Nonlinear Covariance Steering using Variational Gaussian Process\n Predictive Models

In this work, we consider the problem of steering the first two moments of\nthe uncertain state of an unknown discrete-time stochastic nonlinear system to\na given terminal distribution in finite time. Toward that goal, first, a\nnon-parametric predictive model is learned from a set of available training\ndata points using stochastic variational Gaussian process regression: a\npowerful and scalable machine learning tool for learning distributions over\narbitrary nonlinear functions. Second, we formulate a tractable nonlinear\ncovariance steering algorithm that utilizes the Gaussian process predictive\nmodel to compute a feedback policy that will drive the distribution of the\nstate of the system close to the goal distribution. In particular, we implement\na greedy covariance steering control policy that linearizes at each time step\nthe Gaussian process model around the latest predicted mean and covariance,\nsolves the linear covariance steering control problem, and applies only the\nfirst control law. The state uncertainty under the latest feedback control\npolicy is then propagated using the unscented transform with the learned\nGaussian process predictive model and the algorithm proceeds to the next time\nstep. Numerical simulations illustrating the main ideas of this paper are also\npresented.\n