# The Power of Linear Controllers in LQR Control

arXiv: https://arxiv.org/abs/2002.02574 · HTML: https://mlsanity.com/paper/2002.02574/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2020.

## Built on by (most cited first, 10 of 10)

- [Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds](https://mlsanity.com/paper/2111.07018/) (2021)
- [Regret-optimal Estimation and Control](https://mlsanity.com/paper/2106.12097/) (2021)
- [Competitive Control](https://mlsanity.com/paper/2107.13657/) (2021)
- [Optimal Dynamic Regret in LQR Control](https://mlsanity.com/paper/2206.09257/) (2022)
- [Best of Both Worlds in Online Control: Competitive Ratio and Policy Regret](https://mlsanity.com/paper/2211.11219/) (2022)
- [Predictive Linear Online Tracking for Unknown Targets](https://mlsanity.com/paper/2402.10036/) (2024)
- [Implications of Regret on Stability of Linear Dynamical Systems](https://mlsanity.com/paper/2211.07411/) (2022)
- [Regret-Optimal LQR Control](https://mlsanity.com/paper/2105.01244/) (2021)
- [Sub-linear Regret in Adaptive Model Predictive Control](https://mlsanity.com/paper/2310.04842/) (2023)
- [Self-Adaptive Learning and Model Predictive Control for Tracking Unknown Dynamics with No Regret](https://mlsanity.com/paper/2607.26370/) (2026)

## Abstract (arXiv)

The Linear Quadratic Regulator (LQR) framework considers the problem of regulating a linear dynamical system perturbed by environmental noise. We compute the policy regret between three distinct control policies: i) the optimal online policy, whose linear structure is given by the Ricatti equations; ii) the optimal offline linear policy, which is the best linear state feedback policy given the noise sequence; and iii) the optimal offline policy, which selects the globally optimal control actions given the noise sequence. We fully characterize the optimal offline policy and show that it has a recursive form in terms of the optimal online policy and future disturbances. We also show that cost of the optimal offline linear policy converges to the cost of the optimal online policy as the time horizon grows large, and consequently the optimal offline linear policy incurs linear regret relative to the optimal offline policy, even in the optimistic setting where the noise is drawn i.i.d from a known distribution. Although we focus on the setting where the noise is stochastic, our results also imply new lower bounds on the policy regret achievable when the noise is chosen by an adaptive adversary.