IN this paper, we control a robotic sensor network to estimate

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Distributed Active State Estimation wit User-Specified Accuracy Carles Freundlic, Student Member, IEEE, Soomin Lee, Member, IEEE, and Micael M. Zavlanos, Member, IEEE arxiv: v2 [cs.ro] 14 Jan 2018 Abstract In tis paper, we address te problem of controlling a network of mobile sensors so tat a set of idden states are estimated up to a user-specified accuracy. Te sensors take measurements and fuse tem online using an Information Consensus Filter (ICF). At te same time, te local estimates guide te sensors to teir next best configuration. Tis leads to an LMI-constrained optimization problem tat we solve by means of a new distributed random approximate projections metod. Te new metod is robust to te state disagreement errors tat exist among te robots as te ICF fuses te collected measurements. Assuming tat te noise corrupting te measurements is zeromean and Gaussian and tat te robots are self localized in te environment, te integrated system converges to te next best positions from were new observations will be taken. Tis process is repeated wit te robots taking a sequence of observations until te idden states are estimated up to te desired user-specified accuracy. We present simulations of sparse landmark localization, were te robotic team acieves te desired estimation tolerances wile exibiting interesting emergent beavior. Index Terms Distributed optimization, Estimation, Cooperative control, Sensor networks I. INTRODUCTION IN tis paper, we control a robotic sensor network to estimate a collection of idden states so tat desired accuracy tresolds and confidence levels are satisfied. We require tat estimation and control are completely distributed, i.e., belief states and control actions are decided upon locally troug pairwise communication among robots in te network. A common assumption in problems like te one discussed erein is tat observations depend linearly on te state and are corrupted by Gaussian noise as well as tat te sensors are self localized [1] [8]. Under tese assumptions, a typical approac to estimate a set of idden states is to use an Information Filter (IF). Te role of te IF is to fuse new observations of te idden states wit a prior Gaussian distribution to create a minimum-variance posterior Gaussian distribution. Typically, te observations are subject to error covariance tat is a function of, e.g., viewing position, angle of incidence, and te sensing modality. Several suc models ave been proposed in te distributed active sensing literature, see, e.g., [1] [3], [5] [8], but regardless of te specific model used for te measurement covariance, te IF combines te data among te nodes in te network in an additive way, making te IF Carles Freundlic, Soomin Lee, and Micael M. Zavlanos are wit te Dept. of Mecanical Engineering and Materials Science, Duke University, Duram, NC 27708, USA {crf29, s.lee, mz61}@duke.edu. Tis work is supported in part by te NSF award CNS # and by ONR under grant #N algoritm readily distributed; see, e.g., Information Consensus Filtering (ICF) [9]. Te dependence of te measurement model on te sensor state as been widely used to obtain planning algoritms tat, given a istory of measurements, determine te next best set of observations of a idden state. Te objective is tat tis new set of observations optimizes an information-teoretic criterion of interest. In tis paper, we coose to maximize te minimum eigenvalue of te posterior information matrix, effectively minimizing te directional variance of every idden state. We express tis requirement by reformulating te problem as a Linear Matrix Inequality (LMI) constrained optimization problem. Specifically, we introduce as many LMI constraints as te number of idden states, tat are coupled wit respect to te sensors via an ICF tat estimates te uncertainty of eac idden state using te sensor measurements. To solve tis problem, we propose a new distributed optimization algoritm, wic we call random approximate projections, tat is robust to te state disagreement errors tat exist among te robots as an ICF fuses te collected measurements. Te ability to andle suc errors is critical in combining distributed estimation and planning in one process. Te proposed distributed random approximate projections metod falls in te class of so called consensus-based optimization algoritms, were te goal is to interleave a decentralized averaging step between local optimization steps so tat te local estimates of all agents simultaneously obtain a network-wide consensus and optimality of te global problem. Existing algoritms require expensive optimization steps or projections on a complicated constraint set, e.g., sets of LMI constraints as in tis paper, at every iteration. Instead, our metod employs a subgradient step in te direction tat minimizes te violation of randomly selected local constraints, significantly reducing computational cost compared to relevant metods tat rely on projection operations. Assuming tat eac LMI constraint is selected wit non-zero probability, we sow tat our proposed algoritm converges almost surely to te next best sensor positions from were a new set of observations minimizes te worst-case uncertainty of te idden states. Tis process is repeated wit te sensors taking a sequence of observations until all te idden states are estimated up to te desired user-specified accuracy. For details on relevant distributed optimization metods, we refer te interested reader to [10] and references terein. Related to our random projection tecnique is te Markov incremental random subgradient metod [11]. Te key difference is in te mecanism tat is used to distribute te

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE computations. In te Markov incremental algoritm, te agents sequentially update a single iterate sequence by passing te iterate to eac oter following a time inomogeneous Markov cain. In our algoritm, eac agent maintains a local iterate sequence and updates tis by communicating wit its neigbors. In two recent papers [12], [13], a dual-decomposition metod is proposed tat can be used for problems similar to te distributed control problem considered ere, wit lower communication overead and increased privacy. Generally, dual metods are well suited for network optimization problems, tey come at increased local cost because of te need to compute te dual function at eac node. Te approaces described in [12], [13] additionally require a special problem structure; specifically, a decomposable objective and/or a star network topology. Applications in te area of distributed sensor planning and estimation tat are directly relevant to our work include SLAM [14], localization [4] [6], coverage [7], [8], mobile target tracking [1] [3], [15], classification [16], and inverse problems for PDE-constrained systems [17], [18]. Compared to state estimation, inverse problems for PDE-constrained systems address bot state and source identification and are particularly difficult to solve. A common caracteristic of te applications discussed above is tat te sensors must collectively reason over te possible posterior distributions resulting from (i) multiple idden states being observed by a single sensor or (ii) individual idden states being observed by several sensors at once. Witout assuming Gaussian distributions, optimization over te possible posteriors can only be done approximately [16]. In tis paper, and in muc of te relevant literature [1] [6], it is assumed tat te posteriors for te idden states are Gaussian. Under tis assumption, typical coices of an objective function to minimize in order to obtain te next best set of observations include te trace, te determinant, or te maximum eigenvalue of te posterior covariance matrix. Intuitively, te trace minimizes te sum of te uncertainties of te idden states, te determinant te volume of te confidence ellipsoids of te idden states, and te maximum eigenvalue te worst-case error. To our knowledge, in te context of te distributed active localization problem, existing works tat control te maximum eigenvalue of te error covariance matrix eiter focus on sensor selection from a discrete set [12], [13], [19] [21], are limited to two sensors [6], or do not factor in simultaneous measurements by oter sensors wen deciding were to go [2], [4], [22]. Metods tat account for nonlinear dynamics and arbitrary error distributions [15], [23] are also relevant to te problem addressed erein. In tese cases, updates to te idden state pdfs may no longer be available in closed form, and so numerical metods are needed to evaluate te expectation integrals. To account for uncertainty in te sensor location and actions, decentralized Partially Observable Markov Decision Process (dec-pomdp) can be used to solve tese problems. Dec-POMDPs ave coupled value functions among te agents, and typically require a centralized planner [24]. At te same time, comparing te possible posterior distributions can be computationally expensive, so tese metods do not usually scale well wit te number of targets and agents. Altoug many of tese works assume tat te robots are self localized wit respect to a common global reference frame [1] [4], [6], [15], [16], tis is certainly not always te case; see, e.g., work on partially observable systems [23], [24] or te SLAM problem [14]. Contributions: In tis paper we propose a distributed estimation and sensor planning metod tat can ensure desired estimation tolerances for large numbers of idden states. To our knowledge, even toug te distributed active sensing literature is well-developed, te ability to control worst-case estimation uncertainty in a decentralized way is new. Tis is possible by combining an ICF for distributed data fusion wit an LMI-constrained optimization problem for sensor planning, for wic we propose a new computationally inexpensive distributed metod tat is robust to te inexact pre-consensus filter data. Coordination among te sensors requires only weak connectivity of te communication grap and can, tus, andle occasional disruption of communication links. Te distributed planning algoritm itself is an extension of te autors previous work [25] and [26]. Te difference is tat te metod developed in tis paper is robust to state disagreement errors tat appear as a result of te information filter. Te ability to andle suc errors is critical in integrating distributed estimation and planning. To te best of our knowledge, tere are no distributed optimization metods tat can andle efficiently LMI constraints under inexact data. It is also wort mentioning tat a wide variety of convex constraints arising in control, suc as quadratic inequalities, inequalities involving matrix norms, Lyapunov inequalities and quadratic matrix inequalities, can be cast as LMIs. Terefore, our developed algoritm can be used for oter, potentially unrelated, control problems involving inexact data. Te rest of te paper is organized as follows. Section II formulates te problem under consideration. Ten, Section III develops te proposed distributed metod to maximize te minimum eigenvalue of te information matrix. Section IV states assumptions and te main convergence result of our proposed algoritm. Section V sows tat te proposed algoritm converges to te optimal point. Section VI provides simulations, and Section VII concludes te paper. II. PROBLEM FORMULATION Consider te problem of estimating m stationary idden states {x i R p } i I using noisy observations from n mobile sensors, were p is te dimension of te idden states and I = {1,..., m}. We assume tat te idden states are sparse so tat tere are no correlations between tem. Denote te locations of te mobile sensors at time t by {r s (t) R q }, were q is te dimension of te sensors configuration space and S = {1,..., n}. Te observation of state i by sensor s at time t is given by y i,s (t) = x i + ζ i,s (t). (1) We assume tat te measurement errors are Normally distributed so tat ζ i,s (t) N ( 0, Q(r s (t), x i ) ), were Q: R q+p Sym + (p, R) denotes te measurement precision matrix, or information matrix (not te covariance), and

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Sym + (p, R) is te set of p p symmetric positive semidefinite real matrices. We also assume tat if signal x i is out of range of sensor s, ten te function Q returns te information matrix 0 p p, corresponding to infinite variance. We provide an example of te measurement information function Q in Section VI. Denote te set of all observations at time t by O(t) {(y i,s (t), Q(r s (t), x i ))} i I, Moreover, let ˆx i (t) denote te estimate of x i at time t and S i (t) denote te information matrix corresponding to te estimate ˆx i (t). We assume tat initial values for te pair (ˆx i (0), S i (0)) are available a priori. Let also ˆx(t) R pm denote te stack vector of all estimates ˆx i (t) and S(t) Sym + (p, R) Sym + (p, R) := ( Sym + (p, R) ) m denote te collection of all information matrices S i (t) at time t. Ten, te global data at time t can be captured by te set D(t) (ˆx(t), S(t)) R pm ( Sym + (p, R) ) m. Given prior data D(t 1) known by all sensors, and observations O(t 1), te current set of data D(t) can be determined in a distributed way using an Information Consensus Filter (ICF) [9] as 1 (D(t 1), O(t 1)) ICF D(t). (2) Ten, te goal is to control te next set of observations O(t) in order to reduce te future uncertainty in D(t + 1). We can acieve tis goal by controlling te mobile sensors and, terefore, te expected information {Q(r s (t), x i (t)} i I, associated wit te observations {y i,s (t)} i I,. Specifically, let r s (t) = r s (t 1) + u s denote te motion model of sensor s, were u s R q is a candidate control input used to drive sensor s to a new position r s (t). Ten, our goal is to maximize te minimum eigenvalues of te information matrices in D(t + 1) tat, for eac idden state i, are given by from D(t) {}}{ S i (t + 1)= S i (t)+ Q(r s(t 1) + u s, from D(t) {}}{ ˆx i (t) ) } {{ } from O(t). (3) We assume tat u s lies in a compact set containing te origin. Denote tis set by U s. Note tat tere is a δ > 0 suc tat te closed ball of radius δ contains U s. Define U U s and let u R qn denote te stack of all controllers u s. Additionally, define te set Γ i I [0, τ i], were τ i represent te given user-specified estimation tresolds for eac idden state i. Define also γ = (γ 1,..., γ m ) R m. Wit te new notation, te problem we consider in tis paper can be defined as max (γ,u) Γ U i I γ i (4a) s.t γ i I S i + Q(r s + u s, ˆx i ) i I, (4b) were we ave dropped dependence on te time t for notational convenience. For te explicit references to time in (4b) see (3). Te symbol denotes an ordering on te negative semidefinite cone, i.e., A B if and only if A B is negative semidefinite. Te great callenge in solving (4) is tat te 1 Explicit details on te mapping (2) are provided in Section III. constraints (4b) require knowledge of D(t), owever, only D(t 1) and tose parts of O(t 1) corresponding to local sensor measurements are available to any given sensor. In wat follows, we propose an integrated approac tat combines te ICF to determine te global data D(t) and te solution of te optimization (4) in one process. We sow tat our proposed metod can andle pre-consensus disagreement errors due to te ICF and converges to te next best set of observations O(t), as desired. III. CONCURRENT ESTIMATION AND SENSOR PLANNING Problem (4) is a nonlinear semidefinite program. To simplify tis problem, we replace te nonlinear function Q(r s +u s, ˆx i ) in (4b) wit its Taylor expansion about te input u s = 0. Specifically, define te function : R m+qn ( R pm ( Sym + (p, R) ) m) I Sym(p, R), were Sym(p, R) denotes te set of symmetric p p dimensional matrices, by ((γ, u); D, i) γ i I S i (5) [ Q(r s, ˆx i ) + ] q jq(r s, ˆx i )[u s ] j. j=1 Te notation j Q(r s, ˆx i ) is te partial derivative of te matrix-valued function Q wit respect to te j-t coordinate of R q+p, evaluated at (r s, ˆx i ). Also, [u s ] j denotes te j-t coordinate of te vector u s, and te semicolon notation in te arguments of separates te decision variables (γ, u) from te parameters D and i. Note tat te value of is negative semidefinite at a point ((γ, u); D, i) if and only if te linearized version of te constraint (4b) is satisfied for γ i using te data D = (ˆx, S). Te effect of te linear approximation is investigated in Section VI, Fig. 4. To simplify notation, collect all decision variables in te vector z = (γ, u) R m+qn. Ten, to decompose te linearized version of (4) over te set of sensors so tat te problem can be solved in a distributed way, let z s denote a copy of z tat is local to sensor s S. Additionally, let f(z s ) = i I γ i,s so tat we may express te global objective as f(z s). Ten, problem (4) wit linearized constraints can be expressed as min f(z s) (6a) {z s} X 0 s.t (z s ; D, i) 0 i I, s S, (6b) were X 0 = Γ U. In problem (6), te objective function (6a) is separable among te sensors and te constraints (6b) are linear in z s and local to every sensor. In tis form, problem (6) can be decomposed and solved in a distributed way. A necessary requirement is tat, at te solution, te local variables z s need to agree for all sensors s, i.e., z s = z v for all s, v S. As discussed in Section II, te main callenge in solving problem (6) is tat te global data in D are not known to te sensors and need to be computed using te ICF concurrently wit te optimization of te decision variables z s. Te ICF is an iterative process by wic every sensor updates a local copy of te global data until all sensors agree on te true values

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Algoritm 1 Concurrent Filtering and Optimization for s S during te interval [t 1, t] Require: Data D s (t 1), measurements {y i,s (t 1)} i I, and z s,0 feasible for (6). 1: (ˆx, S) D s (t 1) 2: Ξ i,s,0 1 n S i + Q(r s, ˆx i ) for all i I 3: ξ i,s,0 ( 1 n S i + Q(r s, ˆx i ) ) y i,s for all i I 4: for k = 1, 2,... do 5: Broadcast z s,k 1, { Ξ i,s,k 1, ξ i,s,k 1 i I } to N s,k 6: p s,k j N s,k [W k ] sj z s,k 1 7: Ξ i,s,k j N s,k [W k ] sj Ξ i,s,k 1 for all i I 8: ξ i,s,k j N s,k [W k ] sj ξ i,s,k 1 for all i I 9: ( ((nξ1,s,k D s,k ) 1 ξ 1,s,k,..., (nξ m,s,k ) 1 ) ξ m,s,k, ) (nξ 1,s,k,..., nξ m,s,k ) 10: v s,k Π X0 (p s,k α k f (p s,k )) 11: Coose random ω s,k I, compute + (v s,k ; D s,k, ω s,k ) and +(v s,k ; D s,k, ω s,k ) 12: β s,k +(v s,k;d s,k,ω s,k ) + (v s,k;d s,k,ω s,k ) 2 13: z s,k Π X0 ( vs,k β s,k + (v s,k ; D s,k, ω s,k ) ) 14: end for 15: return Updated data D s (t) D s,k and control input u s, taken from z s,k in D. Terefore, before te ICF as converged, disagreement errors on te local copies of te data set D exist, wic means tat te gradients and objective function evaluations at every sensor node will not depend on te same data. In oter words, gradients and objective function evaluations will be based on inexact data. An additional callenge is tat te number of LMI constraints can grow very large wit te number of sensors and idden states. We address tese callenges in te remainder of tis section. A. Distributed Information Filtering We solve problem (6) using an iterative approac. Let D s,k denote te copy of te data set D tat is local to sensor s at iteration k N, were N = {1, 2, 3,... } denotes te natural numbers. Additionally, let z s,k denote te decision variables of te sensor s at iteration k. Te first step of te algoritm is for sensor s to broadcast z s,k 1 and a summary of D s,k 1. In Algoritm 1, te data summary for state i is captured by te intermediate information matrix Ξ i,s,k and information vector ξ i,s,k, defined in lines 2 and 3. We refer te reader to [9] for a detailed discussion of Information Consensus Filtering, but we note briefly ere tat if te sensors eac compute Ξ i,s,0 and ξ i,s,0, and carry out te weigted averaging tat is standard in all consensus algoritms, given in lines 7 and 8, ten nξ i,s,k and ξ i,s,k converge geometrically to te consensus posterior information. In oter words, under te ICF, we ave tat, for all s S, {(nξ 1,s,k,..., nξ m,s,k )} k N S for all i I and {( (nξ 1,s,k ) 1 ξ 1,s,k,..., (nξ m,s,k ) 1 ξ m,s,k )}k N ˆx. In particular, let G k = (S, E k ) represent a communication grap suc tat (s, v) E k if and only if sensors s and v can communicate at iteration k. Tis defines te neigbor set, N s,k {s} {v S (v, s) E k. Sensor s receives {z v,k 1, { } } ξ i,v,k, Ξ i,v,k v N i I s,k. Wit tis new information, te sensor updates its own optimization variable and data summary via using weigted averaging. In particular, let W k be an n n row stocastic matrix suc tat [W k ] sv > 0 v N s. Te exact assumptions for W k will be introduced later, wen tey are necessary for te proof of our main result. Te consensus steps are depicted in lines 6, 7, and 8 of Algoritm 1. Ten, te agent can compute D s,k as sown in line 9, concluding te IF portion of iteration k. Note tat, given initial conditions {D s,0 }, te data for any sensor s at time k is bounded by te initial conditions as D s,k max v S (D v,0), (7) were is any norm in te space of data R pm ( Sym++ (p, R) ) m. Tis implies tat te set of possible data is bounded wit respect to any norm in te vector space R pm (Sym(p, R)) m. Note tat, by te same arguments as can be found in [9], under any norm, all sensors data converge to te same consensus value, i.e., D s,k D v,k 0 and D s,k D 0 for all s, v S. B. Random Approximate Projections Te optimization part of te algoritm begins in line 10 by computing a negative subgradient of te f at te local iterate, wic we denote by f (z s,k ), taking a gradient descent step, and projecting te iterate back to te simple constraint set X 0. Te positive step size α k is diminising, non-summable k N α k and square-summable, i.e, α k 0, = and k N α2 k <. Te last step of our metod is not a projection to te feasible set, but instead a subgradient step in te direction tat minimizes te violation of a single LMI constraint. Essentially, eac node selects one of te LMIs randomly, measures te violation of tat constraint, and takes an additional gradient descent step wit step size β s,k in order to minimize tis violation. We call tis process random approximate projection. Specifically, line 11 randomly assigns ω s,k I, effectively coosing to activate ( ;, ω s,k ). Te idea beind line 13 is to minimize a scalar metric tat measures te amount of violation of ( ;, ω s,k ), wic we denote by + ( ;, ω s,k ), wile remaining feasible. Let Π + : Sym(p, R) Sym + (p, R) denote te projection operator onto te set of positive semidefinite matrices, given by Π + X = Ediag [(λ 1 ) +,..., (λ p ) + ] E, (8) were ( ) + max {, 0} and λ 1,..., λ p are te eigenvalues and E is a matrix wit columns e 1,..., e p tat are te corresponding eigenvectors of X. A measure of te positive definiteness of X is tus Π + X F (λ 1 ) (λ p ) 2 +. Tis defines our measure of constraint violation, + (z; D, i) Π + (z; D, i) F. (9) We use special notation to denote te vector of derivatives of + wit respect to te decision variables z

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE evaluated at (z; D, i). In particular, define te gradient + (z; D, i) = ( 1 + (z; D, i),..., m+qn + (z; D, i)) = z + (z; D, i). Defining te function ψ ij : Sym(p, R) R m+qn ( R pm ( Sym + (p, R) ) m) R for notational convenience, te j-t entry of +(z; D, i) is given by j + (z; D, i) = ψ ij ( Π + ( (z; D, i) ), z, D ), (10) were ψ ij (X, z, D) { Tr( j(z,d,i)x) X if X F F > 0 d else (11) and d > 0 is some arbitrary constant. Te function ψ ij is not continuous at X = 0, and we set d 0 for tecnical reasons discussed below. Te step size β s,k = +(v s,k ; D s,k, ω s,k ) + (v s,k ; D s,k, ω s,k ) 2 (12) given in line 12 is a variant of te Polyak step size [27]. Note tat tis is a well defined step size as we let +(v s,k ; D s,k, ω s,k ) = d1 m+qn (cf. Eq (11)) wenever + (v s,k ; D s,k, ω s,k ) = 0 and it is nonzero elsewere by construction. IV. MAIN RESULTS In tis section, we discuss our assumptions and state te main result. Te detailed proofs are deferred to Section V. Te feasible set is X = X 0 {z R m+qn (z; D, i) 0, i I}. Te optimal value and solution set of te problem (6) is f = min X f(z) and X = {z X f(z) = f }. Note tat 0 Γ U and (0; D, i) 0 for all possible D and i, tus tere is always a feasible point. At step k N, D s,k in line 9 of Algoritm 1 may not yet ave converged to D, so we denote te error in te constraint violation by ν s,k = + (z s,k ; D s,k, ω s,k ) + (z s,k ; D, ω s,k ). (13) Similar to te definition in (13), we define te error in te gradient of te constraint violation to be δ s,k = + (z s,k ; D s,k, ω s,k ) + (z s,k ; D, ω s,k ). (14) Te vectors ν s,k and δ s,k will enable us to quantify te joint effect of te function value error ν s,k and te subgradient error δ s,k in studying convergence properties of Algoritm 1. A. Bounds Te norms of ν s,k and δ s,k are bounded, i.e., for some scalars D, N > 0, tere olds for all k N and s S a.s.: ν s,k N and δ s,k D. We calculate te exact values of D and N in Corollaries A.2 and A.7, respectively. Te set X 0 = Γ U is convex and compact. Terefore, tere must exist a constant C z > 0 suc tat for any z 1, z 2 X 0 z 1 z 2 C z. (15) Also, note tat te functions f( ) and (z; D, i) for i I are convex (not necessarily differentiable) over some open set tat contains X 0. A direct consequence of tis and te compactness of X 0 is tat te subdifferentials f(z) and + (z; D, i) are nonempty over X 0, were note tat ere we use te symbol to refer to te subdifferential set and not a partial derivative. It also implies tat te subgradients f (z) f(z) and +(z; D, i) + (z; D, i) are uniformly bounded over te set X 0. Tat is, tere is a scalar L f suc tat for all f (z) f(z) and z X 0, f (z) L f, (16a) and for any z 1, z 2 X 0, f(z 1 ) f(z 2 ) L f z 1 z 2. (16b) Note tat for te objective function f in (6), we ave L f = 1. Also, tere is a scalar L suc tat for all +(z; D, i) + (z; D, i), z X 0, i I, +(z; D, i) L. (17) We calculate te exact value of L in Corollary A.7. From relation (14) we ave for any i I, s S, k N, and z X 0 +(z; D s,k, i) = +(z; D, i) + δ s,k L + D, and for any z 1, z 2 X 0 (18a) + (z 1 ; D s,k, i) + (z 2 ; D s,k, i) (L +D) z 1 z 2. (18b) Te compactness of X 0, te boundedness of te data sequences D s,k, and te continuity of te functions + (z; D, i) for i I imply tat tere exist constants C > 0 suc tat for any s S, k N, i I and z X 0 + (z; D, i) C, (19a) + (z; D s,k, i) = + (z; D, i) + ν s,k C + N, (19b) were te second relation follows from (13). Furtermore, wen + (z; D s,k, i) 0, we ave +(z; D s,k, i) 0. Terefore, tere must exist a constant c > 0 suc tat +(z; D s,k, i) c (20) for all z X 0, s S, k N, and i I. Tis and relation (19b) imply tat β s,k = +(z; D s,k, ω s,k ) +(z; D s,k, ω s,k ) 2 C + N. (21) Note tat wen + (z; D s,k, ω s,k ) = 0, te above bound olds trivially. B. Assumptions In te preceding sections, we ave made extensive use of te function Q: R q R p Sym + (p, R). For our algoritm to converge, we require te following to old true regarding tis information model. Assumption IV.1. We assume tat te information function Q (a) is bounded, (b) is twice subdifferentiable (c) as bounded subdifferentials up to te second order, and c 2

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE (d) as relatively few critical points, i.e., te sets of critical points of Q and its partial derivatives up to te second order are measure zero. Denote te bound on te magnitude of Q by η 0, te bound on its first derivatives by η 1, and te bound on its second derivatives by η 2. Te tecnical restrictions on te information model Q are not overly restrictive in practice. In particular, items (a), (b), and (c) imply tat one cannot obtain infinite information and tat, by canging sensor configuration, te information rate (and te rate of cange of te rate) cannot cange infinitely quickly. Item (d) essentially allows us to distinguis between sensor configurations, i.e., te set of configurations tat offer optimal information rates are relatively sparse wit respect to te space of signal source positions and sensor configurations R q R p. Te next two assumptions are related to te random variables ω s,k. At eac iteration k of te inner-loop, recall tat eac sensor s randomly generates ω s,k I. We assume tat tey are i.i.d. samples from some probability distribution on I. Assumption IV.2. In te k-t iteration of te inner-loop, eac element i of I is generated wit nonzero probability, i.e., for any s S and k N it olds tat π i Pr{ω s,k = i} > 0. Assumption IV.2 is easy to satisfy because I is a finite set. Assumption IV.3. For all s S and k N, tere exists a constant κ > 0 suc tat for all z X 0 dist 2 (z, X ) κe [ 2 +(z; D s,k, ω s,k ) ]. Te upper bound in Assumption IV.3 is known as global error bound and is crucial for te convergence analysis of Algoritm 1. Sufficient conditions for tis bound ave been sown in [28] [31], wic includes te case wen eac function ( ;, i) is linear in z, or wen te feasible set X as a nonempty interior. Assumption IV.4. For all k N, te weigted graps G k = (S, E k, W k ) satisfy: (a) Tere exists a scalar η (0, 1) suc tat [W k ] sj η if j N s,k. Oterwise, [W k ] sj = 0. (b) Te weigt matrix W k is doubly stocastic, i.e., [W k] sj = 1 for all j S and j S [W k] sj = 1 for all s S. (c) Tere exists a scalar Q > 0 suc tat te grap (S, l=0,...,q 1 E t+l ) is strongly connected for any t 1. Tis assumption ensures a balanced communication between sensors. It also ensures tat tere exists a pat from one sensor to every oter sensor infinitely often even if te underlying grap topology is time-varying. C. Main result Our main result sows te almost sure convergence of Algoritm 1. Specifically, te result states tat te sensors asymptotically reac an agreement to a random point z wic is in te optimal set X almost surely (a.s.), as given in te following teorem. Teorem IV.5 (Asymptotic Convergence Under Noise). Let Assumptions IV.1 - IV.4 old. Let also te stepsizes be suc tat k=1 α k = and k=1 α2 k <. Ten, te iterates {z s,k} generated by eac agent s S via Algoritm 1 converge almost surely to te same point in te optimal set X, i.e., for a random point z X lim k z s,k = z, s S a.s. V. CONVERGENCE OF ALGORITHM 1 A. Preliminary Results Lemma V.1 (Non-expansiveness [32]). Let X R d be a nonempty closed convex set. Te function Π X : R d X is nonexpansive, i.e., Π X [a] Π X [b] a b for all a, b R d. Lemma V.2 ( [33, Lemma 10-11, p ]). Let F k {v l, u l, a l, b l } k l=0 denote a collection of nonnegative real random variables for k N { } suc tat E[v k+1 F k ] (1+a k )v k u k +b k for all k {0} N a.s. Assume furter tat {a k } and {b k } are a.s. summable. Ten, we ave a.s. tat (i) {u k } is summable and (ii) tere exists a nonnegative random variable v suc tat {v k } v. In te next lemma, we relate te two iterates p s,k and z s,k in Line 6 and 13 of Algoritm 1. In particular, we sow a relation of p s,k and z s,k 1 associated wit any convex function g wic will be often used in te analysis. For example, g(z) = z a 2 for some a R n or g(z) = dist 2 (z, X ). Lemma V.3 (Convexity and Double Stocasticity). For any convex function g : R n R, we ave g(p s,k) g(z s,k 1) Proof. Te double stocasticity of te weigts plays a crucial role in tis lemma. From te definition of p s,k in Line 6 of Algoritm 1, we obtain [W k ] sj g(z j,k 1 ) g(p s,k ) j S = ( ) [W k ] sj g(z j,k 1 ) = g(z j,k 1 ). j S j S Lastly, for te convergence proof of our algoritm, we use a result from [34], wic ensures successful consensus in te presence of a well beaved disturbance sequence. Lemma V.4 (Perturbed Consensus). Let Assumption IV.4 old. Consider te iterates generated by θ s,k = v S[W k ] sv θ v,k 1 + e s,k, s S, (22) Suppose tere exists a nonnegative nonincreasing scalar sequence {α k } suc tat Ten, for all s, v S, k=1 α k e s,k < for all s S. k=1 α k θ s,k θ v,k <. In addition to te well-known results of Lemmas V.1-V.4, we need te following intermediate results before te main result in Teorem IV.5 can be proven. Te proofs of all of tese Lemmas are deferred to Appendices A and B. In te first

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE two of tem, we posit tat te two error sequences {ν s,k } and {δ s,k } are summable. Lemma V.5 (Summable Constraint Errors). For almost every bounded sequence {z s,k }, te error in te constraint { ν s,k } k N is summable a.s., i.e., k=0 ν s,k <. Lemma V.6 (Summable Constraint Violation Errors). For almost every bounded sequence {z s,k }, te error in te gradient of te constraint { δ s,k } k N is summable a.s., i.e., k=0 δ s,k <. In te next lemma, we relate te two iterates p s,k and z s,k in Line 6 and 13 of Algoritm 1. Lemma V.7 (Basic Iterate Relation). Consider te sequences {z s,k } k=0 and {p s,k} k=0 for s S generated by Algoritm 1. Ten, for any z, ž X, s S and k N, we ave a.s.: z s,k z 2 p s,k z 2 2α k (f(ž) f(z)) + 2 C z(c + N) c 2 δ s,k + 2 C + N c 2 ν s,k + 1 4η p s,k ž 2 + A η,τ α 2 kl 2 f, τ 1 τ(l + D) 2 2 +(p s,k ; D s,k, ω s,k ) were A η,τ = 1 + 4η + τ and η, τ > 0 are arbitrary. Since we use a random approximate projection, we cannot guarantee te feasibility of te sequences p s,k and z s,k for every k N and s S. In te next lemma, we prove tat p s,k for all s S asymptotically acieve feasibility even under te effect of te disturbances {ν s,k } and {δ s,k }. Lemma V.8 (Asymptotic Feasibility Under Noise). Let Assumption IV.2 and IV.3 old. Let α k be square summable Consider te sequences {p s,k } for s S generated by Algoritm 1. Ten, for any s S, te sequence {dist(p s,k, X )} k N is a.s. square summable. In te final Lemma, we sow tat z s,k p s,k eventually converges to zero for all s S. Tis result combined wit Lemma V.8 implies tat te sequences {z s,k } k N also acieve asymptotic feasibility. Lemma V.9 (Network Disagreement Under Noise). Let Assumptions IV.2-IV.4 old. Let te sequence {α k } is suc tat k=1 α2 k <. Define e s,k = z s,k p s,k for all s S and k 1. Ten, we ave a.s.: (a) k=1 e s,k 2 < for all s S, (b) k=1 α k ž s,k ž k < for all s S, were ž s,k = Π X [p s,k ] and ž k = 1 n žs,k. Note tat te sequences {z s,k } k N for s S generated by Algoritm 1 can be represented as relation (22). Tat is, z s,k = [W k] sj z j,k 1 +e s,k for all s S and e s,k = z s,k p s,k. Terefore, from Lemma V.9(a), we ave k=1 α k e s,k = 1 2 k=1 α2 k k=1 e s,k 2 <. Invoking Lemma V.4, we see tat tere is consensus among z s,k for s S. B. Proof of Teorem IV.5 We invoke Lemma V.7 wit ž = ž s,k (Note tat ž s,k is defined as Π X [p s,k ] in Lemma V.9), τ = 4 and η = κ(l + D) 2. We also let z = z for an arbitrary z X. Terefore, for any z X, s S and k N, we ave a.s.: z s,k z 2 p s,k z 2 2α k (f(ž s,k ) f(z )) + 2 C z(c + N) c 2 δ s,k + 2 C + N c 2 ν s,k 1 + 4κ(L + D) 2 dist2 (p s,k, X ) + AαkL 2 2 f 3 4(L + D) 2 2 +(p s,k ; D s,k, ω s,k ), (23) were A = 5+4κ(L +D) 2. From Assumption IV.3, we know tat dist 2 (p s,k, X ) κe [ ] 2 +(p s,k ; D s,k, ω s,k ) F k 1. Denote by F k te sigma-field induced by te istory of te algoritm up to time k, i.e., F k = {z s,0, (ω s,l, 1 l k), s S} for k N, and F 0 = {z s,0, s S}. Taking te expectation conditioned on F k 1 in relation (23), summing tis over s S, and using te above relation, we obtain E [ z s,k z 2 ] F k 1 (24) z s,k 1 z 2 2α k (f(ž s,k ) f(z )) + 2 C z(c + N) E[ δ s,k F k 1 ] c C + N c 2 E[ ν s,k F k 1 ] 1 2κ 2 (L + D) 2 dist 2 (p s,k, X ) + AαknL 2 2 f, were we used Lemma V.3 for te first term on te rigt-and side. Recall tat f(z) = f(z) and ž k 1 n l S žl,k. Using ž k and f, we can rewrite te term (f(ž s,k) f(z )) as follows: (f(ž s,k ) f(z )) = (f(ž s,k ) f( ž k )) + (f( ž k ) f ) (25a) f ( ž k ), ž k ž s,k + (f( ž k ) f ) (25b) f ( ž k ) ž s,k ž k + (f( ž k ) f ) (25c) L f ž s,k ž k + (f( ž k ) f ), (25d) were (25a) follows from adding and subtracting f( ž k ); (25b) follows from te convexity of te function f; (25c) follows from te Scwarz inequality; and (25d) follows from relation (16a) and ž k X X 0. Combining (25d) wit (24), we obtain E [ z s,k z 2 ] F k 1 z s,k 1 z 2 2α k (f( ž k ) f ) + 2 C z(c + N) c 2 E[ δ s,k F k 1 ]

8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE C + N c 2 E[ ν s,k F k 1 ] + L f α k ž s,k ž k + AαknL 2 2 f, were we omitted te negative term. Since ž k X, we ave f( ž k ) f 0. Tus, under te assumption k=0 α2 k < and Lemma V.9(b), te above relation satisfies all te conditions of Lemma V.2. Using tis, we ave te following results. Result 1: For some z X and any s S, te sequence { z s,k z } is convergent a.s. Result 2: k=1 α k(f( ž k ) f ) < a.s. As a direct consequence of Result 1, we know tat te sequences { p s,k z } and { ž s,k z } are also convergent a.s. (Tis is straigtforward from Line 6 of Algoritm 1 and Lemma V.8 wit te knowledge tat dist 2 (p s,k, X ) = p s,k ž s,k 2.) Since ž k z 1 N i S ž s,k z, we furter know tat te sequence { ž k z } is also convergent a.s. As a direct consequence of Result 2, since α k is not summable, lim inf k (f( ž k ) f ) = 0 a.s. From tis relation and te continuity of f, it follows tat te sequence { ž k } must ave one accumulation point in te set X a.s. Tis and te fact tat { ž k z } is convergent a.s. for every z X imply tat Also, by Lemma V.9(b), we ave lim ž k = z a.s. (26) k lim inf k ž s,k ž k = 0 for all s S a.s. (27) Te fact tat { ž s,k z } is convergent a.s. for all s S, togeter wit (26) and (27) implies tat lim k žs,k = z for all s S a.s. (28) Since z s,k ž s,k z s,k p s,k + p s,k ž s,k, by invoking Lemma V.8 and V.9(a), we ave lim k z s,k ž s,k = 0 for all s S a.s. Tis and relation (28) give us te desired result, wic is lim k z s,k = z for all s S a.s. VI. NUMERICAL EXPERIMENTS In tis section, we illustrate Algoritm 1 on a network of n mobile sensors tat collaboratively localize a collection of m landmarks. We use a distributed connectivity maintenance controller [35] to ensure tat te sensor network can sare information and collaborate for all time. Te controller operates by signaling for a strong attractive force wenever a link is going to be deleted tat can break connectivity. Te controller also deletes or replaces weak links as long as connectivity is preserved. A. Sensor Model We assume tat te landmarks live in R 2 and denote te configuration of landmark i by x i. We also assume tat te configuration space of te robot is R 2. Generic models for te measurement covariance matrix for sparse landmark localization ave been proposed [1], [2], [7]. r s ˆ x i î Figure 1. A sensor at r s observes targets it believes to be at ˆx i and ˆx j. Possible confidence regions are sown in grey. Dotted lines are parallel to te global î axis. Red lines guide te eye from te sensor to targets Tese models ave two important caracteristics tat apply to a wide variety of localization sensors: (i) Measurement quality is inversely proportional to viewing distance, and (ii) Te direction of maximum uncertainty is te viewing direction, i.e., te sensors are more proficient at sensing bearing tan range. Te general idea is to use te vector ˆx i (t) r s (t) to construct a diagonal matrix in a coordinate frame relative to te sensor, ten rotate te matrix to a global coordinate frame so tat it can be compared to oter observations. Following te accepted generic sensor models, we express Q(r s, ˆx i ) by its eigenvalue decomposition Q = R φi Λ i Rφ i, were te argument (r s, ˆx i ) of Q, Λ i, and R φi is implicit. Te angle φ i ( π 2, ] π 2 is defined so tat te first column of R φi, wic by convention is [cos φ i sin φ i ], is parallel to te subspace spanned by te vector ˆx i r s. Tis angle is given by φ i = tan 1 (([ˆx i ] 2 [r s ] 2 )/([ˆx i ] 1 [r s ] 1 )), were te subscripts outside te brackets refer to te first and second coordinate of te vector representing te location of te landmark and sensor located at ˆx i and r s. Te eigenvalue matrix is given by Λ i = diag(λ 1,i, λ 2,i ), were λ 1,i = c 0 (/1 + c 1 ˆx i r s 2 ) and λ 2,i = ρλ 1,i. In te body frame of sensor s, te eigenvalues λ 1,i and λ 2,i control te sape of te confidence ellipses for individual measurements of te i-t target. Te parameter c 0 > 0 represents te overall sensor quality and scales te wole region equally, c 1 > 0 controls te sensitivity to dept, and ρ > 1 controls eccentricity of te confidence regions associated wit te measurements. Figure 1 illustrates te measurement model for one sensor and two targets. Since tan 1 is not continuous at zero, we need to impose te following restriction so tat Assumption IV.1 olds. Essentially, we assume tat, for te selected δ > 0, wic recall as an affect on te robot s ability to translate in R 2, we coose a τ i suc tat τ i < c 0 /(1 + c 1 δ). Tus, if te robot is witin δ of a target i, te constraint (z, D, i) is trivially satisfied, and terefore we will never evaluate Q or any of its derivatives in tis region. B. Results Figure 2 sows an example of te resulting active sensing trajectories for a network of four sensors tat cooperatively localize a uniformly random distribution of 100 landmarks in a square 100 m 2 workspace to a desired accuracy of 1 m. Tis i x j j

9 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE m t = t = m t = t = 242 Figure 2. Four sensors localize 100 landmarks. Gray s indicate landmarks wit uncertainty above te tresold, and black s indicate localized landmarks. Gray lines connecting te agents indicate te communication links. corresponds to an eigenvalue tolerance of τ 1 = = τ m = 9 m 2. Te prior distribution on te landmarks was uninformative, i.e., te initial mean estimate and information matrix are given by te initial observations {y i,s (0) i I, s S} and te corresponding information matrices were set to {Q(r s (0), y i,s (0))} i I,. Te parameters of te sensing model Q were set to c 0 = 1 2, c 1 = 10, and ρ = 30. Te feasible control set U was set to U = { u s R 2 u s 1 m }. In te figure, note tat between t = 194 and t = 242, te agent in te bottom rigt moves nort in order to maintain connectivity wile te oter agents move to finis te localization task. It can be seen from te four snapsots of te trajectories in Fig. 2, tat te robots effectively divide and conquer te task based on proximity to unlocalized targets. Fig. 3 sows te evolution of te minimum eigenvalues of te information matrices S i (t) from te simulation sown in Fig. 2. Note tat, even after te minimum eigenvalue associated wit a landmark as surpassed τ, more information is still collected, but it does not affect te control signal. Note also tat te maximal uncertainty in te bottom panel does not go below te desired tresold, even toug te minimal eigenvalue does go above te tresold in te top panel. Tis is expected because we use te 95% confidence interval to generate te tresold τ, tus we expect 5% of landmarks to ave errors larger tan te desired tresold. In tis case study, we selected te termination criterion for Algoritm 1 to be at most 30 iterations. Wen testing te algoritm wit α k = 1/k, we observed tat te relative cange in te iterates z s,k and te data D s,k would often reac levels at or below 10% inside of tirty iterations, so we allowed te message passing to terminate early if tis criterion was met. Figure 4 measures te amount of constraint violation of te original nonlinear constraints for te solution obtained from te linearized version tat we observed in a simulation wit twenty landmarks. For te U and te parameters for te function Q used in tese simulations, te constraint violation was very small, and driven to zero typically for t > t Figure 3. Top panel: Evolution of te minimum eigenvalues. Te center curve is te mean among all 100 targets, wit te grey block extending to te lowest and largest eigenvalues at eac time instant. Te dotted line is at te tolerance τ = 9 m 2. Bottom panel: Evolution of localization error. Te center curve is te mean among all 100 targets, wit te grey block extending to te minimum and maximum errors at eac time instant. Te dotted line is at te desired accuracy of 1 m. Note tat te scale on vertical axis is large because te initial uncertainty is infinite. m iteration Figure 4. Plotting te amount of constraint violation of te original nonlinear constraint tat will occur because we solve a linearized version. Twenty lines are drawn, one for eac landmark in a simulation wit m = 20. Te orizontal axis represents te outer loop iterations t. Lines are drawn to guide te eye We tested larger feasible sets U and (predictably) observed a general increase in te constraint violation, altoug we still observed reasonable robot beaviors. VII. CONCLUSION In tis paper, we addressed te problem of controlling a network of mobile sensors so tat a set of idden states are estimated up to a user-specified accuracy. We proposed a new algoritm tat combines te ICF wit control. Te first major contribution is te capability of controlling worst-case error in te idden states. Te second major contribution is te novel numerical tecnique, wic is robust toward noisy gradients, computationally ligt, can andle large systems of coupled LMIs. We sowed tat our metod converges to te optimal solution and presented simulations for sparse

10 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE landmark localization, were te team was able to acieve te desired estimation tolerances wile exibiting interesting emergent beaviors, notably dividing and conquering te distributed estimation task, observing landmarks from diverse viewing angles. APPENDIX A PROOF OF LEMMAS V.5 AND V.6 To prove Lemmas V.5 and V.6, consider te following. Proposition A.1 (Summable Disagreement in Constraints). Let Assumptions IV.1 - IV.4 old. For any bounded sequence {z s,k } and any i I, k=0 (z s,k; D s,k, ω s,k ) (z s,k ; D, ω s,k ) F <. Proof. In tis proof, we will essentially be comparing te function for te unconverged local data D s,k and te consensus data D. Note tat at some iteration k N, te ICF is actually computing te information summary (ξ 1,s,k,..., ξ m,s,k, Ξ 1,s,k,..., Ξ m,s,k ), wic is used to compute D s,k in line 9 of Algoritm 1. In particular, we will need to refer to te unconverged mean and information matrix, wic is ( ((nξ1,s,k D s,k ) 1 ξ 1,s,k,..., (nξ m,s,k ) 1 ) ξ m,s,k, ) (nξ 1,s,k,..., nξ m,s,k ). In te proof, we will drop te reference to te sensor s, as we only consider one sensor at a time. In oter words, te unconverged data will be written as )( ) ( D k =( (nξ 1,k ) 1 ξ 1,k }{{} ˆx 1,k,...,(nΞ m,k) 1 ξ m,k } {{ } ˆx m,k, nξ 1,k,...,nΞ 1,m }{{}}{{} S 1,k S m,k for k N, i.e., tis is implicitly te data local to sensor s in te proof. For te consensus data D = ((ˆx 1,,..., ˆx m, ), (S 1,,..., S m, )), we also drop te k subscript, i.e., we write te consensus data as D = (ˆx, S) = ((ˆx 1,..., ˆx m ), (S 1,..., S m )). Let us fix an arbitrary i I, and prove te proposition for te sequence ω k = {i, i, i,... }. If te proof olds for tis sequence, it tus olds for te random sequences ω k I because (z k; D k, ω k ) (z k ; D, ω k ) k=0 F (z k ; D k, i) (z k ; D, i) i I F, k=0 and if eac sequence (z k ; D k, i) (z k ; D, i) F is summable, so is teir sum. Recall te definition of from (5). We ave tat (z k ; D k, i) (z k ; D, i) = S i,k S i (29a) + Q (r v, ˆx i,k ) Q (r v, ˆx i ) (29b) v S q + ( j Q(r v, ˆx i,k ) j Q(r v, ˆx i )) [u v ] j, (29c) j=1 were S i and ˆx i are te eventual consensus variables tat compose te consensus data D, i.e., S i = S i, and ˆx i = ˆx i,. Te remainder of te proof will sow tat eac of te constituent terms, specifically, (29a), (29b), and (29c), are summable, and te proof will follow from tat immediately. (29a) For tis term, recall from line 9 of Algoritm 1 tat S i,k nξ i,k, were n is te number of agents in te network. It is well known tat for te consensus sequence {Ξ i,k } k N te errors {Ξ i,k Ξ i, } k N are summable; cf., e.g., [36]. Noting tat S i = nξ i, and S i,k = nξ i,k, we ave te result. (29b) By Assumption IV.1, q+j Q(r v, x i ) F η 1 for all r v, x i and j. Ten, for eac v S and k N, using te taylor expansion of Q(r v, ) about te point ˆx i, we ave tat Q (r v, ˆx i,k ) Q (r v, ˆx i ) F = = p q+jq(r v, ˆx i )[ˆx i,k ˆx i ] j +o ( ˆx i,k ˆx i ) j=1 η 1 ˆx i,k ˆx i 1 + o ( ˆx i,k ˆx i ), wic is summable because ˆx i,k ˆx is te error sequence of a consensus filter, wic is summable. (29c) First, since u v < δ for all v S, for any j = 1,..., p, we ave tat ( j Q(r s, ˆx i,k ) j Q(r s, ˆx i )) [u s ] j F (30) δ j Q(r s, ˆx i,k ) j Q(r s, ˆx i ) F Also due to Assumption IV.1, η 2 is te upper bound for te norm of te second derivative of Q. In particular, let q+l j Q (r v, ˆx i ) denote te mixed second partial derivative of Q wit respect to te l-t component of te x i coordinates and te j-t component of te r v coordinates of te function Q, evaluated at te point (r v, ˆx i ). Tis notation is useful for te following inequality, wic uses te first order Taylor expansion wit respect to te x i coordinates of j Q. In particular, te norm on te rigt and side of (30) is bounded above by p q+l j Q (r v, ˆx i ) [ˆx i,k ˆx i ] l + o ( ˆx i,k ˆx i ) l=1 F ( η 2 ˆx i,k ˆx i 1 + o ( ˆx i,k ˆx i ) ). Te sequence above is summable again because ˆx i,k ˆx i is te error sequence of a consensus filter, wic is summable. Since (29a), (29b), and (29c) are eac summable, teir sum is also a summable sequence, and te proof follows. Proof of Lemma V.5. Suppose ω s,k = {i, i, i,... }. If te proof olds for tis arbitrary i, ten it olds for arbitrary sequences ω s,k I for te same reasons as in Proposition A.1. Let X k = (z s,k ; D s,k, i) and Y k = (z s,k ; D, i). From te definition of ν s,k, ν s,k = + (z s,k ; D s,k, i) + (z s,k ; D, i) (31a) = Π + X k F Π + Y k F (31b) Π + X k Π + Y k F X k Y k F, F (31c) (31d) wic is summable by Proposition A.1. In te above manipulations, (31b) is less tan or equal to (31c) by te reverse triangle inequality and (31c) is less tan or equal to (31d)

11 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE because Sym + (p, R) is a convex set. Tus, ν s,k is elementwise less tan or equal to a summable sequence, and te proof follows. Corollary A.2 (Bound for ν s,k ). Te constant N, wic is te bound for ν s,k, is given by N = 1 n max v S (Ξ i,0 ) v F + m(η 0 + 2η 1 qδ). Proof. From te proof of Lemma V.5, eac term ν s,k is bounded above by X k Y k F, wic, in view of (29), is te sum of tree terms. Te corollary follows by bounding eac term, ten taking te sum of te bounds. We ave tat ν s,k = + (z s,k ; D s,k, ω s,k ) + (z s,k ; D, ω s,k ) X k Y k F S i,k S i F (32a) + Q (r v, ˆx i,k ) Q (r v, ˆx i ) (32b) v S + q j=1 ( jq(r s, ˆx i,k ) j Q(r s, ˆx i ))[u s ] j. (32c) Te first term (32a) is te error sequence of a consensus filter, wic is bounded by te largest initial condition in te network 1 n max v S (Ξ i,0 ) v F. For te second term (32b), recall Assumption IV.1 bounded Q F by η 0. Given tat Q is positive semidefinite, (32b) is bounded by mη 0. Similarly, te tird term is bounded by 2η 1 qδ, and, by taking te sum, te proof follows. Te proof of Lemma V.6 requires te elp of several intermediate results. First, we present two basic results in classical analysis. Bot are direct results of te Implicit Function Teorem; see, e.g., [37]. Corollary A.3. Let p: R n R be a nonzero polynomial function. Ten, te set of roots of p as measure zero in R n. Corollary A.4. Let f : R n+m R m be a smoot function, f surjective, and A R m a measure zero set. Ten, te preimage of A under f is measure zero in R n+m. Proposition A.5 (Set of Reacable Information Matrices tat Are Singular or Have Nonsimple Eigenvalues is Measure Zero). Let Z 1 = {X Sym(p, R) λ i (X) = λ j (X), i j}, Z 2 = {X Sym(p, R) det(x) = 0}. (33a) (33b) Now, recall te function defined in (5) and fix arbitrary i I. Te set is measure zero. Z = {(z, D) (z; D, i) Z 1 Z 2 } (34) Proof. Bot te determinant and te discriminant of te caracteristic polynomial are nonvanising polynomials on Sym(p, R). Tus, Z 1 and Z 2 represent te zero locii of teir respective polynomials, and tey are measure zero sets by Corollary A.3. To complete te proof, we need only to ceck surjectivity of and apply Teorem A.4. Recall te first definition of wit respect to te constant r and te variables γ, S, ˆx, and u, wic were later sortened to te inputs z and te data D. Omitting unnecessary subscripts from te original definition in (5), sligtly abusing notation, we ave (γ, S, ˆx i, u)=γi S s VQ(r q s, ˆx i ) j Q(r s, ˆx i )[u s ] j. j=1 Note tat, in tis notation, : R Sym ++ (p, R) R p+qn Sym(p, R). Since Sym ++ (p, R) is omeomorpic to R p(p+1)/2, witout loss of generality, we may write as a map from R a+b to R b wit a = 1 + p(p + 1)/2 + p + qn and b = p(p + 1)/2. If we write te derivative : R a+b R b as a b a matrix (γ, S, ˆx i, u), te partial derivatives corresponding to te vectorized version of S, i.e., S (γ, S, ˆx i, u) will be I b, so tat (γ, S, ˆx i, u) will be easily be rank b, wic is te required condition for surjectivity of. Proposition A.6 (Bounded Entries of + ). For any input z and data D, te function ψ ij (, z, D), defined in (11), representing te j-t entry of te vector + (z; D, i), is bounded for all i I and j {1,..., m + qn}. Proof. Recall te definition of ψ ij (, z, D): Sym(p, R) R, repeated ere for convenience, { Tr( j(z,d,i)x) X if X ψ ij (X, z, D) F F > 0 d else Note tat j (z, D, i) is a constant matrix for any j because is a linear function of z. To simplify notation, call tis matrix A ij. Let λ = [λ 1 λ p ] denote te vector of eigenvalues of X and v j denote te eigenvector corresponding to λ j. We can write X = p α=1 λ αv α vα. If X 0, ten λ 0 and ψ ij = TrA ijx (35a) TrX 2 = TrA ( p ij α=1 λ ) αv α vα p α=1 = λ αvα A ij v α. (35b) λ p α=1 λ2 α Let B ij diag(v1 A ij v 1,..., vp A ij v p ), and note tat ψ ij = 1 B ij λ/ λ. Note tat te limit of ψ ij (X, z, D) terefore exists for sequences {X k } 0. Wit respect to bounding te magnitude of ψ ij, we ave tat 1 B ij λ p ψ ij = B ij 1 = λ α=1 (v α A ij v α ) 2. Terefore, ψ ij is bounded. Additionally, ψ ij is maximized if {v α } represent a set of ortogonal eigenvectors of A ij, tat is, if X as te same eigenvectors as A ij. In tis case, te bound is tigt and equal to µij, were µij is te vector of eigenvalues of te matrix A ij j (z, D, i). It is left to sow tat A ij is bounded. Tis obvious wen we write out te individual terms of from (5). It can be seen tat if j m, ten A ij = I. Oterwise, it is te matrix of partial derivatives of Q, wic are bounded by Assumption IV.1. Corollary A.7 (Bound for δ s,k ). Let A ij j (z, D, i). Te constant L and te constant D, wic are bounds for +(z, D, i) and δ s,k, respectively, are given by L = µ µ2 m+qn, and D = 2L, were µ j max i I { µij µij } is te vector of eigenvalues of Aij.

12 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Proof. Te j-t entry of + is a restriction of ψ ij to a subset of its domain. Tus, bounds for ψ ij also old for te j-t entry of +. In te proof of Proposition A.6, we saw tat ψ ij is bounded by te 2-norm of te eigenvalues of A ij j (z, D, i). Terefore, L = µ 2 ij. Taking te norm of bot sides of (14), we ave tat δ s,k = + (z s,k ; D s,k, ω s,k ) + (z s,k ; D, ω s,k ). Tus, te j-t entry of δ s,k is bounded by 2 µ j and δ s,k. Taking te norm, we retrieve te result of te Corollary. Proposition A.8 (Bounded Derivatives of ψ ij ). Assume X Sym + (p, R). Ten, te magnitudes of te partial derivatives D ψ ij (X, z, D) and X ψ ij (X, z, D) are pointwise bounded for all i I and j {1,..., m + qn}. Proof. If X = 0, ten ψ ij is constant, and tus te magnitudes of its partial derivatives are zero. Te rest of te proof deals wit X > 0. To simplify notation, we refer to j (z, D, i) by te sortand A ij. Coose ɛ R suc tat X > ɛ > 0. Fix arbitrary i. Since X and A ij are symmetric, for any j we ave tat X ψ ij (X, z, D)= 1 X A ij TrAijX X and X 3 X ψ ij (X, z, D) 2 = X 4 ( A ij 2 X 2 (TrA ij X) 2 ), (36) wic is strictly less tan 1 for all X > A ij. For X A ij, ( A ij /ɛ) 4 is an upper bound for (36). For te term D ψ ij (X, z, D), note tat te only part of ψ ij tat depends on D is A ij. Tus by te cain rule, D ψ ij (X, z, D) = ( Aij ψ ij (X, z, D) ) ( D A ij ). Te first term is given by Aij ψ ij (X, z, D) = 1 X F I 1 ɛ I. For te second term, recall tat D = (ˆx, S). Split te partial derivative into two parts D = ( ˆx, S ) so tat D j (z, D, i) = ( ˆx j (z, D, i), S j (z, D, i)). For any j, S j (z, D, i) = I p, and obviously I p p. Similarly, j (z, D, i) = I p for j m. If j > m, ten ˆx j (z, D, i) results in second mixed partial derivatives of Q, wic we assumed to be bounded in Assumption IV.1. Proposition A.9 (Local Differentiability of Constraint Gradient). Consider an arbitrary z R m+qn and D = (ˆx, S) wit ˆx i R p and S i Sym + (p, R). Suppose (z; D, i) is nonsigular wit distinct eigenvalues. Ten, an open neigborood Ω containing (z; D, i) suc tat ψ ij (Π + ( ), z, D) Ω is differentiable. Proof. Notation Note tat f C (A, B) means tat te n-t derivative of f exists at all points in A for all n N. We also use te notation f Ω to denote te restriction of f to Ω. Rougly speaking, te proof uses a well known result [38] to sow tat, if (z; D, i) as distinct eigenvalues, ten for α = 1,..., p we can find an open neigborood Ω in wic tere are differentiable maps λ α C (Ω α, R) and v α C (Ω α, R p ) tat represent te α-t eigenvalueeigenvector pair of eac X Ω and agree wit te eigenvalues and eigenvectors of (z; D, i). Since ψ ij (Π + ( ), z, Σ) is well beaved in an open neigborood, it is not difficult to arrive at te desired result. Te details of te proof are as follows, and generalize te line of reasoning in [38]. Let ( λ α, v α ) be an eigenvalue-eigenvector pair of (z; D, i). Since (z; D, i) Sym(p, R), we know tat λ α is real valued and distinct. Consider te mapping φ: R R p Sym(p, R) R p+1 given by [ ] (λi X)v φ(λ, u, X) v. (37) v 1 By definition, φ(λ, u, X) = 0 if and only if (λ, v) are a (normalized) eigenvalue-eigenvector pair of X. It is also a well known properties of eigenvalues and eigenvectors tat (λ,u) φ ( λα, v α, (z; D, i) ) is full rank if and only if λ α is simple, wic is true by assumption. By te implicit function teorem, tere is an open neigborood Ω α of (z; D, i) in wic we can find differentiable functions λ α C (Ω α, R) and v α C (Ω α, R p ) suc tat λ α ( (z; D, i)) = λ α and v α ( (z; D, i)) = v α Let Ω + α = {X Ω α λ α (X) > 0}, and (38a) Ω α = {X Ω α λ α (X) < 0}. (38b) If λ α is positive for all α = 1,..., p, every Ω + α is an open set tat contains (z; D, i). Let Ω = α Ω + α, (39) wic is nonempty and open. Wen restricted to Ω, te mapping Π + is te identity map, and tus (at least for tis case) te proof follows if ψ ij (, z, D) Ω is differentiable. To verify tat it is, consider arbitrary X Ω. Recall again te definition of ψ ij (, z, D): Sym(p, R) R from (11). Note tat j (z, D, i) is a constant matrix because is a linear function of z. Call tis matrix A ij. We ave tat ψ ij (X, z, D) = Tr (A ijx) α = λ α(x)vα (X)A ij v α (X), X F α λ2 α(x) wic uses te same algebraic manipulation as te derivation of (35b). By inspection, ψ ij (, z, D) is differentiable at all X Ω, and te result follows. If λ α is negative for all α = 1,..., p, every Ω α is an open set tat contains (z; D, i). Let Ω = α Ω α, (40) wic is again open and nonempty. Note tat Π + projects every matrix X Ω to te matrix of zeros. Te map ψ ij (0, z, D) is tus constant w.r.t. X Ω and is tus differentiable. If (z; D, i) as some positive and negative eigenvalues, ten we can split tem into two groups. Let α index te positive eigenvalues and β denote te negative eigenvalues. Similar to before, let Ω = ( α Ω + ) ( ) α β Ω β, (41) wic is again nonempty, open, and contains (z; D, i). Note tat Π + does not cange λ α or v α for matrices X Ω, i.e., λ α (X) = λ α (Π + (X)), X Ω v α (X) = v α (Π + (X)), X Ω. (42a) (42b)

13 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Te eigenvalues corresponding to β of Π + (X), on te oter and, are always projected to zero. Tus we write tat λ β (Π + (X)) 0, and we ave, omitting some elementary algebra, tat ψ ij (Π + (X), z, D) = Tr (A ijπ + (X)) Π + (X) F = v 1 (X)A ij v 1 (X). By inspection, ψ ij (Π + ( ), z, D) is differentiable at X. Since X was an arbitrary matrix from te set Ω, te result follows. Since (z; D, i) was assumed to be nonsingular, all possibilities of combinations eigenvalues ave been covered, and, for eac case, we found an open set Ω in wic te desired result olds. Proof of Lemma V.6. Recall te definition δ s,k + (z s,k ; D s,k, ω s,k ) + (z s,k ; D, ω s,k ), were {ω s,k } k N I. Consider sequences of te form {i, i, i,... }. We will use te fact tat + (z s,k ; D s,k, i) + (z s,k ; D, i) k N δ s,k k N i I to sow tat tat δ s,k is summable, provided tat eac sequence + (z s,k ; D s,k, i) + (z s,k ; D, i) is summable. To sow tat + (z s,k ; D s,k, i) + (z s,k ; D, i) is summable, we will sow tat te sequence of te entries for an arbitrary entry j {1,..., m + qn} is summable. If tis is true, te norm of te wole vector will be summable as well because a b. if a b. Fix j {1,..., m + qn}. Case 1 If j m, ten tis entry corresponds to te variable from te original problem tat we denoted as γ j. Note tat, for tis coice of j, we ave tat { I p if i = j j (z, D, i) = (43) 0 else for all possible variables z or data D. Terefore, te j-t entry of + (z s,k ; D s,k, i) + (z s,k ; D, i), wic is te difference of two functions j + (z s,k, D s,k, i) j + (z s,k, D, i) = 0 (44) because tese functions are actually te same constant. Case 2 If j > m, ten we are looking at an entry of te gradient wit respect to te control variables u s, wic are non trivial. In tis case, recall from te definition in (11) tat j + (z, D, i) = ψ ij (Π + ( (z; D, i)), z, D). In oter words, we can look at j + as essentially a composition of tree functions: ψ ij, Π +, and. In te preceding lemmas and propositions, we ave establised useful facts about eac of tese constituent functions tat will enable us to complete te proof. Let X k = (z s,k ; D s,k, i) and Y k = (z s,k ; D, i), and note tat X k Y k because is continuous and D s,k D. By Proposition A.5, bot sequences {z s,k, D s,k } k N and {z s,k, D} k N almost surely lie in te preimage of te set of nonsingular matrices tat ave distinct eigenvalues wit respect to te mapping ( ;, i). Let Ω k be an open neigborood of Y k suc tat ψ ij (Π + ( ), z, D) Ωk is differentiable. Suc a neigborood is guaranteed to be available by Proposition A.9 as long as Y k converges, and te limit point is nonsingular and does not ave a simple spectrum. We know tat Y k converges by construction of te decreasing step sizes in Algoritm 1, and w.p.1. tis limit point will be nonsingular and ave a nonsimple spectrum by Proposition A.5. Ten, we can find K N suc tat l K, X l Ω l. By Proposition A.8, te derivatives of ψ ij (Π + (X), z, D) wit respect to bot X and D are bounded. Denote te larger bound by µ l. Proposition A.6 bounds ψ ij globally almost surely. We ave tat + (z s,k ; D s,k, i) + (z s,k ; D, i) = = ψ ij (Π + ( (z s,l ; D s,l, i)), z s,l, D s,l ) ψ ij (Π + ( (z s,l ; D, i)), z s,l, D) = ψ ij (Π + (X l ), z s,l, D s,l ) ψ ij (Π + (Y l ), z s,l, D) = X (ψ ij Π + )(Y l, z s,l, D), X l Y l Sym(p,R) + o( X l Y l ) X (ψ ij Π + )(Y l, z s,l, D) X l Y l + o ( X l Y l ) µ l X l Y l + o ( X l Y l ), were, Sym(p,R) denotes te matrix inner product. Te final term is summable if and only if X l Y l is summable. To verify tat it is, note tat X l Y l = (z s,k ; D s,k, i) (z s,k ; D, i) (45) wic is summable by Proposition A.1. APPENDIX B PROOF OF LEMMAS V.7-V.9 Proof of Lemma V.7. From Line 13 of Algoritm 1, te following cain of relations old for any feasible point z X X 0 a.s.: z s,k z 2 v s,k z β s,k +(v s,k ; D s,k, ω s,k ) 2 v s,k z 2 2β s,k +(v s,k ; D s,k, ω s,k ), v s,k z (46a) + β 2 s,k +(v s,k ; D s,k, ω s,k ) 2 (46b) v s,k z 2 2β s,k +(v s,k ; D, ω s,k ), v s,k z +2β s,k δ s,k v s,k z +β 2 s,k +(v s,k ; D s,k, ω s,k ) 2 v s,k z 2 2β s,k + (v s,k ; D, ω s,k ) + 2β s,k δ s,k v s,k z + β 2 s,k +(v s,k ; D s,k, ω s,k ) 2, (46c) (46d) were (46a) follows from te nonexpansiveness of te projection operator; (46b) follows from te expansion of 2 ; (46c) follows from relation (14) and te Scwarz inequality; and (46d) follows from te convexity of te function + ( ;, ω s,k ), and + (z; D, ω s,k ) = 0 due to te feasibility of te point z,

14 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE i.e., + (v s,k ; D, ω s,k ) +(v s,k ; D, ω s,k ), v s,k z. From relation (13) and te definition of β s,k in (12), we furter ave 2β s,k + (v s,k ; D, ω s,k ) + β 2 s,k +(v s,k ; D s,k, ω s,k ) 2 = 2β s,k + (v s,k ; D s,k, ω s,k ) + 2β s,k ν s,k + β 2 s,k +(v s,k ; D s,k, ω s,k ) 2 = 2β s,k ν s,k 2 +(v s,k ; D s,k, ω s,k ) +(v s,k ; D s,k, ω s,k ) 2 2β s,k ν s,k 2 +(v s,k ; D s,k, ω s,k ) (L + D) 2, were te last inequality follows from relation (18a). Combining tis wit inequality (46d), we obtain for any z X a.s.: z s,k z 2 v s,k z 2 + 2β s,k δ s,k v s,k z (47) + 2β s,k ν s,k 2 +(v s,k ; D s,k, ω s,k ) (L + D) 2. For te last term on te rigt-and side, we can rewrite + (v s,k ; D s,k, ω s,k ) = + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k ) + + (p s,k ; D s,k, ω s,k ). Terefore, 2 +(v s,k ; D s,k, ω s,k ) 2 + (p s,k ; D s,k, ω s,k ) ( + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k )) + 2 +(p s,k ; D s,k, ω s,k ). (48) Regarding te first term on te rigt-and side of (48), te following cain of relations olds: 2 + (p s,k ; D s,k, ω s,k ) ( + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k )) 2 + (p s,k ; D s,k, ω s,k ) + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k ) (49a) 2(L + D) v s,k p s,k + (p s,k ; D s,k, ω s,k ) 2α k (L + D)L f + (p s,k ; D s,k, ω s,k ) τα 2 k(l + D) 2 L 2 f 1 τ 2 +(p s,k ; D s,k, ω s,k ), (49b) (49c) (49d) were (49a) follows from te Scwarz inequality; (49b) follows from relation (18b); (49c) follows from Line 10 of Algoritm 1; and (49d) follows from using 2 a b τa τ b2 wit a = α k (L + D)L f and b = + (p s,k ; D s,k, ω s,k ), and τ > 0 is arbitrary. Using relations (48)-(49) in (47), we obtain for all z X a.s.: z s,k z 2 v s,k z 2 + 2β s,k δ s,k v s,k z (50) τ 1 + 2β s,k ν s,k τ(l + D) 2 2 +(p s,k ; D s,k, ω s,k ) + ταkl 2 2 f. Similarly to (46a)-(46d), from Line 10 of Algoritm 1, te following cain of relations old for any z X X 0 a.s.: v s,k z 2 p s,k z α k f (p s,k ) 2 (51a) p s,k z 2 2α k f (p s,k ), p s,k z + α 2 k f (p s,k ) 2 p s,k z 2 2α k (f(p s,k ) f(z)) + α 2 kl 2 f, (51b) (51c) were (51a) follows from te nonexpansive projection; (51b) follows from te expansion of 2 ; (51c) follows from te convexity of te function f and relation (16a). For te second term on te rigt-and side, we furter ave for any ž X : 2α k (f(p s,k ) f(z)) = 2α k (f(p s,k ) f(ž) + f(ž) f(z)) (52a) 2α k f (ž), p s,k ž 2α k (f(ž) f(z)) (52b) 2α k L f p s,k ž 2α k (f(ž) f(z)) (52c) 4ηαkL 2 2 f + 1 4η p s,k ž 2 2α k (f(ž) f(z)), (52d) were (52a) follows from adding and subtracting f(ž); (52b) follows from te convexity of te function f; (52c) follows from te Scwarz inequality; and (52d) follows from using 2 a b 4ηa η b2 wit a = α k L f and b = p s,k ž. Combining (52d) in (51c), we ave for any z, ž X X 0 a.s.: v s,k z 2 p s,k z 2 2α k (f(ž) f(z)) + 1 4η p s,k ž 2 + (1 + 4η)α 2 kl 2 f. Substituting tis inequality in relation (50), we obtain z s,k z 2 p s,k z 2 2α k (f(ž) f(z)) + 2β s,k δ s,k v s,k z + 2β s,k ν s,k + 1 4η p s,k ž 2 + A η,τ αkl 2 2 f τ 1 τ(l + D) 2 2 +(p s,k ; D s,k, ω s,k ), were A η,τ = 1+4η +τ. From relation (15), we ave v s,k z C z. Using tis and te upper estimate for β s,k in (21) for bounding te tree error terms completes te proof. Proof of Lemma V.8. We use Lemma V.7 wit ž = z = Π X [p s,k ]. Terefore, for any s S, and k 1, we obtain a.s.: z s,k Π X [p s,k ] 2 p s,k Π X [p s,k ] 2 (53) + 2 C z(c + N) c 2 δ s,k + 2 C + N c 2 ν s,k + 1 4η p s,k Π X [p s,k ] 2 + A η,τ αkl 2 2 f τ 1 τ(l + D) 2 2 +(p s,k ; D s,k, ω s,k ), were A η,τ = 1 + 4η + τ and η, τ > 0 are arbitrary. By te definition of te projection, we ave dist(p s,k, X ) = p s,k Π X [p s,k ] and dist(z s,k, X ) = z s,k Π X [z s,k ] z s,k Π X [p s,k ]. Upon substituting tese estimates in relation (53), we obtain dist 2 (z s,k, X ) dist 2 (p s,k, X ) (54) + 2 C z(c + N) c 2 δ s,k + 2 C + N c 2 ν s,k + 1 4η dist2 (p s,k, X ) + A η,τ αkl 2 2 f τ 1 τ(l + D) 2 2 +(p s,k ; D s,k, ω s,k ).

15 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE Taking te expectation conditioned on F k 1 and noting tat p s,k is fully determined by F k 1, we ave for any s S and k 1 a.s.: E [ dist 2 ] (z s,k, X ) F k 1 dist 2 (p s,k, X ) (55) +2 C z(c + N) c 2 E[ δ s,k F k 1 ]+2 C + N c 2 E[ ν s,k F k 1 ] + 1 4η dist2 (p s,k, X ) + A η,τ αkl 2 2 f τ 1 τ(l + D) 2 E [ 2 ] +(p s,k ; D s,k, ω s,k ) F k 1. We now coose τ = 4, η = κ(l + D) 2 and use Assumption IV.3 to yield E [ dist 2 ] (z s,k, X ) F k 1 dist 2 (p s,k, X ) +2 C z(c + N) c 2 E[ δ s,k F k 1 ]+ 2 C + N c 2 E[ ν s,k F k 1 ] 1 2κ(L + D) 2 dist2 (p s,k, X ) + Aα 2 kl 2 f, were A = 5 + 4κ(L + D) 2. Finally, by summing over all s S and using Lemma V.3 wit (x) = dist 2 (x, X ), we arrive at te following relation: E [ dist 2 ] (z s,k, X ) F k 1 dist 2 (z s,k 1, X ) (56) + 2 C z(c + N) E[ δ s,k F k 1 ] c C + N c 2 E[ ν s,k F k 1 ] 1 2κ(L + D) 2 dist 2 (p s,k, X ) + AαknL 2 2 f. Terefore, from k 0 α2 k < and Lemmas V.5 and V.6, all te conditions in te convergence teorem (Teorem V.2) are satisfied and te desired result follows. Proof of Lemma V.9(a). From Line 6-10 and 13 of Algoritm 1, we define e s,k z s,k p s,k for s S and k 0. Hence, e s,k can be viewed as te perturbation tat we make on p s,k after te network consensus step in Line 6 of Algoritm 1. Consider e s,k, for wic we can write e s,k z s,k v s,k + v s,k p s,k = ΠX0 [ vs,k β s,k +(v s,k ; D s,k, ω s,k ) ] v s,k + Π X0 [p s,k α k f (p s,k )] p s,k +(v s,k ; D s,k, ω s,k ) c + α k L f, (57) were te second inequality follows from fact tat p s,k, v s,k X 0, te nonexpansiveness of te projection operator, relations (16a) and (20). Applying (a + b) 2 2a 2 + 2b 2 in inequality (57), we ave for all s S and k 1, e s,k (v s,k ; D s,k, ω s,k ) c 2 + 2α 2 kl 2 f (58) For te term 2 +(v s,k ; D s,k, ω s,k ) in (58), we ave te following cain of relations: 2 +(v s,k ; D s,k, ω s,k ) = ( + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k )) 2 (59) + 2 +(p s,k ; D s,k, ω s,k ) (p s,k ; D s,k, ω s,k ) ( + (v s,k ; D s,k, ω s,k ) + (p s,k ; D s,k, ω s,k )) (L + D) 2 v s,k p s,k (p s,k ; D s,k, ω s,k ) (60) + 2(L + D) v s,k p s,k + (p s,k ; D s,k, ω s,k ) α 2 k(l + D) 2 L 2 f + 2 +(p s,k ; D s,k, ω s,k ) (61) + 2α k (L + D)L f + (p s,k ; D s,k, ω s,k ) 2α 2 k(l + D) 2 L 2 f + 2 +(p s,k ; D s,k, ω s,k ), (62) were (59) follows from adding and subtracting + (p s,k ; D s,k, ω s,k ); (60) follows from relation (18b); (61) follows from Line 10 of Algoritm 1; (62) follows from using te relation 2 a b a 2 + b 2 wit a = α k (L + D)L f and b = + (p s,k ; D s,k, ω s,k ). By combining (58) and (62), we obtain e s,k 2 2αkL 2 2 f (1 + 2(L + D) 2 ) c (p s,k ; D s,k, ω s,k ) c 2 2αkL 2 2 f (1 + 2(L + D) 2 ) c 2 + 2L2 dist2 (p s,k, X ) c 2, were te last inequality is from (17). Terefore, from Lemma V.8 and k=1 α2 k <, we conclude tat k=1 e s,k 2 < for all s S a.s. Proof of Lemma V.9(b). Te proof of tis part of Lemma will coincide wit [26, Lemma 6(b)]. REFERENCES [1] T. H. Cung, J. W. Burdick, and R. M. Murray, A decentralized motion coordination strategy for dynamic target tracking, in IEEE Int. Conf. on Robotics and Automation (ICRA), 2006, pp [2] P. Jalalkamali and R. Olfati-Saber, Information-driven self-deployment and dynamic sensor coverage for mobile sensor networks, in IEEE American Control Conf. (ACC), 2012, pp [3] C. Freundlic, P. Mordoai, and M. M. Zavlanos, Hybrid control for mobile target localization wit stereo vision, in IEEE Conf. on Decision and Control (CDC), 2013, pp [4], Optimal pat planning and resource allocation for active target localization, in IEEE American Control Conf. (ACC), 2015, pp [5], A ybrid control approac to te next-best-view problem using stereo vision, in IEEE Int. Conf. on Robotics and Automation (ICRA), 2013, pp [6] J. Vander Hook, P. Tokekar, and V. Isler, Algoritms for cooperative active localization of static targets wit mobile bearing sensors under communication constraints, IEEE Trans. on Robotics, vol. 31, no. 4, pp , [7] A. Simonetto and T. Keviczky, Distributed multi-target tracking via mobile robotic networks: a localized non-iterative SDP approac, in IEEE Conf. on Decision and Control (CDC), 2011, pp [8] J. Derenick, J. Spletzer, and A. Hsie, An optimal approac to collaborative target tracking wit performance guarantees, J. Intelligent and Robotic Systems, vol. 56, no. 1-2, pp , [9] A. T. Kamal, J. Farrell, and A. K. Roy-Cowdury, Information weigted consensus, in IEEE Conf. on Decision and Control (CDC), 2012, pp [10] A. Nedić and A. Olsevsky, Distributed optimization over time-varying directed graps, IEEE Transactions on Automatic Control, vol. 60, no. 3, pp , 2015.

16 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE 2017 [11] B. Joansson, M. Rabi, and M. Joansson, A randomized incremental subgradient metod for distributed optimization in networked systems, J. Optimization, vol. 20, no. 3, pp , [12] H. Jamali-Rad, A. Simonetto, X. Ma, and G. Leus, Distributed sparsityaware sensor selection, IEEE Trans. on Signal Processing, vol. 63, no. 22, pp , [13] A. Simonetto and H. Jamali-Rad, Primal recovery from consensusbased dual decomposition for distributed convex optimization, vol. 168, no. 1, pp , [14] E. D. Nerurkar, K. J. Wu, and S. I. Roumeliotis, C-KLAM: Constrained keyframe-based localization and mapping, in IEEE Int. Conf. on Robotics and Automation (ICRA), 2014, pp [15] G. Huang, K. Zou, N. Trawny, and S. I. Roumeliotis, A bank of maximum a posteriori (MAP) estimators for target tracking, IEEE Trans. on Robotics, vol. 31, no. 1, pp , [16] B. J. Julian, M. Angermann, M. Scwager, and D. Rus, Distributed robotic sensor networks: An information-teoretic approac, Int. J. Robotics Researc, vol. 31, no. 10, pp , [17] M. Patan, Optimal sensor networks sceduling in identification of distributed parameter systems. Springer Science & Business Media, 2012, vol [18] R. Kodayi-mer, W. Aquino, and M. M. Zavlanos, Model-based sparse source identification, in IEEE American Control Conf. (ACC). [19] S. Josi and S. Boyd, Sensor selection via convex optimization, IEEE Trans. on Signal Processing, vol. 57, no. 2, pp , [20] J. Ranieri, A. Cebira, and M. Vetterli, Near-optimal sensor placement for linear inverse problems, IEEE Trans. on Signal Processing, vol. 62, no. 5, pp , [21] S. P. Cepuri and G. Leus, Compression scemes for time-varying sparse signals, in 48t Asilomar Conf. on Signals, Systems and Computers. IEEE, 2014, pp [22] M. Patan, Decentralized mobile sensor routing for parameter estimation of distributed systems, IFAC Proceedings Volumes, vol. 42, no. 20, pp , [23] M. T. Spaan, T. S. Veiga, and P. U. Lima, Decision-teoretic planning under uncertainty wit information rewards for active cooperative perception, Int. Found. for Autonomous Agents and Multiagent Sys. (IFAAMAS), pp. 1 29, [24] S. Omidsafiei, A.-a. Aga-moammadi, C. Amato, and J. P. How, Decentralized control of partially observable markov decision processes using belief space macro-actions, in IEEE Int. Conf. on Robotics and Automation (ICRA), [25] S. Lee and A. Nedic, Distributed random projection algoritm for convex optimization, IEEE J. Selected Topics in Signal Processing, vol. 7, pp , April [26] S. Lee and M. Zavlanos, Approximate projections for decentralized optimization wit SDP constraints, 2015, ttp://arxiv.org/abs/ [27] B. Polyak, Minimization of unsmoot functionals, USSR Computational Matematics and Matematical Pysics, vol. 9, no. 3, pp , [28] F. Faccinei and J.-S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York, [29] A. Lewis and J.-S. Pang, Error bounds for convex inequality systems, in Generalized Convexity. Kluwer Academic Publisers, 1996, pp [30] H. H. Bauscke and J. M. Borwein, On projection algoritms for solving convex feasibility problems, SIAM Rev., vol. 38, no. 3, pp , Sep [31] L. Gubin, B. Polyak, and E. Raik, Te metod of projections for finding te common point of convex sets, USSR Computational Matematics and Matematical Pysics, vol. 7, no. 6, pp. 1 24, [32] D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex analysis and optimization. Atena Scientific, [33] B. Polyak, Intro. to optimization. Optimization software, Inc., Publications division, New York, [34] S. S. Ram, A. Nedic, and V. V. Veeravalli, A new class of distributed optimization algoritms: application to regression of distributed data, Optimization Metods and Software, vol. 27, no. 1, pp , [35] B. M. M. Zavlanos, M. B. Egerstedt, and G. J. Pappas, Grap-teoretic connectivity control of mobile robot networks, Proceedings of te IEEE, vol. 99, no. 9, pp , [36] A. Nedic, A. Olsevsky, A. Ozdaglar, and J. Tsitsiklis, Distributed subgradient metods and quantization effects, in Proceedings of 47t IEEE Conference on Decision and Control, December 2008, pp [37] J. E. Marsden and M. J. Hoffman, Elementary classical analysis. Macmillan, [38] J. R. Magnus, On differentiating eigenvalues and eigenvectors, Econometric Teory, vol. 1, no. 2, pp , Carles Freundlic (S 13) received a B.A. in pysics from Middlebury College (2010), an M.Eng. in mecanical engineering from Stevens Institute of Tecnology (2013), and a PD in mecanical engineering from Duke University (2017), were e studied control teory, differential geometry, and computer vision. Currently, e is a Sr. Software Engineer at Tesla. He currently works on cooperative indoor SLAM for mobile cameras and large-scale supply cain estimation and control. Soomin Lee (S 07 M 13) is a Researc Engineer at Georgia Institute of Tecnology. Se was formerly a Postdoctoral Associate in Mecanical Engineering and Materials Science at Duke University. Se received er P.D. in Electrical and Computer Engineering from te University of Illinois, UrbanaCampaign (2013). Se received two master s degrees from te Korea Advanced Institute of Science and Tecnology in Electrical Engineering, and from te University of Illinois at Urbana-Campaign in Computer Science. In 2009, se was an assistant researc officer at te Advanced Digital Science Center (ADSC) in Singapore. Her interests include teoretical optimization (convex, non-convex, online and stocastic), distributed control and optimization of dynamic networks, umanrobot interactions and macine learning. Micael Zavlanos (S 05 M 09) received te Diploma in mecanical engineering from te National Tecnical University of Atens (NTUA), Atens, Greece, in 2002, and te M.S.E. and PD degrees in Electrical and Systems Engineering from te University of Pennsylvania, Piladelpia, PA, in 2005 and 2008, respectively. From 2008 to 2009 e was a Post-Doctoral Researcer in te Dept. of Electrical and Systems Engineering at te University of Pennsylvania. He ten joined te Stevens Institute of Tecnology, Hoboken, NJ, as an Assistant Professor of Mecanical Engineering, were e remained until Currently, e is an assistant professor of Mecanical Engineering and Materials Science at Duke University, Duram, NC. He also olds a secondary appointment in te Dept. of Electrical and Computer Engineering. His researc interests include a wide range of topics in te emerging discipline of networked systems, wit applications in robotic, sensor, communication, and biomolecular networks. He is particularly interested in ybrid solution tecniques, on te interface of control teory, distributed optimization, estimation, and networking. Dr. Zavlanos is a recipient of te 2014 Office of Naval Researc Young Investigator Program (YIP) Award, te 2011 National Science Foundation Faculty Early Career Development (CAREER) Award, and a finalist for te Best Student Paper Award at CDC 2006.