Minimizing Convergence Error in Multi-Agent Systems Via Leader Selection: A Supermodular Optimization Approach

Abstract

In a leader-follower multi-agent system (MAS), the leader agents act as control inputs and influence the states of the remaining follower agents. The rate at which the follower agents converge to their desired states, as well as the errors in the follower agent states prior to convergence, are determined by the choice of leader agents. In this paper, we study leader selection in order to minimize convergence errors experienced by the follower agents, which we define as a norm of the distance between the follower agents' intermediate states and the convex hull of the leader agent states. By introducing a novel connection to random walks on the network graph, we show that the convergence error has an inherent supermodular structure as a function of the leader set. Supermodularity enables development of efficient discrete optimization algorithms that directly approximate the optimal leader set, provide provable performance guarantees, and do not rely on continuous relaxations. We formulate two leader selection problems within the supermodular optimization framework, namely, the problem of selecting a fixed number of leader agents in order to minimize the convergence error, as well as the problem of selecting the minimum-size set of leader agents to achieve a given bound on the convergence error. We introduce algorithms for approximating the optimal solution to both problems in static networks, dynamic networks with known topology distributions, and dynamic networks with unknown and unpredictable

Submodularity in Dynamics and Control of Networked Systems

“…This structure holds for, e.g., the metrics of [108,38,34]. For these metrics, we have the following optimality result.…”

“…For such systems, controllability must be jointly considered alongside other performance metrics such as error due to link noise and convergence error. Current algorithms for selecting leader nodes, however, either focus on one of these performance criteria [82,51,34], or on guaranteeing controllability [85], leaving joint optimization of performance and controllability as an open problem.…”

“…In [85], a polynomial-time graph matching algorithm was introduced for selecting a minimum-size set of input nodes to satisfy structural controllability. The problem of selecting input nodes for controllability was further considered in [35,107,118] for the case where all matrix entries are either zero or are free, independent parameters. For the case where all entries of the system matrices are fully known, input selection algorithms and optimality bounds were derived in [102].…”

“…Second, the analysis of [4,138] assumes that each node can compute the impact of each of its actions on the objective function at each time instant. Since the computation of many common objective functions requires global knowledge of the network topology and current value of the set [35,57,73], we do not make this assumption.…”

“…Under our approach, a subset of nodes (e.g., a set of nodes to act as caches [57], leaders [35], or active sensors [73]) is selected by each node deciding, in a distributed fashion, whether to join the set based on the current and previous values of the objective function. In the case of unconstrained optimization, we introduce a probabilistic algorithm in which each node decides, at each tick of its internal clock, whether to join or exit the set based on the previously observed utility.…”