Scientific machine learning applied to real-time magnetic actuator dynamic modeling

In this article, we examine the potential of employing scientific machine learning techniques to develop an accurate dynamic real-time model of an axial electromagnetic actuator. We focus on one side of the axial AMB, specifically a fixed electromagnet with a single coil, and aim to model the magnetic flux applied to a fixed part as a function of the input voltage to the coil. Real-time capability is defined by the models ability to operate at a frequency of at least 14Khz on a standard computer CPU. To achieve this goal, we compare two methods for modeling dynamical systems: Neural Ordinary Differential Equations (NODE), a purely data-driven approach, and Universal Differential Equations (UDE), which integrates physics-based structures into the model. In this study, the physics-based structure employed is a Cauer Ladder Network (CLN), representing the system as a linear electrical equivalent circuit. A finite element model is constructed, and the parameters of the CLN are directly extracted from the finite element matrices. The UDE discussed in this article seeks to adjust the resistances and inductances of the CLN using a neural network based on the states of the ordinary differential equation. Training and testing datasets are generated from the finite element model. Both NODE and UDE are trained using stochastic gradient descent with a quadratic loss against the training datasets, and their accuracy is evaluated using testing datasets. Our findings demonstrate that the UDE provides an accurate and stable dynamical model, delivering superior accuracy and convergence speed compared to NODE.