We consider the problem of robustness in large consensus networks that occur in many areas such as distributed optimization. Robustness, in this context, is the scaling of performance measures, e.g. H2-norm, as a function of network dimension. We provide a formal framework to quantify the relation between such performance scaling and the convergence speed of the network. Specifically, we provide upper and lower bounds for the convergence speed in terms of robustness and discuss how these bounds scale with the network topology. The main contribution of this work is that we obtain tight bounds, that hold regardless of network topology. The work here also encompasses some results in convergence time analysis in previous literature.
Consensus of autonomous agents is a benchmark problem in cooperative control. In this paper, we consider standard continuous-time averaging consensus policies (or Laplacian flows) over time-varying graphs and focus on robustness of consensus against communication delays. Such a robustness has been proved under the assumption of uniform quasi-strong connectivity of the graph. It is known, however, that the uniform connectivity is not necessary for consensus. For instance, in the case of undirected graph and undelayed communication consensus requires a much weaker condition of integral connectiv…
In this paper, we first propose a novel maneuvering technique compatible with displacement-consensus-based formation controllers. We show that the formation can be translated with an arbitrary velocity by modifying the weights in the consensus Laplacian matrix. In fact, we demonstrate that the displacement-consensus-based formation control is a particular case of our more general method. We then uncover robustness issues with undesired steady-state motions and resultant distorted shapes in undirected displacement-consensus-based formation control. In particular, these issues are triggered when…
We revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the well-known Gradient Tracking and with distributed integral control. Ov…
We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady state population variance of states under the noisy consensus dynamics with edge weights from the unit interval. We derive tight upper and lower bounds on the structural robustness of networks based on the average distance between nodes and the average node degree. Using the proposed bounds, we characterize the networks with different types of robustness scaling under increasing size. Furthermore, w…
Consensus of autonomous agents is a benchmark problem in multi-agent control. In this paper, we consider continuous-time averaging consensus policies (or Laplacian flows) and their discrete-time counterparts over time-varying graphs in presence of unknown but bounded communication delays. It is known that consensus is established (no matter how large the delays are) if the graph is periodically, or uniformly quasi-strongly connected (UQSC). The UQSC condition is often believed to be the weakest sufficient condition under which consensus can be proved. We show that the UQSC condition can actual…
This paper examines the consensus problem on time-varying matrix-weighed undirected networks. First, we introduce the matrix-weighted integral network for the analysis of such networks. Under mild assumptions on the switching pattern of the time-varying network, necessary and/or sufficient conditions for which average consensus can be achieved are then provided in terms of the null space of matrix-valued Laplacian of the corresponding integral network. In particular, for periodic matrix-weighted time-varying networks, necessary and sufficient conditions for reaching average consensus is obtain…
In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks. To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matri…
A networked system can be made resilient against adversaries and attacks if the underlying network graph is structurally robust. For instance, to achieve distributed consensus in the presence of adversaries, the underlying network graph needs to satisfy certain robustness conditions. A typical approach to making networks structurally robust is to strategically add extra links between nodes, which might be prohibitively expensive. In this paper, we propose an alternative way of improving network's robustness, that is by considering heterogeneity of nodes. Nodes in a network can be of different …
The association of weights in a distributed consensus protocol quantify the trust that an agent has on its neighbors in a network. An important problem in such networked systems is the uncertainty in the estimation of trust between neighboring agents, coupled with the losses arising from mistakenly associating wrong amounts of trust with different neighboring agents. We introduce a probabilistic approach which uses the historical data collected in the network, to determine the level of trust between each agent. Specifically, using the finite history of the shared data between neighbors, we obt…
This paper investigates the problem of securing exponentially fast consensus (exponential consensus for short) for identical agents with finite-dimensional linear system dynamics over dynamic network topology. Our aim is to find the weakest possible conditions that guarantee exponentially fast consensus using a Lyapunov function consisting of a sum of terms of the same functional form. We first investigate necessary conditions, starting by examining the system (both agent and network) parameters. It is found that controllability of the linear agents is necessary for reaching consensus. Then, t…
This paper considers the distributed robust suboptimal consensus control problem of linear multi-agent systems, with both H2 and H_infty performance requirements. A novel two-step complementary design approach is proposed. In the first step, a distributed control law is designed for the nominal multi-agent system to achieve consensus with a prescribed H2 performance. In the second step, an extra control input, depending on some carefully chosen residual signals indicating the modeling mismatch, is designed to complement the H2 performance by providing robustness guarantee in terms of H_infty r…
This paper examines the cluster consensus problem of multi-agent systems on matrix-weighted switching networks. Necessary and/or sufficient conditions under which cluster consensus can be achieved are obtained and quantitative characterization of the steady-state of the cluster consensus are provided as well. Specifically, if the underlying network switches amongst finite number of networks, a necessary condition for cluster consensus of multi-agent system on switching matrix-weighted networks is firstly presented, it is shown that the steady-state of the system lies in the intersection of the…
The existing cryptosystem based approaches for privacy-preserving consensus of networked systems are usually limited to those with undirected topologies. This paper proposes a new privacy-preserving algorithm for networked systems with directed topologies to reach confidential consensus. As a prerequisite for applying the algorithm, a structural consensus problem is formulated and the solvability conditions are discussed for an explicitly constructed controller. The controller is then implemented with encryption to achieve consensus while avoiding individual's information leakage to external e…
In this paper, we study the robustness of the event-triggered consensus algorithms against frequency-domain uncertainties. It is revealed that the sampling errors resulted by event triggering are essentially images of linear finite-gain $\mathcal{L}_2$-stable operators acting on the consensus errors of the sampled states and the event-triggered mechanism is equivalent to a negative feedback loop introduced additionally to the feedback system. In virtue of this, the robust consensus problem of the event-triggered network systems subject to additive dynamic uncertainties and network multiplicati…
This paper proposes model reduction approaches for consensus network systems based on a given clustering of the underlying graph. Namely, given a consensus network system of time-scaled agents evolving over a weighted undirected graph and a graph clustering, a parameterized reduced consensus network system is constructed with its edge weights and nodal time-scales as the parameters to be optimized. H-infinity- and H-2-based optimization approaches are proposed to select the reduced network parameters such that the corresponding approximation errors, i.e., the H-infinity- and H-2-norms of the e…
In this work, inspired by secret sharing schemes, we introduce a privacy-preserving approach for network consensus, by which all nodes in a network can reach an agreement on their states without exposing the individual state to neighbors. With the privacy degree defined for the agents, the proposed method makes the network resistant to the collusion of any given number of neighbors, and protects the consensus procedure from communication eavesdropping. Unlike existing works, the proposed privacy-preserving algorithm is resilient to node failures. When a node fails, the method offers the possib…
This paper provides a control protocol for the robust output feedback consensus of networked heterogeneous nonlinear negative-imaginary (NI) systems. Heterogeneous nonlinear output strictly negative-imaginary (OSNI) controllers are applied in positive feedback according to the network topology to achieve output feedback consensus. The main contribution of this paper is extending the previous studies of the robust output feedback consensus problem for networked heterogeneous linear NI systems to nonlinear NI systems. Output feedback consensus is proved by investigating the internal stability of…
This paper presents new classes of consensus protocols with fixed-time convergence, which enable the definition of an upper bound for consensus state as a parameter of the consensus protocol, ensuring its independence from the initial condition of the nodes. We demonstrate that our methodology subsumes current classes of fixed-time consensus protocols that are based on homogeneous in the bi-limit vector fields. Moreover, the proposed framework enables for the development of independent consensus protocols that are not needed to be homogeneous in the bi-limit. This proposal offers extra degrees…
The Evolution of Network Entropy in Classical and Quantum Consensus Dynamics
Research School of Engineering, The Australian National University（澳大利亚国立大学工程研究学院）；School of Engineering and Information Technology, University of New South Wales（新南威尔士大学工程与信息技术学院）
Volt-VAR control (VVC) is a critical application in active distribution network management system to reduce network losses and improve voltage profile. To remove dependency on inaccurate and incomplete network models and enhance resiliency against communication or controller failure, we propose consensus multi-agent deep reinforcement learning algorithm to solve the VVC problem. The VVC problem is formulated as a networked multi-agent Markov decision process, which is solved using the maximum entropy reinforcement learning framework and a novel communication-efficient consensus strategy. The p…
