Primal-dual Learning for the Model-free Risk-constrained Linear Quadratic Regulator

TL;DR

This paper introduces a model-free, data-driven primal-dual approach to risk-aware linear quadratic regulation, ensuring safety constraints while optimizing control performance in unknown systems.

Contribution

It develops a novel primal-dual learning framework for risk-constrained LQR problems, establishing strong duality and convergence properties without requiring a known model.

Findings

Proven strong duality in the non-convex Lagrangian landscape.

Developed a convergent random search algorithm for dual function learning.

Validated the approach through simulation results.

Abstract

Risk-aware control, though with promise to tackle unexpected events, requires a known exact dynamical model. In this work, we propose a model-free framework to learn a risk-aware controller with a focus on the linear system. We formulate it as a discrete-time infinite-horizon LQR problem with a state predictive variance constraint. To solve it, we parameterize the policy with a feedback gain pair and leverage primal-dual methods to optimize it by solely using data. We first study the optimization landscape of the Lagrangian function and establish the strong duality in spite of its non-convex nature. Alongside, we find that the Lagrangian function enjoys an important local gradient dominance property, which is then exploited to develop a convergent random search algorithm to learn the dual function. Furthermore, we propose a primal-dual algorithm with global convergence to learn the optimal policy-multiplier pair. Finally, we validate our results via simulations.