Robust Nonlinear System Identification in Reproducing Kernel Hilbert Spaces via Scenario Optimization

TL;DR

This paper introduces a kernel-based approach for nonlinear system prediction using scenario optimization, providing violation guarantees without prior norm bounds, demonstrated on obstacle avoidance.

Contribution

It develops a finite-dimensional RKHS approximation method combined with scenario optimization to ensure prediction reliability in nonlinear systems.

Findings

Finite-dimensional RKHS approximation scales with kernel smoothness and input dimension.

Scenario optimization yields violation guarantees without prior norm bounds.

Method successfully applied to obstacle-avoidance task.

Abstract

This paper proposes a method for constructing one-step prediction tubes for nonlinear systems using reproducing kernel Hilbert spaces. We approximate a bounded reproducing kernel Hilbert space (RKHS) hypothesis set by a finite-dimensional subspace using bounds based on n-widths and a greedy algorithm for basis reduction. For kernels whose native spaces are norm-equivalent to Sobolev spaces, we derive how the required basis size scales with kernel smoothness and input dimension. This finite-dimensional representation enables the use of convex scenario optimization to obtain violation guarantees for the learned predictor without requiring an a priori bound on the true system's RKHS norm or Lipschitz constant. The method is demonstrated on an obstacle-avoidance task. We also discuss the main limitations of the current analysis, including dimensional scaling and dependence on i.i.d. data.