On the Sample Complexity of the Linear Quadratic Regulator (1710.01688v3)

Abstract: This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are nearly optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.

• The paper introduces Coarse-ID control that estimates system dynamics from trial data and incorporates uncertainty into robust controller design.
• It integrates least squares estimation with random matrix theory to derive tight non-asymptotic error bounds for LQR problems.
• Empirical results show that the approach stabilizes systems with fewer data points, promoting safer and more reliable data-driven control.

Sample Complexity of the Linear Quadratic Regulator

This paper addresses a critical problem in control theory, known as the Linear Quadratic Regulator (LQR), specifically in scenarios where system dynamics are unknown. The authors propose a methodology, termed Coarse-ID control, to effectively handle the LQR problem by estimating system dynamics from experimental trials and incorporating this estimation uncertainty into the control design.

Methodology and Approach

The proposed Coarse-ID control involves a three-stage process:

1. Model Estimation: Utilizing supervised learning techniques, a nominal model of the system is obtained from trial data. This phase leverages least squares to approximate system dynamics, addressing estimation error through random matrix theory.
2. Model Uncertainty Quantification: Statistical tools, possibly complemented by prior knowledge, are employed to derive probabilistic bounds on the deviation between the estimated and actual system dynamics.
3. Robust Controller Design: A robust optimization problem is solved using System Level Synthesis (SLS), which integrates the nominal model with the uncertainty estimates. This step ensures that the designed controller offers stable and robust performance.

A significant contribution is the integration of contemporary random matrix techniques and non-asymptotic statistics to provide tight error bounds on model estimation. The approach extends System Level Synthesis to design controllers that perform optimally even in the presence of model uncertainties by solving a quasiconvex optimization problem.

Key Results and Findings

Empirical evidence shows that the Coarse-ID control approach effectively stabilizes systems in conditions where simpler methods fail due to unaddressed uncertainty. The end-to-end error bounds derived in the paper optimally scale with the number of parameters, shedding light on properties like closed-loop sensitivity and control magnitude.

Furthermore, the authors demonstrate experimental results where Coarse-ID control consistently stabilizes the system with fewer data points, surpassing naive estimation-based methods. Estimation error bounds capitalize on the dynamics' least controllable modes, and the design of stable controllers is facilitated via finite sample guarantees.

Implications and Future Directions

The paper provides a theoretically robust framework for LQR problems with unknown dynamics. The Coarse-ID control strategy can guide upcoming developments in safe, data-driven control for applications like autonomous vehicles and sensor networks. The insights into integrating modern statistical methodologies with control synthesis have broader implications for control theory and potential intersections with machine learning.

For future research, the authors suggest exploring a few directions: leveraging their framework in model predictive control, adaptive control, and nonlinear systems. Further refinement of estimation techniques or integrating additional constraints could enhance practical robustness without performance trade-offs.

Conclusion

This paper presents a sophisticated yet practical approach to addressing LQR problems with unknown dynamics. By coupling rigorous statistical estimation with robust control synthesis, it advances the field's understanding of navigating uncertainties in dynamic systems. The Coarse-ID control stands out in its applicability to modern, uncertain environments, promising safer and more reliable automation systems.

Overall, this research underscores the importance of systematically addressing uncertainties in control design and sets a high bar for future work in robust, data-informed control systems.