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An Iterative Bayesian Approach for System Identification based on Linear Gaussian Models

Alexandros E. Tzikas, Mykel J. Kochenderfer

TL;DR

This work develops a Bayesian framework for system identification that leverages a linear Gaussian approximation around the current parameter estimate and first-order derivatives to efficiently update parameters. It combines an online covariance calibration mechanism with an informative-input design strategy, selecting inputs to minimize the posterior uncertainty of the parameters. The approach is model-agnostic with respect to the parametric family and extends to nonlinear dynamics via a Taylor-based convex optimization, while also providing a diagnostic via model-mismatch covariance to assess model suitability. Empirical results on linear, Henon-map, and unicycle dynamics demonstrate convergent parameter estimates, effective active input design, and a practical criterion for detecting when the chosen model family is inadequate.

Abstract

We tackle the problem of system identification, where we select inputs, observe the corresponding outputs from the true system, and optimize the parameters of our model to best fit the data. We propose a practical and computationally tractable methodology that is compatible with any system and parametric family of models. Our approach only requires input-output data from the system and first-order information of the model with respect to the parameters. Our approach consists of two modules. First, we formulate the problem of system identification from a Bayesian perspective and use a linear Gaussian model approximation to iteratively optimize the model's parameters. In each iteration, we propose to use the input-output data to tune the covariance of the linear Gaussian model. This online covariance calibration stabilizes fitting and signals model inaccuracy. Secondly, we define a Gaussian-based uncertainty measure for the model parameters, which we can then minimize with respect to the next selected input. We test our method with linear and nonlinear dynamics.

An Iterative Bayesian Approach for System Identification based on Linear Gaussian Models

TL;DR

This work develops a Bayesian framework for system identification that leverages a linear Gaussian approximation around the current parameter estimate and first-order derivatives to efficiently update parameters. It combines an online covariance calibration mechanism with an informative-input design strategy, selecting inputs to minimize the posterior uncertainty of the parameters. The approach is model-agnostic with respect to the parametric family and extends to nonlinear dynamics via a Taylor-based convex optimization, while also providing a diagnostic via model-mismatch covariance to assess model suitability. Empirical results on linear, Henon-map, and unicycle dynamics demonstrate convergent parameter estimates, effective active input design, and a practical criterion for detecting when the chosen model family is inadequate.

Abstract

We tackle the problem of system identification, where we select inputs, observe the corresponding outputs from the true system, and optimize the parameters of our model to best fit the data. We propose a practical and computationally tractable methodology that is compatible with any system and parametric family of models. Our approach only requires input-output data from the system and first-order information of the model with respect to the parameters. Our approach consists of two modules. First, we formulate the problem of system identification from a Bayesian perspective and use a linear Gaussian model approximation to iteratively optimize the model's parameters. In each iteration, we propose to use the input-output data to tune the covariance of the linear Gaussian model. This online covariance calibration stabilizes fitting and signals model inaccuracy. Secondly, we define a Gaussian-based uncertainty measure for the model parameters, which we can then minimize with respect to the next selected input. We test our method with linear and nonlinear dynamics.

Paper Structure

This paper contains 14 sections, 23 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Problem Formulation
  4. Proposed Method
  5. Parameter Estimation
  6. Informative Input Design
  7. Fine-tuning
  8. Algorithm
  9. Results
  10. Linear System
  11. Hénon map dynamics
  12. Unicycle Model
  13. Model Mismatch
  14. Conclusion

Figures (5)

  • Figure 1: Mean $\log \det \Sigma_\mathrm{model\ error}$ and 1-standard deviation interval over time for the parametric model family \ref{['eq:bad_1']} and the Hénon dynamics. The fact that $\log \det \Sigma_\mathrm{model\ error}$ plateaus higher than its initial value indicates that this is an inadequate parametric model family.
  • Figure 2: Mean $\log \det \Sigma_\mathrm{model\ error}$ and 1-standard deviation interval over time for the parametric model family \ref{['eq:lin_model']} and the Hénon dynamics. The fact that $\log \det \Sigma_\mathrm{model\ error}$ plateaus higher than its initial value indicates that this is an inadequate parametric model family.
  • Figure 3: Results for the linear dynamics.
  • Figure 4: Results for the Hénon dynamics.
  • Figure 5: Results for the unicycle dynamics.