Asymptotic Analysis Based Greedy Method for Threshold-Based Distributed Optimization of Persistent Monitoring on Graphs
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We consider the optimal multi-agent persistent monitoring problem defined for a team of agents traversing on a set of nodes (targets) interconnected according to a fixed graph topology. The underlying objective is to minimize a measure of mean overall node state uncertainty evaluated over a finite interval. The solution to this problem involves each agent's trajectory defined both by the sequence of nodes to be visited and the amount of time to be spent at each node. In literature, for this problem, a class of distributed threshold-based parametric controllers has been proposed where agent transitions from one node to the next are controlled via enforcing thresholds on the respective node uncertainties. Under such a policy, the behavior of the agent-target system is a hybrid dynamic system, which enables the use of Infinitesimal Perturbation Analysis (IPA) to find the optimal threshold parameters in an on-line manner using gradient descent. However, due to the non-convexity of the associated objective function, the used initial thresholds in the gradient descent scheme directly affects the final optimal solution (controller). Therefore, initial thresholds should be carefully chosen to achieve a better optimal solution. To overcome this initialization challenge, we extensively analyze the asymptotic steady-state behavior of the agent-target hybrid system under periodic agent trajectories. Then, based on the obtained theoretical results, we propose a computationally efficient offline greedy approach to generate a reasonably well-performing initial thresholds. In almost all the considered cases, it was observed that the initial thresholds provided by the proposing greedy technique is optimal (still local) and performs better than the locally optimal threshold policies given by the IPA techniques (with randomly chosen initial threshold).
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