Finite-Time Regret Bounds for Recursive Adaptive Control with Matched Uncertainty

Core Concepts

Recursive proximal learning and recursive least squares with exponential forgetting achieve finite regret in discrete-time adaptive control with matched uncertainty, scaling with the time required to satisfy a persistence of excitation condition.

Abstract

The content discusses the problem of efficient adaptive control for discrete-time nonlinear systems with matched uncertainty. The key highlights are: The authors consider a discrete-time system with matched uncertainty, where the unknown dynamics can be parameterized by an unknown parameter vector and a known feature matrix. They propose a novel recursive proximal learning (RPL) algorithm and analyze its performance in terms of asymptotic stability and regret. RPL is shown to achieve finite regret scaling with the time required to satisfy a weak persistence of excitation (PE) condition. The authors also analyze the well-established recursive least squares with exponential forgetting (RLSFF) algorithm in this setting. RLSFF is shown to achieve finite regret under a stronger PE condition, with a similar bound to RPL. The regret bounds for both RPL and RLSFF consist of three terms: a constant term, an exponentially decaying term, and a linear term in the PE time. The linear term arises from the non-expansive properties of the parameter estimates during the initial non-PE phase. The authors demonstrate the performance of the proposed algorithms on a discrete-time model reference adaptive control (MRAC) numerical example, showcasing the superior tracking and regret performance of RPL compared to RLSFF and a command governor-based MRAC controller.

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Key Insights Distilled From

On the Regret of Recursive Methods for Discrete-Time Adaptive Control with Matched Uncertainty

by Aren Karapet... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.02023.pdf

Deeper Inquiries

How can the proposed methods be extended to handle inexact basis matrices or bounded, non-stochastic noise in the system dynamics

To extend the proposed methods to handle inexact basis matrices or bounded, non-stochastic noise in the system dynamics, the algorithms can be modified to incorporate robust estimation techniques. For inexact basis matrices, the parameter estimation update rules can be adjusted to account for uncertainties in the regressor matrices. This can involve introducing regularization terms or robust optimization criteria to mitigate the effects of inaccuracies in the basis matrices. Additionally, for bounded, non-stochastic noise, the adaptive controllers can be augmented with robust control strategies that can handle disturbances within known bounds. Techniques such as robust model predictive control or disturbance rejection can be integrated into the adaptive control framework to enhance robustness against noise and uncertainties in the system dynamics.

Can the authors develop goal-oriented controllers that directly minimize the objective cost, rather than the estimation cost, to potentially achieve better regret bounds

Developing goal-oriented controllers that directly minimize the objective cost, rather than the estimation cost, can potentially lead to improved regret bounds and better overall performance of the adaptive controllers. By formulating the control problem as an optimization task to directly minimize the objective cost function, the controllers can focus on achieving the desired control objectives while simultaneously adapting to system uncertainties. This approach can lead to more efficient and effective control strategies that prioritize achieving the control objectives over minimizing the estimation error. By optimizing the objective cost directly, the controllers can potentially achieve better regret bounds and enhance the overall closed-loop stability and performance of the system.

What are the implications of the regret bounds on the closed-loop stability and performance of the adaptive controllers beyond just the state tracking error

The implications of the regret bounds on the closed-loop stability and performance of the adaptive controllers go beyond just the state tracking error. The finite regret bounds provide insights into the trade-off between adaptation and performance in the control system. By quantifying the additional cost incurred by the controller over a horizon due to partial model knowledge, the regret bounds offer a measure of the controller's performance relative to an optimal benchmark. Understanding the regret bounds can help in assessing the robustness, convergence properties, and overall efficiency of the adaptive controllers. Additionally, the regret bounds can provide valuable information on the controller's ability to adapt to changing system dynamics while maintaining stability and achieving the desired control objectives.