Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Modelling real world systems involving humans such as biological processes\nfor disease treatment or human behavior for robotic rehabilitation is a\nchallenging problem because labeled training data is sparse and expensive,\nwhile high prediction accuracy is required from models of these dynamical\nsystems. Due to the high nonlinearity of problems in this area, data-driven\napproaches gain increasing attention for identifying nonparametric models. In\norder to increase the prediction performance of these models, abstract prior\nknowledge such as stability should be included in the learning approach. One of\nthe key challenges is to ensure sufficient flexibility of the models, which is\ntypically limited by the usage of parametric Lyapunov functions to guarantee\nstability. Therefore, we derive an approach to learn a nonparametric Lyapunov\nfunction based on Gaussian process regression from data. Furthermore, we learn\na nonparametric Gaussian process state space model from the data and show that\nit is capable of reproducing observed data exactly. We prove that stabilization\nof the nominal model based on the nonparametric control Lyapunov function does\nnot modify the behavior of the nominal model at training samples. The\nflexibility and efficiency of our approach is demonstrated on the benchmark\nproblem of learning handwriting motions from a real world dataset, where our\napproach achieves almost exact reproduction of the training data.\n