Discovering differential governing equations of hysteresis dynamic systems by data-driven sparse regression method

Hysteresis phenomenon widely exists in various metamaterials and smart actuators. Governing equations can describe and predict the static/dynamic behavior of the systems with hysteresis property. However, the hysteresis force cannot be measured nor explicitly expressed by state variables, which brings great challenge for the model reconstruction of the hysteresis systems, especially when the nonlinear restoring and damping forces also exist. In this paper, a data-driven method is proposed to reconstruct the model of the systems with both hysteresis and nonlinearity properties from dynamic information. From the proposed method, the linear, nonlinear, and hysteresis forces, can be separately reconstructed based on the data generation with incremental generation of dynamics signals under supervision. Facing to the challenge for the functional representation of hysteresis, based on an agent model, the function library is successfully constructed. Next, for the sparsity and accuracy of the reconstruction model, the sparse regression method is generalized to identify all the nonlinear terms and coefficients. Once linear, nonlinear and hysteresis terms are figured out, the discovery of differential governing equations of hysteresis dynamic systems is completed. Three numerical examples are carried out to demonstrate the effectiveness and capability of the proposed data-driven method in the dynamic systems with different nonlinearities, dimensions and hysteresis; and the model reconstruction for Tachi-Miura polyhedron (TMP) origami structure, which possesses both hysteresis nonlinearity and geometric nonlinearity, is shown in experiments. The proposed model reconstruction method realizes the reconstruction of constitutive relation and governing equations of nonlinear hysteresis systems based on compressive sensing from dynamics, which demonstrates that great benefit of dynamic data. The model reconstruction method also provides an accurate estimation method for constitutive equation for metamaterials, robotic joint, isolation system, flexibility deployable structures, etc.