Sampling-Horizon Neural Operator Predictors for Nonlinear Control under Delayed Inputs

1. Problem Statement

The paper addresses the control of nonlinear systems subject to two practical constraints:

1. Input Delays: The control input $U(t)$ affects the system dynamics with a constant but potentially long delay $D$.
2. Sampled Measurements: State measurements $X(t)$ are not continuously available but are only observed at discrete sampling instants $\{T_i\}$. These measurements may be subject to noise, irregular intervals, or temporary dropouts.

The Core Challenge:
Standard predictor feedback is a known strategy to compensate for input delays by predicting the future state $X(t+D)$ and applying a nominal control law to this prediction. However, traditional predictor feedback requires solving an implicit Ordinary Differential Equation (ODE) online. This involves numerical integration and successive approximations, leading to:

• High computational cost: Making real-time implementation difficult, especially for stiff dynamics or large delays.
• Incompatibility with sampled data: Most theoretical predictor formulations assume continuous state availability, whereas real-world sensors provide discrete, potentially non-uniform samples.

2. Methodology

The authors propose a Neural Operator (NO)-based framework to approximate the predictor feedback law offline, enabling fast online inference. They introduce two distinct architectures to handle the hybrid nature of sampled measurements and delayed inputs.

A. Mathematical Formulation

The system is modeled as:
$$\dot{X}(t) = f(X(t), U(t-D))$$
The control law uses a hybrid predictor approach where the input is $U(t) = \kappa(\hat{Z}(t))$, and $\hat{Z}(t)$ is the predicted state trajectory.

B. Two Proposed Designs

The paper distinguishes between two implementation strategies based on sampling assumptions:

Case 1: Sampling-Horizon Prediction Operator (Uniform Sampling)

• Assumption: Measurements are taken at uniform intervals ($T_{i+1} - T_i = h$).
• Mechanism: A single neural operator $\hat{\mathcal{M}}$ maps the current state $X(T_i)$ and input history $U_i$ directly to the predicted state trajectory $\hat{Z}(s)$ over the entire next sampling interval $[T_i, T_{i+1})$.
• Advantage: The error bound scales directly with the neural operator approximation error.
• Limitation: Strictly requires uniform sampling.

Case 2: Predictor Approximation (Bounded/Non-Uniform Sampling)

• Assumption: Measurements are available at non-uniform intervals, provided the gap between samples is bounded ($T_{i+1} - T_i \leq h$).
• Mechanism: A neural operator $\hat{\mathcal{P}}$ approximates only the predictor (calculating the initial condition $\hat{Z}(T_i)$ based on $X(T_i)$ and input history). The state trajectory $\hat{Z}(s)$ between samples is then generated by numerically integrating the closed-loop ODE $\dot{\hat{Z}} = f(\hat{Z}, \kappa(\hat{Z}))$.
• Advantage: Flexible; accommodates irregular sampling and missing data.
• Limitation: The approximation error is amplified by the continuous ODE evolution between samples, requiring a more accurate neural operator to maintain stability.

C. Theoretical Foundation

• Neural Operators: The authors utilize architectures (specifically Fourier Neural Operators) designed to learn mappings between function spaces, allowing them to approximate the infinite-dimensional predictor operators.
• Stability Analysis: Using Lyapunov analysis and Input-to-State Stability (ISS) theory, the authors prove Semi-Global Practical Stability.
  • The system state converges to a residual bound determined by the neural operator approximation error ($\epsilon$).
  • Trade-off: Case 2 offers sampling flexibility but amplifies the error by a factor of $e^{C_f(1+C_\kappa)h}$, whereas Case 1 has a direct error scaling but rigid sampling requirements.

3. Key Contributions

1. Novel Architectures: Development of two specific neural-operator predictor-feedback designs that bridge the gap between theoretical predictor feedback and practical sampled-data systems.
2. Rigorous Stability Guarantees: Establishment of semi-global practical stability with explicit bounds dependent on the neural operator approximation error and sampling characteristics.
3. Sampling Flexibility vs. Accuracy Trade-off: A formal analysis revealing that relaxing the uniform sampling assumption (Case 2) necessitates higher approximation accuracy to compensate for error propagation during the inter-sample interval.
4. Computational Efficiency: Demonstration that replacing numerical integration with neural operator inference drastically reduces computational latency.

4. Numerical Results

The methods were validated on a 6-link nonlinear robotic manipulator (xArm6) with delayed torque control.

• Setup: The system had a state dimension of 12 (position and velocity for 6 links) and a delay $D=0.2s$. Training data was generated using a high-precision numerical predictor.
• Performance:
  • Tracking: Both neural operator approaches successfully tracked a sinusoidal reference trajectory, comparable to a high-precision numerical baseline.
  • Robustness: The controllers handled Gaussian noise in measurements and, in Case 2, randomized non-uniform sampling intervals.
  • Speedup: The neural operators achieved a 25× computational speedup compared to the baseline numerical predictor.
    • Neural Operator inference: ~1 ms (GPU).
    • Numerical integration: ~25 ms.

5. Significance

This work is significant for the deployment of advanced control systems in real-world robotics and industrial automation where:

• Latency is critical: The 25× speedup makes predictor feedback viable for high-frequency control loops that were previously computationally prohibitive.
• Data is imperfect: The ability to handle non-uniform sampling (Case 2) makes the control strategy robust against network jitter, sensor dropouts, or asynchronous hardware.
• Safety is paramount: The paper provides rigorous mathematical guarantees (stability bounds) for data-driven control, moving beyond "black-box" heuristics to certified neural control architectures.

In summary, the paper successfully translates the theoretical power of predictor feedback into a practical, computationally efficient, and robust framework for nonlinear systems with delays and sampled measurements.