system is obtained under appropriate assumptions. It is shown that the augmented linear iteration can solve the distributed averaging problem faster than the original linear iteration, but the adjustable parameter must be chosen carefully. The optimal choice of the parameter and the corresponding fastest rate of convergence are also provided in closed form. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction There has been considerable interest recently in developing algorithms for distributing information among the members of a group of sensors or mobile autonomous agents via local interac- tions. Notable among these are those algorithms intended to cause such a group to reach a consensus in a distributed manner (Jadbabaie, Lin, & Morse, 2003; Blondel, Hendrichkx, Olshevsky, & Tsitsiklis, 2005; Moreau, 2005; Olfati-Saber & Murray, 2004; Ren & Beard, 2005). Consensus problems have a long history in statis- tics (DeGroot, 1974) and computer science (Pease, Shostak, & Lamport, 1980) and have a tradition in systems and control theory starting with the early work of Tsitsiklis (1984), Tsitsiklis, Bertse- kas, and Athans (1986). Many different problems in various disci- plines of science and engineering are closely related to consensus problems. They include subjects such as flocking (Jadbabaie et al., 2003), rendezvousing (Lin, Morse, & Anderson, 2003), formation (Fax & Murray, 2004), and synchronization (Freris, Graham, & Kumar, 2011). For a survey on the most recent works in this area see (Olfati-Saber, Fax, & Murray, 2007). One particular type of consensus process which has received much attention lately is called distributed averaging (Xiao & Boyd, 2004). In the development of algorithms to perform distributed averaging, one typically encounters linear update equations of the form x i ðt þ 1Þ¼ X j2N i a ij x j ðtÞ; i 2f1; 2; ... ; ng; t P 1 ð1Þ where n > 1 is an integer, x i (t) is a real scalar, N i is a subset of the set {1, 2, ... , n} containing i, and the a ij are constants satisfying a ij ¼ 0; j R N i ; i 2f1; 2; ... ; ng, X n i¼1 a ij ¼ 1; j 2f1; 2; ... ; ng and X n j¼1 a ij ¼ 1; i 2f1; 2; ... ; ng Typically these are the update equations for a group of n autono- mous agents with labels 1 to n; in this case N i is the set of labels of agent i’s neighbors including itself. By introducing an n-vector x whose ith entry is x i , (1) can be written as xðt þ 1Þ¼ AxðtÞ; t P 1 ð2Þ where A is a real-valued n  n matrix whose row and column sums all equal 1. It is clear that any such matrix must have an eigenvalue at value 1 and moreover that x(t) will converge to a finite limit just in case all remaining eigenvalues lie strictly inside of the unit circle in the complex plane. 1 In such cases, the rate at which x(t) converges {in the worst case} is determined by A’s ‘‘sub-spectral radius’’ where by the sub-spectral radius we mean the second largest among the magnitudes of the n eigenvalues of A. In the sequel we call any real 1367-5788/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2011.10.005 q An earlier version of this paper was presented at the IFAC Workshop on 50 Years of Nonlinear Control and Optimization (London, UK, September 30-October 1, 2010), dedicated to David Q. Mayne on the occasion of his 80th birthday. ⇑ Corresponding author. E-mail addresses: ji.liu@yale.edu (J. Liu), as.morse@yale.edu (A.S. Morse). 1 As a consequence of the constraints on the a ij , it is easy to see that if x(t) converges, in the limit each of its entries must equal the average 1 n P n i¼1 xð1Þ. We will not make use of this fact in this paper. Annual Reviews in Control 35 (2011) 160–165 Contents lists available at SciVerse ScienceDirect Annual Reviews in Control journal homepage: www.elsevier.com/locate/arcontrol