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A guided residual search for nonlinear state-space identification

Merijn Floren, Jan Swevers

TL;DR

This work improves the reliability and efficiency of the estimation process by decomposing the overall optimization problem into a sequence of tractable subproblems.

Abstract

Parameter estimation of nonlinear state-space models from input-output data typically requires solving a highly non-convex optimization problem prone to slow convergence and suboptimal solutions. This work improves the reliability and efficiency of the estimation process by decomposing the overall optimization problem into a sequence of tractable subproblems. Based on an initial linear model, nonlinear residual dynamics are first estimated via a guided residual search and subsequently refined using multiple-shooting optimization. Experimental results on two benchmarks demonstrate competitive performance relative to state-of-the-art black-box methods and improved convergence compared to naive initialization.

A guided residual search for nonlinear state-space identification

TL;DR

This work improves the reliability and efficiency of the estimation process by decomposing the overall optimization problem into a sequence of tractable subproblems.

Abstract

Parameter estimation of nonlinear state-space models from input-output data typically requires solving a highly non-convex optimization problem prone to slow convergence and suboptimal solutions. This work improves the reliability and efficiency of the estimation process by decomposing the overall optimization problem into a sequence of tractable subproblems. Based on an initial linear model, nonlinear residual dynamics are first estimated via a guided residual search and subsequently refined using multiple-shooting optimization. Experimental results on two benchmarks demonstrate competitive performance relative to state-of-the-art black-box methods and improved convergence compared to naive initialization.
Paper Structure (15 sections, 1 theorem, 14 equations, 5 figures, 2 tables)

This paper contains 15 sections, 1 theorem, 14 equations, 5 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. Mitigating distribution shift using multiple shooting
  4. METHODOLOGY
  5. Bilevel guided residual search
  6. Inner problem
  7. Outer problem
  8. Parametric learning of the nonlinear residual
  9. Final optimization using multiple shooting
  10. RESULTS
  11. Silverbox benchmark system
  12. Algorithmic performance
  13. Test data performance
  14. F-16 ground vibration test
  15. CONCLUSIONS

Key Result

Theorem 1

Let $x(n+1)=f(x(n),\,u(n))$ denote the unified state-update equation corresponding to eq:mod_x4, eq:mod_z4, and eq:mod_w4, learned in a one-step fashion from the samples $\{x(n),\,u(n)\}_{n=0}^{N-1}$. Moreover, assume $f : \mathbb{R}^{n_x} \times \mathbb{R}^{n_u} \to \mathbb{R}^{n_x}$ is uniformly L

Figures (5)

  • Figure 1: Schematic overview of the nllfr structure.
  • Figure 2: nllfr simulation nrmse over the iterations of the final optimization stage, for Silverbox scenarios \ref{['scenario:S1']} to \ref{['scenario:S4']}. The guided residual search provides a favorable initialization, and multiple shooting outperforms single shooting.
  • Figure 3: Best-performing nllfr model from \ref{['scenario:S1']} on the Silverbox arrowhead (left) and multisine (right) test data, compared to the bla model. The dashed vertical line marks the start of the extrapolation region.
  • Figure 4: Distribution shift over the F-16 neural network training iterations. The static neural network loss \ref{['eq:nn_opti_cost']} steadily decreases, but the simulation error of the resulting nllfr model quickly deteriorates and exhibits high volatility.
  • Figure 5: F-16 simulation nrmse during the final optimization stage. Despite a higher initial error due to distribution shift (see Fig. \ref{['fig:f16_dist_shift']}), the proposed method starts in a favorable parameter region and converges rapidly, while the linear-only initialization is trapped in a local minimum.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2