Reduced-Order Gaussian Processes for Partially Unknown Nonlinear Control Systems

In this work, we develop a scheme for constructing continuous approximations (referred to as abstractions) of a class of discrete-time control systems with partially unknown dynamics. The abstraction, itself a nonlinear discrete-time control system (preferably with a significantly lower dimension than the original one) can be used as a substitute in the controller design process. The technique consists of using data sampled from the concrete system to find a lower dimensional subspace of its state space (which we call the active subspace), and constructing an abstraction candidate using Gaussian process (GP) regression. We derive sufficient conditions under which the GP candidate is shown to be the abstraction of the original system while quantifying the error bound between the output of the abstraction and that of the concrete system. A numerical example is presented to illustrate the effectiveness of this approach.