Efficient Non-Asymptotic Identification of Linear Switching Systems

Core Concepts

This paper presents a novel data-driven approach to simultaneously achieve desirable control and system identification objectives in linear switching systems. The proposed algorithm leverages recent advances in non-asymptotic analysis of linear least-square methods to efficiently identify the unknown system parameters within a finite number of steps.

Abstract

The paper focuses on the problem of linear system identification in the setting where the underlying partially-observed linear dynamical system lies within a finite collection of known candidate models. The authors first consider the problem of identification from a given trajectory, which reduces to identifying the index of the true model with high probability. They characterize the finite-time sample complexity of this problem by leveraging recent advances in the non-asymptotic analysis of linear least-square methods. Next, the authors consider the switching control of linear systems, where there is a candidate controller for each of the candidate models and data is collected through interaction of the system with a collection of potentially destabilizing controllers. They develop a dimension-dependent criterion that can detect those destabilizing controllers in finite time. By leveraging these results, they propose a data-driven switching strategy that identifies the unknown parameters of the underlying system and provide a non-asymptotic analysis of its performance. The key contributions are: A least-square-based method for linear model identification with prior knowledge that the ground truth is contained in a finite collection of candidate models, yielding a dimension-free sample complexity bound. An instability detection criterion that quantitatively bounds the finite-time input-to-output gain of a stable linear system. A data-driven algorithm for linear system identification in switching control with non-asymptotic guarantees. Implications of the non-asymptotic guarantees on the classical method of estimator-based supervisory control.

Stats

The paper does not contain any explicit numerical data or statistics. The technical results are presented in the form of theoretical bounds and guarantees.

Quotes

"We develop a dimension-dependent criterion that can detect those destabilizing controllers in finite time." "By leveraging these results, we propose a data-driven switching strategy that identifies the unknown parameters of the underlying system and provide a non-asymptotic analysis of its performance."

Key Insights Distilled From

A least-square method for non-asymptotic identification in linear switching control

by Haoyuan Sun,... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08120.pdf

Deeper Inquiries

How can the proposed techniques be extended to handle nonlinear switching systems?

The proposed techniques for non-asymptotic system identification in linear switching control can be extended to handle nonlinear switching systems by incorporating nonlinear dynamics and controllers into the analysis. One approach could be to use nonlinear system identification methods, such as kernel methods or neural networks, to model the nonlinear dynamics of the system. This would involve adapting the least-square-based approach to handle the nonlinear relationships between inputs and outputs in the system. Additionally, the instability detection criterion could be modified to account for the nonlinear behavior of the system. Instead of relying solely on observability and the explosive growth of states, the criterion could incorporate nonlinear stability analysis techniques to detect unstable behavior in the system. This would involve analyzing the Lyapunov stability of the closed-loop dynamics under different controllers and models. Overall, extending the proposed techniques to handle nonlinear switching systems would require a combination of nonlinear system identification methods, stability analysis techniques, and adaptive control strategies to effectively identify and control the system in a non-asymptotic manner.

What are the implications of the transient behaviors on the performance of estimator-based supervisory control, and how can this be addressed?

Transient behaviors in a system can have significant implications on the performance of estimator-based supervisory control. These transients can introduce delays, oscillations, and instability in the system, affecting the accuracy of the estimations and control actions. In the context of the proposed techniques, the presence of transients can lead to challenges in identifying the true system parameters and selecting the appropriate controller. To address the impact of transient behaviors on estimator-based supervisory control, one approach is to incorporate robust control techniques that can handle uncertainties and disturbances caused by transients. This could involve designing controllers with adaptive mechanisms to adjust to changing system dynamics and mitigate the effects of transients on the control performance. Furthermore, implementing predictive control strategies that anticipate and compensate for transient behaviors can improve the overall performance of the system. By modeling and predicting the transient responses of the system, the controller can proactively adjust its actions to minimize the impact of transients on the control performance.

Can the instability detection criterion be further improved to reduce the dependency on the problem dimensions?

Yes, the instability detection criterion can be further improved to reduce the dependency on the problem dimensions by incorporating more sophisticated stability analysis techniques and model-based approaches. One way to reduce the dependency on problem dimensions is to utilize model order reduction techniques to simplify the system dynamics and focus on the most critical states and inputs that contribute to instability. Additionally, leveraging advanced control theory methods, such as robust control and adaptive control, can help in detecting and mitigating instability without relying heavily on the problem dimensions. By designing controllers that are robust to uncertainties and disturbances, the system can maintain stability even in the presence of varying system parameters and dimensions. Furthermore, incorporating data-driven approaches, such as machine learning algorithms, to analyze system behavior and detect instability patterns can provide a more dimension-independent criterion for instability detection. By learning from data and identifying instability indicators, the criterion can adapt to different system configurations and reduce its reliance on specific problem dimensions.