# Risk-Constrained Linear-Quadratic Regulators

arXiv: https://arxiv.org/abs/2004.04685 · HTML: https://mlsanity.com/paper/2004.04685/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2020.

## Built on by (most cited first, 8 of 8)

- [Fast Policy Learning for Linear Quadratic Control with Entropy Regularization](https://mlsanity.com/paper/2311.14168/) (2023)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [Policy Gradient Methods for the Cost-Constrained LQR: Strong Duality and Global Convergence](https://mlsanity.com/paper/2406.03734/) (2024)
- [Policy Gradient-based Model Free Optimal LQG Control with a Probabilistic Risk Constraint](https://mlsanity.com/paper/2403.16767/) (2024)
- [Deterministic Policy Gradient Primal-Dual Methods for Continuous-Space Constrained MDPs](https://mlsanity.com/paper/2408.10015/) (2024)
- [Communication-aware Wide-Area Damping Control using Risk-Constrained Reinforcement Learning](https://mlsanity.com/paper/2509.23620/) (2025)
- [Actor-Critic Learning for Risk-Constrained Linear Quadratic Regulation](https://mlsanity.com/paper/2510.22267/) (2025)
- [Model-free LQG Control with Chance Constraints](https://mlsanity.com/paper/2605.31310/) (2026)

## Abstract (arXiv)

We propose a new risk-constrained reformulation of the standard Linear Quadratic Regulator (LQR) problem. Our framework is motivated by the fact that the classical (risk-neutral) LQR controller, although optimal in expectation, might be ineffective under relatively infrequent, yet statistically significant (risky) events. To effectively trade between average and extreme event performance, we introduce a new risk constraint, which explicitly restricts the total expected predictive variance of the state penalty by a user-prescribed level. We show that, under rather minimal conditions on the process noise (i.e., finite fourth-order moments), the optimal risk-aware controller can be evaluated explicitly and in closed form. In fact, it is affine relative to the state, and is always internally stable regardless of parameter tuning. Our new risk-aware controller: i) pushes the state away from directions where the noise exhibits heavy tails, by exploiting the third-order moment (skewness) of the noise; ii) inflates the state penalty in riskier directions, where both the noise covariance and the state penalty are simultaneously large. The properties of the proposed risk-aware LQR framework are also illustrated via indicative numerical examples.