Differential Neural Network Identifiers for Periodic Systems, a Floquet’s Theory Approach

Abstract

The precise modeling of dynamic systems with periodic trajectories is required to describe diverse systems in mechanical, electrical, and many other disciplines. Nevertheless, the modeling task based on traditional methodologies could be complicated, considering the specific nature of periodic motions in actual systems. Differential neural networks (DNNs) are modeling tools for dynamic systems that can be useful for developing precise representations of periodic systems. This study presents the design of a novel family of DNN identifiers that could reproduce the trajectories of periodic systems with an uncertain mathematical model. The suggested DNN identifiers may produce an approximate model with periodic properties similar to the system under analysis exhibiting an one-period convergence of DNN weights. The fundamentals of the Floquet’s theory drive the design of the learning laws to ensure the reproduction of the periodic properties in the DNN. The design of a controlled Lyapunov function allows the learning laws to be derived for the DNN weights whose evolution depends on the positive definite solution of a periodic differential Lyapunov equation. Several numerical evaluations on periodic systems confirmed the modeling performance of the proposed identifier when the approximation performance is compared with traditional DNN identifiers using sigmoidal functions.