Learning Stable Nonlinear Dynamical Systems with External Inputs using Gaussian Mixture Models

Learning nonlinear dynamics from data is particularly useful in control design for complex systems since developing analytical models might be intractable. This paper presents an algorithm to learn a class of stable nonlinear dynamical systems from data. To accommodate for the variability in data, Gaussian mixture models (GMM) are used to represent the nonlinear dynamics in the state space. The algorithm can learn dynamical systems with time-varying external inputs and multiple equilibria. To ensure that the trajectories generated by the learned model converge to the desired equilibrium point, the parameters of the GMMs are learned subject to the constraints derived from partial contraction analysis. Given the initial conditions and the steady-state inputs, a separate GMM is trained to predict the equilibrium point at which the trajectories converge. Experimental results illustrate the proposed algorithm's ability to accurately learn the nonlinear dynamics of a counter-flow heat exchanger model.