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 $\epsilon$. 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.

翻译：本研究首次为非线性系统时滞自适应控制中的近似预测器建立了严格的稳定性保证，解决了实际应用中无法获得精确预测器的关键挑战。我们分析了两种情形：(i) 当驱动输入可直接测量时；(ii) 当输入需在线估计时。对于可测量输入情形，我们证明了半全局实际渐近稳定性，其显式界与近似误差 $\epsilon$ 成正比。对于不可测输入情形，我们证明了局部实际渐近稳定性，其吸引域显式依赖于初始时滞估计和预测器近似误差。为连接理论与实践，我们证明了神经算子——一类灵活的基于神经网络的近似器——能够实现任意小的近似误差，从而满足我们稳定性定理的条件。在两个非线性基准系统——生物蛋白质激活/抑制模型和微生物生长恒化器模型——上的数值实验验证了我们的理论结果。特别地，我们的数值模拟证实了近似预测器下的稳定性，凸显了神经算子强大的泛化能力，并展示了相较于基线定点方法高达15倍的计算加速。