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arxiv: 2512.21096 · v4 · pith:IRHPCOIDnew · submitted 2025-12-24 · 🧮 math.OC

Identification with Orthogonal Basis Functions: Convergence Speed, Asymptotic Bias, and Rate-Optimal Pole Selection

Jiayun Li , Yiwen Lu , Yilin Mo , Jie Chen This is my paper

Pith reviewed 2026-05-21 17:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords orthogonal basis functionssystem identificationpole selectionasymptotic biasTsuji pointsChebyshev constantH2 normLTI systems
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The pith

Selecting Tsuji points as model poles makes OBF identification asymptotically optimal with exponentially decaying bias.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to select poles for orthogonal basis functions when identifying linear time-invariant systems so that the asymptotic bias under the H2 norm is minimized. It derives that the bias decreases exponentially with the number of basis functions and that the optimal poles are the Tsuji points. A sympathetic reader would care because this provides a way to achieve the best possible identification accuracy for a given model order without prior knowledge of the true system poles. The analysis also supplies the explicit convergence rate and bias bounds that depend on the pole locations.

Core claim

The paper claims that by choosing the model poles to be the Tsuji points, the pole selection algorithm achieves the fundamental lower bound on the worst-case asymptotic identification bias in the H2 sense. Consequently, the bias decreases exponentially as the number of basis functions grows, at a rate determined by the hyperbolic Chebyshev constant. This optimality holds asymptotically for stable LTI systems.

What carries the argument

The Tsuji points, which serve as the optimal locations for the poles of the orthogonal basis functions to minimize the maximum possible H2 bias over all true systems.

If this is right

  • The OBF identification algorithm has an analytically derivable convergence rate.
  • The asymptotic bias admits an explicit bound based on the true and model poles.
  • The proposed selection is robustly optimal for the worst-case bias under H2 criterion.
  • The exponential decay rate of the bias is set by the hyperbolic Chebyshev constant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the optimality holds, engineers could use fewer basis functions to reach a target accuracy in system models for control design.
  • This result links system identification to classical approximation theory through the use of Chebyshev constants in rational functions.
  • Extensions might include applying the same pole selection to nonlinear or time-varying systems for similar bias reductions.

Load-bearing premise

The true system is stable, linear, and time-invariant with all poles strictly inside the unit circle.

What would settle it

Simulate the identification for a specific stable system with known poles, compute the H2 bias using Tsuji points, and check whether it equals or approaches the predicted lower bound while decreasing exponentially with added basis functions.

Figures

Figures reproduced from arXiv: 2512.21096 by Jiayun Li, Jie Chen, Yilin Mo, Yiwen Lu.

Figure 1
Figure 1. Figure 1: Visualization of the introduced initialization strategy [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Asymptotic relative H2 approximation bias of the nominal system (39) under Gaussian perturbations of the true system poles, visualized using a multi-kernel density estimate (KDE). x(m) y(m) 0 × 1 2 3 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The shape of the region considered in the diffusion [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Computation time versus the number of basis poles q for different pole selection algorithms, evaluated on pole regions D = [−0.95, 0.95] (top) and D = [−0.99, 0.99] (bottom). Lemma 12 (Hyperbolic Chebyshev Constant of real in￾tervals; see page 278 in [30]). The hyperbolic Chebyshev constant of the real interval [0, ρ], ρ < 1 is τ ([0, ρ]) = exp    − π 2 K p 1 − ρ 2  K(ρ)    , where K(ρ) = Z 1 0 dx … view at source ↗
Figure 6
Figure 6. Figure 6: Online relative H2 identification error of the system (39) (left) and the diffusion process (right). Results are shown for OBF-based identification using 10 basis functions with Tsuji points, as well as for the Ho-Kalman algorithm. The plot reports 100 Monte Carlo experiments, where the solid line denotes the mean identification error at each time step and the shaded area indicates the range of errors acro… view at source ↗
Figure 7
Figure 7. Figure 7: Conformal mapping from the real interval [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original abstract

This paper is concerned with performance analysis and pole selection problem in identifying linear time-invariant (LTI) systems using orthogonal basis functions (OBFs), a system identification approach that consists of solving least-squares problems and selecting poles within the OBFs. Specifically, we analyze the convergence properties and asymptotic bias of the OBF algorithm, and propose a pole selection algorithm that robustly minimizes the worst-case identification bias, with the bias measured under the $\mathcal{H}_2$ error criterion. Our results include an analytical expression for the convergence rate and an explicit bound on the asymptotic identification bias, which depends on both the true system poles and the preselected model poles. Furthermore, we demonstrate that the pole selection algorithm is asymptotically optimal, achieving the fundamental lower bound on the identification bias. The algorithm explicitly determines the model poles as the so-called Tsuji points, and the asymptotic identification bias decreases exponentially with the number of basis functions, with the rate of decrease governed by the hyperbolic Chebyshev constant. Numerical experiments validate the derived bounds and demonstrate the effectiveness of the proposed pole selection algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first derives an explicit analytical expression for the asymptotic H2 identification bias that depends on both the unknown true-system poles and the chosen model poles within the orthogonal basis functions. It then constructs a pole-selection procedure (Tsuji points) that minimizes the worst-case value of this same expression and verifies that the resulting bias decays at the rate given by the hyperbolic Chebyshev constant. Because the claimed optimality is simply the statement that the chosen poles attain the infimum of the derived bias functional over all admissible pole placements, the argument does not reduce the result to its own inputs by definition, nor does it rely on a self-citation chain or a fitted parameter renamed as a prediction. The convergence-rate and bias bounds are obtained from standard least-squares analysis on the OBF expansion and are therefore independent of the subsequent optimality claim. The overall derivation remains self-contained with respect to the stated H2 error criterion for stable LTI systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of orthogonal rational functions and H2 norms; no new free parameters or invented entities are introduced in the abstract. The optimality result is grounded in an external lower bound whose derivation is not shown here.

axioms (2)
  • domain assumption The true plant is a stable discrete-time LTI system whose poles lie inside the unit disk.
    Required for the bias expression and exponential decay claim to hold.
  • domain assumption The H2 error is the appropriate worst-case performance measure for the identification bias.
    Stated as the criterion under which the pole-selection algorithm is optimal.

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