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Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data

Lei Xin, Baike She, Qi Dou, George Chiu, Shreyas Sundaram

TL;DR

This work addresses identifying a linearized model of a nonlinear, time-invariant system from data collected under initialization constraints. It introduces a deterministic, multi-trajectory data acquisition scheme coupled with regularized least squares to estimate the linear part Θ = AB around the origin, while bounding the nonlinear remainder and noise. The authors prove finite-sample guarantees, revealing a trade-off between nonlinear bias (controlled by the initialization radius q and center m) and measurement noise, with a novel O(1/N^{1/4}) learning rate when the feasible region contains the origin. Numerical experiments corroborate the theory and show advantages over single-trajectory, i.i.d. input schemes, as well as practical benefits of regularization and region design.

Abstract

The identification of a linear system model from data has wide applications in control theory. The existing work that provides finite sample guarantees for linear system identification typically uses data from a single long system trajectory under i.i.d. random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear, given that there is a certain constraint on the region where one can initialize the experiments. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound shows that one can consistently learn the linearized dynamics, and demonstrates a trade-off between the error due to nonlinearity and the error due to noise. We validate our results through numerical experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d. random inputs, when nonlinearity does exist.

Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data

TL;DR

This work addresses identifying a linearized model of a nonlinear, time-invariant system from data collected under initialization constraints. It introduces a deterministic, multi-trajectory data acquisition scheme coupled with regularized least squares to estimate the linear part Θ = AB around the origin, while bounding the nonlinear remainder and noise. The authors prove finite-sample guarantees, revealing a trade-off between nonlinear bias (controlled by the initialization radius q and center m) and measurement noise, with a novel O(1/N^{1/4}) learning rate when the feasible region contains the origin. Numerical experiments corroborate the theory and show advantages over single-trajectory, i.i.d. input schemes, as well as practical benefits of regularization and region design.

Abstract

The identification of a linear system model from data has wide applications in control theory. The existing work that provides finite sample guarantees for linear system identification typically uses data from a single long system trajectory under i.i.d. random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear, given that there is a certain constraint on the region where one can initialize the experiments. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound shows that one can consistently learn the linearized dynamics, and demonstrates a trade-off between the error due to nonlinearity and the error due to noise. We validate our results through numerical experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d. random inputs, when nonlinearity does exist.
Paper Structure (16 sections, 8 theorems, 40 equations, 5 figures, 2 algorithms)

This paper contains 16 sections, 8 theorems, 40 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation and System Identification Algorithm
  4. Theoretical Analysis
  5. Intermediate results
  6. Main Results
  7. Numerical Examples
  8. System with mild nonlinearity and $m=\textbf{0}$
  9. System with strong nonlinearity and $m\neq \textbf{0}$
  10. Effects of regularization
  11. Conclusion
  12. Appendix
  13. Auxiliary Results
  14. Proof of Lemma \ref{['lemma:PE']}
  15. Proof of Lemma \ref{['bound noise']}
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose that Algorithm algo1 is used to generate data, and Assumption ass:Safety_set holds. Let $N\geq 4(n+p)$. Then we have the following inequalities

Figures (5)

  • Figure 1: System identification error and bound with different $q$, mild nonlinearity
  • Figure 2: System identification error using a single trajectory with different $\sigma_{u}$, mild nonlinearity
  • Figure 3: System identification error under initialization error
  • Figure 4: System identification error using Algorithms \ref{['algo1']}-\ref{['algo2']} with different $m,q$. $N=10000$, strong nonlinearity
  • Figure 5: System identification error using different regularization parameter $\lambda$

Theorems & Definitions (12)

  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 3
  • Proposition 1
  • Lemma 4
  • ...and 2 more