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

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.