Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons

1. Problem Statement

The paper addresses the control of linear systems subject to time-varying delays in both the input channel ($D_1(t)$) and the measurement channel ($D_2(t)$).

• The Core Challenge: While predictor feedback is a robust method for handling delays, its exact implementation requires the inverse delay mapping (specifically, the prediction horizon $\psi(t) = \phi_1^{-1}(t) - t$, where $\phi_1(t) = t - D_1(t)$).
• The Bottleneck: For time-varying delays, this inverse mapping rarely has a closed-form analytical solution. Consequently, exact implementation is impossible, and the system must rely on approximations.
• The Gap: Existing literature often assumes exact implementation or focuses on state-feedback. This work tackles the output-feedback scenario (where only delayed measurements are available) and rigorously analyzes the stability of the system when the prediction horizon is approximated.

2. Methodology

The authors propose a three-stage control architecture and two distinct methods for approximating the critical prediction horizon.

A. Control Architecture (Output-Feedback Design)

The proposed controller combines an observer and a predictor:

1. Observer: Reconstructs the delayed state $Z(\phi_2(t))$ from the delayed measurement $Y(t)$.
2. State Propagation: Propagates the estimated state forward to the current time $t$ using plant dynamics to obtain $\hat{Z}(t)$.
3. Predictor: Propagates the current state estimate to a future time $t + \psi(t)$ to generate $\hat{P}(t)$, which is used in the control law $U(t) = K\hat{P}(t)$.

B. Approximation Schemes for the Prediction Horizon ($\psi$)

Since $\psi(t)$ cannot be computed exactly, the authors propose two approaches to approximate the inverse delay operator:

1. Numerical Integration (ODE-based):

   • Reformulates the inverse delay problem as an Ordinary Differential Equation (ODE): $\dot{\psi}(t) = \frac{\dot{D}(t+\psi(t))}{1-\dot{D}(t+\psi(t))}$.
   • Solves this ODE using standard numerical methods (e.g., Explicit Euler or Runge-Kutta).
   • Pros: Model-based, no training data required.
   • Cons: Computationally expensive for long time horizons; error bounds grow with the time horizon $T$.

2. Neural Operator Approximation (Data-Driven):

   • Treats the inverse delay mapping as an operator learning problem: $D(\cdot) \mapsto \psi(\cdot)$.
   • Uses Neural Operators (specifically Fourier Neural Operators, FNO) to learn this mapping from a dataset of delay functions and their corresponding exact inverses.
   • Pros: Extremely fast inference (GPU-accelerated); acts as a surrogate for the expensive numerical integration.
   • Cons: Requires offline training; accuracy depends on the training distribution.

3. Key Contributions

The paper makes three primary theoretical and practical contributions:

1. Operator-Theoretic Framework:

   • The authors establish that the inverse delay mapping is Lipschitz continuous on compact sets.
   • They prove that both the numerical integration method and the neural operator approach can achieve arbitrary approximation accuracy over compact sets, providing a unified mathematical foundation for analyzing these approximations.

2. Robust Stability Analysis:

   • The paper derives sufficient conditions for Global Exponential Stability of the closed-loop system.
   • Using a backstepping transformation and Lyapunov-Krasovskii functionals, they prove that if the approximation error of the prediction horizon ($|\psi(t) - \hat{\psi}(t)|$) is sufficiently small (bounded by $\epsilon^*$), the system remains stable despite the approximation error and the presence of both input and measurement delays.
   • This is the first output-feedback predictor design for linear systems with both time-varying input and measurement delays that accounts for horizon approximation errors.

3. Practical Implementation & Trade-offs:

   • The authors provide a comprehensive numerical study comparing the Euler method, Runge-Kutta (RK4), and the Neural Operator (FNO).
   • They demonstrate that while numerical methods are accurate, they are computationally heavy. The Neural Operator offers a massive speedup (up to 21x faster than RK4) with negligible loss in control performance, provided the approximation error stays within the theoretical stability bounds.

4. Results

• Theoretical Guarantees: Theorem 4 proves that the closed-loop system is globally exponentially stable if the prediction horizon error is bounded by a specific threshold $\epsilon^*$. Corollary 1 extends this to infinite time horizons by stitching together finite-time stability intervals.
• Numerical Validation:
  • A simulation was performed on an unstable linear system with oscillatory, time-varying delays.
  • Accuracy: The Neural Operator (FNO) achieved a validation error of $9 \times 10^{-3}$ and successfully stabilized the system.
  • Efficiency:
    • Euler Method: ~12 ms per evaluation.
    • RK4: ~42.5 ms per evaluation.
    • Neural Operator (FNO): ~2.0 ms per evaluation (approx. 6x faster than Euler, 21x faster than RK4).
  • The results confirm that the Neural Operator can replace computationally intensive numerical solvers in real-time control loops without compromising stability.

5. Significance

This work is significant for several reasons:

• Bridging Theory and Practice: It solves the "implementation gap" in predictor feedback. Theoretical predictor designs are often dismissed because the inverse delay is unsolvable; this paper provides rigorous, implementable approximations with guaranteed stability.
• Advancement in Learning-Based Control: It moves beyond using neural networks merely as black-box controllers. Instead, it uses Neural Operators as high-fidelity surrogates for specific mathematical operators (inverse delay), integrating them into a rigorous control-theoretic framework with proven stability margins.
• Output-Feedback Complexity: By handling both input and measurement delays simultaneously in an output-feedback setting, the work addresses a more realistic and complex scenario than previous state-feedback-only studies.
• Real-Time Viability: The demonstration that neural operators can accelerate control computations by an order of magnitude makes advanced predictor feedback viable for high-frequency, latency-sensitive cyber-physical systems.