Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invertibility of Discrete-Time Linear Systems with Sparse Inputs

Kyle Poe, Enrique Mallada, Rene Vidal

TL;DR

The paper addresses the problem of recovering input sequences from outputs for discrete-time linear systems when inputs are sparse. It develops two geometric invariants, the weakly unobservable subspace arrangements $\mathcal{V}(s)$ and the strongly reachable subspace arrangements $\mathcal{T}(s)$, and provides comprehensive necessary-and-sufficient conditions for left invertibility with sparse inputs through geometric, rank-based, and spectral characterizations, including a generalized Rosenbrock matrix. An illustrative network example demonstrates how these characterizations apply in practice and how invertibility outcomes depend on sparsity patterns. The framework establishes a foundation for sparse-input inversion and connects to related areas such as switched systems and unknown-input observers, with potential impact on sparse control and structured inversion problems.

Abstract

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.

Invertibility of Discrete-Time Linear Systems with Sparse Inputs

TL;DR

The paper addresses the problem of recovering input sequences from outputs for discrete-time linear systems when inputs are sparse. It develops two geometric invariants, the weakly unobservable subspace arrangements and the strongly reachable subspace arrangements , and provides comprehensive necessary-and-sufficient conditions for left invertibility with sparse inputs through geometric, rank-based, and spectral characterizations, including a generalized Rosenbrock matrix. An illustrative network example demonstrates how these characterizations apply in practice and how invertibility outcomes depend on sparsity patterns. The framework establishes a foundation for sparse-input inversion and connects to related areas such as switched systems and unknown-input observers, with potential impact on sparse control and structured inversion problems.

Abstract

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.
Paper Structure (20 sections, 21 equations, 1 figure)

This paper contains 20 sections, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Linear Systems
  5. Subspace Arrangements
  6. Invertibility of Linear Systems with Sparse Inputs
  7. Geometric Characterization
  8. Rank-Based Characterization
  9. Spectral Characterization
  10. Example: Network with Edge Attacks
  11. $\alpha = 0$, Geometric Characterization
  12. $\alpha = 1$, Rank-Based Characterization
  13. $\alpha = 0$, Spectral Characterization
  14. Discussion
  15. Bounding constants associated with $\mathcal{T}(s), \mathcal{V}(s)$
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The system $\Sigma_\alpha$ depicted as a network. Solid black arrows indicate a weight of $+1$, dashed indicates $-1$. Red vertices are inputs, and others are states. Blue indicates the vertex state is included in the measurement $\bm{C}_\alpha\bm{x}$, and green indicates inclusion when $\alpha = 1$.

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof