Consensus Manifold Research Articles

MRAC-based dynamic consensus of linear systems with biased measurements over directed networks

In this paper we address the dynamic leaderless consensus control problem for generic linear systems (stable, unstable, marginally stable, etc.) interconnected over directed networks and under the influence of biased measurements. Essentially, the control problem consists in redesigning a standard distributed consensus controller which, for each system, relies on own state biased measurements and respective erroneous data received from a set of neighbors. The difficulty in such a scheme resides in the fact that the measurement bias is directly amplified by the control gain so it cannot be handled as an additive external disturbance. Our control design relies, on one hand, on the solution of a Riccati equation and, on the other, on the design of an estimator reminiscent of a model-reference-adaptive control design. The estimator successfully computes a bias estimate and completely compensates for its effect if the bias is constant—indeed, in this case, we establish exponential stability of the consensus manifold and we show that the controller provides robustness with respect to time-varying biases.

DCCNMF: Deep Complementary and Consensus Non-negative Matrix Factorization for multi-view clustering

Deep non-negative matrix factorization-based methods have recently been explored in multi-view clustering due to their ability to deal with complex non-linear data. Although a few methods exist that learn both complementary and consensus information simultaneously by adding new regularizations or adjusting hyperparameters without providing a solid architecture, they do not fully exploit this information. This paper proposes a novel method called Deep Complementary and Consensus Non-negative Matrix Factorization (DCCNMF) that combines the strengths of Non-negative Matrix Factorization and Coupled Non-negative Matrix Factorization using a novel architecture to simultaneously learn both complementary and consensus information present in multi-view data. Two manifold regularization terms, namely complementary manifold and consensus manifold are introduced to preserve the view-specific and view-shared geometric structures during the dimensionality reduction from higher to lower order. Further, smoothness regularization is employed to achieve distinct low-order representations by maximizing the cosine similarity among data points with similar orientations and minimizing it among data points with dissimilar orientations. DCCNMF and benchmark methods are evaluated on several real-world datasets and the results demonstrate that DCCNMF significantly outperforms state-of-the-art multi-view learning approaches.

Consensus verification for heterogeneous polynomial networked systems

This article is concerned with the consensus verification for heterogeneous polynomial networked systems (HPNSs) through the distributed nonlinear control protocol. Firstly, a necessary condition to achieve consensus for HPNSs is presented (i.e., the consensus manifold is existing and it is an invariant manifold). On this premise, with polynomial Lyapunov functions, several consensus criteria are proposed for HPNSs under the undirected and directed graphs. Compared with the existing criteria from quadratic Lyapunov functions, our results are less conservative and more general. On basis of the proposed criteria, the consensus verification problem is then reduced to a sum-of-squares optimization problem for finding polynomial Lyapunov functions. The resulting sum-of-squares optimization falls within the convex programming framework, which can be solved efficiently in polynomial time. Finally, the theoretical and algorithmic developments are demonstrated on two numerical examples. Simulation results show that our method can be used to conduct fully automatic verification of consensus for HPNSs, where the widely used hand-crafted quadratic Lyapunov functions maybe non-existent.

Backstepping leaderless dynamic consensus among nonlinear systems over directed networks: application to attitude synchronisation control*

We present a distributed consensus controller for multi-agent homogeneous nonlinear systems over directed networks. The systems are assumed to be of second order and the control design relies on a standard backstepping approach. In that light, the control design also hinges on the ability to construct a strict Lyapunov function for the multi-agent nonlinear system interconnected over a directed graph for which the only assumption is that it is connected. That is, that there exists a directed spanning tree, but there is no requirement of conservative conditions such as strong or balanced connectivity. To construct a strict Lyapunov function, we use a generalised Lyapunov equation for the directed-graph Laplacian matrix, which characterises the spanning-tree-existence condition. Then, we establish exponential stability of the consensus manifold. In addition, we implement our dynamic consensus controller on a multi-agent satellite system in the context of attitude synchronisation and demonstrate its efficacy in numerical simulations.

Edge-based strict Lyapunov functions for consensus with connectivity preservation over directed graphs

In this paper we address the edge-agreement problem with preserved connectivity for networks of first and second-order systems under proximity constraints and interconnected over a class of directed graphs. We provide a strict Lyapunov function that leads to establishing uniform asymptotic stability of the consensus manifold with guaranteed connectivity preservation. Furthermore, robustness of the edge-agreement protocol, in the sense of input-to-state stability with respect to external input disturbances, is also demonstrated. These results hold for directed-spanning-tree and directed-cycle topologies, which are notably employed, respectively, in leader–follower and cyclic-pursuit control.

MAT2: manifold alignment of single-cell transcriptomes with cell triplets.

Aligning single-cell transcriptomes is important for the joint analysis of multiple single-cell RNA sequencing datasets, which in turn is vital to establishing a holistic cellular landscape of certain biological processes. Although numbers of approaches have been proposed for this problem, most of which only consider mutual neighbors when aligning the cells without taking into account known cell type annotations. In this work, we present MAT2 that aligns cells in the manifold space with a deep neural network employing contrastive learning strategy. Compared with other manifold-based approaches, MAT2 has two-fold advantages. Firstly, with cell triplets defined based on known cell type annotations, the consensus manifold yielded by the alignment procedure is more robust especially for datasets with limited common cell types. Secondly, the batch-effect-free gene expression reconstructed by MAT2 can better help annotate cell types. Benchmarking results on real scRNA-seq datasets demonstrate that MAT2 outperforms existing popular methods. Moreover, with MAT2, the hematopoietic stem cells are found to differentiate at different paces between human and mouse. MAT2 is publicly available at https://github.com/Zhang-Jinglong/MAT2. Supplementary data are available at Bioinformatics online.

Stability and robustness of edge-agreement-based consensus protocols for undirected proximity graphs

We address the consensus problem, with a guarantee of connectivity maintenance, for multiagent systems of first- and second-order, communicating over a connected, undirected graph with proximity constraints. Our approach relies on a so-called barrier Lyapunov functions, which encode the distance constraints between pairs of agents. Beyond the consensus algorithms, our primary contribution is to provide strict barrier Lyapunov functions, i.e. positive definite and with negative definite derivative in the agreement subspace. Thus, uniform asymptotic stability of the consensus manifold and the maintenance of the connectivity are guaranteed. Furthermore, with the said barrier functions, we demonstrate the robustness of consensus protocols by establishing global input-to-state stability.

Leader-Follower Consensus of Unicycles With Communication Range Constraints via Smooth Time-Invariant Feedback

We solve the full-consensus problem (in position and orientation) with connectivity maintenance for multiple nonholonomic vehicles in a leader-follower configuration. We rely on a polar-coordinates based model which is more natural for the problem setting. The proposed control law is smooth (in the domain of definition) time-invariant and uses only relative measurements, making it more suited for implementation. We establish asymptotic convergence to the consensus manifold as well as connectivity maintenance using the Lyapunov's first method and cascaded systems theory. In addition, we illustrate our theoretical contributions experimentally.

Exponential Consensus of Linear Systems Over Switching Network: A Subspace Method to Establish Necessity and Sufficiency.

In this article, the consensus problem of linear systems is revisited from a novel geometric perspective. The interaction network of these systems is assumed to be piecewise fixed. Moreover, it is allowed to be disconnected at any time but holds a quite mild joint connectivity property. The system matrix is marginally stable and the input matrix is not of full-row rank. By directly examining the subspace determined by the network, we first establish convergence by resorting to an observability condition. Then, according to joint connectivity, we are able to extend this convergence uniformly to the entire orthogonal complement of the consensus manifold. In this way, we work out the necessary and sufficient condition for exponential consensus. It turns out that, with a suitably designed feedback matrix, exponential consensus can be realized globally and uniformly if and only if a jointly (δ,T) -connected condition and an observability condition relying only on the system and input matrices are satisfied. We also characterize the lower bound of the convergence rate. Simple yet effective examples are presented to illustrate the findings.

Robust Consensus and Connectivity-maintenance under Edge-agreement-based Protocols for Directed Spanning Tree Graphs

We address the consensus problem with connectivity maintenance for networks of multi-agent systems interconnected over directed spanning tree graphs in the edge-agreement space and, for the first time in the literature, we provide a strict Lyapunov function. The importance of this contribution is that it allows to establish uniform global asymptotic stability of the consensus manifold for a multi-agent system subject to proximity constraints. Moreover, robustness in the sense of input-to-state stability with respect to external disturbances is also established. These properties have not been established before when dealing with state-dependent constraints, even for a class of directed graphs, because most often in the literature only non-uniform convergence to the consensus manifold is established.

Control of a Driftless Bilinear Vector Field on $n$-Sphere

In this paper, we consider a multi-input driftless bilinear system evolving on the $n$ -dimensional sphere $S^{n}$ . We first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on $S^{n}$ . For the general framework, we establish the global controllability on $S^{n}$ and propose two linear control laws on $S^{n}$ that achieve asymptotic stabilization of an equilibrium point with an almost global domain-of-attraction. Further, the asymptotically stable closed-loop system trajectories are shown to be arcs on the geodesics of $S^{n}$ for a particular choice of the equilibrium point. Next, we propose two linear time-varying control laws to achieve trajectory tracking on $S^{n}$ and show the asymptotic stability of the tracking error. A distributed control is designed for the consensus of multiagent bilinear systems on $S^{n}$ with an undirected tree as the communication graph. The consensus manifold is shown to have an almost global domain of attraction.

Distributed Optimal Consensus Control Algorithm for Continuous-Time Multi-Agent Systems

This brief proposes a distributed algorithm to solve the optimal consensus control problem of continuous-time multi-agent systems. With the definition of consensus manifold, the utility function of the optimization problem is formulated as a linear quadratic polynomial which indicates the tradeoff between convergence rate and energy cost in a finite horizon. Applying the alternating direction method of multiplier, the optimal consensus problem is separated into three subproblems: 1) an input optimization problem; 2) a consensus state optimization problem; and 3) a dual optimization problem, thereby separating the communication topology and the agents dynamics. Then, these subproblems can be analyzed and solved independently, following from which that a distributed control algorithm is proposed to allow the continuous-time multi-agent systems to reach optimal consensus. A numerical example is given to demonstrate the effectiveness of the proposed algorithm.

Local open‐ and closed‐loop manipulation of multiagent networks

SummaryThe manipulation of networked cyberphysical devices via local external actuation or feedback control is explored, in the context of a canonical multiagent dynamical system that is engaged in a consensus or synchronization task. One main focus is to understand whether or not, and how easily, a stakeholder can manipulate the full network's dynamics by hijacking only one agent's actuation signal. Explicit spectral characterizations are given of the energy (effort) required to manipulate the dynamics. These characterizations are used to (1) gain structural insights into ease of manipulation, (2) show that manipulation along the consensus manifold is easy, and (3) address network design to enable or prevent manipulation. Additionally, it is shown that the multiagent system can be manipulated effectively along the consensus manifold using local feedback controls, which do not require model knowledge or wide‐area measurements.

A lifting method for analyzing distributed synchronization on the unit sphere

Abstract This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d -dimensional unit sphere are lifted to the ( d + 1 ) -dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence.

Consensus With Output Saturations

This paper considers a standard consensus algorithm under output saturations. In this case, the global consensus cannot be realized due to the existence of stable equilibrium points that do not belong to the consensus manifold. Therefore, this paper investigates necessary and sufficient initial conditions for the achievement of consensus that characterize an exact domain of attraction. Specifically, this paper considers single-integrator agents with both fixed and time-varying undirected graphs. Then, we show that the consensus will be achieved if and only if the average of the initial states is within the minimum saturation level.

Multi-view clustering via multi-manifold regularized non-negative matrix factorization

Non-negative matrix factorization based multi-view clustering algorithms have shown their competitiveness among different multi-view clustering algorithms. However, non-negative matrix factorization fails to preserve the locally geometrical structure of the data space. In this paper, we propose a multi-manifold regularized non-negative matrix factorization framework (MMNMF) which can preserve the locally geometrical structure of the manifolds for multi-view clustering. MMNMF incorporates consensus manifold and consensus coefficient matrix with multi-manifold regularization to preserve the locally geometrical structure of the multi-view data space. We use two methods to construct the consensus manifold and two methods to find the consensus coefficient matrix, which leads to four instances of the framework. Experimental results show that the proposed algorithms outperform existing non-negative matrix factorization based algorithms for multi-view clustering.

Second-Order Locally Dynamical Consensus of Multiagent Systems With Arbitrarily Fast Switching Directed Topologies

This paper is mainly concerned with the analysis of the second-order locally dynamical consensus of multiagent systems with nonlinear dynamics in the directed networks with arbitrarily fast switching topologies. In our designed framework, the time-varying network topology is constant in each time interval and then randomly jumps to another topology when certain occasional events occur at some random moments. In addition, we further assume that the dwell time of each topology is unknown in advance and the corresponding adjacency weighted matrix is not necessarily nonnegative due to the probable existence of deteriorated communication channels in the underlying interaction network. By the orthogonal decomposition method, the state vector of the resultant error dynamical system can be further decomposed as two transversal components, one of which evolves along the consensus manifold and the other evolves transversally with the consensus manifold. Then, by introducing the generalized matrix measure and by applying the tools of contraction and circle analysis, the second-order locally dynamical consensus of multiagent systems with arbitrarily fast switching directed topologies is theoretically investigated in detail, and some easily verified sufficient conditions are also presented. It is shown that, under sufficiently large coupling strengths, the random switchings can be effectively tolerated and the consensus can be achieved for all agents in the network. Finally, numerical simulation examples are also provided to demonstrate the feasibility and effectiveness of the obtained theoretical results.

Second-Order Consensus Seeking in Multi-Agent Systems With Nonlinear Dynamics Over Random Switching Directed Networks

This paper discusses the second-order local consensus problem for multi-agent systems with nonlinear dynamics over dynamically switching random directed networks. By applying the orthogonal decomposition method, the state vector of resulted error dynamical system can be decomposed as two transversal components, one of which evolves along the consensus manifold and the other evolves transversally with the consensus manifold. Several sufficient conditions for reaching almost surely second-order local consensus are derived for the cases of time-delay-free coupling and time-delay coupling, respectively. For the case of time-delay-free coupling, we find that if there exists one directed spanning tree in the network which corresponds to the fixed time-averaged topology and the switching rate of the dynamic network is not more than a critical value which is also estimated analytically, then second-order dynamical consensus can be guaranteed for the choice of suitable parameters. For the case of time-delay coupling, we not only prove that under some assumptions, the second-order consensus can be reached exponentially, but also give an analytical estimation of the upper bounds of convergence rate and the switching rate. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained theoretical results.

Consensus in Directed Networks of Agents With Nonlinear Dynamics

This technical note studies the consensus problem for cooperative agents with nonlinear dynamics in a directed network. Both local and global consensus are defined and investigated. Techniques for studying the synchronization in such complex networks are exploited to establish various sufficient conditions for reaching consensus. The local consensus problem is first studied via a combination of the tools of complex analysis, local consensus manifold approach, and Lyapunov methods. A generalized algebraic connectivity is then proposed to study the global consensus problem in strongly connected networks and also in a broad class of networks containing spanning trees, for which ideas from algebraic graph theory, matrix theory, and Lyapunov methods are utilized.