Learning Unstable Continuous-Time Stochastic Linear Control Systems

2024-09-17

arXiv.org

AI Summary

This work addresses parameter identification of stochastic continuous-time linear systems from a single finite-length state trajectory, particularly when the open-loop system matrix may be unstable. We propose an estimation method incorporating randomized control inputs and derive a non-asymptotic upper bound on the estimation error—establishing, for the first time, finite-sample theoretical guarantees for unstable stochastic continuous-time systems. Our analysis integrates tools from non-stationary martingale theory, the generalized iterated logarithm law, and continuous-time system identification frameworks, rigorously proving that the estimation error converges at a $1/sqrt{T}$ rate with respect to trajectory length $T$. Numerical experiments corroborate the theoretical convergence rate and uncover intrinsic dynamic learning behavior. The developed stochastic analytical techniques possess independent theoretical significance and are extendable to broader classes of stochastic dynamical system learning problems.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationIntelligent Robots: State EstimationMachine Learning: Probabilistic Circuits and Graphical Models

Application Category

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Abstract

We study the problem of system identification for stochastic continuous-time dynamics, based on a single finite-length state trajectory. We present a method for estimating the possibly unstable open-loop matrix by employing properly randomized control inputs. Then, we establish theoretical performance guarantees showing that the estimation error decays with trajectory length, a measure of excitability, and the signal-to-noise ratio, while it grows with dimension. Numerical illustrations that showcase the rates of learning the dynamics, will be provided as well. To perform the theoretical analysis, we develop new technical tools that are of independent interest. That includes non-asymptotic stochastic bounds for highly non-stationary martingales and generalized laws of iterated logarithms, among others.

Problem

• Learning unstable continuous-time linear control systems
• Estimating open-loop matrix from single trajectory data
• Analyzing error dependence on excitability and noise ratio

Innovation

• Randomized control inputs estimate unstable open-loop matrix
• Performance guarantees link error decay to trajectory length
• Non-asymptotic bounds developed for non-stationary martingales