Rational Continuous Neural Network Identifier for Singular Perturbed Systems With Uncertain Dynamical Models

This study aims at designing a robust nonparametric identifier for a class of singular perturbed systems (SPSs) with uncertain mathematical models. The identifier structure uses a novel identifier based on a differential neural network (DNN) with rational form, which can take into account the multirate nature of SPS. The identifier uses a mixed learning law including a rational formulation of neural networks which is useful to solve the identification of the fast dynamics in the SPS dynamics. The rational form of the design is proposed in such a way that no-singularities (denominator part of the rational form never touches the origin) are allowed in the identifier dynamics. A proposed control Lyapunov function and a nonlinear parameter identification methodology yield to design the learning laws for the class of novel rational DNN which appears as the main contribution of this study. A complementary matrix inequality-based optimization method allows to get the smallest attainable convergence invariant region. A detailed implementation methodology is also given in the study with the aim of clarifying how the proposed identifier can be used in diverse SPSs. A numerical example considering the dynamics of the enzymatic-substrate-inhibitor system with uncertain dynamics is showing how to apply the DNN identifier using the multirate nature of the proposed DNN identifier for SPSs. The proposed identifier is compared to a classical identifier which is not taking into account the multirate nature of SPS. The benefits of using the rational form for the identifier are highlighted in the numerical performance comparison based on the mean square error (MSE). This example justifies the ability of the suggested identifier to reconstruct both the fast and slow dynamics of the SPS.

Automated summary

This study aims to design a robust nonparametric identifier for singular perturbed systems (SPSs) with uncertain mathematical models. The identifier uses a novel differential neural network (DNN) structure, which takes into account the multirate nature of SPS. A rational form of the DNN and a mixed learning law are proposed to solve the identification of the fast dynamics in SPS. The study also proposes a control Lyapunov function and a nonlinear parameter identification methodology for the design of the learning laws. A complementary matrix inequality-based optimization method is used to obtain the smallest attainable convergence invariant region. A numerical example of an enzymatic-substrate-inhibitor system is provided to demonstrate the application of the DNN identifier. The benefits of using the rational form for the identifier in terms of mean square error (MSE) are highlighted in the comparison with a classical identifier.