Necessary and Sufficient Conditions for Distributed Averaging with State Constraints Mahmoud El Chamie

Abstract— Distributed averaging algorithms for multi-agent systems have recently gained a significant amount of interest. In many cases, maximizing the convergence rate of these algorithms leads to rapid changes in the system’s state, which may not be desirable or physically possible. This paper derives necessary and sufficient conditions to guarantee that the states of the system are always contained within a polytopic region during the convergence process. These constraints prevent sudden or steep changes in the state variables. In addition to providing these conditions, we provide a convex design for the controller that can achieve the fastest convergence while satisfying the constraints. The design problem is formulated as a semi-definite program (SDP) which can be solved using any standard interior-point method SDP solver, and the solution can thus be efficiently computed.

I. I NTRODUCTION Large scale decentralized systems, such as multi-agent systems, rely on distributed algorithms to opperate efficiently and effectively. Distributed algorithms, e.g., consensus-based algorithms such as distributed averaging [1]–[7], are robust to failure of individual agents and provide a scalable implementation of the system as only local information is required. Hence consensus problems in multi-agent systems have gained increasing attention for their interdisciplinary applications such as sensor fusion [1], [5], robot formation [8], rendezvous problems [9], flocking [10], to name a few. In a typical consensus process within a multi-agent system framework, the agents send an estimate of a certain variable of interest (such as location, speed, temperature, etc.) to their neighbors. This local information is then processed by each agent to compute a new estimate of the variable, which is driven by the mismatch between neighbors’ estimates. The local information is then used to enable distributed decision-making that achieves a global objective as a result of decisions based on local interactions. The system is usually represented by a connectivity network modeled as a graph with vertices and links. Each vertex is associated with an agent in the system, and a link in the graph corresponds to a communication capability between the corresponding two agents. Earlier research on multi-agent systems has focused on communication constraints on the network topology (e.g., time-varying connectivity [11], [12], delays [13], quantization [3], [14], [15]) with limited attention to state constraints on the variables of interest (e.g., ensuring that the distance between agents or their relative velocities is always within acceptable limits). In this paper, we address the latter Mahmoud El Chamie ([email protected]) and Behc¸et Ac¸ıkmes¸e ([email protected]) are with the University of Washington, Department of Aeronautics and Astronautics, Seattle, WA 98195, USA.

Behc¸et Ac¸ıkmes¸e

constraints that are enforced on the system variables, i.e., state constraints. With the growing interest in multi-agent cooperation (e.g., rendezvous problems [9], [16]), there is an increasing need for incorporating state constraints into the consensus models to capture physical limitations and mission safety requirements [17]. Related literature on consensus with state constraints studies the projection of the state of an agent on a closed convex set, with the sets being separable [18], i.e., each agent’s state has a separate convex set. As an alternative approach to projection, [19] uses the idea of repulsive and attractive components for convergence with state constraints. [20] and [21] tackle the state constrained problems by allowing agents to discard the estimates of neighbors that fall out of their convex sets. [22] studies a leader-follower consensus problem with input saturation constraints. A common assumption in previous works is that they consider decoupled state constraints, i.e., each agent’s constraint set is independent from other states in the network. In this paper, however, we consider joint state constraints. State constraints are also related to the literature of model-predictive control (MPC) [23], [24], safety constraints in Markov chains [25], [26], and constrained Markov Decision Processes (MDP) [27], [28]. In [2], necessary and sufficient conditions are given for the convergence of the system to consensus without considering constraints on the states. In many cases, to achieve fast convergence, fast changes in the states are required. But some systems are limited by the physical environment, that would not allow such changes or can cause damage to the system. For example, if the states correspond to the velocities of agents, it might not be physically possible to suddenly increase the velocity or reduce it in short time, thus constraints on the rate of change of velocity are required. The contribution of this paper is twofold. First we provide additional necessary and sufficient conditions, to those derived in [2], to guarantee that the states are always within polytopic regions1 during the convergence process. These constraints prevent sudden or steep changes in the states variables. Second, we provide a convex design of the controller that can achieve the fastest convergence while satisfying the constraints. The design problem is formulated as a semi-definite program (SDP) which can be solved using any standard interior-point method SDP solver, and the solution can thus be efficiently computed. 1 Polytopic regions are expressed as {x ∈ Rn : Ax ≤ b} for any given matrix A and vector b with conformable dimensions.

II. P RELIMINARIES Let R be the set of real numbers. 1n (or simply 1) is the vector of all n elements equal to one. (.)T is the transpose of a matrix. For two matrices A and B, A  B if and only if B − A is positive semi-definite, and A ≤ B is the element-wise inequality. Let G = (V, E) be an undirected graph where V = {1, . . . , n} is the set of vertices and E = {1, . . . , m} is the set of edges. Every link l ∈ E is associated with its incident vertices (i, j), denoted by l ∼ (i, j). Let Qm,n be the incidence matrix of the graph, i.e., every link l ∼ (i, j) corresponds to a row in this matrix such that Q(l, i) = +1 and Q(l, j) = −1. Each agent has a scalar xi (0) ∈ R. Let x(k) ∈ Rn be the vector of all scalars where k is a discrete time index. The dynamics of the networked system is given by the following averaging iteration: x(k + 1) = M x(k)

(1)

where M n,n is a weight matrix that satisfies M ∈ M, where

M 1 = 1, Mij = 0 if (i, j) ∈ / E}. (2) The objective of a distributed averaging algorithm is to design a weight matrix2 M that drives the system in equation (1) to consensus on the average of initial Pn states, i.e., limk→∞ x(k) = xave 1, where xave = n1 i=1 xi (0). Note that the algorithm is considered “distributed” because the value of xi (k + 1) only depends on the values of xj (k) where j is a neighbor of i in the graph G, denoted as j ∈ Ni . In particular, equation (1) can be written as follows: X xi (k+1) = xi (k)+ Mij (xj (k)−xi (k)) for i = 1, . . . , n. j∈Ni

It is worth noting that the design of M can be achieved both in a decentralized or centralized methods. The decentralized design methods have the advantage of matching the distributed nature of the implemented algorithm, and hence they allow online adaptation as the communication network evolves. However, they can suffer from slower convergence speeds as compared to centralized designs. It is well known that a necessary and sufficient condition for M ∈ M for convergence of the system to the average is 1 T 11 ) < 1, n

III. P ROBLEM F ORMULATION In most existing research on distributed averaging, the values x(k), k = 0, 1, . . . , are allowed to be unconstrained. In practical applications, the physical system imposes constraints on the states. Consider the following general linear constraints: U x(k) ≤ b (4) where U and b are arbitrary, but fixed matrices. We will focus in this paper on the following state constraints: |xi (k) − xj (k)| ≤ dij , for all (i, j) ∈ E

M = {M ∈ Rn,n : M = M T ,

ρ(M −

For a distributed weight selection algorithm, most of the algorithms use sufficient conditions for convergence. In ¯ ij ≥ δ for all (i, j) ∈ E where δ ≥ 0 particular, if M ¯ − is a positive scalar, then it can be shown that ρ(M 1 T n 11 ) < 1 for any connected graph G. Based on this fact, the well-known local algorithms, e.g., Metropolis and maximum degree [5], provide convergence guarantees, but without providing guarantees on the speed of convergence, i.e., the convergence to consensus can be very slow.

(3)

where ρ(.) is the spectral radius of a matrix, i.e., its largest eigenvalue in magnitude. Condition (3) is the basis of the design algorithm for distributed averaging given in [2] that selects M by formulating an SDP to minimize ρ(M − n1 11T ) subject to M ∈ M. Note that in the conditions characterized by M, the weight matrix is allowed to have negative entries, in fact, the fastest weight matrix usually does have negative entries. 2 Note that we are considering here only static graphs and the matrix M is a fixed matrix that do not change during the distributed averaging process.

(5)

where dij ∈ R+ are given non-negative real numbers, but the methodology applies to the general constraints (4). Let dm,1 be the vector defined by dk = dij if (i, j) ∼ k. The constraints in (5) can be expressed as linear inequalities (4) by using the incidence matrix Qm,n as follows:     Q d U 2m,n = and b2m,1 = (6) −Q d Next section provides some motivating examples for this class of constraints. In summary, we consider with the following synthesis problem. Problem Formulation. Given a graph G = (V, E), and a system whose networked dynamics is governed by (1). Find a fixed weight matrix M ∈ M such that 1.) limk→∞ x(k) = xave 1 (i.e., system (1) converges to the average) 2.) Constraints (4) are satisfied for all k ≥ 0 While it is not explicitly mentioned in the problem formulation, the initial condition should satisfy the constraints otherwise the problem is not feasible, i.e., we assume |xi (0) − xj (0)| ≤ dij for all (i, j) ∈ E. Note that due to condition 1 above, we have limk→∞ |xi (k) − xj (k)| = 0. Therefore, there is an iteration k0 such that for k ≥ k0 , |xi (k) − xj (k)| ≤ dij . Hence condition 1 implies that condition 2 is satisfied at some time, even if condition 2 is not imposed. So the second condition requires that k0 = 0, i.e., the constraints are also satisfied during the transients, which is a harder constraint. A. Motivating Examples We present two applications where state constraints are inherited from the physical system. The first example is on electric power grids where distributed averaging has been used as a model [29], [30]. In this example, the states xi (k)

are voltages at given circuit points in the electrical circuit and the weights on links measure the resistance across these points. In these networks, it may be desirable to bound the amplitude of the traversing current on a link because of the physical capacity of wires. Since the current between any two reference points in a simple electrical network is proportional to |Vi − Vj | by Ohm’s law, it may be desirable to design the resistance matrix (the weight matrix) that guarantees |xi (k)− xj (k)| ≤ cij for all k and any two connected points (i, j), where cij is the physical capacity of the link (i, j). In the second example, we consider a system of networked vehicular robots where an edge of the graph represents a communication link with another neighboring robot. A communication link between two vehicles exists if their relative distance is contained within the communication radius of both agents. The states in this application are the position coordinates of each agent. In the rendezvous problem [9], [16], local communication among robots allows them to meet at the centroid of their initial configuration. Therefore, it is necessary to insure that ||xi (k) − xj (k)||∞ ≤ r for all k ≥ 0 to preserve communication connectivity. Fig. 1a shows a multi-robot network where the objective is to move all the agents to the centroid of the initial configuration (rendezvous). Note that the communication range is limited, so the agents should always stay within this range otherwise the link is broken. We assume that the network is established (each node discovers its neighbors and establishes the connections) before the agents start moving. Thus during the process, no new connections are established even if two agents happen to be within the communication range. However, if the range is violated, ||xi (k0 ) − xj (k0 )||∞ > r at a given iteration k0 , then (i, j) is removed from the set of edges E for all k ≥ k0 . Classical consensus algorithms for the rendezvous problems do not take communication range constraints into account in designing the motion policy in these scenarios. Fig. 1b shows that classical consensus algorithms can cause the links in the communication network to break and thus the network cannot achieve the desired rendezvous because it leaves one node separated from the others. Hence the proposed state constraints ensure the integrity of the underlying communication network connectivity as the agent position evolves in time. (1) (p) Note that here xi (k) = [xi (k), . . . , xi (k)]T is a vector of p coordinates as compared to the scalar assumption in this paper, however, without loss of generality we can deal with each dimension in xi as being a separate system because the constraints are defined using the infinity norm, i.e., ||xi (k) − xj (k)||∞ ≤ r is equivalent to having for l = (l) (l) 1, . . . p, |xi (k) − xj (k)| ≤ r with x(l) (k + 1) = M x(l) (k).

Connectivity network for multi-agent systems with connectivity radius r = 0.5

1

1

Connectivity network for multi-agent systems with connectivity radius r = 0.5

Agents Rendezvous Point (centroid of initial configuration)

Agents Rendezvous Point (centroid of initial configuration)

dij > 0.5 i

0

j

0 1 0

0

(a) Initial Configuration

1

(b) Disconnected Network

Fig. 1: Network of multi-agent systems spread in a square area. The links in the network indicate that the neighbor is within the communication range of an agent. The weight matrix M is designed using a classical consensus weight selection algorithm (e.g., Metropolis weights). Starting from an initial network configuration given in Fig. 1a, the agents move to locations where the distance between two of the nodes is more than the communication range. This results in the disconnection of one node from the other part of the network and the rendezvous is not achieved.

Fig. 2: Geometric representation of the set X when n = 2. The invariance property guarantees that M x ∈ X if x ∈ X .

1 corresponding to the largest eigenvalue of M . Formally, given any vector y(k) whose dynamics is given by (1), the dynamics of x(k) = a(y(k) + c1) where a, c ∈ R also follows (1), i.e., x(k + 1) = M x(k), with the same convergence properties as y(k). Therefore, without loss of generality, we can work directly with a normalized system having 0 ≤ x(k) ≤ 1 (simply by normalizing using c = 1 ). Consider the − mini yi (0) and a = maxi yi (0)−min i yi (0) n following set X ⊆ R defined as follows: X = {x ∈ Rn : U x ≤ b, x ≥ 0, x ≤ 1}.

(7)

The set X defines a polytopic region. The following definition characterizes some properties of the set X .

IV. M AIN R ESULTS This section presents the necessary and sufficient conditions for choosing a matrix M ∈ M to satisfy the convergence property and the connectivity constraints requirements given in the Problem Formulation. Notice first that given the dynamics in equation (1), the system is invariant under scaling and translation in the direction of the eigenvector

Definition 1 (Invariance Property). We say that a system with a weight matrix M is invariant with respect to a set X if for any x ∈ X , we have M x ∈ X . For any system having the dynamics (1), if the system starts from an initial point x(0) ∈ X , the invariance property guarantees that x(k) ∈ X for all k ≥ 0. Fig. 2 gives a

geometric representation of the set X and an illustration of the invariance property. The following proposition gives necessary and sufficient conditions on a matrix M for the system to satisfy an invariance property. Proposition 1. A system with a weight matrix M is invariant with respect to the set X = {x ∈ Rn : U x ≤ b, 0 ≤ x ≤ 1} if and only if there exist two matrices Y12m,2m and Y22m,n such that U M ≤ Y1 U + Y2

(8)

Y1 b + Y2 1n ≤ b

(9)

Y1 ≥ 0, Y2 ≥ 0, M ≥ 0, M 1 ≤ 1 where U

2m,n

and b

2m,1

(10)

are given in (6).

Proof. First note that all inequalities are element-wise, and equations (8)-(10) are linear inequalities in M , Y1 , and Y2 . They will be used for a convex synthesis of M . The necessity part of the theorem ensures that these inequalities are not restrictive, and they provide a full characterization of the set of feasible matrices M that satisfy the invariance property. Let us show first that equations (8)-(10) are sufficient for having the invariance property. Let M be a matrix that satisfies equations (8)-(10) with a given two matrices Y1 and Y2 . We have to show that for any x ∈ X we get M x ∈ X , i.e., U M x ≤ b, M x ≥ 0, and M x ≤ 1. Let x ∈ X , then U M x ≤ (Y1 U + Y2 )x

(11)

= Y1 U x − Y1 b + Y1 b + Y2 x − Y2 1n + Y2 1n (12) = Y1 (U x − b) + Y2 (x − 1n ) + Y1 b + Y2 1n (13) {z } | {z } | | {z } ≤0

≤0

≤b

≤b

(14)

where in (11) we used equation (8) from the proposition and the fact that x ≥ 0. The inequalities under the expressions in (13) follow directly from the fact that x ∈ X and (9). Finally, (14) holds because Y1 and Y2 are nonnegative matrices given by (10). M x ≥ 0 and M x ≤ 1 are due to M ≥ 0 and M 1 ≤ 1 respectively. This ends the sufficiency proof. We prove now that (8)-(10) are indeed necessary for the invariance property to hold. We need to find two matrices Y1 and Y2 that satisfy the equations in the proposition. Let M be a matrix such that U M x ≤ b for all x ∈ X . Thus for i = 1, . . . , 2m we can write max xT (M T U T ei ) ≤ bi , x∈X

(15)

where ei ∈ R2m is the vector of all zeros except having a value 1 in the i-th element, so M T U T ei is simply the i-th column of the matrix M T U T . Since X is a set of linear equalities and inequalities, then the left hand side of (15) can be written as a linear program (LP) with primal variable x. Let fi∗ := maxx∈X xT (M T U T ei ), then the dual of the linear program, miny∈Yi y T v where v is a vector deduced from X , has the same optimal cost fi∗ . Using duality theory of linear programming, we have: Yi = {y ∈ R2m+n,1 : [U T I]y ≥ M T U T ei , y ≥ 0} (16)



 b and v = . Note that Yi is a function of i because it has 1n ei in the equations. Therefore, we have for i = 1, . . . , 2m min y T v = fi∗

y∈Yi

≤ bi

(17)

where (17) follows from (15). Thus for i = 1, . . . , 2m there T exists yi ∈ Yi such that yiT v ≤ bi . Let Y1 = [← yi1 →] be the 2m × 2m matrix formed by having its rows equal to T where yi1 is the vector of the first 2m elements in yi , yi1 T and let Y2 = [← yi2 →] where yi2 is the vector of the last n elements in yi , then Y1 and Y2 satisfy equations (8) and (10) because of the definition of Yi , and Y1 and Y2 satisfy (9) because yiT v ≤ bi . Note also that M ≥ 0 and M 1 ≤ 1 because M x ≥ 0 and M x ≤ 1 for all x ∈ X . This ends the proof. Proposition 1 provides necessary and sufficient conditions on M so that the constraints (4) are satisfied. The next theorem summarizes the main technical result of this paper by giving necessary and sufficient conditions for convergence of distributed averaging with state constraints. Theorem 1. The necessary and sufficient conditions on a matrix M ∈ M to guarantee convergence to consensus while satisfying the state constraints (i.e., satisfying the two requirements stated in the Problem Formulation) are the following: there exist two matrices Y12m,2m and Y22m,n such that U M ≤ Y1 U + Y2

(18)

Y1 b + Y2 1n ≤ b

(19)

Y1 ≥ 0, Y2 ≥ 0, M ≥ 0 (20) 1 T (21) ρ(M − 11 ) < 1 n Proof. Equations (18)-(20) are necessary and sufficient for constraint satisfaction as given by Proposition 1. The other equation, (21), provides the necessary and sufficient condition for convergence as given by [2]. Combining both, leads to the necessary and sufficient conditions for constrained consensus problem formulation. Remark: The necessary and sufficient conditions are applied for the class of symmetric matrices (i.e., M = M T ). We will show in the next section a semi-definite program for convex synthesis of fastest M . However, even if the graph is undirected, we can still relax the symmetry assumption, and the necessary and sufficient conditions would still hold after adding an additional constraint 1T M = 1T to Theorem 1. But in contrast to the symmetric case, the feasible set for M would not be convex due to the nonconvex function ρ(M − n1 11T ) and thus a convex synthesis cannot be obtained.  V. FASTEST C ONVERGENCE WITH C ONSTRAINTS In this section, we formulate a semi-definite program for the convex synthesis of the fastest distributed averaging algorithm that satisfies the constraints. It is well-known that

e(k) ≤ Cλ2 (M )k ,

(22)

where C is a constant and λ2 (M ) is the second largest eigenvalue in magnitude of M . Since λ2 (M ) = ρ(M − 1 1 T T n 11 ), then ρ(M − n 11 ) is the convergence speed metric. By minimizing this metric, the speed of convergence of the system is maximized. As a result, the fastest convergence with state constraints would be: 1 minimize ρ(M − 11T ) M,Y1 ,Y2 n subject to U M ≤ Y1 U + Y2

State variable of an agent in the network (xi (k))

for a symmetric matrix M , all its eigenvalues are real and we can write the error from consensus e(k) = ||x(k) − xave 1||2 as follows:

1

fast unconstrained design (FDLA [2]) fast design with state constraints (SDP)

0.9

0.8

0.7

0.6

xave 0.5

0.4

0

5

10

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25

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35

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Fig. 3: The figure shows that unconstrained FDLA design can have steep changes in the state variable of the agent. The constrained design proposed in this paper provides smooth variability in the state variable at the expense of slower convergence.

Y1 b + Y2 1n ≤ b Y1 ≥ 0, Y2 ≥ 0, M ≥ 0, M = M T The above optimization problem is a convex optimization when M is symmetric because the function ρ(M − 1 1 T T n 11 ) = ||M − n 11 ||2 is the L-2 matrix induced norm (all norms are convex functions), and all constraints are linear (in-)equalities. This problem can be solved using a semidefinite program (SDP) as follows: minimize s M,Y1 ,Y2 ,s

1 subject to − sI  M − 11T  sI n U M ≤ Y1 U + Y2

(23)

Y1 b + Y2 1n ≤ b Y1 ≥ 0, Y2 ≥ 0, M ≥ 0, M = M T M 1 = 1, Mij = 0 if (i, j) ∈ /E where s ∈ R is an auxiliary variable that corresponds to the second largest eigenvalue in magnitude of M , I is the identity matrix, and  is the semi-definite comparison operator (i.e., A  B if and only if B − A is positive semi-definite) as compared to ≤ which is element-wise. Note that a similar semi-definite program was given in [2] but without the equations corresponding to Proposition 1. The SDP here would give the same result as the SDP in [2] if the fastest convergent matrix M for unconstrained consensus turns out to be a feasible solution of the consensus with state constraints. However, in many cases it is not, so we need to sacrifice convergence speed in order for the system to satisfy state constraints (e.g., maintain connectivity). Note that the SDP (23) is always solvable because it has a feasible solution {M = I, Y1 = I, Y2 = 0, s = 1}, but it is required that s < 1 for the convergence condition (21) to be satisfied. VI. S IMULATIONS In the simulations we compare two design algorithms: the first is the fastest distributed linear averaging (FDLA) that was given in [2], which allows to design a fast converging weight matrix M without taking into account the constraints.

Relative difference in states of neighboring agents (|xi (k) − xj (k)|)

M 1 = 1, Mij = 0 if (i, j) ∈ /E 0.35

fast unconstrained design (FDLA [2]) fast design with state constraints (SDP) Upper bound constraint dij

0.3 0.25 0.2 0.15 0.1 0.05 0

0

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Fig. 4: This figure shows that the FDLA design can violate upper bound constraints in the system. While both constrained and unconstrained designs would eventually guarantee the convergence of the relative distance to 0, the SDP in this paper ensures that the relative state difference would not violate its upper bound constraint specification.

The second design algorithm is the one developed in this paper by the SDP program in equation (23), which gives the fastest weight matrix M that satisfy the state constraints. We consider a network of 10 nodes forming a random geometric graph G = (V, E) (a graph where nodes are distributed randomly in a unit square area, and any two nodes within a connectivity radius r are connected by a link). Each node is then given an initial value in [0, 1], and we associate with every link (i, j) ∈ E an upper bound constraint dij . We compare the performance of the system having dynamics (1) where the weight matrix is designed using the two approaches mentioned earlier. Fig. 3 shows that the FDLA weight matrix gives faster convergence but the state changes abruptly between iter-

ations. The SDP proposed in this paper, however, gives smoother curve but at the expense of slower convergence. Fig. 4 shows that the FDLA approach violates the upper bound constraints of the system. These constraints can correspond to physical capacity limits in the network. For example, in electrical networks, exceeding those capacity constraints can lead to high currents traversing the wires which can cause damage in the network. The SDP approach in this paper, leads to relative states that ensure that the capacity limits are not violated. VII. C ONCLUSION In this paper, we have studied the problem of designing a weight matrix for distributed averaging algorithms that provides convergence guarantees and also ensures that the states do not violate prescribed polytopic constraints. We have given the necessary and sufficient conditions for the convergence with state constraints, and formulated the design problem as a semi-definite program (SDP) to speed up the convergence. For decentralized weight selection algorithms, it is well known that local algorithms (such as the Metropolis weights) guarantee convergence while sacrificing speed of convergence for the benefit of a local design. For future work, it would be interesting to design a decentralized weight selection algorithm (similar to the Metropolis weights) that can, in addition to convergence, guarantee the satisfaction of state constraints as those given by (5). ACKNOWLEDGMENTS This research was supported in part by the National Science Foundation (NSF) Grant No. CNS-1624328, the Defense Advanced Research Projects Agency (DARPA) Grant No. D14AP00084, and the Jet Propulsion Laboratory (JPL) Contract No. RSA-1539790. R EFERENCES [1] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Randomized gossip algorithms,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2508–2530, 2006. [2] L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” Systems & Control Letters, vol. 53, no. 1, pp. 65–78, 2004. [3] A. Nedi´c, A. Olshevsky, A. Ozdaglar, and J. N. Tsitsiklis, “On distributed averaging algorithms and quantization effects,” IEEE Transactions on Automatic Control, vol. 54, no. 11, pp. 2506–2517, 2009. [4] J. Liu, S. Mou, A. S. Morse, B. D. O. Anderson, and C. Yu, “Deterministic gossiping,” Proceedings of the IEEE, vol. 99, no. 9, pp. 1505–1524, 2011. [5] K. Avrachenkov, M. El Chamie, and G. Neglia, “A local average consensus algorithm for wireless sensor networks,” in Proceedings of IEEE International Conference on Distributed Computing in Sensor Sytems and Workshops, pp. 1–6, 2011. [6] M. El Chamie, G. Neglia, and K. Avrachenkov, “Distributed weight selection in consensus protocols by Schatten norm minimization,” IEEE Transactions on Automatic Control, vol. 60, no. 5, pp. 1350– 1355, 2015. [7] B. Ac¸ıkmes¸e, M. Mandi´c, and J. L. Speyer, “Decentralized observers with consensus filters for distributed discrete-time linear systems,” Automatica, vol. 50, no. 4, pp. 1037–1052, 2014. [8] F. Bullo, J. Cort´es, and S. Mart´ınez, Distributed Control of Robotic Networks. Applied Mathematics Series, New York: Princeton University Press, 2009.

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