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A Novel Online Pseudospectral Method for Approximation of Nonlinear Systems Dynamics

Arian Yousefian, Avimanyu Sahoo, Vignesh Narayanan

TL;DR

The paper tackles online identification of unknown nonlinear system dynamics under aperiodic sampling. It develops an online Chebyshev pseudospectral framework with sliding windows, where the dynamics are approximated as time-based Chebyshev polynomials and coefficients are learned via least squares using aperiodic samples; an adaptive node-design guarantees a prescribed approximation accuracy. An adaptive state estimator is integrated, enforcing continuity across window transitions by adjusting coefficients to match derivatives up to the window order. Theoretical guarantees show bounded parameter and state estimation errors, and simulations on the Stuart--Landau oscillator demonstrate significant sampling efficiency and accurate state reconstruction, highlighting the method's potential for real-time nonlinear system identification and control design.

Abstract

This note presents an online pseudospectral method for system identification using Chebyshev polynomial basis under aperiodic sampling. The system dynamics are approximated piecewise by introducing a sliding time window. The number of sampling instants (Chebyshev nodes) within each sliding window is selected dynamically based on a proposed node-selection criterion that guarantees desired approximation accuracy. The system states are measured at these aperiodic instants and used to estimate the coefficients of the basis polynomials using least squares. An adaptive state estimator is also proposed to reconstruct the continuous states using the approximated dynamics. The boundedness of the parameter and state estimation errors is proven analytically and validated numerically.

A Novel Online Pseudospectral Method for Approximation of Nonlinear Systems Dynamics

TL;DR

The paper tackles online identification of unknown nonlinear system dynamics under aperiodic sampling. It develops an online Chebyshev pseudospectral framework with sliding windows, where the dynamics are approximated as time-based Chebyshev polynomials and coefficients are learned via least squares using aperiodic samples; an adaptive node-design guarantees a prescribed approximation accuracy. An adaptive state estimator is integrated, enforcing continuity across window transitions by adjusting coefficients to match derivatives up to the window order. Theoretical guarantees show bounded parameter and state estimation errors, and simulations on the Stuart--Landau oscillator demonstrate significant sampling efficiency and accurate state reconstruction, highlighting the method's potential for real-time nonlinear system identification and control design.

Abstract

This note presents an online pseudospectral method for system identification using Chebyshev polynomial basis under aperiodic sampling. The system dynamics are approximated piecewise by introducing a sliding time window. The number of sampling instants (Chebyshev nodes) within each sliding window is selected dynamically based on a proposed node-selection criterion that guarantees desired approximation accuracy. The system states are measured at these aperiodic instants and used to estimate the coefficients of the basis polynomials using least squares. An adaptive state estimator is also proposed to reconstruct the continuous states using the approximated dynamics. The boundedness of the parameter and state estimation errors is proven analytically and validated numerically.
Paper Structure (14 sections, 7 theorems, 74 equations, 5 figures)

This paper contains 14 sections, 7 theorems, 74 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on Chebyshev Polynomial-based Approximation and Problem Statement
  3. Chebyshev Approximation on the Standard Interval
  4. Chebyshev Expansion on a Shifted Interval
  5. Problem Formulation
  6. Proposed Online PS Framework
  7. Online PS Method for System Identification
  8. Minimum Nodes for Desired Approximation Accuracy
  9. Adaptive State Estimator And Node Design
  10. Adaptive State Estimator Design
  11. Adaptive Node Selection Approach
  12. Simulation Results
  13. Conclusions
  14. Appendix

Key Result

Lemma 1

Define $\mathscr{E}^w_{\max}=\max_{t^{w-1}<t\le t^{w}}\|F^w(t)-\hat{F}^w(t)\|$. For the dynamics eqPNnnnnn1 and its piecewise Chebyshev approximation eqPNnn1 on $(t^{w-1},t^{w}]$, it holds that where $0<C_w<\frac{2D_w^{(M_w+1)}}{\sqrt{6\pi}}\bigl(\frac{\tau}{4}\bigr)^{M_w+1}$, $M_w\ge2$, and $\alpha_{m,w}=\frac{M_w+1}{e}$ with Euler’s number $e\approx2.718$.

Figures (5)

  • Figure 1: Schematic of proposed approach.
  • Figure 2: Convergence of the Chebyshev approximation $\hat{F}(t)$ to the true dynamics $F(x(t))$.
  • Figure 3: Number of nodes in each time window for approximation.
  • Figure 4: Convergence of Chebyshev approximation $\hat{x}(t)$ to system state $x$.
  • Figure 5: Convergence of the state estimation error $\tilde{x}(t)$.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Theorem 1
  • Remark 4
  • Remark 5
  • Proposition 1
  • Lemma 2
  • Theorem 2
  • ...and 4 more