Convergence Analysis of Recursive Least Squares for Identifying Higher-Order Input-Output Models

Belangrijkste concepten

When the order of the identified input-output model is higher than the true system, the regressor of recursive least squares is not persistently exciting, and standard convergence guarantees do not apply. This work analyzes the convergence of the identified model in this case, showing that it converges to the higher-order model equivalent to the true system that minimizes the regularization term.

Samenvatting

This work focuses on the online identification of discrete-time input-output models, also called infinite impulse response (IIR) or autoregressive moving average (ARMA) models, using recursive least squares (RLS).

The key insights are:

1. It introduces the notion of equivalence between input-output models of different orders. Two models are equivalent if they produce the same outputs under the same inputs and initial conditions.

2. It analyzes the case where the order of the identified model is higher than the true system. In this case, the regressor of RLS is not persistently exciting, so standard convergence guarantees do not apply.

3. It shows that, under persistent excitation of a modified regressor, the identified higher-order model converges to the higher-order model that is equivalent to the true system and minimizes the regularization term of RLS.

4. It provides necessary and sufficient conditions for the reducibility of an input-output model, i.e., the existence of an equivalent lower-order model. This is related to the case of model order mismatch.

The analysis provides insights into the behavior of RLS-based identification when the model order is not known a priori, which is common in practical applications.

Statistieken

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Citaten

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Belangrijkste Inzichten Gedestilleerd Uit

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

by Brian Lai,De... om arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.10850.pdf

Diepere vragen

What are some practical applications where the model order is not known a priori, and the results of this work could be useful

One practical application where the model order is not known a priori and the results of this work could be useful is in the field of signal processing for communication systems. In wireless communication, the channel characteristics are often unknown and can vary over time, making it challenging to determine the appropriate model order for system identification. By using the analysis presented in this work, adaptive algorithms can be developed to identify the system model order in real-time based on the incoming data. This can improve the efficiency and accuracy of communication systems by dynamically adjusting the model order to match the changing channel conditions.

How could the analysis be extended to consider the case where the order of the identified model is lower than the true system

To extend the analysis to consider the case where the order of the identified model is lower than the true system, one approach could be to investigate the concept of model reduction techniques. By applying model reduction methods, such as balanced truncation or modal truncation, the high-order identified model can be reduced to a lower-order model while preserving key system dynamics. This reduced-order model can then be compared to the true system to analyze the convergence behavior and identify any discrepancies or errors introduced by the model order mismatch. By incorporating model reduction techniques into the analysis, a more comprehensive understanding of the effects of model order mismatch can be achieved.

Are there alternative identification methods that can handle model order mismatch more robustly than recursive least squares

There are alternative identification methods that can handle model order mismatch more robustly than recursive least squares. One such method is the subspace identification algorithm, which is based on the concept of state-space modeling and can handle model order mismatch by directly estimating the state-space matrices of the system. Subspace identification methods, such as the Extended Kalman Filter (EKF) or the Canonical Variate Analysis (CVA), are particularly effective in situations where the true system order is unknown or differs from the identified model order. These methods utilize advanced mathematical techniques to estimate the system dynamics and parameters, even in the presence of model order mismatch, making them more robust and accurate for system identification tasks.