Actor-Critic Learning for Risk-Constrained Linear Quadratic Regulation

In this paper, we investigate the infinite-horizon risk-constrained linear quadratic regulator problem (RC-QR), which augments the classical LQR formulation with a statistical constraint on the variability of the system state to incorporate risk awareness, a key requirement in safety-critical control applications. We propose an actor-critic learning algorithm that jointly performs policy evaluation and policy improvement in a model-free and online manner. The RC-QR problem is first reformulated as a max-min optimization problem, from which we develop a multi-time-scale stochastic approximation scheme. The critic employs temporal-difference learning to estimate the action-value function, the actor updates the policy parameters via a policy gradient step, and the dual variable is adapted through gradient ascent to enforce the risk constraint.