Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systems

TL;DR

This paper develops finite-time system identification and adaptive control algorithms for unknown ARX systems, achieving near-optimal regret bounds in stochastic linear dynamical systems with convex costs.

Contribution

It introduces finite-time learning guarantees and new adaptive control algorithms for ARX systems with provable regret bounds under different cost functions.

Findings

Achieves $ ilde{O}( oot{T}{} )$ regret with epoch-based updates.

Attains $ ilde{O}( oot{T}{} )$ regret with continuous model updates.

Provides finite-time guarantees for system identification in ARX systems.

Abstract

Autoregressive exogenous (ARX) systems are the general class of input-output dynamical systems used for modeling stochastic linear dynamical systems (LDS) including partially observable LDS such as LQG systems. In this work, we study the problem of system identification and adaptive control of unknown ARX systems. We provide finite-time learning guarantees for the ARX systems under both open-loop and closed-loop data collection. Using these guarantees, we design adaptive control algorithms for unknown ARX systems with arbitrary strongly convex or convex quadratic regulating costs. Under strongly convex cost functions, we design an adaptive control algorithm based on online gradient descent to design and update the controllers that are constructed via a convex controller reparametrization. We show that our algorithm has $\tilde{\mathcal{O}}(\sqrt{T})$ regret via explore and commit approach and if the model estimates are updated in epochs using closed-loop data collection, it attains the optimal regret of $\text{polylog}(T)$ after $T$ time-steps of interaction. For the case of convex quadratic cost functions, we propose an adaptive control algorithm that deploys the optimism in the face of uncertainty principle to design the controller. In this setting, we show that the explore and commit approach has a regret upper bound of $\tilde{\mathcal{O}}(T^{2/3})$, and the adaptive control with continuous model estimate updates attains $\tilde{\mathcal{O}}(\sqrt{T})$ regret after $T$ time-steps.