multiplex pi control for consensus in networks of heterogeneous linear agents in this paper we propose a multiplex proportional integral approach for solving consensus problems in networks of heterogeneous nodes dynamics affected by constant disturbances the proportional and integral actions are deployed on two different layers across the network each with its own topology sufficient conditions for convergence are derived that depend upon the structure of the network the parameters and topologies characterizing the control layers and the node dynamics the effectiveness of the theoretical results is illustrated numerically on a set of representative examples including a power network model

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

distributed static output feedback control for state synchronization in networks of identical lti systems this paper studies state synchronization of multi agent systems with disturbances using distributed static output feedback opfb control the bounded l 2 gain synchronization problem using distributed static opfb is defined and solved the availability of only output measurements restricts the local controls design while the communication graph topology restricts global information flow among the agents it is shown here that these two types of restriction can be dealt with in a symmetric manner and lead to two similar conditions guaranteeing the existence of bounded l 2 gain static opfb one condition is on the output measurement matrix on a local scale and the other on the graph laplacian matrix on a global scale under additional conditions a distributed two player zero sum game using static opfb is also solved and leads to distributed nash equilibrium on the communication graph as a special case the static opfb globally optimal control is given a new class of digraphs satisfying the above condition on the graph laplacian is studied the synchronizing region for distributed static opfb control is exposed and found to be conical different than the infinite right half plane synchronizing region for distributed state feedback

coordination of multi agent systems on interacting physical and communication topologies abstract a new framework is given for coordination of multi agent systems that are interconnected by a physical coupling digraph the edges of this graph represent physical couplings between agents that are fixed due to dynamical interactions on top of the physical graph distributed control protocols are designed where the allowed communications between agents for control purposes are prescribed by a second fixed communication digraph the physical and communication digraphs are generally different and the combination of these two graphs forms a cyber physical system the interactions between physical and communication graphs are the focus of this paper we consider different interactions between two graphs including the case when their pinned laplacian commutes the case of the communication graph with diagonalizable pinned laplacian and the case of two general graphs moreover within each graph the relations between the agents can be either collaborative or antagonistic to capture this the theory of bipartite consensus is used coordination protocols for different cases are designed that are distributed with respect to the communication graph and overcome the detrimental effects of the signed physical graph the proposed control methods are illustrated by simulation examples

asymptotic stability of a mechanical robotics model with damping and delay in this paper we study the asymptotic stability of a mechanical robotics model with damping and delay this model yields a certain linear third order delay differential equation in proving our results we make use of pontryagin s theory for quasi polynomials

robust synchronization of uncertain linear multi agent systems this paper deals with robust synchronization of uncertain multi agent networks given a network with for each of the agents identical nominal linear dynamics we allow uncertainty in the form of additive perturbations of the transfer matrices of the nominal dynamics the perturbations are assumed to be stable and bounded in h norm by some a priori given desired tolerance we derive state space formulas for observer based dynamic protocols that achieve synchronization for all perturbations bounded by this desired tolerance it is shown that a protocol achieves robust synchronization if and only if each controller from a related finite set of feedback controllers robustly stabilizes a given single linear system our protocols are expressed in terms of real symmetric solutions of certain algebraic riccati equations and inequalities and also involve weighting factors that depend on the eigenvalues of the graph laplacian for undirected network graphs we show that within the class of such dynamic protocols a guaranteed achievable tolerance can be obtained that is proportional to the quotient of the second smallest and the largest eigenvalue of the laplacian we also extend our results to additive nonlinear perturbations with l2 gain bounded by a given tolerance

stability criteria for certain high odd order delay differential equations in this paper we study the asymptotic stability of the zero solution of odd order linear delay differential equations of the formy 2 m 1 t j 02ma jy j t j 02mb jy j t t where a j and b j are certain constants and m 1 here t 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

optimal design for synchronization of cooperative systems state feedback observer and output feedback this technical note studies synchronization of identical general linear systems on a digraph containing a spanning tree a leader node or command generator is considered which generates the desired tracking trajectory a framework for cooperative tracking control is proposed including full state feedback control observer design and dynamic output feedback control the classical system theory notion of duality is extended to networked systems it is shown that unbounded synchronization regions that achieve synchronization on arbitrary digraphs containing a spanning tree can be guaranteed by using linear quadratic regulator based optimal control and observer design methods at each node

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

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

properties of the third order trinomial differential equations with delay argument abstract in this paper we study properties of the third order trinomial delay differential equation y t p t y t g t y t 0 by transforming this equation to binomial second third order differential equation employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation

pinning control of scale free dynamical networks abstract recently it has been demonstrated that many large complex networks display a scale free feature that is their connectivity distributions have the power law form in the present work control of a scale free dynamical network by applying local feedback injections to a fraction of network nodes is investigated the specifically and randomly pinning schemes are considered the specifically pinning of the most highly connected nodes is shown to require a significantly smaller number of local controllers as compared to the randomly pinning scheme the method is applied to an array of chua s oscillators as an example

fuzzy adaptive fault tolerant control of multi agent systems with interactions between physical coupling graph and communication graph abstract this paper considers the fuzzy fault tolerant control problem for multi agent systems with interactions between physical coupling graph and communication graph the dynamics of agents are interconnected by a signed physical coupling graph and the control protocols are computed by using the information flow prescribed in a signed communication graph a novel fuzzy adaptive fault tolerant control method for agents with antagonistic interactions to achieve bipartite consensus is proposed firstly in the presence of actuator bias faults and unknown nonlinear functions different from the traditional method to estimate the weight matrix only the weight vector needs to be estimated by exchanging the order of weight vectors and fuzzy basis functions in the fuzzy logic systems finally a simulation example is given to illustrate the effectiveness of the proposed controller

on the consensus and bipartite consensus in high order multi agent dynamical systems with antagonistic interactions abstract the aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents under the assumption that their mutual interactions can be described by a weighted signed connected and structurally balanced communication graph this amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative and hence represented by nonnegative weights while interactions between agents belonging to opposite groups are antagonistic and hence represented by nonpositive weights in this framework bipartite consensus can always be reached under the stabilizability assumption on the state space model describing the dynamics of each agent on the other hand nontrivial standard consensus may be achieved only under very demanding requirements both on the laplacian associated with the communication graph and on the agents description in particular consensus may be achieved only if there is a sort of equilibrium between the two groups both in terms of cardinality and in terms of the weights of the conflicting interactions amongst agents

distributed control of networked dynamical systems static feedback integral action and consensus this paper analyzes distributed control protocols for first and second order networked dynamical systems we propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain and a sum of nonlinear interaction functions by using integral lyapunov functions we prove the stability of the proposed control protocols and explicitly characterize the equilibrium set we also propose a distributed proportional integral pi controller for networked dynamical systems the pi controllers successfully attenuate constant disturbances in the network we prove that agents with single integrator dynamics are stable for any integral gain and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double integrator dynamics throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites power systems and building temperature control

existence properties and trajectory specification of generalised multi agent flocking a class of flocking algorithms for second order multi agent networks with single virtual leaders are developed by including more weighting and navigating matrices in this paper flocking existence

cooperative optimal control for multi agent systems on directed graph topologies this note brings together stability and optimality theory to design distributed cooperative control protocols that guarantee consensus and are globally optimal with respect to a positive semi definite quadratic performance criterion a common problem in cooperative optimal control is that global optimization problems generally require global information which is not available to distributed controllers optimal control for multi agent systems is complicated by the fact that the communication graph topology interplays with the agent system dynamics in the note we use an inverse optimality approach together with partial stability to consider the cooperative consensus and pinning control agents with identical linear time invariant dynamics are considered communication graphs are assumed directed and having fixed topology structured quadratic performance indices are derived that capture the topology of the graph which allows for global optimal control that is implemented using local distributed protocols a new class of digraphs is defined that admits a distributed solution to the global optimal control problem namely those with simple graph laplacian matrices

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

distributed consensus tracking for multiple uncertain nonlinear strict feedback systems under a directed graph in this brief we study the distributed consensus tracking control problem for multiple strict feedback systems with unknown nonlinearities under a directed graph topology it is assumed that the leader s output is time varying and has been accessed by only a small fraction of followers in a group the distributed dynamic surface design approach is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader the function approximation technique using neural networks is employed to compensate unknown nonlinear terms induced from the controller design procedure from the lyapunov stability theorem it is shown that the consensus errors are cooperatively semiglobally uniformly ultimately bounded and converge to an adjustable neighborhood of the origin

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

fourier tauberian theorems and applications abstract the main aim of the paper is to present a general version of the fourier tauberian theorem for monotone functions this result together with berezin s inequality allows us to obtain a refined version of the li yau estimate for the counting function of the dirichlet laplacian which implies the li yau estimate itself and at the same time the asymptotic results obtained by the variational method

oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations the equation to be considered iswhere pi t 0 i n and q t are continuous and positive on some half line a it is known that always has strictly monotone nonoscillatory solutions defined on a so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of in this paper sufficient conditions are given for this situation to hold for the structure of the solution space of is also studied

brief paper distributed adaptive control for synchronization of unknown nonlinear networked systems this paper is concerned with synchronization of distributed node dynamics to a prescribed target or control node dynamics a design method is presented for adaptive synchronization controllers for distributed systems having non identical unknown nonlinear dynamics and for a target dynamics to be tracked that is also nonlinear and unknown the development is for strongly connected digraph communication structures a lyapunov technique is presented for designing a robust adaptive synchronization control protocol the proper selection of the lyapunov function is the key to ensuring that the resulting control laws thus found are implementable in a distributed fashion lyapunov functions are defined in terms of a local neighborhood tracking synchronization error and the frobenius norm the resulting protocol consists of a linear protocol and a nonlinear control term with adaptive update law at each node singular value analysis is used it is shown that the singular values of certain key matrices are intimately related to structural properties of the graph

leader follower consensus of time varying nonlinear multi agent systems a distributed consensus protocol is developed for a class of homogeneous time varying nonlinear multi agent systems in this paper the agents dynamics are supposed to be in the strict feedback form and satisfy lipschitz conditions with time varying gains it is firstly shown that the leader follower consensus problem is equivalent to a conventional stabilization control problem of augmented high dimensional multi variable systems by introducing an appropriate state transformation the stabilization control problem is then converted into a design problem of an appropriate time varying parameter which plays a key role to cope with the time varying nonlinear terms in the concerned systems it is proved that the exponential consensus of the multi agent system can be achieved with the proposed consensus protocol a simulation example is given to illustrate the effectiveness of the proposed consensus protocol

consensus problems on networks with antagonistic interactions in a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents expresses by a communication graph with nonnegative weights the question we ask in this paper is the following is it possible to achieve a form of agreement also in presence of antagonistic interactions modeled as negative weights on the communication graph the answer to this question is affirmative on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign necessary and sufficient conditions are obtained to describe cases in which this is possible these conditions have strong analogies with the theory of monotone systems linear and nonlinear laplacian feedback designs are proposed

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros