Invertibility of Discrete-Time Linear Systems with Sparse Inputs

作     者：Poe, Kyle Mallada, Enrique Vidal, René 

作者机构：Applied Mathematics and Computational Science Group University of Pennsylvania PA19104 United States Department of Electrical and Computer Engineering Johns Hopkins University MD21218 United States Departments of Electrical and Systems Engineering Radiology Computer and Information Science Statistics and Data Science University of Pennsylvania PA19104 United States 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Linear systems 

摘      要：One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparse-input setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control. © 2024, CC BY.