Online Control with Adversarial Disturbances (1902.08721v1)

Abstract: We study the control of a linear dynamical system with adversarial disturbances (as opposed to statistical noise). The objective we consider is one of regret: we desire an online control procedure that can do nearly as well as that of a procedure that has full knowledge of the disturbances in hindsight. Our main result is an efficient algorithm that provides nearly tight regret bounds for this problem. From a technical standpoint, this work generalizes upon previous work in two main aspects: our model allows for adversarial noise in the dynamics, and allows for general convex costs.

• The paper presents a regret minimization framework that adapts control strategies to adversarial disturbances in linear dynamical systems.
• It introduces an efficient algorithm achieving an O(√T) regret bound, ensuring robust performance under arbitrary convex cost functions.
• The study proposes a disturbance-action policy to approximate linear controls with strongly stable matrices for resilient system management.

Online Control with Adversarial Disturbances

The paper "Online Control with Adversarial Disturbances" presents an innovative approach to managing the control of linear dynamical systems subjected to adversarial disturbances. This paper diverges from traditional methods that largely focus on stochastic noise by considering an online control procedure aiming to minimize regret. The notion of regret here centers on achieving performance nearly as effective as if the controller had full knowledge of the disturbances in hindsight.

Problem Formulation and Objectives

The authors address the challenge of controlling a linear dynamical system expressed by the dynamics equation $x_{t+1} = Ax_t + Bu_t + w_t$. Here, $x_t$ and $u_t$ denote the state and control, respectively, while $w_t$ represents the disturbance. The system suffers a cost $c(x_t, u_t)$ at each timestep, which in this paper, can be any convex cost function. This marks a generalization of standard control problems, which often restrict costs to quadratic forms. The adversary in the proposed framework selects both the cost functions and the disturbances, highlighting the necessity to develop a control strategy resilient under adversarial conditions.

Key Contributions

1. Regret Minimization Framework: The paper extends the domain of robust control by employing the regret minimization methodology from online learning. This concept is pivotal as it shifts towards a more robust control framework that adapts to unknown, possibly adversarial disturbances.
2. Algorithmic Advancement: The paper introduces an efficient algorithm that achieves an $O(\sqrt{T})$ regret bound, a significant improvement in this setting. A noteworthy advancement is the capability to handle bounded adversarial disturbances and arbitrary convex costs, whereas preceding approaches often required stochastic assumptions or quadratic costs.
3. Disturbance-Action Policy Class: By proposing a disturbance-action policy, the authors provide a coherent method to approximate any linear control policy with strongly stable matrices, ensuring the systems can be managed effectively even under challenging conditions.

Methodology

The approach employs techniques from online convex optimization (OCO) with memory. A tailored online gradient descent (OGD) algorithm manages the control strategy, and this choice leverages the advances in OCO to handle the inherent complexity of adversarial dynamics. The policy class utilized in this framework is structured to manipulate the system dynamically, maintaining effectiveness while reducing computational burdens. The authors bolster this approach with rigorous theorems and proofs to ensure theoretical robustness in their claims.

Implications and Future Directions

The research provides a comprehensive framework that could recalibrate the landscape of system control under uncertainty. The practical implications are substantial, extending to various fields such as robotics, aerospace, and automated system management, where adaptability to unforeseen disruptions is critical. On a theoretical level, this work provokes further exploration into blending adversarial minimization techniques with control theory. Potential extensions could involve exploring non-linear systems, optimizing computational efficiency further, or elaborating on hybrid approaches that combine statistical noise with adversarial settings.

This paper's contributions lay a robust foundation as the field gravitates towards more resilient and adaptable control systems amid unpredictable environments. As AI and control systems become increasingly integral to complex decision-making processes, frameworks like these will be crucial to ensuring efficacy and reliability.