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

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.

Figures (6)

• Figure 1: Neural-NPV framework. Within Neural-NPV, a PD controller $\pi(x,\theta)$ (to stabilize the NPV system) and a PD Lyapunov function $V(x,\theta)$ (to certify the stability of the closed-loop system) are jointly trained to minimize the Lyapunov loss via supervised learning. ROI: Region of interest; ROA: Region of attraction
• Figure 2: Robust ROA comparison between SOS-NPV and Neural-NPV with the same input constraint for the inverted pendulum system.
• Figure 3: Trajectories of states from ten random initial points inside $\mathcal{X}$ under $\theta(t) = 0.6 + 0.4\cos(t)$ (top), the Lyapunov function (middle), control input (bottom). Note that the input constraint $|u| \leq 3$ is respected for all cases.
• Figure 4: 3D slice of the robust ROA $\Lambda^{\rho}_V$ for the quadrotor in $x$-$y$-$z$ space
• Figure 5: Ten random trajectories of the state (top) and the corresponding Lyapunov function (bottom) for the quadrotor

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Remark 1