# On the Certainty-Equivalence Approach to Direct Data-Driven LQR Design

arXiv: https://arxiv.org/abs/2109.06643 · HTML: https://mlsanity.com/paper/2109.06643/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2021.

## Built on by (most cited first, 5 of 5)

- [Policy Gradient Adaptive Control for the LQR: Indirect and Direct Approaches](https://mlsanity.com/paper/2505.03706/) (2025)
- [Stability-Certified On-Policy Data-Driven LQR via Recursive Learning and Policy Gradient](https://mlsanity.com/paper/2403.05367/) (2024)
- [Regret Analysis of Policy Optimization over Submanifolds for Linearly Constrained Online LQG](https://mlsanity.com/paper/2403.08553/) (2024)
- [Adaptive Control of Unknown Linear Switched Systems via Policy Gradient Methods](https://mlsanity.com/paper/2601.03016/) (2026)
- [Data-Driven Continuous-Time Linear Quadratic Regulator via Closed-Loop and Reinforcement Learning Parameterizations](https://mlsanity.com/paper/2604.27922/) (2026)

## Abstract (arXiv)

The linear quadratic regulator (LQR) problem is a cornerstone of automatic control, and it has been widely studied in the data-driven setting. The various data-driven approaches can be classified as indirect (i.e., based on an identified model) versus direct or as robust (i.e., taking uncertainty into account) versus certainty-equivalence. Here we show how to bridge these different formulations and propose a novel, direct, and regularized formulation. We start from indirect certainty-equivalence LQR, i.e., least-square identification of state-space matrices followed by a nominal model-based design, formalized as a bi-level program. We show how to transform this problem into a single-level, regularized, and direct data-driven control formulation, where the regularizer accounts for the least-square data fitting criterion. For this novel formulation we carry out a robustness and performance analysis in presence of noisy data. Our proposed direct and regularized formulation is also amenable to be further blended with a robust-stability-promoting regularizer. In a numerical case study we compare regularizers promoting either robustness or certainty-equivalence, and we demonstrate the remarkable performance when blending both of them.