Learning a Stability Filter for Uncertain Differentially Flat Systems using Gaussian Processes

Many physical system models exhibit a structural property known as differential flatness. Intuitively, differential flatness allows us to separate the system's nonlinear dynamics into a linear dynamics component and a nonlinear term. In this work, we exploit this structure and propose using a nonparametric Gaussian Process (GP) to learn the unknown nonlinear term. We use this GP in an optimization problem to optimize for an input that is most likely to feedback linearize the system (i.e., cancel this nonlinear term). This optimization is subject to input constraints and a stability filter, described by an uncertain Control Lyapunov Function (CLF), which probabilistically guarantees exponential trajectory tracking when possible. Furthermore, for systems that are control-affine, we choose to express this structure in the selection of the kernel for the GP. By exploiting this selection, we show that the optimization problem is not only convex but can be efficiently solved as a second-order cone program. We compare our approach to related works in simulation and show that we can achieve similar performance at much lower computational cost.