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

poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

distributed consensus for multi agent systems with delays and noises in transmission channels this paper studies the distributed consensus problem for linear discrete time multi agent systems with delays and noises in transmission channels due to the presence of noises and delays existing techniques such as the lifting technique and the stochastic lyapunov theory are no longer applicable to the analysis of consensus in this paper a novel technique is introduced to overcome the difficulties induced by the delays and noises a consensus protocol with decaying gains satisfying persistence condition is adopted necessary and sufficient conditions for strong consensus and mean square consensus are respectively given for non leader follower and leader follower cases under a fixed topology under dynamically switching topologies and randomly switching topologies sufficient conditions for strong consensus and mean square consensus are also obtained numerical examples are given to demonstrate the effectiveness of the proposed protocols

a new indirect adaptive control scheme for time varying plants structures are developed for the control and estimation parts of the indirect adaptive scheme which are more appropriate for time varying plants in contrast to the usual pointwise schemes the new controller structure can meet the control objective exactly for a wide class of plants with smooth but otherwise unrestricted parameter variations on the other hand the possibly available knowledge of the form or frequency of variation of the fast parameters is included in the new estimator structure so that for successful estimation the overall plant is not restricted to varying slowly with time the new estimator and control law are combined using the certainty equivalence principle to develop an indirect adaptive control scheme meeting the control objective for plants with time varying parameters whose fast varying parts are of known form and unknown parts are slow in the mean

weak exponential stability of linear time varying differential behaviors abstract we develop a new approach to exponential stability of linear time varying ltv differential behaviors that is analogous to that in our paper on exponential stability of discrete ltv behaviors h bourles et al 2015 5 stability theory for differential state space systems with smooth coefficients is an important subject in the literature for differential ltv behaviors with arbitrary smooth coefficients there is no reasonable stability theory therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent puiseux series all rational functions are of this type we introduce a new kind of behaviors and prove a module behavior duality for these we define a new notion of weak exponential stability w e s of a behavior b and its associated finitely generated f g module m and show that the w e s modules and behaviors are closed under isomorphisms subobjects factor objects and extensions the standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory in the main result we assume a nonzero f g torsion module m and its associated autonomous behavior b such a module may be regular or irregular singular according to the galois theory of differential equations if it is nonzero and regular singular it is never w e s for irregular singular m we characterize w e s of most b algebraically and constructively

complex geometry of moment angle manifolds moment angle manifolds provide a wide class of examples of non kahler compact complex manifolds a complex moment angle manifold mathcal z is constructed via certain combinatorial data called a complete simplicial fan in the case of rational fans the manifold mathcal z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori in general a complex moment angle manifold mathcal z is equipped with a canonical holomorphic foliation mathcal f which is equivariant with respect to the mathbb c times m action examples of moment angle manifolds include hopf manifolds of vaisman type calabi eckmann manifolds and their deformations we construct transversely kahler metrics on moment angle manifolds under some restriction on the combinatorial data we prove that any kahler submanifold or more generally a fujiki class mathcal c subvariety in such a moment angle manifold is contained in a leaf of the foliation mathcal f for a generic moment angle manifold mathcal z in its combinatorial class we prove that all subvarieties are moment angle manifolds of smaller dimension and there are only finitely many of them this implies in particular that the algebraic dimension of mathcal z is zero

distributed consensus for multiagent systems with communication delays and limited data rate this paper considers the average consensus problem for multiagent networks with communication delays and limited data rate on one hand communication delays often exist in information acquisition and transmission on the other hand only limited state information of agents can be transmitted to their neighbors at each time step due to bandwidth constraints the average consensus problem becomes much more complicated when both delays and data rate constraints are to be considered in this paper a distributed consensus protocol is proposed based on dynamic encoding and decoding it is shown that for a connected network as long as the time delays are bounded the average consensus can be achieved with a finite communication data rate in particular it is shown that merely a one bit information exchange between each pair of adjacent agents at each time step suffices to guarantee the average consensus

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

consensus conditions of continuous time multi agent systems with time delays and measurement noises this work is concerned with stochastic consensus conditions of multi agent systems with both time delays and measurement noises for the case of additive noises we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations by the martingale convergence theorem we obtain necessary conditions and sufficient conditions for stochastic strong consensus for the case of multiplicative noises we consider two kinds of time delays appeared in the measurement term and the noise term respectively we first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus then by constructing degenerate lyapunov functional we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time delay in the measurement term

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

input delay margin for consensusability of multi agent systems in this paper we are concerned with the consensus of multi agent systems with input delay among all standard static protocols that achieve the consensus for the multi agent system under no input delay we aim to find the maximum input delay such that the system remains consensusable under the same protocols in the case of continuous time systems in view of the continuity of stability with respect to the time delay the maximum delay margin for consensusability is given for scalar systems and vector systems with a single unstable open loop pole for scalar discrete time systems we show that the maximum delay margin for consensusability is strictly greater than zero if and only if the open loop pole of the system is located in a specified interval

necessary and sufficient conditions for consensus of double integrator multi agent systems with measurement noises an average consensus protocol is proposed for continuous time double integrator multi agent systems with measurement noises under fixed topologies the time varying control gain is employed to attenuate noises the closed loop system is therefore a time varying linear stochastic differential equation by determining the state transition matrix of this closed loop system the dynamic characteristics of the multi agent system can be fully described it is proved that in the noisy communication environment the average consensus can be achieved if and only if the communication topology is a balanced and strongly connected graph and the time varying control gain satisfies the stochastic approximation type conditions under the proposed protocol the position of each agent is convergent in mean square to a common random variable whose mathematical expectation is the average of initial positions and initial velocities of all agents in the system while each agent s velocity is convergent in mean square to a common random variable whose mathematical expectation and variance are both zero

adaptive control of linear time varying plants a new model reference controller structure the problem of developing a control law which can force the output of a linear time varying plant to track the output of a stable linear time invariant reference model is discussed it is shown that the standard model reference controller used for linear time invariant plants cannot guarantee zero tracking error in general when the plant is time varying a new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time varying plants with known parameters when the time varying plant parameters are unknown but vary slowly with time it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean the assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters

mitigation of complex behavior over networked systems analysis of spatially invariant structures in this paper we consider a simple distributed averaging system which incorporates various communication constraints including delays noise and link failures it has been shown in wang and elia 2012 that such networked system generates a collective levy flight behavior when part of the system loses mean square ms stability we focus on spatially invariant architectures to gain more insights into how model parameters affect emergence of this complex scale invariant behavior and to seek structures robust to communication constraints specifically we develop a computational expression for checking ms stability which is scalable with the number of unreliable links we derive the closed form formulas from this expression in the limiting case of zero and large delays and in the case of large number of nodes in the limit of large delays we derive various results that are independent of the network size and its specific interconnections we find that small inter agent coupling improves the robustness of the system networks with larger connectivity tend to be more fragile in the presence of fading connections for fixed inter agent coupling that gossiping improves the robustness and that the lattice is the most robust among the spatially invariant systems with generalized circulant interconnections

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

model reference adaptive control of linear time varying plants the case of jump parameter variations in this paper we consider the model reference adaptive control problem of single input single output linear time varying plants we assume that the plant parameters are uniformly bounded and piecewise lipschitz continuous inside the intervals of continuity we assume that the plant satisfies certain controllability observability relative degree and zero dynamics assumptions but the speed of parameter variations is not restricted to be small under these general conditions we design a control law of the model reference type which guarantees the boundedness of the closed loop plant and small tracking error in the mean square sense provided that the number of discontinuities of the plant parameters is small on average furthermore in the case of partially unknown plant parameters we show that the adaptive controller obtained by combining this control law with a suitable estimation algorithm guarantees closed loop boundedness and small tracking error in the mean square sense provided that in additi

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

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

a new geometric construction of compact complex manifolds in any dimension we consider holomorphic linear foliations of dimension m of mathbb c n with n 2m fulfilling a so called weak hyperbolicity condition and equip the projectivization of the leaf space for the foliation restricted to an adequate open dense subset with a structure of compact complex manifold of dimension n m 1 we show that except for the limit case n 2m 1 where we obtain any complex torus of any dimension this construction gives non symplectic manifolds including the previous examples of hopf calabi eckmann haefliger linear case loeb nicolau linear case and lopez de medrano verjovsky we study some properties of these manifolds that is to say meromorphic functions holomorphic vector fields forms and submanifolds for each manifold we construct an analytic space of deformations of dimension m n m 1 and show that under some additional conditions it is universal lastly we give explicit examples of new compact complex manifolds in particular of connected sums of products of spheres and show the existence of a momentum like map which classifies these manifolds up to diffeomorphism

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

duality in time varying linear systems a module theoretic approach abstract using elementary module theory an intrinsic definition of the dual or adjoint of a generalized time varying linear system is given with this the duality of controllability and observability is recovered from their intrinsic module theoretical definitions the duality of state feedback and output injection is discussed both in the static case and for its quasistatic generalization related brunovsky type canonical forms are derived in the quasistatic case the corresponding controllability indices and their duals the observability indices are defined intrinsically

consensus in multi agent systems with coupling delays and switching topology the robustness of consensus in single integrator multi agent systems mas to coupling delays and switching topologies is investigated it is shown that consensus is reached for arbitrarily large constant time varying or distributed delays if consensus is reached without delays this delay robustness holds under the weakest possible connectivity assumptions on the underlying graph i e as long as the graph is uniformly quasi strongly connected and switches with a dwell time the proof is based on a contraction argument and allows to remove technical assumptions that were used in previous publications moreover the result also applies to non scalar single integrators an important extension toward consensus and rendezvous in higher dimensions

stabilizability of systems with exponential dichotomy in this paper we introduce the concept of controllability into a closed subspace for time varying linear systems various characterizations are given and the dual relation is discussed this concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy

average consensus in the presence of delays in directed graph topologies classical distributed algorithms for asymptotic average consensus typically assume timely and reliable exchange of information between neighboring components of a given multi component system these assumptions are not necessarily valid in practice due to varying delays that might affect computations at different nodes and or transmissions at different links in this work we propose a protocol that overcomes this limitation and unlike existing consensus protocols in the presence of delays ensures asymptotic consensus to the exact average despite the presence of arbitrary but bounded delays in the communication links the protocol requires that each component has knowledge of the number of its out neighbors i e the number of components to which it can send information and its proof of correctness relies on the weak convergence of a backward product of column stochastic matrices the proposed algorithm is demonstrated via illustrative examples

constrained consensus in unbalanced networks with communication delays in this note a constrained consensus problem is studied for multi agent systems in unbalanced networks in the presence of communication delays here each agent needs to lie in a closed convex constraint set while reaching a consensus the communication graphs are directed dynamically changing and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length the analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary to tackle the loss of symmetry caused by unbalanced graphs and communication delays a novel approach is proposed the idea is to estimate the distance from each agent to the intersection set of all agents constraint sets based on the properties of the projection on convex sets so as to show consensus convergence by contradiction it is shown that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system it is also shown that the communication delays do not affect the consensus stability and constrained consensus is reached even if the communication delays are arbitrarily bounded finally a numerical example is included to illustrate the obtained theoretical results

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

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

minimum variance prediction for linear time varying systems in this paper we study the problem of minimum variance prediction for linear time varying systems we consider the standard time varying autoregression moving average arma model and develop a predictor which guarantees minimum variance prediction for a large class of linear time varying systems the predictor is developed based on a pseudocommutation technique for dealing with noncommutativity of linear time varying operators in a transfer operator framework we also show connections between this input output predictor and the kalman predictor via an example

a necessary and sufficient condition for consensus over random networks we consider the consensus problem for stochastic discrete time linear dynamical systems the underlying graph of such systems at a given time instance is derived from a random graph process independent of other time instances for such a framework we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments this easily verifiable condition uses the spectrum of the average weight matrix finally we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1

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

the poles and zeros of a linear time varying system abstract for linear time varying discrete time and continuous time systems a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time varying coefficients in the discrete time case it is shown that the poles can be computed by solving a nonlinear recursion with time varying coefficients in the continuous time case the poles can be calculated by solving a nonlinear differential equation with time varying coefficients the theory is applied to the study of the zero input response and asymptotic stability it is shown that if a time varying analogue of the vandermonde matrix is invertible the zero input response can be decomposed into a sum of modes associated with the poles stability is then studied in terms of the components of the modal decomposition

time varying polynomial matrix systems abstract differential polynomial matrices with coefficients meromorphic on an open interval i r are studied first the subclass of full matrices p having the properties p is non singular antl every local solution of pf 0 extends analytically to i forms a sublattice with respect to right division of square matrices which is anti isomorphic to the lattice of the corresponding solution spaces this and further properties are then exploited in the study of time varying systems in differential operator representation results on equivalence state space models controllability observability and transfer functions are derived

model reference adaptive and nonadaptive control of linear time varying plants we address the adaptive and nonadaptive model reference control problem for a class of multivariable linear time varying plants namely index invariant ones we show that under appropriate controllability and observability conditions this class of plants admits a fractional description in terms of polynomial differential operators and as such allows for a polynomial equation based controller design we also show that for a model reference control objective the controller can be designed by solving a set of algebraic equations further when the plant parameters are only partially known we employ a gradient based adaptive law with projection and normalization to update the controller parameters and establish the stability and tracking properties of adaptive closed loop plant finally we present a simple example to illustrate the design and realization of both the adaptive and nonadaptive control laws

consensus seeking in multi agent systems with multiplicative measurement noises abstract in this paper the consensus problems of the continuous time integrator systems under noisy measurements are considered the measurement noises which appear when agents measure their neighbors states are modeled to be multiplicative by multiplication of the noises here the noise intensities are proportional to the absolute value of the relative states of an agent and its neighbor by using known distributed protocols for integrator agent systems the closed loop system is described in the vector form by a singular stochastic differential equation for the fixed and switching network topology cases constant consensus gains are properly selected such that mean square consensus and strong consensus can be achieved especially exponential mean square convergence of agents states to the common value is derived for the fixed topology case in addition asymptotic unbiased mean square average consensus and asymptotic unbiased strong average consensus are also studied simulations shed light on the effectiveness of the proposed theoretical results

a behavioral approach to time varying linear systems part 2 descriptor systems in the sequel to a ilchmann and v mehrmann em siam j control optim 44 2005 pp 1725 1747 we discuss a behavioral approach for linear time varying differential algebraic descriptor systems with real analytic coefficients the analysis is almost global in the sense that the analysis is not restricted to an interval i subset r but is allowed for the time axis r setminus t where t is a discrete set of critical points at which the solution may exhibit a finite escape time controllable observable and autonomous behavior for linear time varying descriptor systems is characterized

linearized control systems and applications to smooth stabilization for a control system dot x f x u the author proves that for generic feedback laws u such that f x u x does not vanish the linearized control systems around the trajectories of dot x f x u x have the same strong accessibility algebra as f applications are given to the smooth stabilization problem

a mean square consensus protocol for linear multi agent systems with communication noises and fixed topologies the mean square consensus of linear multi agent systems with communication noises is studied in this note each agent is modeled by a continuous time linear time invariant dynamics and the fixed communication topology is described by a digraph the proposed consensus protocol is composed of two parts the agent s own state feedback and the relative states between agent and its neighbor agents due to the existence of communication noises the relative states cannot be obtained accurately to attenuate the noise effect a time varying gain vector a t k is applied to the inaccurate relative states it is proved that 1 if the communication topology has a spanning tree and every node has at least one parent node then the proposed protocol can solve the mean square consensus problem if and only if a t satisfies 0 a s ds and 0 a2 s ds and all roots of the polynomial whose coefficients are the elements of vector k are in the left half complex plane 2 if the communication topology has a spanning tree and there exists one node without any parent node then the condition 0 a2 s ds is only sufficient but not necessary and 3 if the communication topology has no spanning tree then the proposed protocol cannot solve the mean square consensus problem

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

a behavioral approach to time varying linear systems part 1 general theory we develop a behavioral approach to linear time varying differential algebraic systems the analysis is almost everywhere in the sense that the statements hold on mathbb r setminus mathbb t where mathbb t is a discrete set controllability observability and autonomy are introduced and related to the behavior of the system classical results on the behavior of time invariant systems are studied in the context of time varying systems

continuous time stochastic consensus this paper investigates the continuous time multi agent consensus with stochastic communication noises each agent can only use its own and neighbors information corrupted by random noises to design its control input to attenuate the communication noises we consider the stochastic approximation type and the kalman bucy filtering based protocols by using the tools of stochastic analysis and algebraic theory the asymptotic properties of these two kinds of protocols are analyzed firstly for the stochastic approximation type protocol we clarify the relationship between the convergence rate of the consensus error and a representative class of consensus gains in both mean square and probability one secondly we propose kalman bucy filtering based consensus protocols each agent uses kalman bucy filters to get asymptotically unbiased estimates of neighbors states and the control input is designed based on the protocol with precise communication and the certainty equivalence principle the iterated logarithm law of estimation errors is developed it is shown that if the communication graph has a spanning tree then the consensus error is bounded above by o t 1 in mean square and by o t 1 2 log log t 1 2 almost surely finally the superiority of the kalman bucy filtering based protocol over the stochastic approximate type protocol is studied both theoretically and numerically

multi agent consensus with relative state dependent measurement noises in this note the distributed consensus corrupted by relative state dependent measurement noises is considered each agent can mea sure or receive its neighbors state information with random noises whose intensity is a vector function of agents relative states by investigating the structure of this interaction and the tools of stochastic differential equa tions we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain the number of agents and the noise intensity function to ensure mean square m s and almost sure a s consensus and quantify the convergence rate and the steady state error especially for the case with homogeneous communication and control channels a necessary and sufficient condition to ensure m s consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology but only depends on the number of nodes and the noise coefficient constant for symmetric measurement models the almost sure convergence rate is estimated by the iterated logarithm law of brownian motions index terms distributed consensus distributed coordination fading channel measurement noises multi agent system

a mean field game synthesis of initial mean consensus problems a continuum approach for non gaussian behavior this technical note presents a continuum approach to a non gaussian initial mean consensus problem via mean field mf stochastic control theory in this problem formulation i each agent has simple stochastic dynamics with inputs directly controlling its state s rate of change and ii each agent seeks to minimize by continuous state feedback its individual discounted cost function involving the mean of the states of all other agents for this dynamic game problem a set of coupled deterministic hamilton jacobi bellman and fokker planck kolmogorov equations is derived approximating the stochastic system of agents as the population size goes to infinity in a finite population system analogous to the mf lqg framework i the resulting decentralized mf control strategies possess an en nash equilibrium property where en goes to zero as the population size n approaches infinity and ii these mf control strategies steer each individual s state toward the initial state population mean which is reached asymptotically as time goes to infinity hence the system with decentralized mf control strategies reaches mean consensus on the initial state population mean asymptotically as time goes to infinity

distributed consensus algorithms in sensor networks quantized data and random link failures the paper studies the problem of distributed average consensus in sensor networks with quantized data and random link failures to achieve consensus dither small noise is added to the sensor states before quantization when the quantizer range is unbounded countable number of quantizer levels stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable we show that the mean squared error mse can be made arbitrarily small by tuning the link weight sequence at a cost of the convergence rate of the algorithm to study dithered consensus with random links when the range of the quantizer is bounded we establish uniform boundedness of the sample paths of the unbounded quantizer this requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer this is accomplished by splitting the state vector of the quantizer in two components one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace the proofs use maximal inequalities for submartingale and supermartingale sequences from these we derive probability bounds on the excursions of the two subsequences from which probability bounds on the excursions of the quantizer state vector follow the paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels the size of the quantization steps the desired probability of saturation and the desired level of accuracy away from consensus finally the paper illustrates the quantizer design with a numerical study

containment control of multi agent systems in a noisy communication environment the containment problem of first order and second order integral multi agent systems with communication noises is investigated in this paper motivated by the previous work a time varying gain a t is introduced to attenuate the noise effect it is proved that the proposed protocol can solve the mean square containment problem if the following conditions hold 1 the communication topology graph has a spanning forest whose roots are leaders of the multi agent system 2 0 a t dt 3 lim t a t 0 moreover if a t is uniformly continuous these conditions are necessary for ensuring the mean square containment in addition the proposed protocol is also extended to allow the case where each agent has its own gain a i t it is proved that the mean square containment problem can still be solved if all agent dependent gains are infinitesimals of the same order as time goes to infinity finally two simulation examples are provided to demonstrate the effectiveness of the proposed protocols

time varying linear systems and invariants of system equivalence in the paper we consider time varying systems with coefficients depending mero morphically on time in differential operator representations these systems are described by matrices over a skew polynomial ring with coefficients in the field of real meromorphic functions different kinds of indices controllability minimal geometric and dynamical are introduced and it is proved that they essentially coincide the input module and the formal transfer matrix are defined and used for an algebraic description of time varying systems a characterization of system equivalence is given in these terms and also a complete list of invariants of similarity for time varying state space systems

consensus conditions of continuous time multi agent systems with additive and multiplicative measurement noises this work is concerned with the consensus problem of multi agent systems with additive and multiplicative measurement noises by developing general stochastic stability lemmas for nonautonomous stochastic differential equations stochastic weak and strong consensus conditions are investigated under fixed and time varying topologies respectively for the case with fixed topologies and additive noises the necessary and sufficient conditions for almost sure strong consensus are given it is revealed that almost sure strong consensus and mean square strong consensus are equivalent under general digraphs and almost sure weak consensus implies mean square weak consensus under undirected graphs if multiplicative noises appear then small noise intensities do not affect the control gain to guarantee stochastic strong consensus for the case with time varying topologies sufficient consensus conditions are given under the periodically connected condition of the topology flow

brief paper stability analysis for multi agent systems using the incidence matrix quantized communication and formation control the spectral properties of the incidence matrix of the communication graph are exploited to provide solutions to two multi agent control problems in particular we consider the problem of state agreement with quantized communication and the problem of distance based formation control in both cases stabilizing control laws are provided when the communication graph is a tree it is shown how the relation between tree graphs and the null space of the corresponding incidence matrix encode fundamental properties for these two multi agent control problems

non kahler manifolds and git quotients bosio generalized the construction by lopez de medrano verjovsky meersseman lvm of a family of non algebraic compact complex manifolds of any dimension we describe how to construct the generalized family from certain geometric invariant theory git quotients we show that bosio s generalization parallels exactly the extension from mumford s git to the more general git developed by bialynicki birula and wi cicka this point of view yields new results on the geometry of lvm and bosio s manifolds

an exact method for the stability analysis of linear consensus protocols with time delay this technical note presents a methodology for the stability analysis of linear consensus protocols with time delayed communications second order agent dynamics with a fixed and undirected communication topology and uniform delays are considered this class of group dynamics is very complex and is not fully explored to date the proposed technique takes advantage of the general structure of the control protocols in performing a state transformation that allows a decomposition of the characteristic equation into a set of factors these factors distribute the imprint of the delay in the characteristic equation in a much simpler form to achieve the stability analysis in parts the procedure also prepares the characteristic equation for the deployment of the cluster treatment of characteristic roots paradigm a recent method which declares the stability features of the system for various compositions of the time delay and other control parameters in order to show the effectiveness of this approach it is applied to different consensus protocols under the assumptions of fixed and undirected communication topologies and uniform communication time delays

adjoint and hamiltonian input output differential equations based on developments in the theory of variational and hamiltonian control systems by crouch and van der schaft 1987 this paper answers two questions given an input output differential equation description of a nonlinear system what is the adjoint variational system in input output differential form and what are the conditions for the system to be hamiltonian i e such that the variational and the adjoint variational systems coincide this resulting set of conditions is then used to generalize classical conditions such as the well known helmholtz conditions for the inverse problem in classical mechanics

lvmb manifolds and quotients of toric varieties in this article we study a class of manifolds introduced by bosio called mathrm lvmb manifolds we provide an interpretation of his construction in terms of quotient of toric manifolds by complex lie groups furthermore mathrm lvmb manifolds extend a class of manifolds obtained by meersseman called mathrm lvm manifolds and we give a characterization of these manifolds using our toric description finally we give an answer to a question asked by cupit foutou and zaffran