Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors

2025-09-30

AI Summary

Existing delay-adaptive control methods for nonlinear systems with unknown time delays lack rigorous stability guarantees and high-precision predictor design. Method: This paper proposes a neural-operator-based delay-adaptive approximate predictor, applicable to both measurable and unmeasurable input scenarios. Contribution/Results: It provides the first theoretical foundation for such predictors: (i) quantitatively characterizing the relationship between prediction error bounds and the domain of attraction; and (ii) proving uniform high-accuracy approximation capability in function space. The closed-loop system achieves semi-global or local practical asymptotic stability, depending on the scenario. Extensive simulations on two nonlinear benchmark systems—biological protein regulation and microbial cultivation—demonstrate controllable prediction errors, strong generalization, and up to 15× higher computational efficiency compared to conventional fixed-point methods. The approach effectively bridges the gap between theoretical rigor and engineering practicality.

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Abstract

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error $ε$. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.

Problem

• Stabilizing nonlinear systems with unknown delays
• Establishing stability guarantees for approximate predictors
• Addressing delay-adaptive control with neural operator approximations

Innovation

• Delay-adaptive neural operator approximate predictors
• Stability guarantees for nonlinear systems with delays
• Neural operators achieve small approximation errors