Online Learning of Stabilizing Controllers Using Noisy Input-Output Data and Prior Knowledge

This paper presents online prior-knowledge-based data-driven approaches for verifying stability and learning a stabilizing dynamic controller for linear stochastic input-output systems. The system is modeled in an autoregressive exogenous (ARX) framework to accommodate cases where states are not fully observable. A key challenge addressed in this article is online stabilizing open-loop unstable systems, where collecting sufficient data for controller learning is impractical due to the risk of failure. To mitigate this, the proposed method integrates uncertain prior knowledge, derived from system physics, with limited available data. Inspired by set-membership system identification, the prior knowledge set is dynamically updated as new data becomes available, reducing conservatism over time. Unlike traditional approaches, this method bypasses explicit system identification, directly designing controllers based on current knowledge and data. A connection between ARX models and behavior theory is established, providing necessary and sufficient stability conditions using strict lossy S-Lemma. Quadratic difference forms serve as a framework for Lyapunov functions, and robust dynamic controllers are synthesized via linear matrix inequalities. The methodology is validated through simulations, including an unstable scalar system visualizing the integration of prior knowledge and data, and a rotary inverted pendulum demonstrating controller effectiveness in a nonlinear, unstable setting.