Performance and design of cycles in consensus networks☆
Introduction
The consensus protocol has recently emerged as a canonical model for studying networked dynamic systems. The simplicity of the model, most often presented as a collection of single integrators interacting over a communication graph, reveals a deep connection between its dynamic behavior and the underlying properties of the graph [1]. The use of consensus models, even beyond its analytical elegance, is its practical relevance in applications ranging from optimization and sensor fusion to problems in formation control and distributed control and estimation [2], [3], [4], [5], [6]. It is precisely the utility and simplicity of this model that has pushed research in this area to pursue a more general control theory for networked dynamic systems.
Within the framework of this setup, the use of graph theory as a tool for analysis is recognized as the correct mathematical abstraction to study these systems. Perhaps the most celebrated result in this direction is the relationship of the algebraic connectivity of a graph, sometimes referred to as the Fiedler eigenvalue [7], to the convergence rate of the dynamic system. The importance of this property has been extended in many directions, including convergence analysis for random graphs [8] and graphs with communication delays and switching topologies [9]. Various modifications to the consensus protocol have also led to more general system theoretic notions such as controllability and observability [10], [11] and input–output properties [12], [13], [14].
While the analysis of consensus systems has matured, work related to the design of consensus protocols and the characterization of system performance beyond that of its convergence rate have not been as deeply studied. As the consensus protocol becomes more integrated into complex real-world networks, the importance of design from an optimality perspective becomes crucial. A fundamental challenge for the design of consensus networks, however, relates to the computational complexity of solving combinatorial problems. A common approach to this problem is to consider optimization over weighted graphs or other convex relaxations [15], [16], [17]. However, most results that deal with the design of networks focus on the optimization of the Fiedler eigenvalue, aligning with the mainstream results related to connectivity and convergence rates [18], [19].
An important extension of these works, therefore, is the introduction of more general notions of system performance in coordination with the design of these networks. A first step in this direction is to include exogenous inputs in the form of noises and disturbances into the consensus protocol. Such models have been considered in [17], [20], [21] where noises were introduced in either the process or measurement of the consensus protocol. This more general model of the consensus protocol better reflects their use in real-world systems. Indeed, multi-agent systems relying on relative sensing, such as formation flying or sensor fusion, will depend on sensors and actuators that are imperfect.
Noises in the consensus protocol lead to a random walk of the agreement value, and attempts to compensate for this includes the design of edge weights [17], or the introduction of a time-varying gain on the control to effectively reject the noise asymptotically [20]. When the consensus protocol is corrupted by noise, the system norm provides a measure of how the noises affect the asymptotic deviation of each node’s state from a consensus state. This performance metric has been used to study, for example, the leader selection problem for consensus networks [22]. The performance of large scale networked systems and fractal graphs were considered in [23], [24]. Other works have considered this metric for consensus networks over directed graphs [25], spanning trees [26], and biochemical networks [27].
However, many of these results do not exploit the underlying combinatorial properties of the graph as it relates to these metrics. We emphasize a distinction between spectral properties of the graph, i.e. the eigenvalues of the Laplacian matrix, and combinatorial properties of a graph such as path lengths and cycles. Indeed, spectral properties introduce a layer of abstraction to the underlying graph that makes more tangible design issues, such as edge costs and distances, less intuitive.
A thorough treatment in this direction was recently given in [28] via the introduction of the Edge Laplacian and its corresponding edge agreement problem. The edge Laplacian is a variant of the graph Laplacian that provides a more transparent understanding of how spanning trees and cycles affect certain algebraic properties of a graph. When the consensus protocol is analyzed using this construction, clear graph theoretic interpretations of the norm of the system can be derived [28].
An unresolved question from [28], however, was the precise role that cycles play in the performance of consensus networks. The main result from [28] showed that the performance is determined by the inverse of a matrix related to the cycle structure of the graph. However, the precise structure of this matrix was not considered, and this work contributes in that regard. It is well known that the addition of cycles in a graph will increase its algebraic connectivity [29],1 however, a similar result has not been found for other performance metrics. A fundamental contribution of this work, therefore, is an analytic characterizations of how cycles affect the performance of consensus networks. In this direction, we provide new interpretations for the role of cycles as related to algebraic properties of the edge Laplacian, and dynamic properties of the consensus protocol. In particular, we show that the norm of the consensus protocol always improves with the addition of cycles. We provide an exact characterization of how the addition of one or two cycles improves the performance. The improvement is proportional to the inverse of the cycle lengths and also related to the number of edges that are shared between cycles. This establishes a strong connection between combinatorial properties of the underlying graph and dynamic properties of the corresponding protocol.
The analytic results are then used to formulate a synthesis problem for consensus networks. The problem considers the task of adding a fixed number of edges to an existing consensus network over a spanning tree that leads to the greatest improvement in performance. A first approach to this problem leads to a mixed-integer program, generally considered a hard problem to solve due to its combinatorial nature. This combinatorial problem can be formulated as an -optimization problem. By combining graph theoretic insights with results from compressed sensing [30], [31], we reformulate the problem as a reweighted -optimization problem. The -optimization problems are well known to achieve sparse solutions [32], [33], [34] and we show that this convex relaxation of the original mixed-integer problem gives very good results. We also highlight how the weighting mechanism used in the formulation provides an important tuning parameter for design. The analytic results of this work provide intuition for appropriate weighting functions, including cycle length weighting and cycle correlation weighting. This formulation allows to consider additional performance criteria including maximizing the algebraic connectivity of the graph. These results are demonstrated via some simulation examples.
This paper is organized as follows. Section 2 reviews the fundamental properties of the edge Laplacian and provides results relating to algebraic properties of this matrix and cycles in this graph. The edge agreement problem is given in Section 3. These results are then applied in Section 4 to derive the performance of the agreement protocol as a function of the edges in the graph. The synthesis problem and formulation is given in Section 5, and numerical simulations are presented in Section 6. Finally, we offer some concluding remarks in Section 7.
The notation used is standard. The set of real numbers is denoted by . For a vector , its transpose is given by and the th component by ; the th element of a matrix is denoted . The trace of a matrix is denoted . The null space and range space of a matrix is denoted as and respectively. A symmetric matrix is positive definite (semi-definite) if all its eigenvalues are positive (non-negative), and is denoted ; the linear matrix inequality is equivalent to . The cardinality of a set is denoted as . The norm of a vector is defined as . The -norm of a vector is defined as , the number of non-zero elements in the vector . Note that in fact, the -norm is not a true norm, but it is commonly referred to as a norm in the literature and we adopt that convention in this work.
Section snippets
Cycles and the edge Laplacian
As discussed in the introduction, graph theory plays a central role in the analysis of consensus networks. In particular, the consensus protocol is described in terms of a certain algebraic representation of the underlying communication graph known as the (graph) Laplacian. The work presented in here, however, relies on an alternative representation that we term the edge Laplacian [28]. In this section, we review the construction of the edge Laplacian and focus the presentation on the role
The edge agreement problem
The standard consensus model is based on a collection of single integrator agents that exchange relative state information over a communication graph to generate a control. The model is usually presented as an autonomous system with no noises or disturbances [1], Here, the vector is the concatenated state of each agent, and is the Laplacian matrix.
A two-port interpretation of the consensus protocol provides a framework for considering the presence of exogenous
Performance of cycles
In this work, we consider the performance of the edge agreement problem (10), and in particular focus on how the addition of cycles impacts this measure. Analyzing the consensus protocol with this metric is meaningful in the sense that it provides a measure of how noises can affect the asymptotic deviation of each agent from a consensus configuration; i.e., it studies the effect of noises on the relative states in the consensus protocol. As discussed in [17], [20], noises can lead to a
Design of cycles
The results of Section 4 provide a clear analytic picture of how cycles impact the performance of the edge agreement problem. A trivial conclusion that one may arrive at is to always use the complete graph. The complete graph also, for example, has the largest algebraic connectivity and thus is desirable for other performance indicators. However, as is common in real-world engineering applications, it may not be feasible to implement a consensus network with all possible
Simulation example
In this section we demonstrate the design procedure described in Section 5 with a few numerical examples. For each example we will work with the same spanning tree graph on nodes, generated randomly in MATLAB; see Fig. 4. The performance and algebraic connectivity for this graph can be determined as and and for the complete graph and . The longest cycle in this graph is determined by its diameter, .
For this example, there are
Concluding remarks
This work provided a characterization of how cycles impact the performance of consensus networks. This analysis was facilitated by using an edge variant of the graph Laplacian matrix, termed the essential edge Laplacian. This matrix representation of the graph leads to a deeper insight of the role cycles play for certain algebraic properties. When applied to the corresponding edge agreement problem, the role of cycles were related to the performance of the system. In particular, the purely
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The authors thank the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology and the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems”.