asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

exponential convergence of a distributed algorithm for solving linear algebraic equations abstract in a recent paper a distributed algorithm was proposed for solving linear algebraic equations of the form a x b assuming that the equation has at least one solution the equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix a b the current estimates of the equation s solution generated by each of its neighbors and nothing more neighbor relationships are represented by a time dependent directed graph n t whose vertices correspond to agents and whose arcs characterize neighbor relationships sufficient conditions on n t were derived under which the algorithm can cause all agents estimates to converge exponentially fast to the same solution to a x b these conditions were also shown to be necessary for exponential convergence provided the data about a b available to the agents is non redundant the aim of this paper is to relax this non redundant assumption this is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand the conditions are based on a new notion of graph connectivity an improved bound on the convergence rate is also derived

distributed linear estimation over sensor networks we consider a network of sensors in which each node may collect noisy linear measurements of some unknown parameter in this context we study a distributed consensus diffusion scheme that relies only on bidirectional communication among neighbour nodes nodes that can communicate and exchange data and allows every node to compute an estimate of the unknown parameter that asymptotically converges to the true parameter at each time iteration a measurement update and a spatial diffusion phase are performed across the network and a local least squares estimate is computed at each node the proposed scheme allows one to consider networks with dynamically changing communication topology and it is robust to unreliable communication links and failures in measuring nodes we show that under suitable hypotheses all the local estimates converge to the true parameter value

jordan product maps commuting with the λ aluthge transform abstract let b h be the algebra of all bounded linear operators acting on a hilbert space h the main purpose in this paper is to obtain a characterization of bijective maps b h b k k hilbert space satisfying the following condition a b a b for all a b b h where t stands the aluthge transform of the operator t b h and a b 1 2 a b b a is the jordan product of a and b we prove that a bijective map satisfies the above condition if and only if there exists a unitary linear bounded operator u h k such that has the form a u a u for all a b h

reaching a consensus in a dynamically changing environment convergence rates measurement delays and asynchronous events this paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied vicsek consensus problem the paper also uses graph theoretic constructions to solve modified versions of the vicsek problem in which there are measurement delays asynchronous events or a group leader in all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous delay free leaderless case

observer design for tracking consensus in second order multi agent systems fractional order less than two this technical note studies the leader following tracking consensus problem of a class of multi agent systems where the dynamics of the leader is described by second order systems in order to track the states of the leader observers for the followers are designed by fractional order multi agent systems where the relative velocity information is unavailable it is interestingly found that the followers can track the leader with second order dynamics even if the fractional order is less than two by only using the position information of the local neighbors which is different from the existing results a novel fractional order observer is first proposed whose order is surprisingly less than the original leader system it is also shown that leader following consensus can be ensured if some carefully selected followers are informed and the relative position based protocols are appropriately designed with fractional order being between one and two a necessary and sufficient condition for the leader following consensus in multi agent systems without control input delay is proposed the results are then extended to the case with constant control input delay it is found that in both cases the real and imaginary parts of the eigenvalues of the augmented laplacian matrix of the topology play an important role in achieving consensus

distributed impulsive group consensus in second order multi agent systems under directed topology this paper investigates the impulsive group consensus problems of second order multi agent systems under directed network topology with acyclic partition two distributed impulsive sampled data protocols are presented to realise group consensus for both cases with leaders and leaderless and then some simple yet generic criteria on convergence for such algorithms are established it is shown that for the case with leaderless a feature of the present protocols is to give an explicit expression of group consensus states in terms of the initial values of the agents which allows us to develop a simple approach yielding the desired group or cluster consensus while for the case of virtual leaders a pinning like tracking technique is effectively designed to regulate all the agents access to the virtual leaders respectively for the purpose of practical control strategy finally simulation results are presented to illustrate the effectiveness of the theoretical analysis

product commuting maps with the λ aluthge transform let h and k be two hilbert spaces and b h be the algebra of all bounded linear operators from h into itself the main purpose of this paper is to obtain a characterization of bijective maps b h b k satisfying the following condition a b ab f orall a b b h where t stands the aluthge transform of the operator t b h more precisely we prove that a bijective map satisfies the above condition if and only if a u au for all a b h for some unitary operator u h k

asynchronous rendezvous analysis via set valued consensus theory this paper presents the design and analysis result for asynchronous rendezvous control of multiagent systems with continuous time dynamics and intermittent interactions the protocol designing strategies only impose weak restrictions on anticipated way point sets from which the way points are selected and path planning of each agent and can be applied to the networks of arbitrary dimensional subsystems explicitly the anticipated way point sets are in a polytope like form and the path between any two consecutive way points is required to be included within the minimum convex region covering the two associated anticipated way point sets under the assumption of directed and switching interaction topology and the assumption of intermittent and asynchronous interactions with time varying delays we perform the set valued consensus analysis on the evolution of anticipated way point sets with respect to update times and provide mild sufficient conditions for the solvability of the asynchronous rendezvous problem the proof techniques rely much on graph theory and nonnegative matrix theory the obtained result extends greatly the existing work in the literature and several examples demonstrate its broad potential applications particularly additional distributed control rules different from the circumcenter algorithm are devised for network connectivity maintenance

on the optimal parameter of the composite laplacian quadratics function abstract recently the composite quadratic lyapunov function has been extended to study of multi agent systems leading to the so called composite laplacian quadratics clqs function compared with quadratic lyapunov functions the clqs function can yield a larger convergence region and is particularly useful in stabilization of multi agent systems with complex dynamics such as differential inclusions in the definition of the clqs function an optimal vector parameter plays a critical role in determining the value of the clqs function and in constructing stabilization laws derived from the clqs function this paper focuses on the properties of the optimal parameter of the clqs function the uniqueness of the optimal parameter is established a distributed computation approach is further proposed which is useful in computing the optimal parameter the robustness issue of the optimal parameter is also investigated for a multi agent system described by linear differential inclusions finally a numerical example is provided to validate the proposed theoretical results

on the multichannel rendezvous problem fundamental limits optimal hopping sequences and bounded time to rendezvous one of the fundamental problems in a cognitive radio network known as the multichannel rendezvous problem is for two secondary users to find a common channel that is not blocked by primary users the basic idea for solving such a problem in most works in the literature is for the two users to select their own channel hopping sequences and then rendezvous when they both hop to a common unblocked channel at the same time in this paper we focus on the fundamental limits of the multichannel rendezvous problem and formulate such a problem as a constrained optimization problem where the selection of the random hopping sequences of the two secondary users must satisfy certain constraints we derive various lower bounds for the expected respectively maximum time to rendezvous under certain constraints for some of these lower bounds we are also able to construct optimal channel hopping sequences that achieve the lower bounds inspired by the constructions of quorum systems and relative difference sets our constructions of the channel hopping sequences are based on the mathematical theories of finite projective planes orthogonal latin squares and sawtooth sequences the use of such theories in the constructions of channel hopping sequences appear to be new and better than other existing schemes in terms of minimizing the expected respectively maximum time to rendezvous

block eigenvalues of block compound matrices abstract we obtain some results about the block eigenvalues of block compound matrices and additive block compound matrices assuming that a certain block vandermonde matrix is nonsingular we generalize known results for scalar compound and additive compound matrices

randomized optimal consensus of multi agent systems in this paper we formulate and solve a randomized optimal consensus problem for multi agent systems with stochastically time varying interconnection topology the considered multi agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min z r email protected i 1 nf i z in which the optimal solution set of objective function f i can only be observed by agent i itself at each time step simply determined by a bernoulli trial each agent independently and randomly chooses either taking an average among its neighbor set or projecting onto the optimal solution set of its own optimization component both directed and bidirectional communication graphs are studied connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions the convergence analysis is carried out using convex analysis we compare the randomized algorithm with the deterministic one via a numerical example the results illustrate that a group of autonomous agents can reach an optimal opinion by each node simply making a randomized trade off between following its neighbors or sticking to its own opinion at each time step

coordination and geometric optimization via distributed dynamical systems this paper discusses dynamical systems for disk covering and sphere packing problems we present facility location functions from geometric optimization and characterize their differentiable properties we design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems the resulting dynamical systems promise to be of use in coordination problems for networked robots in this setting the distributed control laws correspond to local interactions between the robots the technical approach relies on concepts from computational geometry nonsmooth analysis and the dynamical system approach to algorithms

network flows that solve linear equations we study distributed network flows as solvers in continuous time for the linear algebraic equation mathbf z mathbf h mathbf y each node i has access to a row mathbf h i mathrm t of the matrix mathbf h and the corresponding entry z i in the vector mathbf z the first consensus projection flow under investigation consists of two terms one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the mathbf h i and z i the second projection consensus flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback without dwell time assumption on switching graphs we prove that all node states converge to a common solution of the linear algebraic equation if there is any the convergence is global for the consensus projection flow while local for the projection consensus flow in the sense that the initial values must lie on the affine subspaces if the linear equation has no exact solutions we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the consensus projection flow for a fixed bidirectional graph semi global convergence to approximate least squares solutions is also demonstrated for switching balanced directed graphs under suitable conditions it is also shown that the projection consensus flow drives the average of the node states to the least squares solution with a complete graph numerical examples are provided as illustrations of the established results

global bounded synchronization of general dynamical networks with nonidentical nodes this note addresses the problem of synchronization for general dynamical networks with nonidentical nodes the coupling strength outer coupling configuration and inner connection in such networks are all time varying neither an equilibrium for each node nor a synchronization manifold is assumed to exist an estimate of the convergence domain for a general class of time varying nonlinear systems is given by introducing the average dynamics of all nodes and based on this estimate a criterion of global synchronization in the sense of boundedness of the maximum state deviation between nodes is developed an explicit bound of the maximum state deviation between nodes is obtained by the maximum difference between each node dynamics and the average dynamics the proposed criterion is an extension of several related synchronization criteria for the case of identical nodes to the case of nonidentical nodes

the deformed consensus protocol this paper studies a generalization of the standard continuous time consensus protocol obtained by replacing the laplacian matrix of the communication graph with the so called deformed laplacian the deformed laplacian is a second degree matrix polynomial in the real variable s which reduces to the standard laplacian for s equal to unity the stability properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected and directed graphs and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems examples and simulation results are provided to illustrate our theoretical findings

on the solutions of tridiagonal linear systems tridiagonal linear systems are of special importance being appeared in space researches and also in several well known algorithms there are many methods for solving a tridiagonal linear system most of them are suffering from the accumulation of the round off errors in this paper a new method will be given to overcome the defects of the other methods also two direct methods for solving a system of linear equations having matrices of quaternion entries as coefficients and independent terms are proposed second kind chebyshev polynomials in a matrix argument are used as a tool to do certain calculations

homogeneous networks of non introspective agents under external disturbances h almost synchronization this paper addresses the problem of h almost synchronization for homogeneous networks of general linear agents subject to external disturbances and under directional communication links agents are presumed to be non introspective i e agents are not aware of their own states or outputs and the only available information for each agent is a network measurement that is a linear combination of relative outputs under a certain set of conditions a family of dynamic protocols is developed such that the impact of disturbances on the synchronization error dynamics expressed in terms of the h norm of the corresponding closed loop transfer function is reduced to any arbitrarily small value

synchronization in complex networks with stochastically switching coupling structures synchronization in complex networks with time dependent coupling and stochastically switching coupling structure is discussed a novel approach investigating synchronization based on the scramblingness property of the coupling matrix is proposed some sufficient condition for a network with general time varying coupling structure to reach complete synchronization is provided based on the general theorem networks with stochastically switching coupling structures is investigated in particular two kinds of stochastic switching coupling networks are addressed a independent and identically distributed switching processes and b markov jump processes in both cases some sufficient condition for almost sure synchronization of the networks is given also numerical simulations are provided to illustrate the theoretical results

consensus of multiagent systems with distance dependent communication networks in this paper we study the consensus problem of discrete time and continuous time multiagent systems with distance dependent communication networks respectively the communication weight between any two agents is assumed to be a nonincreasing function of their distance first we consider the networks with fixed connectivity in this case the interaction between adjacent agents always exists but the influence could possibly become negligible if the distance is long enough we show that consensus can be reached under arbitrary initial states if the decay rate of the communication weight is less than a given bound second we study the networks with distance dependent connectivity it is assumed that any two agents interact with each other if and only if their distance does not exceed a fixed range with the validity of some conditions related to the property of the initial communication graph we prove that consensus can be achieved asymptotically third we present some applications of the main results to opinion consensus problems and formation control problems finally several simulation examples are presented to illustrate the effectiveness of the theoretical findings

h consensus of second order multi agent systems with asymmetric delays abstract this paper is concerned with the problem of consensus in the h sense for second order continuous time multi agent systems with multiple asymmetric time varying delays by using a model transformation approach and matrix theory we establish several conditions in terms of linear matrix inequalities such that consensus of multi agent systems can be achieved in the h sense the feasibility of the consensus conditions is also analyzed as an application we consider the case of intermittent measurement between agents numerical examples are presented to illustrate the theoretical results which can be applied to the case of negative information weights

brief paper observer design via immersion and invariance for vision based leader follower formation control the paper introduces a new vision based range estimator for leader follower formation control based upon the immersion and invariance i i methodology the proposed reduced order nonlinear observer is simple to implement easy to tune and achieves global asymptotical convergence of the observation error to zero observability conditions for the leader follower system are analytically derived by studying the singularity of the extended output jacobian the stability of the closed loop system arising from the combination of the range estimator and an input state feedback controller is proved by means of lyapunov arguments simulation experiments illustrate the theory and show the effectiveness of the proposed designs

brief paper a novel approach to coordination of multiple robots with communication failures via proximity graph in this paper we investigate the coordination problem of multiple robots with limited communication ranges and communication failures a novel rendezvous algorithm via proximity graph is developed so that the robot group can achieve rendezvous when the communication links satisfy an ergodic assumption the convergence proof of the algorithm is established based on the tools from rooted graph theory the effectiveness of the algorithm is illustrated by numerical examples in a 3d space

equilibria and steering laws for planar formations abstract this paper presents a lie group setting for the problem of control of formations as a natural outcome of the analysis of a planar two vehicle formation control law the vehicle trajectories are described using the planar frenet serret equations of motion which capture the evolution of both the vehicle position and orientation for unit speed motion subject to curvature steering control the set of all possible relative equilibria for arbitrary g invariant curvature controls is described where g se 2 is a symmetry group for the control law and a global convergence result for the two vehicle control law is proved an n vehicle generalization of the two vehicle control law is also presented and the corresponding relative equilibria for the n vehicle problem are characterized work is on going to discover stability and convergence results for the n vehicle problem

the multi agent rendezvous problem part 2 the asynchronous case this paper is concerned with the collective behavior of a group of n 1 mobile autonomous agents labelled 1 through n which can all move in the plane each agent is able to continuously track the positions of all other agents currently within its sensing region where by an agent s sensing region we mean a closed disk of positive radius r centered at the agent s current position the multi agent rendezvous problem is to devise local control strategies one for each agent which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location this paper describes a family of unsynchronized strategies for solving the problem correctness is established appealing to the concept of analytic synchronization

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

preservers of λ aluthge transforms abstract let m and n be arbitrary von neumann algebras for any a in m or in n let a denote the aluthge transform of a suppose that m has no abelian direct summand we prove that every bijective map m n satisfying a b a b for all a b m for a fixed 0 1 maps the hermitian part of m onto the hermitian part of n i e m s a n s a and its restriction m s a m s a n s a is a jordan isomorphism if we also assume that x i y x i y for all x y m s a then there exists a central projection p c in m such that p c m is a complex linear jordan isomorphism and 1 p c m is a conjugate linear jordan isomorphism given two complex hilbert spaces h and k with dim h 2 we also show that every bijection b h b k satisfying a b a b for all a b b h must be a complex linear or a conjugate linear isomorphism

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

target assignment in robotic networks distance optimality guarantees and hierarchical strategies we study the problem of multi robot target assignment to minimize the total distance traveled by the robots until they all reach an equal number of static targets in the first half of the paper we present a necessary and sufficient condition under which true distance optimality can be achieved for robots with limited communication and target sensing ranges moreover we provide an explicit non asymptotic formula for computing the number of robots needed to achieve distance optimality in terms of the robots communication and target sensing ranges with arbitrary guaranteed probabilities the same bounds are also shown to be asymptotically tight in the second half of the paper we present suboptimal strategies for use when the number of robots cannot be chosen freely assuming first that all targets are known to all robots we employ a hierarchical communication model in which robots communicate only with other robots in the same partitioned region this hierarchical communication model leads to constant approximations of true distance optimal solutions under mild assumptions we then revisit the limited communication and sensing models by combining simple rendezvous based strategies with a hierarchical communication model we obtain decentralized hierarchical strategies that achieve constant approximation ratios with respect to true distance optimality results of simulation show that the approximation ratio is as low as 1 4

nonsmooth coordination and geometric optimization via distributed dynamical systems emerging applications for networked and cooperative robots motivate the study of motion coordination for groups of agents for example it is envisioned that groups of agents will perform a variety of useful tasks including surveillance exploration and environmental monitoring this paper deals with basic interactions among mobile agents such as move away from the closest other agent or move toward the furthest vertex of your own voronoi polygon these simple interactions amount to distributed dynamical systems because their implementation requires only minimal information about neighboring agents we characterize the close relationship between these distributed dynamical systems and the disk covering and sphere packing cost functions from geometric optimization our main results are as follows i we characterize the smoothness properties of these geometric cost functions ii we show that the interaction laws are variations of the nonsmooth gradient of the cost functions and iii we establish various asymptotic convergence properties of the laws the technical approach relies on concepts from computational geometry nonsmooth analysis and nonsmooth stability theory

robustness of stability through necessary and sufficient lyapunov like conditions for systems with a continuum of equilibria abstract the equivalence between robustness to perturbations and the existence of a continuous lyapunov like mapping is established in a setting of multivalued discrete time dynamics for a property sometimes called semistability this property involves a set consisting of lyapunov stable equilibria and surrounded by points from which every solution converges to one of these equilibria as a consequence of the main results this property turns out to always be robust for continuous nonlinear dynamics and a compact set of equilibria preliminary results on reachable sets limits of solutions and set valued lyapunov mappings are included

distributed algorithms for reaching consensus on general functions this paper presents analysis and design results for distributed consensus algorithms in multi agent networks we consider continuous consensus functions of the initial state of the network agents under mild smoothness assumptions we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus this characterization is the building block to obtain various design results for networks with weighted directed interconnection topologies we first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus we apply this result to the family of weighted power mean functions and characterize the exponential convergence properties of the resulting algorithms we establish the validity of these results for scenarios with switching interconnection topologies finally we conclude with two discontinuous distributed algorithms that achieve respectively max and min consensus in finite time

state agreement for continuous time coupled nonlinear systems two related problems are treated in continuous time first the state agreement problem is studied for coupled nonlinear differential equations the vector fields can switch within a finite family associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems generalizing the work of moreau under the assumption that the vector fields satisfy a certain subtangentiality condition it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time the proof uses nonsmooth analysis second the rendezvous problem for kinematic point mass mobile robots is studied when the robots fields of view have a fixed radius the circumcenter control law of ando e ieee trans robotics automation 15 1999 pp 818 828 is shown to solve the problem the rendezvous problem is a kind of state agreement problem but the interaction structure is state dependent

agreeing asynchronously this paper formulates and solves a version of the widely studied vicsek consensus problem in which each member of a group of n 1 agents independently updates its heading at times determined by its own clock it is not assumed that the agents clocks are synchronized or that the ldquoeventrdquo times between which any one agent updates its heading are evenly spaced nor is it assumed that heading updates must occur instantaneously using the concept of ldquoanalytic synchronizationrdquo together with several key results concerned with properties of ldquocompositionsrdquo of directed graphs it is shown that the conditions under which a consensus is achieved are essentially the same as those applicable in the synchronous case provided the notion of an agent s neighbor between its event times is appropriately defined however in sharp contrast with the synchronous case where for analysis an n dimensional state space model is adequate for the asynchronous version of the problem a 2n dimensional state space model is required it is explained how to analyze this model despite the fact that unlike the synchronous case the stochastic matrices involved do not have all positive diagonal entries

on the estimation of the consensus rate of convergence in graphs with persistent interconnections abstractthe aim of the current article is to establish myriad convergence rate estimates to consensus for time varying graphs with persistent interaction several novel analysis methodologies for consensus protocols employing the notions of persistence of excitation and lyapunov functions are provided the estimates are compared with each other and existing literature numerical simulations on test examples are illustrated to support the theoretical findings

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper