Consensus and Cooperation in Networked Multi-Agent Systems

Journal: Proceedings of the IEEE

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Authors:

Abstract:

*This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory.***We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms.**A brief introduction is provided on networked systems with non-local information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small world effects on the speed of consensus algorithms and cooperative control of multi vehicle formations.

Notes:

*In networks of agents (or dynamic systems), “consensus” means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents. A “consensus algorithm” (or protocol) is an interaction rule that specifies the information exchange between an agent and all of its (nearest) neighbors on the network*(pp 215)- In my work, this is agreement on heading and velocity

*Graph Laplacians are an important point of focus of this paper. It is worth mentioning that the second smallest eigenvalue of graph Laplacians called algebraic connectivity quantifies*(pp 216)**the speed of convergence of consensus algorithms**.- So, graph Laplacians are related to adjacency matrices? That has some implications about meaning and flocking, I think???? (A Short Tutorial on Graph Laplacians, Laplacian Embedding, and Spectral Clustering, Graphs and Graph Laplacians)

*More recently, there has been a tremendous surge of interest among researchers from various disciplines of engineering and science in problems related to multi-agent networked systems with close ties to consensus problems. This includes subjects such as consensus [26]–[32], collective behavior of flocks and swarms [19], [33]–[37], sensor fusion [38]–[40], random networks [41], [42], synchronization of coupled oscillators [42]–[46], algebraic connectivity of complex networks [47]–[49], asynchronous distributed algorithms [30], [50], formation control for multi-robot systems [51]–[59], optimization-based cooperative control [60]–[63], dynamic graphs [64]–[67], complexity of coordinated tasks [68]–[71], and consensus-based belief propagation in Bayesian networks [72], [73].*(pp 216)- That is a dense lit review. How did they order it thematically?

*A byproduct of this framework is to demonstrate that seemingly different consensus algorithms in the literature [10],**[12]–[15] are closely related.*(pp 216)*To understand the role of cooperation in performing coordinated tasks, we need to distinguish between unconstrained and constrained consensus problems.*(pp 217)**An unconstrained consensus problem is simply the alignment problem in which it suffices that the state of all agents asymptotically be the same**. In contrast, in distributed computation of a function f(z), the**state of all agents has to asymptotically become equal to f(z), meaning that the consensus problem is constrained. We refer to this constrained consensus problem as the f-consensus problem.**- Normal exploring/flocking/stampeding is
**unconstrained**. Herding adds constraint, though it’s dynamic.**The variables that have to be manipulated in the case of constraint**to result in the same amount of consensus are probably what’s interesting here. Examples could be*how ‘loud’ does the herder have to be?*Also,*how ‘primed’ does the population have to be to accept herding?*

- Normal exploring/flocking/stampeding is
*…cooperation can be informally interpreted as “giving consent to providing one’s state and following a common protocol that serves the group objective.”*(pp 217)*Formal analysis of the behavior of systems that involve more than one type of agent is more complicated, particularly, in presence of adversarial agents in noncooperative games [79], [80].*(pp 217)

*The reason matrix theory [81] is so widely used in analysis of consensus algorithms [10], [12], [13], [14], [15], [64] is primarily due to the structure of P in (4) and its connection to graphs.*(pp 218)*The role of consensus algorithms in particle based flocking is for an agent to achieve velocity matching with respect to its neighbors. In [19], it is demonstrated that flocks are networks of dynamic systems with a dynamic topology. This topology is a proximity graph that depends on the state of all agents and is determined locally for each agent, i.e., the topology of flocks is a state dependent graph. The notion of state-dependent graphs was introduced by Mesbahi [64] in a context that is independent of flocking.*(pp 218)- They leave out heading alignment here. Deliberate? Or is heading alignment just another variant on velocity

*Consider a network of decision-making agents with dynamics ẋ*(pp 219)_{i}= u_{i}interested in reaching a consensus via local communication with their neighbors on a graph G = (V, E). By reaching a consensus, we mean**asymptotically converging to a one-dimensional agreement space**characterized by the following equation: x_{1}= x_{2}= … = x_{n }*A dynamic graph G(t) = (V, E(t)) is a graph in which the set of edges E(t) and the adjacency matrix A(t) are time-varying. Clearly, the set of neighbors N*(pp 219)_{i}(t) of every agent in a dynamic graph is a time-varying set as well. Dynamic graphs are useful for describing the network topology of mobile sensor networks and flocks [19].- (pp 220)
- algebraic connectivity of a graph: The
**algebraic connectivity**(also known as*Fiedler value*or*Fiedler eigenvalue*) of a graph*G*is**the second-smallest eigenvalue of the Laplacian matrix of**.*G*^{[1]}**This eigenvalue is greater than 0 if and only if**This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph.*G*is a connected graph.**The magnitude of this value reflects how well connected the overall graph is. It has been used in analysing the robustness and synchronizability of networks**. (wikipedia) (pp 220) *According to Gershgorin theorem [81], all eigenvalues of L in the complex plane are located in a closed disk centered at delta + 0j*(pp 220)**with a radius of delta, the maximum degree of a graph**- This is another measure that I can do of the nomad/flock/stampede structures combined with DBSCAN. Each agent knows what agents it is connected with, and we know how many agents there are. Each agent row should just have the number of agents it is connected to.

*In many scenarios, networked systems can possess a dynamic topology that is time-varying due to node and link failures/creations, packet-loss [40], [98], asynchronous consensus [41], state-dependence [64], formation reconfiguration [53], evolution [96], and flocking [19], [99].*(pp 226)**Networked systems with a dynamic topology are commonly known as switching networks**.(pp 231)**Conclusion**: A theoretical framework was provided for analysis of consensus algorithms for**networked multi-agent systems with fixed or dynamic topology**and directed information flow. The**connections between consensus problems and several applications were discussed that include synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation**. The role of “cooperation” in distributed coordination of networked autonomous systems was clarified and the effects of lack of cooperation was demonstrated by an example. It was demonstrated that notions such as**graph Laplacians, nonnegative stochasticmatrices, and algebraic connectivity of graphs and digraphs play an instrumental role in analysis of consensus algorithms**. We proved that algorithms introduced by Jadbabaie et al. and Fax and Murray are identical for graphs with n self-loops and are both special cases of the consensus algorithm of Olfati-Saber and Murray. The notion of Perron matrices was introduced as the discrete-time counterpart of graph Laplacians in consensus protocols. A number of fundamental spectral properties of Perron matrices were proved. This led to**a unified framework for expression and analysis of consensus algorithms in both continuous-time and discrete-time**. Simulation results for reaching a consensus in small-worlds versus lattice-type nearest-neighbor graphs and cooperative control of multivehicle formations were presented.