Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adaptive and robust experimental design for linear dynamical models using Kalman filter

Arno Strouwen, Bart M. Nicolaï, Peter Goos

Abstract

Current experimental design techniques for dynamical systems often only incorporate measurement noise, while dynamical systems also involve process noise. To construct experimental designs we need to quantify their information content. The Fisher information matrix is a popular tool to do so. Calculating the Fisher information matrix for linear dynamical systems with both process and measurement noise involves estimating the uncertain dynamical states using a Kalman filter. The Fisher information matrix, however, depends on the true but unknown model parameters. In this paper we combine two methods to solve this issue and develop a robust experimental design methodology. First, Bayesian experimental design averages the Fisher information matrix over a prior distribution of possible model parameter values. Second, adaptive experimental design allows for this information to be updated as measurements are being gathered. This updated information is then used to adapt the remainder of the design.

Adaptive and robust experimental design for linear dynamical models using Kalman filter

Abstract

Current experimental design techniques for dynamical systems often only incorporate measurement noise, while dynamical systems also involve process noise. To construct experimental designs we need to quantify their information content. The Fisher information matrix is a popular tool to do so. Calculating the Fisher information matrix for linear dynamical systems with both process and measurement noise involves estimating the uncertain dynamical states using a Kalman filter. The Fisher information matrix, however, depends on the true but unknown model parameters. In this paper we combine two methods to solve this issue and develop a robust experimental design methodology. First, Bayesian experimental design averages the Fisher information matrix over a prior distribution of possible model parameter values. Second, adaptive experimental design allows for this information to be updated as measurements are being gathered. This updated information is then used to adapt the remainder of the design.
Paper Structure (21 sections, 29 equations, 6 figures, 1 algorithm)

This paper contains 21 sections, 29 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Modeling the Information Content of a Dynamical Experiment
  3. The Model
  4. Parameter Estimation
  5. The Fisher Information Matrix
  6. D-optimal Experimental Design
  7. Locally Optimal Experimental Design
  8. Robust Experiments
  9. Adaptive Experiments
  10. Concept
  11. Computational Challenges
  12. Predictive Control
  13. Final Algorithm
  14. Numerical details
  15. Case Studies
  16. ...and 6 more sections

Figures (6)

  • Figure 1: Schematic representation of mass spring damper system.
  • Figure 2: Comparison of optimal inputs compared to random inputs and the corresponding output behaviors.
  • Figure 3: Online maximum likelihood estimates for both the optimal experiment and the random experiment. The optimal experiment converges faster to the true parameters.
  • Figure 4: Likelihood at the end of the experiment, evaluated at $100$ values of $\bm \theta$, drawn from $p(\bm \theta)$. The likelihood decreases sharply away from the true parameters for the optimal experiment, unlike in the random experiment. The maximum likelihood estimate of the optimal experiment is much closer to the true value than the random experiment.
  • Figure 5: $100$ experiments are performed for different combinations of $e$, $N$ and $T$ in Algorithm \ref{['algo']}. The mean and standard deviation of the online maximum likelihood estimates over these experiments are tracked.
  • ...and 1 more figures