Dynamical Pattern Recognition of Univariate Time Series from the Structural Stability Via Deterministic Learning

In order to achieve dynamical pattern recognition of univariate time series from the perspective of structural stability, this paper proposes a method based on Extended State Observer (ESO) and deterministic learning for addressing the issue of recognizing the structural stability of the topological structure of dynamical patterns in univariate time series. The ESO is capable of reconstructing the system states and their unknown dynamics during the observation phase, while the deterministic learning effectively resolves the challenges associated with obtaining and modeling the derivative information of the dynamics. During the learning phase, the ESO is employed to recover state trajectories from sampled output signals. These trajectories utilize the regressed trajectories as inputs, satisfying the partially persistent excitation (PE) condition, thereby accurately approximating the unknown dynamics through the estimated regressed trajectories. Subsequently, we establish a novel recognition error system that approximates the first-order derivatives of the system dynamics using a finite difference method, thus avoiding the need for re-modeling the test patterns and enabling the identification of dynamic behaviors from the topological structural perspective. Finally, we theoretically demonstrate that the residuals of the recognition error system reflect the structural stability differences in system dynamics between the training and test patterns, and simulation further substantiate the effectiveness and accuracy of the proposed method.