Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Neural network based control of unknown nonlinear systems via contraction analysis

Hao Yin, Claudio De Persis, Bayu Jayawardhana, Santiago Sanchez Escalonilla Plaza

TL;DR

This work links contraction analysis with neural-ODE modeling to control unknown nonlinear systems. By approximating unknown drift dynamics with a feedforward NN inside a Neural ODE, and enforcing incremental sector bounds on activations along with contraction constraints on weights, the authors derive convex LMIs that certify NODE contractivity. When the NODE is not contractive, they design a contractive NN controller that combines a linear stabilizing term with a neural compensation to ensure trajectories of the true system converge to a neighborhood of the unknown equilibrium, whose size scales with the NODE approximation error. The approach is demonstrated on a Hopfield single-layer pendulum and a multilayer wheeled-vehicle path follower, achieving quantified neighborhood convergence and highlighting a practical optimization framework to minimize the steady-state bound. The results offer a data-driven, contractivity-based route to stabilizing unknown nonlinear systems without requiring explicit equilibrium knowledge, with potential extensions to PINN-based Lyapunov constructions for broader applicability.

Abstract

This paper studies the design of neural network (NN)-based controllers for unknown nonlinear systems, using contraction analysis. A Neural Ordinary Differential Equation (NODE) system is constructed by approximating the unknown draft dynamics with a feedforward NN. Incremental sector bounds and contraction theory are applied to the activation functions and the weights of the NN, respectively. It is demonstrated that if the incremental sector bounds and the weights satisfy some non-convex conditions, the NODE system is contractive. To improve computational efficiency, these non-convex conditions are reformulated as convex LMI conditions. Additionally, it is proven that when the NODE system is contractive, the trajectories of the original autonomous system converge to a neighborhood of the unknown equilibrium, with the size of this neighborhood determined by the approximation error. For a single-layer NN, the NODE system is simplified to a continuous-time Hopfield NN. If the NODE system does not satisfy the contraction conditions, an NN-based controller is designed to enforce contractivity. This controller integrates a linear component, which ensures contraction through suitable control gains, and an NN component, which compensates for the NODE system's nonlinearities. This integrated controller guarantees that the trajectories of the original affine system converge to a neighborhood of the unknown equilibrium. The effectiveness of the proposed approach is demonstrated through two illustrative examples.

Neural network based control of unknown nonlinear systems via contraction analysis

TL;DR

This work links contraction analysis with neural-ODE modeling to control unknown nonlinear systems. By approximating unknown drift dynamics with a feedforward NN inside a Neural ODE, and enforcing incremental sector bounds on activations along with contraction constraints on weights, the authors derive convex LMIs that certify NODE contractivity. When the NODE is not contractive, they design a contractive NN controller that combines a linear stabilizing term with a neural compensation to ensure trajectories of the true system converge to a neighborhood of the unknown equilibrium, whose size scales with the NODE approximation error. The approach is demonstrated on a Hopfield single-layer pendulum and a multilayer wheeled-vehicle path follower, achieving quantified neighborhood convergence and highlighting a practical optimization framework to minimize the steady-state bound. The results offer a data-driven, contractivity-based route to stabilizing unknown nonlinear systems without requiring explicit equilibrium knowledge, with potential extensions to PINN-based Lyapunov constructions for broader applicability.

Abstract

This paper studies the design of neural network (NN)-based controllers for unknown nonlinear systems, using contraction analysis. A Neural Ordinary Differential Equation (NODE) system is constructed by approximating the unknown draft dynamics with a feedforward NN. Incremental sector bounds and contraction theory are applied to the activation functions and the weights of the NN, respectively. It is demonstrated that if the incremental sector bounds and the weights satisfy some non-convex conditions, the NODE system is contractive. To improve computational efficiency, these non-convex conditions are reformulated as convex LMI conditions. Additionally, it is proven that when the NODE system is contractive, the trajectories of the original autonomous system converge to a neighborhood of the unknown equilibrium, with the size of this neighborhood determined by the approximation error. For a single-layer NN, the NODE system is simplified to a continuous-time Hopfield NN. If the NODE system does not satisfy the contraction conditions, an NN-based controller is designed to enforce contractivity. This controller integrates a linear component, which ensures contraction through suitable control gains, and an NN component, which compensates for the NODE system's nonlinearities. This integrated controller guarantees that the trajectories of the original affine system converge to a neighborhood of the unknown equilibrium. The effectiveness of the proposed approach is demonstrated through two illustrative examples.

Paper Structure

This paper contains 13 sections, 11 theorems, 72 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Main result
  4. Contractive neural ordinary differential equation
  5. ‌Hopfield (Single-layer) neural network case
  6. ‌Multilayer neural network case
  7. Neural network based controller design
  8. ‌‌Hopfield (Single-layer) neural network case
  9. ‌Multilayer neural network case
  10. Simulation setup
  11. ‌Hopfield (Single-layer) neural network case
  12. ‌Multilayer neural network case
  13. CONCLUSION

Key Result

Lemma II.3

Assume $\phi^i$ is incremental sector bounded by $[K_1^i, K_2^i]$. Then, the following inequality holds for all $x_j\in \mathcal{D}$, where $Q_1=\mathrm{BD}(K_1^1W^1, \dots, K_1^{k}W^{k})$, $Q_2=\mathrm{BD}(K_2^1W^1, \dots, K_2^{k}W^{k})\in \mathbb{R}^{\sum_{i=0}^{k-1} m_{i+1}\times \sum_{i=0}^{k-1} m_{i}}$.

Figures (2)

  • Figure 1: The plot shows the trajectories of the controlled system, $x_i$ in Example \ref{['example2']} initialized at $\left[11\right]^{\top}$, $\left[-1-1\right]^{\top}$, $\left[-1.51.5\right]^{\top}$, $\left[1.5-1.5\right]^{\top}$.
  • Figure 2: The plot shows the trajectories of the controlled system, $x_i$ in Example \ref{['example3']} initialized at $\left[0.80.8\right]^{\top}$, $\left[-0.8-0.8\right]^{\top}$, $\left[-0.50.5\right]^{\top}$, $\left[0.5-0.5\right]^{\top}$.

Theorems & Definitions (20)

  • Definition II.1
  • Lemma II.3
  • Definition III.1
  • Theorem III.2
  • Theorem III.3
  • Theorem III.4
  • Remark III.5
  • Theorem III.6
  • Proposition III.7
  • Proposition III.8
  • ...and 10 more