Learning Stable Controlled Dynamical Systems via Input-Contraction Neural Differential Models

Technical Summary: Learning Stable Controlled Dynamical Systems via Input-Contraction Neural Differential Models

Problem Statement
The paper addresses a critical limitation in data-driven system identification: the inability of existing Neural Ordinary Differential Equations (Neural ODEs) to guarantee structural stability when modeling non-autonomous systems subject to time-varying control inputs. While Neural ODEs have successfully parameterized autonomous vector fields, their extensions to controlled systems (e.g., Neural CDEs, ICODE) often treat external inputs heuristically. Consequently, these models lack rigorous closed-loop stability guarantees under generic, non-vanishing input sequences. This deficiency leads to error accumulation, state divergence, and extreme sensitivity to external perturbations during long-horizon rollouts, rendering them unreliable for safety-critical applications in robotics and industrial control.

Methodology: The Input-Contraction Neural Differential Model (ICNDM)
The authors propose the Input-Contraction Neural Differential Model (ICNDM), a framework designed to learn continuous-time representations of controlled dynamical systems while ensuring incremental exponential convergence. The methodology integrates three core components:

1. Input-Embedded Architecture: Unlike standard concatenation approaches, ICNDM employs a dedicated neural input encoder, $E_\phi(u)$, to map raw control inputs into a latent space. This decouples high-frequency control signals from direct algebraic coupling with the state vector, stabilizing gradient variations during stiff ODE integration.
2. Joint Neural Metric Optimization: The framework simultaneously learns the non-autonomous neural vector field, $f_\theta(x, E_\phi(u))$, and a generalized Riemannian contraction metric, $M_\omega(x)$. To ensure the metric remains uniformly positive definite, the authors enforce a structural constraint using a localized Cholesky decomposition ($M_\omega(x) = L_\omega(x)L_\omega(x)^T + \epsilon I$).
3. Regularized Objective Function: The training objective combines a standard trajectory prediction loss with two structural regularizers:
   • A soft-spectral barrier penalty ($L_c$) that enforces the contraction condition $\Psi(x, u) \preceq -\beta M_\omega(x)$, where $\Psi$ is the generalized non-autonomous contraction matrix involving the state Jacobian and the time derivative of the metric.
   • A Frobenius norm regularizer ($L_u$) on the input Jacobian to limit the sensitivity of the vector field to high-frequency input variations.

Theoretical Contributions
The paper provides formal theoretical guarantees for the proposed architecture:

• Incremental Exponential Convergence: Under the assumption that the spectral contraction loss is minimized, the authors prove that any two trajectories driven by the same input sequence converge exponentially, with the error bound decaying as $e^{-\beta t/2}$.
• Input-to-State Contraction (ISC): The authors derive sufficient conditions for input-to-state contraction under bounded external disturbances. They establish that the tracking mismatch between trajectories driven by different inputs is globally bounded by a term dependent on the input difference and the system's Lipschitz constant, ensuring robustness against input perturbations.

Experimental Results
The performance of ICNDM was evaluated against five state-of-the-art baselines (Standard Neural ODE, Controlled Neural ODE, ICODE, ControlSynth, and Differential Neural Networks) across two domains:

1. Nonlinear Chaotic Benchmarks: Tests on Duffing and Van der Pol oscillators driven by multi-frequency sinusoidal inputs demonstrated that ICNDM significantly outperforms baselines in long-horizon predictions. While standard models exhibited substantial error accumulation (e.g., MSE rising to $1.89 \times 10^{-1}$ for Standard NODE on the Duffing oscillator over 200 steps), ICNDM maintained stable bounds (MSE of $1.02 \times 10^{-3}$).
2. Physical PMSM Drive System: Validation on a Permanent Magnet Synchronous Motor (PMSM) dataset confirmed the model's practical utility. In 500-step rollouts of the quadrature axis current, unconstrained models drifted significantly from ground truth, whereas ICNDM tracked the true dynamics accurately.
3. Robustness to Noise: Experiments injecting Gaussian noise into input channels showed that ICNDM degrades gracefully as noise levels increase, whereas baseline models suffered rapid error growth. This behavior empirically validates the theoretical Input-to-State Contraction properties.

Significance and Claims
The paper claims to bridge the gap between input-aware modeling and structural stability properties. By explicitly modeling time-varying control actions while guaranteeing input-to-state contraction through joint metric optimization, ICNDM offers a solution to the instability inherent in unconstrained neural differential equations. The authors assert that this framework provides rigorous asymptotic guarantees for highly nonlinear controlled systems, a capability previously restricted to autonomous formulations or those assuming constant inputs. The work positions ICNDM as a robust tool for system identification in dynamic environments, with demonstrated superiority in reducing long-horizon drift and enhancing resilience to input perturbations.