Leading Impulse Response Identification via the Weighted Elastic Net Criterion

TL;DR

This paper introduces a new weighted elastic net criterion for impulse response identification in linear systems, ensuring zero tail estimates with high probability, supported by theoretical analysis and numerical validation.

Contribution

It proposes a novel regularization-based method that guarantees the tail of the impulse response estimate is zero, improving low-order model identification.

Findings

The method accurately identifies zero tail responses with high probability.

Numerical experiments confirm theoretical predictions.

Comparison shows competitive performance with state-of-the-art techniques.

Abstract

This paper deals with the problem of finding a low-complexity estimate of the impulse response of a linear time-invariant discrete-time dynamic system from noise-corrupted input-output data. To this purpose, we introduce an identification criterion formed by the average (over the input perturbations) of a standard prediction error cost, plus a weighted l1 regularization term which promotes sparse solutions. While it is well known that such criteria do provide solutions with many zeros, a critical issue in our identification context is where these zeros are located, since sensible low-order models should be zero in the tail of the impulse response. The flavor of the key results in this paper is that, under quite standard assumptions (such as i.i.d. input and noise sequences and system stability), the estimate of the impulse response resulting from the proposed criterion is indeed identically zero from a certain time index (named the leading order) onwards, with arbitrarily high probability, for a sufficiently large data cardinality. Numerical experiments are reported that support the theoretical results, and comparisons are made with some other state-of-the-art methodologies.