The Existence and Uniqueness of Solutions for Kernel-Based System Identification

TL;DR

This paper proves that kernel-based system identification, using RKHS, guarantees existence and uniqueness of solutions under broad conditions, ensuring the method's mathematical soundness.

Contribution

It establishes that the key assumption of continuity in kernel-based identification holds in general cases, confirming the approach's well-defined nature.

Findings

Continuity of convolution operators is guaranteed when the kernel is integrable and input is bounded.

The optimization problem in kernel-based identification is strongly convex.

Kernel-based system identification admits a unique solution under broad conditions.

Abstract

The notion of reproducing kernel Hilbert space (RKHS) has emerged in system identification during the past decade. In the resulting framework, the impulse response estimation problem is formulated as a regularized optimization defined on an infinite-dimensional RKHS consisting of stable impulse responses. The consequent estimation problem is well-defined under the central assumption that the convolution operators restricted to the RKHS are continuous linear functionals. Moreover, according to this assumption, the representer theorem hold, and therefore, the impulse response can be estimated by solving a finite-dimensional program. Thus, the continuity feature plays a significant role in kernel-based system identification. This paper shows that this central assumption is guaranteed to be satisfied in considerably general situations, namely when the kernel is an integrable function and the input signal is bounded. Furthermore, the strong convexity of the optimization problem and the continuity property of the convolution operators imply that the kernel-based system identification admits a unique solution. Consequently, it follows that kernel-based system identification is a well-defined approach.