Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parallelizable Parametric Nonlinear System Identification via tuning of a Moving Horizon State Estimator

Léo Simpson, Jonas Asprion, Simon Muntwiler, Johannes Köhler, Moritz Diehl

TL;DR

A novel optimization-based approach for parametric nonlinear system identification that enhances performance for nonlinear systems but also enables learning from multiple trajectories with unknown initial states, broadening its applicability in practical scenarios.

Abstract

This paper introduces a novel optimization-based approach for parametric nonlinear system identification. Building upon the prediction error method framework, traditionally used for linear system identification, we extend its capabilities to nonlinear systems. The predictions are computed using a moving horizon state estimator with a constant arrival cost. Eventually, both the system parameters and the arrival cost are estimated by minimizing the sum of the squared prediction errors. Since the predictions are induced by the state estimator, the method can be viewed as the tuning of a state estimator, based on its predictive capacities. The present extension of the prediction error method not only enhances performance for nonlinear systems but also enables learning from multiple trajectories with unknown initial states, broadening its applicability in practical scenarios. Additionally, the novel formulation leaves room for the design of efficient and parallelizable optimization algorithms, since each output prediction only depends on a fixed window of past actions and measurements. In the special case of linear time-invariant systems, we show an important property of the proposed method which suggests asymptotic consistency under reasonable assumptions. Numerical examples illustrate the effectiveness and practicality of the approach, and one of the examples also highlights the necessity for the arrival cost.

Parallelizable Parametric Nonlinear System Identification via tuning of a Moving Horizon State Estimator

TL;DR

A novel optimization-based approach for parametric nonlinear system identification that enhances performance for nonlinear systems but also enables learning from multiple trajectories with unknown initial states, broadening its applicability in practical scenarios.

Abstract

This paper introduces a novel optimization-based approach for parametric nonlinear system identification. Building upon the prediction error method framework, traditionally used for linear system identification, we extend its capabilities to nonlinear systems. The predictions are computed using a moving horizon state estimator with a constant arrival cost. Eventually, both the system parameters and the arrival cost are estimated by minimizing the sum of the squared prediction errors. Since the predictions are induced by the state estimator, the method can be viewed as the tuning of a state estimator, based on its predictive capacities. The present extension of the prediction error method not only enhances performance for nonlinear systems but also enables learning from multiple trajectories with unknown initial states, broadening its applicability in practical scenarios. Additionally, the novel formulation leaves room for the design of efficient and parallelizable optimization algorithms, since each output prediction only depends on a fixed window of past actions and measurements. In the special case of linear time-invariant systems, we show an important property of the proposed method which suggests asymptotic consistency under reasonable assumptions. Numerical examples illustrate the effectiveness and practicality of the approach, and one of the examples also highlights the necessity for the arrival cost.
Paper Structure (17 sections, 1 theorem, 32 equations, 5 figures)

This paper contains 17 sections, 1 theorem, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Prediction Error Method with Moving Horizon State Estimation
  4. Moving Horizon State Estimation
  5. Constant Arrival Cost
  6. Prediction Error Method
  7. Special case: Linear Time-Invariant Systems
  8. Notation
  9. Definition of $\eta^\star_N$
  10. Stochastic interpretation of the predictions
  11. Covariance between prediction errors and past data
  12. Linearity of the Prediction Errors
  13. Conclusion
  14. Numerical Examples
  15. A Simple LTI Case Study
  16. ...and 2 more sections

Key Result

Theorem 1

Assume the data is generated by the LTI system equation_LTIsmall under Assumption assum:iid and with parameters $\theta = \theta^\star$. Then there exists some $\eta^\star_N$ such that $(\theta^\star, \eta^\star_N)$ minimizes the expected value of the objective function in equation_condensed, i.e.

Figures (5)

  • Figure 1: Trajectory estimation (orange dashed line) using MHE with a constant arrival cost and a horizon of $4$ steps and one-step-ahead prediction errors (red arrows). The system is a harmonic oscillator $\ddot{x} = - x$ with discrete measurements (green dots). The MHE is computed for a model mismatch $\ddot{x} = - \omega^2 x$ with $\omega = 0.7$.
  • Figure 2: Parameter estimates with and without constant arrival cost tuning for the toy example \ref{['equation_counterex']}. For each data instance, the estimates given by the proposed approach are represented by a green cross, and the ones given by the naive approach are the purple dots. The shaded areas depict, for each data length and for both methods, the interval in which the $10$ parameter estimates lie.
  • Figure 3: Simulation of the Lorenz attractor for parameters $\theta = 1030^\top$.
  • Figure 4: Partially measured Lorenz attractor system with process and measurement noise, for $\theta = 1030^\top$.
  • Figure 5: Comparison between ground truth model parameters and estimates based on simulation data of the Lorenz attractor.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • proof