Distributed Coordination of Multi-agent Networks

This chapter introduces notations used in the book, algebraic graph theory background, algebra and matrix theory background, linear and nonlinear system theory background, nonsmooth analysis background, and time-delay system theory background.

This chapter overviews recent research results in distributed multi-agent coordination. Distributed coordination of multiple autonomous agents, including unmanned aerial vehicles (UAVs), unmanned ground vehicles (UGVs), and unmanned underwater vehicles (UUVs), has been a very active research topic in the systems and controls society. The recent research results in distributed multi-agent coordination are roughly categorized as consensus, distributed formation control, distributed optimization, distributed task assignment, distributed estimation and control, and intelligent coordination. A short discussion is given to propose several future research directions and problems that deserve further investigation.

This chapter introduces a collective periodic motion coordination problem. Coordinated periodic motions play an important role in applications involving multi-agent networks with repetitive movements such as cooperative patrol, mapping, sampling, or surveillance. We introduce two types of algorithms. For the first type, we introduce Cartesian coordinate coupling to existing distributed consensus algorithms for respectively, single-integrator dynamics and double-integrator dynamics, to generate collective motions, namely, rendezvous, circular patterns, and logarithmic spiral patterns in the three-dimensional space. It is shown that the interaction graph and the value of the Euler angle in the case of single-integrator dynamics and the interaction graph, the damping gain, and the value of the Euler angle in the case of double-integrator dynamics affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound in the case of double-integrator dynamics, and the Euler angle is below, equal, or above a critical value, the agents will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane normal to the Euler axis. For the second type, we introduce coupled second-order linear harmonic oscillators with local interaction to generate synchronized oscillatory motions. We analyze convergence conditions under, respectively, directed fixed and switching interaction graphs. It is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions. The theoretical result is also applied to synchronized motion coordination in multi-agent systems as a proof of concept.

This chapter introduces a collective tracking problem in the presence of a dynamic leader. The problem has applications in formation flying, body guard, and target tracking. We solve the distributed collective tracking problem via a variable structure approach when there exists a dynamic leader who is a neighbor of only a subset of a group of followers, all followers have only local interaction, and only partial measurements of the states of the leader and the followers are available. In the context of collective tracking, we focus on both coordinated tracking and swarm tracking algorithms. The objective of coordinated tracking is that a group of followers intercepts a dynamic leader with local interaction. The objective of swarm tracking is that a group of followers moves cohesively with a dynamic leader while avoiding inter-agent collision with local interaction. Both single-integrator dynamics and double-integrator dynamics are considered. Several simulation examples are presented as a proof of concept.

This chapter introduces a containment control problem, where a group of followers is driven by a group of leaders to be in the convex hull spanned by the leaders. The problem has applications in hazardous material handling, search and rescue, and cooperative transport. We consider three cases, namely, containment control with multiple stationary leaders, containment control with multiple dynamic leaders, and containment control with swarming behavior. Simulation results are presented to illustrate the theoretical results.

This chapter moves from point models primarily adopted in distributed multi-agent coordination to more realistic Lagrangian models. A class of mechanical systems including autonomous vehicles, robotic manipulators, and walking robots are Lagrangian systems. We focus on fully-actuated Lagrangian systems. We first study distributed leaderless coordination algorithms for networked Lagrangian systems. The objective is to drive a team of agents modeled by Euler–Lagrange equations to achieve desired relative deviations on their vectors of generalized coordinates with local interaction. We then study distributed coordinated regulation and distributed coordinated tracking algorithms in the presence of a leader for networked Lagrangian systems under the constraints that the leader is a neighbor of only a subset of the followers and the followers have only local interaction. In the case of coordinated regulation, the leader has a constant vector of generalized coordinates. In the case of coordinated tracking, the leader has a varying vector of generalized coordinates. In both cases, the objective is to drive the vectors of generalized coordinates of a team of followers modeled by Euler–Lagrange equations to approach that of a leader. Simulation results show the effectiveness of the proposed algorithms.

This chapter moves from integer-order dynamics to fractional-order dynamics motivated by real-world phenomena. We first study distributed coordination of networked fractional-order systems under a directed fixed interaction graph. We show sufficient conditions on the interaction graph and the fractional order such that coordination is achieved. The coordination equilibrium is also given explicitly. We then study distributed coordination of networked fractional-order systems under a directed switching interaction graph. The convergence conditions on both the interaction graph and the fractional order are presented. We finally propose fractional-order coordination algorithms with absolute/relative damping and study the conditions on the interaction graph and the control gains such that coordination is achieved under a directed fixed interaction graph. Simulation examples are presented as a proof of concept.

This chapter considers distributed multi-agent coordination in a sampled-data setting. We first study a distributed sampled-data coordinated tracking algorithm where a group of followers with single-integrator dynamics interacting with their neighbors at discrete-time instants intercepts a dynamic leader who is a neighbor of only a subset of the followers. We propose a proportional-derivative-like discrete-time algorithm and study the condition on the interaction graph, the sampling period, and the control gain to ensure stability under directed fixed interaction and give the quantitative bound of the tracking errors. We then study convergence of two distributed sampled-data coordination algorithms with respectively, absolute damping and relative damping for double-integrator dynamics under undirected/directed fixed interaction. We show necessary and sufficient conditions on the interaction graph, the sampling period, and the control gain such that coordination is achieved using these two algorithms by using matrix theory, bilinear transformation, and Cauchy theorem. We finally study convergence of the two distributed sampled-data coordination algorithms with respectively, absolute damping and relative damping for double-integrator dynamics under directed switching interaction. We derive sufficient conditions on the interaction graph, the sampling period, and the control gain to guarantee coordination by using the property of infinity products of row-stochastic matrices. Simulation results are presented to show the effectiveness of the theoretical results.

This chapter considers the optimality aspect in distributed multi-agent coordination. We study optimal linear coordination algorithms for multi-agent systems with single-integrator dynamics in both continuous-time and discrete-time settings from a linear quadratic regulator perspective. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal state feedback gain matrix in both continuous-time and discrete-time settings. It is shown that the optimal gain matrix is a nonsymmetric Laplacian matrix corresponding to a complete directed graph. In addition, we show that any symmetric Laplacian matrix is inverse optimal with respect to a properly chosen cost function. With the interaction-related cost function, we derive the optimal scaling factor for a prespecified symmetric Laplacian matrix associated with an undirected interaction graph in both continuous-time and discrete-time settings. Illustrative examples are given as a proof of concept.

This chapter considers time delays in distributed multi-agent coordination. The time delays are inevitable in networked systems. Time-domain and frequency-domain approaches are used to study leaderless and leader-following coordination algorithms with communication and input delays under a directed interaction graph. We consider both the single-integrator and double-integrator dynamics and present stability or boundedness conditions. Several interesting phenomena are analyzed and explained. Simulation results are presented to support the theoretical results.