Direct Adaptive Learning of the Linear Quadratic Regulator from Online Closed-Loop Data

Core Concepts

This paper proposes a novel data-enabled policy optimization (DeePO) method for direct adaptive learning of the linear quadratic regulator (LQR) from online closed-loop data. The method uses a new policy parameterization based on the sample covariance, which enables efficient use of data and equivalence to the certainty-equivalence LQR. DeePO achieves global convergence via a projected gradient dominance property, and provides non-asymptotic guarantees showing sublinear regret decay and bias scaling with signal-to-noise ratio.

Abstract

The paper addresses the problem of direct adaptive learning of the linear quadratic regulator (LQR) from online closed-loop data. It proposes a novel data-enabled policy optimization (DeePO) method with the following key components: A new policy parameterization based on the sample covariance of input-state data. This parameterization has a constant dimension independent of data length, and is shown to be equivalent to the indirect certainty-equivalence LQR. A DeePO algorithm that directly updates the policy using projected gradient descent on the parameterized LQR cost. The authors prove a projected gradient dominance property, enabling global convergence guarantees. An adaptive learning framework where DeePO is used to recursively update the control policy from online closed-loop data. The authors provide non-asymptotic regret bounds showing sublinear decay in time and bias scaling with signal-to-noise ratio, independent of noise statistics. The key advantages of the proposed approach are its computational and sample efficiency compared to existing indirect adaptive methods and zeroth-order policy optimization. The authors validate the theoretical results through simulations.

Stats

The following sentences contain key metrics or figures: The average regret of the LQR cost is upper-bounded by two terms signifying a sublinear decrease in time O(1/√T) plus a bias scaling inversely with signal-to-noise ratio (SNR). Under a proper stepsize and a sufficiently large SNR, the gain sequence {Kt} is stabilizing.

Quotes

"The average regret of the LQR cost is upper-bounded by two terms signifying a sublinear decrease in time O(1/√T) plus a bias scaling inversely with signal-to-noise ratio (SNR), which are independent of the noise statistics." "If the SNR scales as O(T), then the regret bound is simply O(1/√T)."

Key Insights Distilled From

Data-Enabled Policy Optimization for Direct Adaptive Learning of the LQR

by Feir... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2401.14871.pdf

Deeper Inquiries

How can the proposed DeePO method be extended to handle time-varying or nonlinear systems

To extend the proposed DeePO method to handle time-varying or nonlinear systems, we can incorporate adaptive mechanisms that account for system dynamics changes. For time-varying systems, the covariance parameterization and gradient descent updates can be adjusted to adapt to changing system matrices over time. This adaptation can involve updating the covariance matrix estimation and policy parameters dynamically based on the evolving system behavior. Additionally, for nonlinear systems, the policy parameterization can be modified to accommodate nonlinearity in the system dynamics. This may involve using more complex function approximators or neural networks to represent the policy, allowing for the optimization of nonlinear control policies through data-driven methods.

What are the potential limitations or drawbacks of the covariance-based policy parameterization compared to other data-driven LQR formulations

While the covariance-based policy parameterization offers advantages such as efficient use of data and recursive updates, it also has potential limitations compared to other data-driven LQR formulations. One limitation is the assumption of linearity in the system dynamics, which may restrict the applicability of the method to linear systems only. Nonlinear systems may not be accurately represented by the covariance-based parameterization, leading to suboptimal control performance. Additionally, the dimensionality of the covariance matrix can grow with the size of the data, potentially increasing computational complexity and memory requirements. This scalability issue may limit the method's efficiency for large-scale systems or long-term data collection scenarios.

Can the ideas in this work be applied to other control problems beyond the LQR, such as model-free reinforcement learning for general nonlinear systems

The ideas presented in this work can be applied to other control problems beyond the LQR, such as model-free reinforcement learning for general nonlinear systems. By extending the covariance-based policy parameterization and DeePO method to nonlinear control problems, it is possible to learn optimal control policies for a wide range of nonlinear dynamical systems. This extension may involve using more advanced function approximators, such as deep neural networks, to represent the policy in a nonlinear form. The adaptive learning framework can be tailored to handle the complexities of nonlinear dynamics and optimize control policies based on data-driven approaches. Overall, the concepts of data-enabled policy optimization and direct adaptive learning can be generalized to various control problems, offering a versatile and efficient approach to model-free control in nonlinear systems.