[Paper Review] Meta-Adaptive Nonlinear Control: Theory and Algorithms

Guanya Shi, Kamyar Azizzadenesheli+3 more|arXiv (Cornell University)|Jun 11, 2021

Advanced Bandit Algorithms Research64 references19 citations

TL;DR

This paper introduces Online Meta-Adaptive Control (OMAC), a novel framework that unifies online representation learning with adaptive control theory to enable fast, stable adaptation in nonlinear systems under adversarial disturbances and unknown environment-dependent dynamics. It provides the first non-asymptotic end-to-end convergence guarantee for multi-task nonlinear control, with empirical improvements over conventional adaptive control in pendulum and drone control under varying wind conditions.

ABSTRACT

We present an online multi-task learning approach for adaptive nonlinear control, which we call Online Meta-Adaptive Control (OMAC). The goal is to control a nonlinear system subject to adversarial disturbance and unknown $ extit{environment-dependent}$ nonlinear dynamics, under the assumption that the environment-dependent dynamics can be well captured with some shared representation. Our approach is motivated by robot control, where a robotic system encounters a sequence of new environmental conditions that it must quickly adapt to. A key emphasis is to integrate online representation learning with established methods from control theory, in order to arrive at a unified framework that yields both control-theoretic and learning-theoretic guarantees. We provide instantiations of our approach under varying conditions, leading to the first non-asymptotic end-to-end convergence guarantee for multi-task nonlinear control. OMAC can also be integrated with deep representation learning. Experiments show that OMAC significantly outperforms conventional adaptive control approaches which do not learn the shared representation, in inverted pendulum and 6-DoF drone control tasks under varying wind conditions.

Motivation & Objective

• Address the challenge of enabling autonomous robots to rapidly adapt to new environmental conditions, such as varying wind or terrain.
• Bridge the gap between representation learning and control theory by integrating online representation learning with control-theoretic stability guarantees.
• Provide end-to-end theoretical convergence guarantees—specifically sublinear cumulative control error—for multi-task nonlinear control under unknown, environment-dependent dynamics.
• Enable faster adaptation in new tasks by learning a shared representation of environment-dependent dynamics across multiple tasks.
• Integrate OMAC with deep representation learning to improve empirical performance while maintaining theoretical guarantees.

Proposed method

• Propose a hierarchical optimization framework with a meta-adapter to learn a shared representation of environment-dependent dynamics across tasks.
• Use an inner-adapter to perform environment-specific control updates based on the shared representation, enabling fast adaptation per environment.
• Formulate the control problem as a sequence of online optimization problems over time steps (inner iterations) and environments (outer iterations).
• Derive theoretical convergence bounds under convexity assumptions (jointly and element-wise convex) using online convex optimization tools.
• Integrate deep neural networks as the representation learner in OMAC, enabling end-to-end differentiable learning with control guarantees.
• Design an interaction protocol where the environment selects conditions adversarially at the start of each outer iteration, and the controller adapts online using observed state and control feedback.

Experimental results

Research questions

• RQ1 Can a unified framework be developed that combines online representation learning with control-theoretic stability guarantees for nonlinear systems?
• RQ2 Does learning a shared representation of environment-dependent dynamics lead to faster adaptation and improved control performance in multi-task settings?
• RQ3 Can non-asymptotic convergence guarantees (e.g., sublinear cumulative error) be established for multi-task nonlinear adaptive control?
• RQ4 How does the integration of deep representation learning affect the empirical performance and generalization of adaptive control in complex environments?
• RQ5 What is the role of environment diversity in enabling effective meta-learning for control, and how does it impact theoretical convergence?

Key findings

• OMAC achieves the first non-asymptotic end-to-end convergence guarantee for multi-task nonlinear adaptive control under jointly and element-wise convex assumptions.
• In experiments, OMAC significantly outperforms conventional adaptive control methods that do not learn a shared representation, particularly in inverted pendulum and 6-DoF drone control under varying wind conditions.
• The bi-convex variant of OMAC achieves performance comparable to deep learning-based OMAC but with substantially fewer parameters, demonstrating sample efficiency.
• In the pendulum task, OMAC (convex) performs well due to the superposition of $ c $-invariant and $ c $-dependent dynamics, indicating robustness to structural dynamics complexity.
• Theoretical analysis shows that environment diversity is critical for performance, as supported by Corollary 4 and Theorem 6, which imply improved convergence with diverse task distributions.
• OMAC with deep representation learning further improves empirical performance, demonstrating the scalability and practical viability of the framework in complex control tasks.