Directed Graph Topology Research Articles

Distributed coordinated tracking control of multiple Euler–Lagrange systems by state and output feedback

This study addresses the distributed state and output feedback coordinated tracking problem for multiple Euler–Lagrange systems with a moving leader in the presence of non-linear uncertainties under a directed topology graph. The authors first design a distributed coordinated tracking control scheme by full-state feedback, where each follower can only share information with its neighbours. Next, the adaptive neural networks are applied to deal with the model uncertainties due to their superior approximation capability. To deal with the absence of velocity sensors, then they propose a distributed output feedback coordinated tracking control law combined with a high-gain observer. The uniform ultimate boundedness of all the state errors can be guaranteed in the sense of Lyapunov stability theory. Furthermore, the leaderless synchronisation problem is solved using the proposed control schemes. Finally, simulation examples are presented to illustrate the feasibility of the theoretical results.

Guaranteed performance design for distributed bounded containment control of networked uncertain underactuated surface vessels

This paper presents a predefined performance design approach for distributed containment control of networked heterogeneous underactuated surface vessels (USVs) under a directed graph topology. It is assumed that the position information of multiple dynamic leaders is informed to only a subset of the followers consisting of USVs and all nonlinearities and external forces in each USV follower are unknown. The local control law for each USV follower is recursively designed using the predefined performance design technique where auxiliary variables are presented to deal with underactuated constraints in the predefined performance design framework and the approach angles capable of the tracking of both straight and curved line are employed. Compared with the existing literature on the control of multiple ocean vehicles with uncertainties, the proposed distributed control methodology is simply designed without using any function approximators and adaptive techniques to estimate uncertainties or their bounds. It is shown that the distributed containment errors are preserved within the predefined bounds of transient and steady-state performance in the Lyapunov sense.

Tensor Decompositions for Identifying Directed Graph Topologies and Tracking Dynamic Networks

Directed networks are pervasive both in nature and engineered systems, often underlying the complex behavior observed in biological systems, microblogs and social interactions over the web, as well as global financial markets. Since their structures are often unobservable, in order to facilitate network analytics, one generally resorts to approaches capitalizing on measurable nodal processes to infer the unknown topology. Structural equation models (SEMs) are capable of incorporating exogenous inputs to resolve inherent directional ambiguities. However, conventional SEMs assume full knowledge of exogenous inputs, which may not be readily available in some practical settings. The present paper advocates a novel SEM-based topology inference approach that entails factorization of a three-way tensor, constructed from the observed nodal data, using the well-known parallel factor (PARAFAC) decomposition. It turns out that second-order piecewise stationary statistics of exogenous variables suffice to identify the hidden topology. Capitalizing on the uniqueness properties inherent to high-order tensor factorizations, it is shown that topology identification is possible under reasonably mild conditions. In addition, to facilitate real-time operation and inference of time-varying networks, an adaptive (PARAFAC) tensor decomposition scheme which tracks the topology-revealing tensor factors is developed. Extensive tests on simulated and real stock quote data demonstrate the merits of the novel tensor-based approach.

Distributed output feedback consensus tracking prescribed performance control for a class of non‐linear multi‐agent systems with unknown disturbances

This paper studies the problem of distributed output feedback consensus tracking control for leader–following non-linear multi-agent systems in strict-feedback form under a directed graph topology with prescribed performance requirement. The unknown external disturbances are considered in the followers. First, the authors design reduced-order observers to estimate the unmeasured state variables of the followers online. Then, the novel controllers are designed with prescribed performance control based on backstepping method. The dynamic surface control technique is used to deal with the calculating explosion problem. It is strictly proved that the resulting whole system is stable in the sense of semi-globally uniformly ultimately boundedness and both transient and steady-state performance of consensus tracking errors are preserved based on Lyapunov stability theory. Finally, a simulation example is presented to verify the effectiveness of the proposed techniques.

Sampled-Data-Based Consensus and $L_{2}$ -Gain Analysis for Heterogeneous Multiagent Systems.

This paper is concerned with the sampled-data-based consensus problem of heterogeneous multiagent systems under directed graph topology with communication failure. The heterogeneous multiagent system consists of first-order and second-order integrators. Consensus of the heterogeneous multiagent system may not be guaranteed if the communication failure always happens. However, if the frequency and the length of the communication failure satisfy certain conditions, consensus of the considered system can be reached. In particular, we introduce the concepts of communication failure frequency and communication failure length. Then, with the help of the switching technique and the Lyapunov stability theory, sufficient conditions are derived in terms of linear matrix inequalities, which guarantees that the heterogeneous multiagent system not only achieves consensus but also maintains a desired L2 -gain performance. A simulation example is given to show the effectiveness of the proposed method in this paper.

Distributed adaptive coordination control for uncertain nonlinear multi-agent systems with dead-zone input

In this paper, a distributed design scheme is developed for consensus tracking control of multi-agent system with nonlinear input under a weighted directed graph topology. Each agent is modeled by a strict-feedback nonlinear system with unknown nonlinear dynamics and unknown external disturbances. The time-varying leader node only gives commands to a small portion of the followers. By using backstepping technique and neural networks method, adaptive distributed controllers for each follower node are constructed, which only require relative state information between themselves and their neighbors. The proposed controllers and adaptive laws guarantee that the tracking errors between all followers and the leader convergence to a small neighborhood of the origin. Moreover, by employing the maximum norm of the unknown neural network weight vectors as the estimated parameter, the algorithm proposed in this paper contains only N(N represents the number of the followers) adaptive parameters that need to be updated online. The number of online learning parameters is independent of the number of the neural networks׳ nodes, which reduces the computation burden significantly. Finally, a numerical example demonstrates the effectiveness of the proposed approach.

Distributed Adaptive Neural Network Output Tracking of Leader-Following High-Order Stochastic Nonlinear Multiagent Systems With Unknown Dead-Zone Input.

This paper studies the problem of distributed output tracking consensus control for a class of high-order stochastic nonlinear multiagent systems with unknown nonlinear dead-zone under a directed graph topology. The adaptive neural networks are used to approximate the unknown nonlinear functions and a new inequality is used to deal with the completely unknown dead-zone input. Then, we design the controllers based on backstepping method and the dynamic surface control technique. It is strictly proved that the resulting closed-loop system is stable in probability in the sense of semiglobally uniform ultimate boundedness and the tracking errors between the leader and the followers approach to a small residual set based on Lyapunov stability theory. Finally, two simulation examples are presented to show the effectiveness and the advantages of the proposed techniques.

Distributed finite-time containment control for second-order nonlinear multi-agent systems

This paper considers the finite-time containment control problem for a class of second-order nonlinear multi-agent systems with external disturbances under the directed topology graph. The considered nonlinear systems are composed of multiple dynamic leaders, and they have only a subset of the followers with their accessing to the leaders. Firstly, by using the one-hop neighbors’ information, a second-order finite-time distributed observer estimator protocol is constructed to achieve accurate estimation of the weighted average of the velocities and positions of the leaders. Furthermore, it is proved that the proposed estimator protocol can guarantee accurate estimation of the weighted average of the velocities and positions of the leaders in the finite-time. Then, we design a new continuous nonlinear control protocol by means of the nonsingular terminal sliding mode. It is shown that by using proposed control protocol, the states of the followers can converge to the convex hull spanned by those of the leaders in the finite-time. In addition, the finite convergence time can be also explicitly estimated. Finally, the effectiveness of the obtained theoretical results is illustrated by the numerical simulation.

Adaptive group consensus in uncertain networked Euler–Lagrange systems under directed topology

This paper investigates the adaptive group consensus of networked Euler–Lagrange systems with parametric uncertainties under directed topology graph. A novel decomposition approach is developed by using both algebraic graph theory and matrix theory. Three distributed adaptive group consensus protocols are proposed for the cases of topology graphs with acyclic partition and balanced couple, respectively. Some necessary and sufficient conditions for solving group consensus problems are established. It is shown that for the case of directed acyclic graphs, the group consensus can always be guaranteed by the structure of acyclic interaction topology. In particular, an explicit expression of group consensus states can be obtained using the proposed integral protocol, which can be used to develop a unified approach yielding the desired group consensus. For the case of directed balanced couple graphs, a simple algebraic criterion for ensuring group consensus is presented in terms of the eigenvalue computation of Laplacian matrix and thus can be easily applied in practice. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed control methodologies.

Containment control of general multi-agent systems with directed random switching topology

This paper devotes itself to the containment tracking problems of general time-varying high-order multi-agent systems (MAS). In this paper, the directed graph (digraph) topologies of the MAS are random Markovian switching, and the dynamics are not merely confined to be linear. The control protocol that only based on the neighbor׳s states is designed. The containment tracking results are obtained in the asymptotic unbiased mean square sense and with bounded error in mean square sense for general linear case and nonlinear case respectively. Simulations are also provided to demonstrate the effectiveness of our results.

Distributed containment control with predefined performance of high‐order multi‐agent systems with unknown heterogeneous non‐linearities

This study presents a predefined performance design approach for the distributed containment control of heterogeneous non-linear strict-feedback systems with multiple dynamic leaders under a directed graph topology where the leaders are neighbours of only a subset of the followers. It is assumed that all followers’ non-linearities unmatched in the control input are unknown. The performance bounds, which characterise the convergence rate and maximum overshoot of control errors, are incorporated into the control error vectors, in order to ensure the predefined transient and steady-state performance of containment control errors. On the basis of these error vectors, the authors design a simple distributed containment control system without employing any function approximators to estimate unknown non-linearities and calculating the repeated derivatives of virtual controllers. It is shown from Lyapunov stability theorem that all followers nearly converge to the dynamic convex hull spanned by the dynamic leaders and the containment control errors are preserved within certain given predefined bounds.

Cooperative synchronization control for agents with control delays: A synchronizing region approach

This paper investigates multi-agent system synchronization in the presence of control signal delays. Agents are assumed to be identical linear time-invariant systems, interacting on a directed graph topology, all having the same control delay. Distributed control of multi-agent systems is complicated by the fact that the communication graph topology interplays with single-agent dynamics. Here a design method based on a synchronizing region is given that decouples the design of local feedback gains from the detailed properties of graph topology. Such extension of the synchronizing region concept to agents with delays is rigorously justified. Delay-dependent synchronizing region is defined and methods are given guaranteeing its estimates. Qualitative properties of delay-dependent synchronizing regions and implications for control design are discussed. It is found that these regions are inherently bounded which restricts the graphs that allow for synchronization under delayed communication. Stronger property of exponential stability with a prescribed convergence rate is presented as a special case.

Synchronised tracking control for multiple strict‐feedback non‐linear systems under switching network

This study addresses a synchronised tracking problem for multiple strict-feedback systems with different non-linear functions under a jointly switching directed graph topology. A group with a stationary leader or with a dynamic leader is considered separately and each follower only shares information with its neighbours locally. For the case with each leader, the authors propose each distributed consensus algorithm based on the dynamic surface design to guarantee the consensus between the followers and the leader in any dimensional space where the leader's velocity and the derivatives of non-linear functions of neighbours are not required to implement the controller even if the order of followers increases. In particular, it is shown from the Lyapunov-based analysis that consensus errors are cooperatively semi-globally uniformly ultimately bounded and the consensus tracking is achieved in the presence of the stationary leader under the switching jointly connected interaction network and in the presence of the dynamic leader under the directed switching network.

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.

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.

Motion Coordination of Thrust-Propelled Underactuated Vehicles in the Presence of Communication Delays

This paper addresses the coordinated control problem of underactuated thrust-propelled vehicles in the presence of irregular communication delays under a directed communication graph topology. We propose a distributed control algorithm that drives the vehicles to a prescribed formation provided that the communication graph contains a spanning tree. Using the multi-dimensional small-gain approach, we show that the proposed control scheme is robust with respect to varying communication delays, which can be discontinuous with unknown upper bounds. Simulations are provided to illustrate the effectiveness of the proposed control algorithm.

Adaptive Estimation Using Multiagent Network Identifiers With Undirected and Directed Graph Topologies

In this paper, we present an adaptive estimation framework predicated on multiagent network identifiers with undirected and directed graph topologies. Specifically, the system state and plant parameters are identified online using N agents implementing adaptive observers with an interagent communication architecture. The adaptive observer architecture includes an additive term which involves a penalty on the mismatch between the state and parameter estimates. The proposed architecture is shown to guarantee state and parameter estimate consensus. Furthermore, the proposed adaptive identifier architecture provides a measure of agreement of the state and parameter estimates that is independent of the network topology and guarantees that the deviation from the mean estimate for both the state and parameter estimates converge to zero. Finally, an illustrative numerical example is provided to demonstrate the efficacy of the proposed approach.

Distributed adaptive consensus tracking of a class of networked non‐linear systems with parametric uncertainties

This study investigates a distributed adaptive consensus tracking problem for multiple non-linear strict-feedback systems with parametric uncertainties under a directed communication graph topology. It is assumed that the information of a leader is available for only a limited subset of followers. The distributed adaptive dynamic surface design approach is presented to design the local controller of each follower to guarantee that all networked followers with unmatched parametric uncertainties synchronise to the leader regardless of a lack of the shared information on the communication links. From the Lyapunov stability theorem, it is shown that the consensus tracking errors are cooperatively semiglobally uniformly ultimately bounded and converge to an adjustable neighbourhood of the origin.

Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form

This paper presents a distributed containment control approach for uncertain nonlinear strict-feedback systems with multiple dynamic leaders under a directed graph topology where the leaders are neighbors of only a subset of the followers. The strict-feedback followers with nonparametric uncertainties are considered and the local adaptive dynamic surface controller for each follower is designed using only neighbors’ information to guarantee that all followers converge to the dynamic convex hull spanned by the dynamic leaders where the derivatives of leader signals are not available to implement controllers, i.e., the position information of leaders is only required. The function approximation technique using neural networks is employed to estimate nonlinear uncertainty terms derived from the controller design procedure for the followers. It is shown that the containment control errors converge to an adjustable neighborhood of the origin.

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.