Suboptimal Performance of Receding Horizon Control with Unknown Linear Systems and Its Applications in Learning-Based Control

Core Concepts

The core message of this article is to provide a novel suboptimality analysis of a nominal receding-horizon linear quadratic (LQ) controller under the joint effect of modeling error, terminal value function error, and prediction horizon. The analysis reveals that for many cases, the prediction horizon can be either 1 or infinity to improve the control performance, depending on the relative difference between the modeling error and the terminal value function error.

Abstract

The article aims to analyze the performance gap between a nominal receding-horizon LQ (RHC) controller and the ideal LQR controller, under the presence of modeling error and terminal value function approximation.
The key highlights and insights are:

For the case with a known system model, the authors show that the performance gap decays quadratically in the terminal value function error and exponentially in the prediction horizon. Moreover, they reveal that choosing a prediction horizon larger than the controllability index can lead to a faster decay rate.

When the system model is unknown, the authors develop a novel perturbation analysis of the Riccati difference equation to characterize the joint effect of modeling error, terminal value function error, and prediction horizon. This analysis extends previous results on the perturbation of the discrete Riccati equation.

Based on the perturbation analysis, the authors derive a novel performance upper bound for the nominal RHC controller with an inexact model. The bound shows that for many cases, the prediction horizon can be either 1 or infinity to achieve better performance, depending on the relative difference between the modeling error and the terminal value function error.

The performance bound is applied to provide novel sample complexity and regret guarantees for nominal RHC controllers in a learning-based setting, where the unknown system is estimated offline or adaptively.

Stats

The article does not contain any explicit numerical data or statistics to support the key logics. The analysis is primarily theoretical, focusing on deriving analytical bounds and characterizing the tradeoffs between different factors.

Quotes

"The core message of this article is to provide a novel suboptimality analysis of a nominal receding-horizon linear quadratic (LQ) controller under the joint effect of modeling error, terminal value function error, and prediction horizon."
"The derived performance upper bound shows how the control performance may vary with changes in the modeling error, the terminal value function error, and the prediction horizon."
"The performance bound is applied to derive a novel suboptimality bound for the nominal receding-horizon LQ controller, where the unknown system is estimated offline. The bound shows the dependence of the control performance on the number of data samples."

Key Insights Distilled From

Suboptimality analysis of receding horizon quadratic control with unknown linear systems and its applications in learning-based control

by Shengling Sh... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2301.07876.pdf

Deeper Inquiries

How can the theoretical bounds derived in this work be extended to handle more general system dynamics, cost functions, and constraints beyond the linear quadratic setting

The theoretical bounds derived in this work for the linear quadratic setting can be extended to handle more general system dynamics, cost functions, and constraints by incorporating non-linearities, non-quadratic cost functions, and non-convex constraints. This extension would involve adapting the perturbation analysis and convergence rate calculations to accommodate the complexities introduced by these generalizations. For non-linear systems, the Riccati difference equation perturbation analysis would need to consider the non-linear dynamics and possibly employ numerical methods for approximation. Similarly, for non-quadratic cost functions, the performance bounds would need to be redefined based on the specific form of the cost function. Constraints could be incorporated by modifying the optimization problem formulation to include the constraints and analyzing their impact on the controller's performance. Overall, the extension to more general settings would require a deeper understanding of control theory principles and advanced mathematical techniques to derive accurate theoretical bounds.

What are the practical implications of the insights on the choice of prediction horizon for different types of systems (e.g., fully-actuated vs. under-actuated)

The insights on the choice of prediction horizon provided in this work have significant practical implications for the design of receding-horizon controllers in real-world applications. For fully-actuated systems where the control inputs are equal to the system's degrees of freedom, choosing a prediction horizon larger than the controllability index can lead to improved control performance. This is because a larger prediction horizon allows for better anticipation of future system behavior and more effective control action planning. On the other hand, for under-actuated systems where the control inputs are fewer than the system's degrees of freedom, the choice of prediction horizon becomes more nuanced. In such cases, a smaller prediction horizon may be sufficient to achieve the desired control performance, as the system dynamics are inherently more constrained. By understanding the relationship between the prediction horizon and system controllability, designers can tailor the receding-horizon controller parameters to suit the specific characteristics of the system, leading to optimized control strategies in real-world applications.

How can these insights guide the design of receding-horizon controllers in real-world applications

The analysis techniques used in this work to study the closed-loop performance of receding-horizon controllers can be extended to analyze the online computational complexity and scalability of these controllers as the prediction horizon and system size increase. By considering the computational requirements of solving the optimization problem at each time step within the receding-horizon framework, one can analyze how the prediction horizon impacts the online computational complexity. As the prediction horizon increases, the optimization problem becomes larger and more computationally intensive, potentially affecting the real-time performance of the controller. Techniques such as complexity analysis, algorithmic efficiency studies, and scalability assessments can be employed to understand how the computational demands scale with the prediction horizon and system size. By incorporating these considerations into the analysis, researchers and practitioners can gain insights into the practical implementation of receding-horizon controllers in systems with varying complexities and sizes.