Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Neural Lyapunov Differentiable Predictive Control

Sayak Mukherjee, Ján Drgoňa, Aaron Tuor, Mahantesh Halappanavar, Draguna Vrabie

TL;DR

This work tackles the computational burden of model predictive control by introducing Neural Lyapunov Differentiable Predictive Control (NLDPC), which jointly learns an explicit neural predictive policy and a neural Lyapunov function within a differentiable programming framework. The approach builds a differentiable closed-loop graph that integrates dynamics, costs, and Lyapunov constraints, enabling offline training via automatic differentiation and gradient-based optimization. It provides probabilistic stability guarantees using Hoeffding-based bounds over sampled initial conditions and demonstrates scalability through two numerical case studies: a double integrator and a PVTOL aircraft model. The results show that the learned Lyapunov function can expand the safe operating region while ensuring constraint satisfaction with reduced online computation, highlighting a practical path toward stability-certified, data-driven MPC for nonlinear systems.

Abstract

We present a learning-based predictive control methodology using the differentiable programming framework with probabilistic Lyapunov-based stability guarantees. The neural Lyapunov differentiable predictive control (NLDPC) learns the policy by constructing a computational graph encompassing the system dynamics, state and input constraints, and the necessary Lyapunov certification constraints, and thereafter using the automatic differentiation to update the neural policy parameters. In conjunction, our approach jointly learns a Lyapunov function that certifies the regions of state-space with stable dynamics. We also provide a sampling-based statistical guarantee for the training of NLDPC from the distribution of initial conditions. Our offline training approach provides a computationally efficient and scalable alternative to classical explicit model predictive control solutions. We substantiate the advantages of the proposed approach with simulations to stabilize the double integrator model and on an example of controlling an aircraft model.

Neural Lyapunov Differentiable Predictive Control

TL;DR

This work tackles the computational burden of model predictive control by introducing Neural Lyapunov Differentiable Predictive Control (NLDPC), which jointly learns an explicit neural predictive policy and a neural Lyapunov function within a differentiable programming framework. The approach builds a differentiable closed-loop graph that integrates dynamics, costs, and Lyapunov constraints, enabling offline training via automatic differentiation and gradient-based optimization. It provides probabilistic stability guarantees using Hoeffding-based bounds over sampled initial conditions and demonstrates scalability through two numerical case studies: a double integrator and a PVTOL aircraft model. The results show that the learned Lyapunov function can expand the safe operating region while ensuring constraint satisfaction with reduced online computation, highlighting a practical path toward stability-certified, data-driven MPC for nonlinear systems.

Abstract

We present a learning-based predictive control methodology using the differentiable programming framework with probabilistic Lyapunov-based stability guarantees. The neural Lyapunov differentiable predictive control (NLDPC) learns the policy by constructing a computational graph encompassing the system dynamics, state and input constraints, and the necessary Lyapunov certification constraints, and thereafter using the automatic differentiation to update the neural policy parameters. In conjunction, our approach jointly learns a Lyapunov function that certifies the regions of state-space with stable dynamics. We also provide a sampling-based statistical guarantee for the training of NLDPC from the distribution of initial conditions. Our offline training approach provides a computationally efficient and scalable alternative to classical explicit model predictive control solutions. We substantiate the advantages of the proposed approach with simulations to stabilize the double integrator model and on an example of controlling an aircraft model.
Paper Structure (16 sections, 2 theorems, 20 equations, 9 figures, 1 algorithm)

This paper contains 16 sections, 2 theorems, 20 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Background
  3. Parametric Optimal Control Problem
  4. Neural Predictive Control Policy
  5. Stability via Neural Lyapunov Functions
  6. Neural Lyapunov Differentiable Predictive Control (NLDPC) Framework
  7. Soft state, control, and Lyapunov constraints
  8. Neural Lyapunov DPC Problem
  9. Neural Lyapunov DPC Computational Graph
  10. Neural Lyapunov DPC Gradients
  11. Simultaneous Learning of Constrained Neural Control Policy and Lyapunov Function
  12. Stability Guarantees
  13. Numerical Case Studies
  14. Example 1: Stabilizing Double Integrator
  15. Example 2: Stabilizing PVTOL aircraft model
  16. ...and 1 more sections

Key Result

Lemma 1

(Lyapunov Characterization) : Using the neural predictive control policy ${\bf u}_k = \pi_{\boldsymbol \theta}({\bf x}_k)$ for discrete time steps $k$, with the Liptsitz continuous closed-loop $f(\bf x_k, {\bf u}_k)$, for the equilibrium $\bf x_\mathcal{O} = \bf 0$, if there exists a set $\mathcal{D then $\bf x_\mathcal{O} = \bf 0$ is an asymptotically stable equilibrium, and $V(.)$ is a control L

Figures (9)

  • Figure 1: Neural Lyapunov DPC methodology.
  • Figure 2: Closed-loop phase portrait with overlaid Lyapunov contours for double integrator.
  • Figure 3: Learned Lyapunov function for double integrator.
  • Figure 4: Stability regions via discrete-time Lyapunov stability condition \ref{['Discrete-time Lyap']} for double integrator.
  • Figure 5: Closed-loop state and control trajectories for double integrator.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 2