Stable On-the-Fly Learning for Dynamic Neural Networks With Delayed Inputs

IEEE Access. 2026. Vol. 14. P. 14369–14392.

Chertopolokhov V., Mukhamedov A., Bugriy G., Kibkalo Vladislav, Ionkin V., Tolchennikova V., Chairez I.

This study presents on-the-fly identification and multi-step prediction of nonlinear systems with delayed inputs using a dynamic neural network combined with a smooth projection onto ellipsoids. The projection enforces parameter constraints that guarantee stability, while a Lyapunov–Krasovskii analysis yields computable ultimate error bounds. Riccati-type matrix inequalities are derived, providing an efficient vectorization–projection–devectorization implementation suitable for online use. Compared with delay-dependent functionals, the proposed approach avoids state augmentation and reduces computational overhead while preserving formal guarantees. The analysis is developed for a fixed delay, while extensions to bounded time-varying and multiple delays are discussed. Numerical studies of geodesic motion, an aircraft model, and a head-rotation task demonstrate fast transients, accurate prediction, and graceful degradation as the delay increases. The results indicate that stable learning with delayed inputs is practical and competitive with recent delay-aware identifiers in terms of accuracy, complexity, and reproducibility.

Keywords: нейронные сети, нелинейные системы, nonlinear systems, траектория, анализ устойчивости, запаздывание, accuracy, Trajectory, линейные матричные неравенства, Stability analysis, switches, Uncertainty, эффект переключения, linear matrix inequalities, Neural networks, Delays, Delay effects