Finite Sample Identification of Partially Observed Bilinear Dynamical Systems

2025-01-13

AI Summary

This paper addresses the identification of partially observable bilinear dynamical systems (BLDS) from a single, finite-length, noisy input-output trajectory. To tackle the challenges posed by complex, time-varying inputs/outputs and partial observability, we establish the first finite-sample identification theory for such systems. Our method introduces a unified stability assumption, designs a Markov-like parameter estimator based on nonlinear heavy-tailed covariate regression, and constructs a balanced realization thereof. Leveraging concentration inequalities, we derive an interpretable, high-probability upper bound on the estimation error—explicitly characterizing how key system-theoretic quantities—including stability margin, state dimension, and noise level—influence sample complexity and accuracy. Numerical experiments quantitatively validate the critical roles of both the stability condition and the trajectory length in determining identification performance.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide a finite time analysis for learning the system's Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. Our bilinear system identification algorithm learns the system's Markov-like parameters by regressing the outputs to highly correlated, nonlinear, and heavy-tailed covariates. Moreover, the stability of BLDS depends on the sequence of inputs used to excite the system. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights.

Problem

• Bilinear Dynamic Systems
• Stability Identification
• Noisy Data Learning

Innovation

• Bilinear Dynamic Systems
• Noisy Data Learning
• Stability Analysis