Core Concepts

Necessary and sufficient conditions for the left invertibility of discrete-time linear systems with sparse inputs, including geometric, rank-based, and spectral characterizations.

Abstract

The content discusses the problem of left invertibility for discrete-time linear systems, where the goal is to recover the input sequence given the output sequence. While many conditions have been established for standard linear systems, the authors focus on the case where the inputs are assumed to be sparse.
The key contributions are:
Introduction of the notions of weakly unobservable and strongly reachable subspace arrangements, which generalize key invariants from classical linear systems theory to the sparse input setting.
Establishment of rank-based conditions for left sparse invertibility, showing that if an inverse exists, it can be realized with finite delay.
Characterization of the invertibility of systems with periodically sparse inputs via the zeros of a generalized Rosenbrock matrix.
Presentation of a concrete example to illustrate the application of these ideas.
The authors conclude by discussing extensions and connections to related problems in sparse control, such as strong observability and unknown input observers.

Stats

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.
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.

Quotes

Nowhere is the importance of this fundamental relationship more apparent than in signal processing, where one observes some transformed or corrupted version of an input signal, and wishes to recover the original signal.
Necessary and sufficient conditions for the finite-horizon case have been established in [13], a problem for which a performant Bayesian recovery algorithm was introduced in [14].

Key Insights Distilled From

by Kyle Poe,Enr... at **arxiv.org** 04-01-2024

Deeper Inquiries

To reduce the computational complexity of verifying the conditions for sparse invertibility, tighter bounds on the size of the weakly unobservable and strongly reachable subspace arrangements can be established. By refining the bounds on the size of these arrangements, we can limit the search space for verifying invertibility conditions, making the process more efficient.
One approach to achieving this is through more precise analysis of the structure of the weakly unobservable and strongly reachable subspaces. By understanding the relationships between these subspaces and the system dynamics more deeply, it may be possible to derive tighter bounds on their sizes. This could involve exploring the interplay between the system matrices and the subspaces to identify patterns that allow for more accurate sizing estimates.
Additionally, leveraging advanced mathematical techniques such as optimization algorithms or convex relaxation methods could help in refining the bounds further. These methods can be used to formulate the problem of determining the size of the subspaces as an optimization task, seeking the most compact representation that satisfies the invertibility conditions. By optimizing the bounds in this way, the computational complexity of verifying sparse invertibility can be significantly reduced.

The connection between sparse input invertibility and switched systems opens up new possibilities for extending the results to more general classes of switched systems. Switched systems involve multiple modes of operation, where the dynamics of the system change based on a discrete switching signal. By relating sparse input invertibility to switched systems, we can leverage the insights and techniques developed for sparse systems to analyze and control a broader range of switched systems.
One implication of this connection is that the characterizations and conditions established for sparse input invertibility can potentially be adapted to analyze the invertibility of switched systems with sparse inputs. This could involve formulating the conditions for invertibility in terms of the switching signals and the corresponding system matrices, similar to how sparse inputs were incorporated into the analysis. By aligning the concepts of sparse input invertibility with the dynamics of switched systems, we can develop a unified framework for studying invertibility in complex dynamical systems.
To extend the results to more general classes of switched systems, researchers can explore the application of sparse input invertibility techniques to different types of switching behaviors and system structures. By considering various modes of operation, different switching sequences, and diverse system dynamics, the insights gained from sparse input invertibility can be generalized to address the invertibility challenges in a wide range of switched systems. This extension would involve adapting the existing conditions and characterizations to accommodate the complexities of switched systems while leveraging the foundational principles established in the context of sparse input invertibility.

The insights from this work on sparse input invertibility can indeed be applied to enhance the performance of practical sparse control algorithms in networked and structured systems. By understanding the conditions for invertibility in systems with sparse inputs, control engineers can design more effective algorithms that leverage the sparsity of the input signals to achieve better control performance. Here are some ways in which these insights can be applied:
Optimized Input Recovery: The conditions for sparse input invertibility provide guidelines for recovering the original input signals from the observed outputs. By incorporating these conditions into sparse control algorithms, engineers can develop more efficient and accurate methods for reconstructing sparse inputs in networked and structured systems.
Fault Detection and Isolation: Sparse input invertibility is closely related to fault detection and isolation in control systems. By utilizing the insights from this work, control algorithms can be enhanced to detect and isolate faults in networked systems by analyzing the sparsity patterns in the input signals and their impact on the system dynamics.
Unknown Input Observers: The concept of unknown input observers, which are used to estimate the system state in the presence of unknown inputs, can benefit from the insights on sparse input invertibility. By considering the sparsity of the inputs and the conditions for invertibility, more robust and accurate unknown input observers can be designed for networked and structured systems.
Adaptive Control Strategies: Sparse input invertibility can inform the development of adaptive control strategies that adjust the control parameters based on the sparsity patterns in the input signals. By dynamically adapting the control algorithms to the sparse nature of the inputs, better performance and robustness can be achieved in networked and structured systems.
Overall, the application of the insights from sparse input invertibility to practical sparse control algorithms can lead to more efficient, reliable, and adaptive control strategies for networked and structured systems, improving their overall performance and robustness.

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