convergence results for the linear consensus problem under markovian random graphs this paper discusses the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph the switching is determined by a finite state markov process each topology corresponding to a state of the process we address the cases where the dynamics of the agents is expressed both in continuous time and in discrete time we show that if the consensus matrices are doubly stochastic average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the markov process is strongly connected the aim of this paper is to show how techniques from the theory of markovian jump linear systems in conjunction with results inspired by matrix and graph theory can be used to prove convergence results for stochastic consensus problems

convergence of consensus models with stochastic disturbances we consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge almost surely to a consensus let a t b t rn n be possibly stochastic nonstationary matrices and x t m t 6 rn 1 be state and perturbation vectors respectively for any consensus algorithm of the form x t 1 a t x t b t m t we provide conditions under which consensus is achieved almost surely i e pr limt x t c1 1 for some c r moreover we show that this general result subsumes recently reported results for specific consensus algorithms classes including sum preserving nonsum preserving quantized and noisy gossip algorithms also provided are the e converging time for any such converging iterative algorithm i e the earliest time at which the vector x t is e close to consensus and sufficient conditions for convergence in expectation to the average of the initial node measurements finally mean square error bounds of any consensus algorithm of the form discussed above are presented

mean field lqg control in leader follower stochastic multi agent systems likelihood ratio based adaptation we study large population leader follower stochastic multi agent systems where the agents have linear stochastic dynamics and are coupled via their quadratic cost functions the cost of each leader is based on a trade off between moving toward a certain reference trajectory which is unknown to the followers and staying near their own centroid on the other hand followers react by tracking a convex combination of their own centroid and the centroid of the leaders we approach this large population dynamic game problem by use of so called mean field mf linear quadratic gaussian lqg stochastic control theory in this model followers are adaptive in the sense that they use a likelihood ratio estimator on a sample population of the leaders trajectories to identify the member of a given finite class of models which is generating the reference trajectory of the leaders under appropriate conditions it is shown that the true reference trajectory model is identified by each follower in finite time with probability one as the leaders population goes to infinity furthermore we show that the resulting sets of mean field control laws for both leaders and adaptive followers possess an almost sure en nash equilibrium property for a system with population n where en goes to zero as n goes to infinity numerical experiments are presented illustrating the results

feedback control of linear multivariable systems with uncertain description in the frequency domain abstract the problem of using approximate reduced order models for control systems design is investigated assuming that both plant and simplified model are driven by the same feedback controller the plant output uncertainty range with respect to the output of the simplified model is analysed and expressed in terms of the given open loop modelling uncertainty and of the frequency domain response of the designed controller the analysis method is based on evaluation of a suitable set of characteristic functions on a standard nyquist d contour therefore it leads to the formulation of additional requirements to be directly included in the set of specifications for the control problem when it is approached by means of frequency domain design techniques in the case where the control design is developed by using time domain techniques it is shown that the well known separation theorem may also allow treatment of the uncertainty reducibility problem in two stages

network based consensus averaging with general noisy channels this paper focuses on the consensus averaging problem on graphs under general imperfect communications we study a particular class of distributed consensus algorithms based on damped updates and using the ordinary differential equation method we prove that the updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes the convergence is not asymptotic in the size of the network our analysis applies to various types of stochastic disturbances to the updates including errors in update calculations dithered quantization and imperfect data exchange among nodes under a suitable stability condition we prove that the error between estimated and true averages is asymptotically gaussian and we show how the asymptotic covariance is specified by the graph laplacian for additive perturbations we show how the scaling of the asymptotic mse is controlled by the spectral gap of the laplacian

stochastic consensus seeking with noisy and directed inter agent communication fixed and randomly varying topologies we consider consensus seeking of networked agents on directed graphs where each agent has only noisy measurements of its neighbors states stochastic approximation type algorithms are employed so that the individual states converge both in mean square and almost surely to the same limit we further generalize the algorithm to networks with random link failures and prove convergence results

stochastic consensus over noisy networks with markovian and arbitrary switches this paper considers stochastic consensus problems over lossy wireless networks we first propose a measurement model with a random link gain additive noise and markovian lossy signal reception which captures uncertain operational conditions of practical networks for consensus seeking we apply stochastic approximation and derive a markovian mode dependent recursive algorithm mean square and almost sure i e probability one convergence analysis is developed via a state space decomposition approach when the coefficient matrix in the algorithm satisfies a zero row and column sum condition subsequently we consider a model with arbitrary random switching and a common stochastic lyapunov function technique is used to prove convergence finally our method is applied to models with heterogeneous quantizers and packet losses and convergence results are proved

consensus conditions of multi agent systems with time varying topologies and stochastic communication noises this paper investigates the average consensus problem of first order discrete time multi agent networks in uncertain communication environments each agent can only use its own and neighbors information to design its control input to attenuate the communication noises a distributed stochastic approximation type protocol is used by using probability limit theory and algebraic graph theory consensus conditions for this kind of protocols are obtained a for the case of fixed topologies a necessary and sufficient condition for mean square average consensus is given which is also sufficient for almost sure consensus b for the case of time varying topologies sufficient conditions for mean square average consensus and almost sure consensus are given respectively especially if the network switches between jointly containing spanning tree instantaneously balanced graphs then the designed protocol can guarantee that each individual state converges both almost surely and in mean square to a common random variable whose expectation is right the average of the initial states of the whole system and whose variance describes the static maximum mean square error between each individual state and the average of the initial states of the whole system

asymptotic properties of consensus type algorithms for networked systems with regime switching topologies this paper is concerned with asymptotic properties of consensus type algorithms for networked systems whose topologies switch randomly the regime switching process is modeled as a discrete time markov chain with a finite state space the consensus control is achieved by using stochastic approximation methods in the setup the regime switching process the markov chain contains a rate parameter e 0 in the transition probability matrix that characterizes how frequently the topology switches on the other hand the consensus control algorithm uses a stepsize m that defines how fast the network states are updated depending on their relative values three distinct scenarios emerge under suitable conditions we show that when 0 e o m a continuous time interpolation of the iterates converges weakly to a system of randomly switching ordinary differential equations modulated by a continuous time markov chain in this case a scaled sequence of tracking errors converges to a system of switching diffusion when 0 email protected email protected the network topology is almost non switching during consensus control transient intervals and hence the limit dynamic system is simply an autonomous differential equation when email protected email protected the markov chain acts as a fast varying noise and only its averaged network matrices are relevant resulting in a limit differential equation that is an average with respect to the stationary measure of the markov chain simulation results are presented to demonstrate these findings

stochastic approximation approach for consensus and convergence rate analysis of multiagent systems in this note we study the consensus problem for multiagent systems with measurement noises different from the existing approach the consensus problem is converted to a root finding problem for which the stochastic approximation theory can be applied by choosing an appropriate regression function we propose a consensus algorithm which is applicable to systems with more general measurement noise processes including stationary autoregressive and moving average arma processes and infinite moving average ma processes further we establish a relationship between the convergence rate and the exponent of the step size of the algorithm particularly strong convergence rate for systems with a leader follower topology is studied

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

distributed linear parameter estimation asymptotically efficient adaptive strategies this paper considers the problem of distributed adaptive linear parameter estimation in multiagent inference networks local sensing model information is only partially available at the agents and interagent communication is assumed to be unpredictable the paper develops a generic mixed time scale stochastic procedure consisting of simultaneous distributed learning and estimation in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly under rather weak assumptions on the statistical model and the interagent communication it is shown that by properly tuning the consensus potential with respect to the innovation potential the asymptotic information rate loss incurred in the learning process may be made negligible as such it is shown that the agent estimates are asymptotically efficient in that their asymptotic covariance coincides with that of a centralized estimator the inverse of the centralized fisher information rate for gaussian

broadcast based consensus with non zero mean stochastic perturbations bolstered by the growing interest in building wireless sensor and ad hoc networks with applications ranging across different engineering disciplines distributed consensus algorithms have recently seen a new revival since their inception in the early 1980s of particular interest is the recently developed broadcast based consensus algorithm which is one special type of randomized consensus algorithms and is amenable to practical implementation in wireless networks this paper focuses on the performance analysis of this broadcast based consensus algorithm in the presence of non zero mean stochastic perturbations it is demonstrated that as the algorithm proceeds the deviation of the node states from their average will converge in expectation to a fixed value which is determined by the laplacian matrix of the network the mixing parameter and the mean of the stochastic perturbations asymptotic upper and lower bounds on the total mean square deviation are derived which describe the range of distances over which the node states deviate from consensus these bounds can facilitate evaluation of the applicability of this algorithm in practice results are also provided on the algorithm s e converging time i e the earliest time at which the deviation is e close to its steady value and on the mean and mean square behaviors of the displacement of node states from their initial states at large iteration number as a special case study performance of the broadcast based consensus algorithm under zero mean stochastic disturbances is analyzed and results regarding its convergence mean square deviation and mean square displacement are given the theoretical results presented in this study hold true regardless of the statistics of the stochastic disturbances and are valid for arbitrary network topology as long as the topology is connected

stochastic approximation for consensus a new approach via ergodic backward products this paper considers both synchronous and asynchronous consensus algorithms with noisy measurements for stochastic approximation based consensus algorithms the existing convergence analysis with dynamic topologies heavily relies on quadratic lyapunov functions whose existence however may be difficult to ensure for switching directed graphs our main contribution is to introduce a new analytical approach we first show a fundamental role of ergodic backward products for mean square consensus in algorithms with additive noise subsequently we develop ergodicity results for backward products of degenerating stochastic matrices converging to a 0 1 matrix via a discrete time dynamical system approach these results complement the existing ergodic theory of stochastic matrices and provide an effective tool for analyzing consensus problems under a joint connectivity assumption our approach may deal with switching topologies delayed measurements and correlated noises and it does not require the double stochasticity condition typically assumed for the existence of quadratic lyapunov functions

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

convergence rates of discrete time stochastic approximation consensus algorithms graph related limit bounds abstract in this paper we study the convergence rates of the discrete time stochastic approximation consensus algorithms over sensor networks with communication noises under general digraphs basic results of stochastic analysis and algebraic graph theory are used to investigate the dynamics of the consensus error and the mean square and sample path convergence rates of the consensus error are both given in terms of the graph and noise parameters especially calculation methods to estimate the mean square limit bounds are presented under balanced digraphs and sufficient conditions on the network topology and the step sizes are given to achieve the fast convergence rate for the sample path limit bounds estimation methods are also presented under undirected graphs

stochastic approximation based consensus dynamics over markovian networks this paper considers consensus problems with random networks a key object of our analysis is a sequence of stochastic matrices which involve markovian switches and decreasing step sizes we establish ergodicity of the backward products of these stochastic matrices the basic technique is to consider the second moment dynamics of an associated markovian jump linear system and exploit its two scale interaction property resulting from the decreasing step sizes the mean square convergence rate of the backward products is also obtained the ergodicity results are used to prove mean square consensus of stochastic approximation algorithms where agents collect noisy information the approach is further applied to a token scheduled averaging model

robustness tests for feedback control systems using multidimensional uncertainty bounds a method is presented for proving the robustness of feedback controllers with respect to the stability requirement and design specifications on the i o behaviour of the closed loop systems the model uncertainties are described by multidimensional upper bounds hence the results are less conservative than those obtained by the singular value approach

decentralized parameter estimation by consensus based stochastic approximation in this paper an algorithm for decentralized multi agent estimation of parameters in linear discrete time regression models is proposed in the form of a combination of local stochastic approximation algorithms and a global consensus strategy an analysis of the asymptotic properties of the proposed algorithm is presented taking into account both the multi agent network structure and the probabilities of getting local measurements and implementing exchange of inter agent messages in the case of non vanishing gains in the stochastic approximation algorithms an asymptotic estimation error covariance matrix bound is defined as the solution of a lyapunov like matrix equation in the case of asymptotically vanishing gains the mean square convergence is proved and the rate of convergence estimated in the discussion the problem of additive communication noise is treated in a methodologically consistent way it is also demonstrated how the consensus scheme in the algorithm can contribute to the overall reduction of measurement noise influence some simulation results illustrate the obtained theoretical results