Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Modelling real world systems involving humans such as biological processes for disease treatment or human behavior for robotic rehabilitation is a challenging problem because labeled training data is sparse and expensive, while high prediction accuracy is required from models of these dynamical systems. Due to the high nonlinearity of problems in this area, data-driven approaches gain increasing attention for identifying nonparametric models. In order to increase the prediction performance of these models, abstract prior knowledge such as stability should be included in the learning approach. One of the key challenges is to ensure sufficient flexibility of the models, which is typically limited by the usage of parametric Lyapunov functions to guarantee stability. Therefore, we derive an approach to learn a nonparametric Lyapunov function based on Gaussian process regression from data. Furthermore, we learn a nonparametric Gaussian process state space model from the data and show that it is capable of reproducing observed data exactly. We prove that stabilization of the nominal model based on the nonparametric control Lyapunov function does not modify the behavior of the nominal model at training samples. The flexibility and efficiency of our approach is demonstrated on the benchmark problem of learning handwriting motions from a real world dataset, where our approach achieves almost exact reproduction of the training data.