Regularized Identification with Internal Positivity Side-Information

Abstract

In this paper, we present an impulse response identification scheme that incorporates the internal positivity side-information of the system. The realization theory of positive systems establishes specific criteria for the existence of a positive realization for a given transfer function. These transfer function criteria are translated to a set of suitable conditions on the shape and structure of the impulse responses of positive systems. Utilizing these conditions, the impulse response estimation problem is formulated as a constrained optimization in a reproducing kernel Hilbert space equipped with a stable kernel, and suitable constraints are imposed to encode the internal positivity side-information. The optimization problem is infinitedimensional with an infinite number of constraints. An equivalent finite-dimensional convex optimization in the form of a convex quadratic program is derived. The resulting equivalent reformulation makes the proposed approach suitable for numerical simulation and practical implementation. A Monte Carlo numerical experiment evaluates the impact of incorporating the internal positivity side-information in the proposed identification scheme. The effectiveness of the proposed method is demonstrated using data from a heating system experiment.

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

“…Note that I is a well-defined integral due to integrability of . From (49) and the triangle inequality, we have…”

“…Moreover, the kernel-based scheme allows the incorporation of various forms of side-information in the identification problem by designing appropriate kernel functions or imposing suitable constraints to the regression problem. The forms of this side-information, studied to date, include stability, relative degree, smoothness of the impulse response, resonant frequencies, external positivity, oscillatory behaviors, steady-state gain, internal positivity, exponential decay of the impulse response, structural properties, internal low-complexity, frequency domain features, and the presence of fast and slow poles [35,30,[36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. While kernel-based system identification has enjoyed considerable progress in the past decade, it is still a thriving area of research with stateof-the-art results and recent studies [52][53][54][55][56][57].…”