Learning-based Sparse Sensing with Performance Guarantees

In this study, we address the challenge of sensor scheduling in discrete-time linear dynamical networks. We propose a novel learning-based rounding method aimed at converting a provided weighted sensor schedule into a sparse, unweighted schedule while preserving a comparable level of observability performance to the original weighted schedule. We introduce the notion of L-systemic performance measures, which enjoy characteristics such as homogeneity, monotonicity, convexity, and Lipschitz continuity, covering a range of well-known measures. We integrate the initialization of the weighted sensor schedule, achieved via a convex relaxation of a combinatorial optimization problem based on an L-systemic measure, into our rounding approach. We show that this produces an unweighted sensor schedule that achieves a
$(1+\epsilon)$
near-optimal approximation solution while ensuring system observability. Our polynomial-time deterministic framework provides a performance guarantee compared to the optimal solution for all types of L-systemic performance measures, including a class of non-submodular metrics. The effectiveness of the theoretical findings is evaluated for a benchmark numerical example in distributed frequency control.