Neural-NPV Control: Learning Parameter-Dependent Controllers and Lyapunov Functions with Neural Networks

TL;DR

Neural-NPV is a learning-based framework that synthesizes parameter-dependent controllers and Lyapunov functions for nonlinear parameter-varying systems, overcoming traditional scalability and conservativeness issues.

Contribution

It introduces a two-stage neural network approach for joint synthesis of controllers and Lyapunov functions, applicable to a broader class of systems than SOS methods.

Findings

Neural-NPV outperforms SOS-based methods in scalability and applicability.

The framework effectively stabilizes systems like inverted pendulum and quadrotor.

It maximizes the robust region of attraction through level-set refinement.

Abstract

Nonlinear parameter-varying (NPV) systems are a class of nonlinear systems whose dynamics explicitly depend on time-varying external parameters, making them suitable for modeling real-world systems with dynamics variations. Traditional synthesis methods for NPV systems, such as sum-of-squares (SOS) optimization, are only applicable to control-affine systems, face scalability challenges and often lead to conservative results due to structural restrictions. To address these limitations, we propose Neural-NPV, a two-stage learning-based framework that leverages neural networks to jointly synthesize a PD controller and a PD Lyapunov function for an NPV system under input constraints. In the first stage, we utilize a computationally cheap, gradient-based counterexample-guided procedure to synthesize an approximately valid PD Lyapunov function and a PD controller. In the second stage, a level-set guided refinement is then conducted to obtain a valid Lyapunov function and controller while maximizing the robust region of attraction (R-ROA). We demonstrate the advantages of Neural-NPV in terms of applicability, performance, and scalability compared to SOS-based methods through numerical experiments involving an simple inverted pendulum with one scheduling parameter and a quadrotor system with three scheduling parameters.