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Negative Imaginary Neural ODEs: Learning to Control Mechanical Systems with Stability Guarantees

Kanghong Shi, Ruigang Wang, Ian R. Manchester

TL;DR

This work addresses stability guarantees for nonlinear mechanical systems controlled by neural models by embedding negative imaginary (NI) properties into a Hamiltonian neural ODE controller (NINODE). It combines Polyak-Łojasiewicz networks (PLNets) and bi-Lipschitz neural blocks to enforce NI behavior and constructs a Lyapunov-based interconnection framework for NI Hamiltonian systems. A mechanical-system specialization provides explicit regularity conditions under which the closed-loop is asymptotically stable, with a nonlinear mass–spring example illustrating stabilization and improved performance over LSTM and linear NI controllers. Taken together, the results offer a principled path to data-driven controllers with formal stability guarantees for safety-critical mechanical systems with colocated actuators and sensors.

Abstract

We propose a neural control method to provide guaranteed stabilization for mechanical systems using a novel negative imaginary neural ordinary differential equation (NINODE) controller. Specifically, we employ neural networks with desired properties as state-space function matrices within a Hamiltonian framework to ensure the system possesses the NI property. This NINODE system can serve as a controller that asymptotically stabilizes an NI plant under certain conditions. For mechanical plants with colocated force actuators and position sensors, we demonstrate that all the conditions required for stability can be translated into regularity constraints on the neural networks used in the controller. We illustrate the utility, effectiveness, and stability guarantees of the NINODE controller through an example involving a nonlinear mass-spring system.

Negative Imaginary Neural ODEs: Learning to Control Mechanical Systems with Stability Guarantees

TL;DR

This work addresses stability guarantees for nonlinear mechanical systems controlled by neural models by embedding negative imaginary (NI) properties into a Hamiltonian neural ODE controller (NINODE). It combines Polyak-Łojasiewicz networks (PLNets) and bi-Lipschitz neural blocks to enforce NI behavior and constructs a Lyapunov-based interconnection framework for NI Hamiltonian systems. A mechanical-system specialization provides explicit regularity conditions under which the closed-loop is asymptotically stable, with a nonlinear mass–spring example illustrating stabilization and improved performance over LSTM and linear NI controllers. Taken together, the results offer a principled path to data-driven controllers with formal stability guarantees for safety-critical mechanical systems with colocated actuators and sensors.

Abstract

We propose a neural control method to provide guaranteed stabilization for mechanical systems using a novel negative imaginary neural ordinary differential equation (NINODE) controller. Specifically, we employ neural networks with desired properties as state-space function matrices within a Hamiltonian framework to ensure the system possesses the NI property. This NINODE system can serve as a controller that asymptotically stabilizes an NI plant under certain conditions. For mechanical plants with colocated force actuators and position sensors, we demonstrate that all the conditions required for stability can be translated into regularity constraints on the neural networks used in the controller. We illustrate the utility, effectiveness, and stability guarantees of the NINODE controller through an example involving a nonlinear mass-spring system.
Paper Structure (8 sections, 4 theorems, 43 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Negative imaginary systems
  4. Bi-Lipschitz and Polyak-Łojasiewicz Neural Networks
  5. Control of an NI Hamiltonian system
  6. The Mechanical System Case
  7. Illustrative Example
  8. Conclusion

Key Result

Proposition 1

wang2024monotone Suppose $\mathcal{G}(x):\mathbb R^n \to \mathbb R^n$ is $\mu$-inverse Lipschitz; i.e., $\mu|x_a-x_b|\leq |\mathcal{G}(x_a)-\mathcal{G}(x_b)|$ for all $x_a,x_b\in \mathbb R^n$. Then with $c>0$ a scalar, is a PLNet that satisfies (eq:PLNet) with $\varrho = \mu^2$.

Figures (4)

  • Figure 1: Closed-loop interconnection of the systems $\mathcal{H}_1$ and $\mathcal{H}_2$ given in (\ref{['eq:Hamiltonian']}) and (\ref{['eq:Hamiltonian2']}), respectively.
  • Figure 2: The top view of a mass-spring system containing three masses, which move rectilinearly on a frictionless floor. All the springs are nonlinear. The displacements of the masses are denoted by $q_1$, $q_2$ and $q_3$, respectively. We apply external forces to the masses and denote them by $F_1$, $F_2$ and $F_3$, respectively.
  • Figure 3: Trajectories of the positions under the control of an LSTM controller and the proposed NINODE controller. The left-column figures show the positions of individual masses for $4\ sec$, while the right-column figures show the Euclidean norm of the position vector $q$ in a log scale.
  • Figure 4: Comparison between the NINODE controller, a linear NI controller and an LSTM controller in terms of the normalized final values of the loss function (\ref{['eq:loss function J']}) after adequate training for different scales of the initial conditions.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3