A data-driven reconstruction method for dynamic systems with multistable property

In this work, Sparse Identification of Nonlinear Dynamics (SINDy) algorithm is generalized to nonlinear dynamic systems with multistable property. This study comes from the application of a bistable bionic joint. In its dynamical model reconstruction, we find that the classical SINDy fails due to the lacking ergodicity with tiny amount of time history signals. Aiming at the accuracy and precision of model reconstruction of multistable nonlinear dynamic systems, a generalized data-driven reconstruction method is proposed based on data assembly principle and sparsification parameter determination technique. The datasets are expanded by different initial conditions for the dynamics ergodic. Besides, to resolve the difficulty on the determination of sparsification parameter, a novel bi-objective optimization scheme considering both sparsity and accuracy of the reconstruction model is constructed. Based on the Pareto front and Knee point theory of the bi-objective optimization scheme, it provides the precisely determination technique for the sparsification parameter range. Two numerical examples are conducted on bistable and quad-stable nonlinear systems to verify the effectiveness of the generalized data-driven reconstruction method. Then, the experimental prototype of the bionic joint with bistable property is carried out, which possesses both geometric and constitutive nonlinearity. Assembling the experimental signals with measurement noise according to the proposed data assembly principle, the generalized reconstruction method proposed in this paper can capture the position of the stable equilibriums and accurately reconstruct the dynamic model of the bistable bionic joint. Since the ergodic dataset can be further extended, proposed reconstruction method for dynamic systems with multistable property has effectiveness and robustness. The generalized data-driven reconstruction method successfully solves the problems of non-ergodic data and undetermined sparsification parameter in the model reconstruction of multistable systems, which has potential applications of multistable nonlinear systems in the fields of robotics, deployable structures, etc.