Deep-learning based KKL chain observer for discrete-time nonlinear systems with time-varying output delay

This paper proposes a Kazantzis–Kravaris–Luenberger (KKL) observer design for discrete-time nonlinear systems whose output is affected by a time-varying measurement delay. Relying on an injective state transformation, a chain of observers is designed in the latent coordinates with exponential stability guarantees through the inverse map in the original coordinates. Moreover, the relationship between the number of sub-predictors and the lower and upper bounds of the delay is derived. The transformations involved in the design of the KKL observer are identified using an unsupervised learning-based approach that relies on neural networks. A disturbance rejection and robustness analysis against measurement noise and neural network approximation error are presented, respectively. Finally, we illustrate the performance and robustness of the proposed learning-based design KKL chain observer through numerical simulations.