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

Abstract: Due to simplicity and strong stability guarantees, predictor feedback methods have stood as a popular approach for time delay systems since the 1950s. For time-varying delays, however, implementation requires computing a prediction horizon defined by the inverse of the delay function, which is rarely available in closed form and must be approximated. In this work, we formulate the inverse delay mapping as an operator learning problem and study predictor feedback under approximation of the prediction horizon. We propose two approaches: (i) a numerical method based on time integration of an equivalent ODE, and (ii) a data-driven method using neural operators to learn the inverse mapping. We show that both approaches achieve arbitrary approximation accuracy over compact sets, with complementary trade-offs in computational cost and scalability. Building on these approximations, we then develop an output-feedback predictor design for systems with delays in both the input and the measurement. We prove that the resulting closed-loop system is globally exponentially stable when the prediction horizon is approximated with sufficiently small error. Lastly, numerical experiments validate the proposed methods and illustrate their trade-offs between accuracy and computational efficiency.

• The paper introduces an operator learning framework using both numerical and neural methods to approximate prediction horizons for delay compensation.
• The paper rigorously establishes exponential stability under bounded approximation errors using Lyapunov analysis and small-gain conditions.
• The paper demonstrates significant computational speed-ups with neural operators, enabling real-time control for systems with complex, time-varying delays.

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

Overview

The paper addresses the implementation and stability analysis of predictor-based output-feedback controllers for finite-dimensional linear systems with time-varying input and measurement delays. The critical difficulty is that, under time-varying delays, computation of the predictor requires inverting a delay operator, which is almost never available in closed form. This work formulates the inverse delay computation as a problem of operator learning, explores numerical and neural operator-based approximation methods, and rigorously establishes exponential stability of the closed-loop system under bounded approximation errors.

System Model and Problem Statement

The class of systems considered is

$$\begin{cases} \dot{Z}(t) = AZ(t) + BU\left(t - D_1(t)\right), \ Y(t) = CZ\left(t - D_2(t)\right), \end{cases}$$

where $Z(t)\in\mathbb{R}^n$ is the system state, $U(t)\in\mathbb{R}^m$ is the control, $Y(t)\in\mathbb{R}^p$ is the measured output, and $D_1$, $D_2$ are differentiable and strictly positive, uniformly bounded time-varying delay functions for actuation and sensing respectively.

Ensuring stability in this context is nontrivial: delays destroy the separation principle and their time-variation prevents closed-form predictor realizations. The practical obstacle is the requirement to compute the prediction horizon, i.e., to access the inverse $\phi^{-1}_1(t)$, where $\phi_1(t) = t - D_1(t)$.

Control Architecture

The output-feedback predictor-controller is realized in three cascaded stages (Figure 1): observer, state reconstruction, and predictor.

Figure 2: Three-stage observer–predictor feedback law and horizons, depicting the flow of delayed measurements, observer estimation, state reconstruction, and prediction.

1. Delayed-Observer: Uses delayed measurements $Y(t)$ to estimate the delayed state $Z(\phi_2(t))$, yielding $Z(t)\in\mathbb{R}^n$0.
2. State Reconstruction: Propagates $Z(t)\in\mathbb{R}^n$1 forward in time using the plant model, producing $Z(t)\in\mathbb{R}^n$2.
3. Prediction: Further propagates $Z(t)\in\mathbb{R}^n$3 to the future time $Z(t)\in\mathbb{R}^n$4, yielding $Z(t)\in\mathbb{R}^n$5, and the control input is $Z(t)\in\mathbb{R}^n$6 with $Z(t)\in\mathbb{R}^n$7 such that $Z(t)\in\mathbb{R}^n$8 is Hurwitz.

These steps are interlinked by the necessity to compute or approximate the solution to $Z(t)\in\mathbb{R}^n$9, i.e., the prediction horizon $U(t)\in\mathbb{R}^m$0.

Approximation of the Prediction Horizon

Operator-Theoretic Formulation

The inverse delay mapping is cast as an operator approximation problem. The inverse operator

$U(t)\in\mathbb{R}^m$1

is shown to be Lipschitz with respect to $U(t)\in\mathbb{R}^m$2 on compact sets. Two approximation strategies are analyzed:

1. Numerical ODE-Based Approximation

The inverse $U(t)\in\mathbb{R}^m$3 satisfies the nonlinear ODE:

$U(t)\in\mathbb{R}^m$4

An explicit Euler discretization is developed, and error bounds are rigorously established in terms of the regularity of $U(t)\in\mathbb{R}^m$5, time horizon length, and ODE step size. The uniform error scales with the time interval and step size, and slow delay variation (small $U(t)\in\mathbb{R}^m$6) leads to smaller errors.

2. Neural Operator Approximation

The paper leverages recent advances in neural operators (e.g., DeepONet, Fourier Neural Operator), which approximate mappings between function spaces. Given the regularity of the inverse delay operator, universal approximation results guarantee for every $U(t)\in\mathbb{R}^m$7 the existence of a neural operator $U(t)\in\mathbb{R}^m$8 such that

$U(t)\in\mathbb{R}^m$9

for compact families $Y(t)\in\mathbb{R}^p$0 of admissible delays. This shifts the bulk of computation offline, and evaluation online reduces to a fast neural inference.

Exponential Stability Analysis

Global exponential stability of the closed-loop system is established under both exact and approximate computation of $Y(t)\in\mathbb{R}^p$1. The main result asserts that if the predictor horizon approximation error remains below a threshold $Y(t)\in\mathbb{R}^p$2 uniform in time, then exponential stability with margin $Y(t)\in\mathbb{R}^p$3 is preserved:

$Y(t)\in\mathbb{R}^p$4

where the Lyapunov functional $Y(t)\in\mathbb{R}^p$5 measures the state, state estimation, and control history energy. The analysis is performed by transforming the system into a backstepping PDE-ODE target form, tracking how horizon approximation errors propagate to boundary terms, and quantifying their effect on Lyapunov decay via small-gain arguments. Sufficient conditions on $Y(t)\in\mathbb{R}^p$6 in terms of controller/observer gains and delay bounds are derived. The performance penalty from approximation manifests as a degradation in the exponential rate.

Importantly, the analysis decouples the stability margin from the internal structure of the approximator (numerical or neural), requiring only a uniform-in-time error bound.

Numerical Experiments

The efficacy of both approximation approaches is validated on an unstable linear system with oscillatory, time-varying, non-analytic delays. A Fourier Neural Operator (FNO) is trained offline on a range of plausible delay profiles to learn $Y(t)\in\mathbb{R}^p$7.

(Figure 1)

Figure 1: Example feedback control response with FNO-approximated prediction horizon. Observer and plant states converge, demonstrating closed-loop stabilization for time-varying delays.

Key results:

• Controller Performance: The neural operator-based predictor yields rapid stabilization even with delays never seen during training, matching the accuracy of ODE-based schemes.
• Computation Time: Evaluation of FNO is $Y(t)\in\mathbb{R}^p$8 faster than Euler and $Y(t)\in\mathbb{R}^p$9 faster than Runge-Kutta on a GPU, but with minor loss in consistency error.
• Trade-off: Neural operator is preferable for real-time, high-rate control when approximation accuracy is within the admissible threshold for closed-loop stability.

Theoretical and Practical Implications

Theoretically, the paper provides an operator-theoretic unification of numerical and learning-based predictor approximation, with general stability guarantees under bounded horizon errors. The framework admits analysis of black-box approximators as long as they fall within the Lyapunov margin, supporting robust digital implementation for time-varying delay compensation.

Practically, the neural operator approach removes the need for online nonlinear equation solving, allowing real-time compensation for systems with complex, time-varying, and even data-driven delay models. The methodology is extendable to higher-order, distributed, and infinite-dimensional systems as the neural operator gains expressivity.

The operator-learning viewpoint—demonstrated here for delay inversion—parallels recent results in neural-approximated PDE backstepping control, adaptation, and robust stabilization for infinite-dimensional and nonlinear systems (Lanthaler et al., 2023, Li et al., 2020), [10374221], [BHAN2025105968], [LV2025112553], [ZHANG2026112809].

Future Directions

The operator-based stability envelope provides a foundation for:

• Extending predictor-feedback to nonlinear, adaptive, or stochastic systems with delays via neural operator surrogacy (Bhan et al., 30 Sep 2025).
• Investigating the effect of neural network architecture choices, generalization, and discretization error in control-theoretic contexts (Lanthaler et al., 2024).
• Integration with modern networked control systems where delays are learned or estimated online.

Conclusion

This work rigorously demonstrates that predictor-based output-feedback control for linear systems subject to time-varying delays can be rendered implementable and robust using either adaptive step ODE solvers or neural operator surrogates for the required inverse delay mapping. Both approaches are shown to yield arbitrary accuracy over finite horizons; closed-loop exponential stability is proved up to an explicit robustness margin. The neural operator solution, benefiting from offline training, enables real-time delay compensation even with high computational restrictions and out-of-model delay profiles, marking a significant advancement for practical implementation of delay-compensating controllers.

Reference: "Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons" (2603.29117)