Statistical Analysis of Block Coordinate Descent Algorithms for Linear Continuous-time System Identification

TL;DR

This work analyzes the statistical properties of block coordinate descent methods for identifying continuous-time MISO and additive SISO systems. It unifies Gauss-Newton and refined instrumental variable descent steps and derives closed-form updates, focusing on asymptotic bias at each descent and conditions for consistency. Under standard persistence of excitation and regularity assumptions, the method is shown to be generically consistent as the sample size grows, with MISO achieving consistency in a single descent iteration while additive SISO may exhibit bias without appropriate iteration or initialization. Simulations corroborate the theory and illustrate identifiability issues in additive setups, providing practical guidance for when and how to apply block coordinate descent in continuous-time system identification.

Abstract

Block coordinate descent is an optimization technique that is used for estimating multi-input single-output (MISO) continuous-time models, as well as single-input single output (SISO) models in additive form. Despite its widespread use in various optimization contexts, the statistical properties of block coordinate descent in continuous-time system identification have not been covered in the literature. The aim of this paper is to formally analyze the bias properties of the block coordinate descent approach for the identification of MISO and additive SISO systems. We characterize the asymptotic bias at each iteration, and provide sufficient conditions for the consistency of the estimator for each identification setting. The theoretical results are supported by simulation examples.

Figures (2)

• Figure 1: Empirical mean of the estimate of each parameter of $G_2^*(p)$ for one descent iteration. The true values are given by $a_{2,1}^*=0.01$, $a_{2,2}^*=0.025$, and $b_{2,0}^*=1$. The estimates of the MISO model setup converge to the true value, while bias is observed in for the additive SISO setup.
• Figure 2: Empirical mean of the estimate of each parameter of $G_2^*(p)$ using the full block coordinate descent method. The true values are given by $a_{2,1}^*=0.01$, $a_{2,2}^*=0.025$, and $b_{2,0}^*=1$. Both the MISO and additive SISO estimators converge to the true values as the sample size grows.

Theorems & Definitions (16)

• Theorem III.1: Global convergence of Algorithm \ref{['algorithm1']}
• proof
• Remark III.1
• Lemma IV.1: Gauss-Newton and SRIVC iterations
• proof
• Theorem IV.2
• proof
• Corollary IV.1
• proof
• Example IV.1