Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of Recursive Least Squares Based Input/Output System Identification with Model Order Mismatch

Brian Lai, Dennis S. Bernstein

TL;DR

This work analyzes online identification of discrete-time IO systems using Recursive Least Squares under model order mismatch. It introduces IO model equivalence across different orders and proves that, with higher-order identification, the estimator converges to the equivalent higher-order model that minimizes RLS regularization when data are persistently exciting. If the regressor is weakly persistently exciting, convergence to this equivalent model is guaranteed; with stronger persistent excitation, the convergence rate scales as $O(1/k)$. These results inform model-order selection in online settings and highlight the role of excitation conditions in achieving reliable convergence.

Abstract

Discrete-time input/output models, also called infinite impulse response (IIR) models or autoregressive moving average (ARMA) models, are useful for online identification as they can be efficiently updated using recursive least squares (RLS) as new data is collected. Several works have studied the convergence of the input/output model coefficients identified using RLS under the assumption that the order of the identified model is the same as that of the true system. However, the case of model order mismatch is not as well addressed. This work begins by introducing the notion of \textit{equivalence} of input/output models of different orders. Next, this work analyzes online identification of input/output models in the case where the order of the identified model is higher than that of the true system. It is shown that, given persistently exciting data, the higher-order identified model converges to the model equivalent to the true system that minimizes the regularization term of RLS.

Convergence of Recursive Least Squares Based Input/Output System Identification with Model Order Mismatch

TL;DR

This work analyzes online identification of discrete-time IO systems using Recursive Least Squares under model order mismatch. It introduces IO model equivalence across different orders and proves that, with higher-order identification, the estimator converges to the equivalent higher-order model that minimizes RLS regularization when data are persistently exciting. If the regressor is weakly persistently exciting, convergence to this equivalent model is guaranteed; with stronger persistent excitation, the convergence rate scales as . These results inform model-order selection in online settings and highlight the role of excitation conditions in achieving reliable convergence.

Abstract

Discrete-time input/output models, also called infinite impulse response (IIR) models or autoregressive moving average (ARMA) models, are useful for online identification as they can be efficiently updated using recursive least squares (RLS) as new data is collected. Several works have studied the convergence of the input/output model coefficients identified using RLS under the assumption that the order of the identified model is the same as that of the true system. However, the case of model order mismatch is not as well addressed. This work begins by introducing the notion of \textit{equivalence} of input/output models of different orders. Next, this work analyzes online identification of input/output models in the case where the order of the identified model is higher than that of the true system. It is shown that, given persistently exciting data, the higher-order identified model converges to the model equivalent to the true system that minimizes the regularization term of RLS.
Paper Structure (7 sections, 16 theorems, 85 equations)

This paper contains 7 sections, 16 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Input/Output Modeling
  3. Equivalence and Reducibility of Input/Output Models
  4. Online Identification of Input/Output Models using Recursive Least Squares
  5. Convergence with Correct Model Order
  6. Convergence with Higher Model Order
  7. Conclusions

Key Result

Proposition 1

Let $k_0 \in {\mathbb N}$. For all $k \ge k_0$, let $u_k \in {\mathbb R}^m$, let $y_{k_0},\hdots,y_{k_0+n-1} \in {\mathbb R}^p$, and, for all $k \ge k_0$, let $y_{k+n} \in {\mathbb R}^p$ be given by eqn: IO Model. Then, for all $k \ge k_0$ and $j \ge 0$, where ${\mathcal{F}}_{n,0} \triangleq {\mathcal{F}}_{n}$, ${\mathcal{G}}_{n,0} \triangleq {\mathcal{G}}_{n}$, and, for all $j \ge 1$, ${\mathcal

Theorems & Definitions (32)

  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 22 more