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Non-causal FIR models from Laurent expansions allow finite-time identification of both stable and unstable LTI systems using closed-loop data and instrumental variables.

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2026-06-30 13:29 UTC pith:YXX5ZWTH

load-bearing objection The paper gives a unified non-causal FIR approach for finite-time identification of stable and unstable LTI systems from closed-loop data. the 2 major comments →

arxiv 2605.24393 v1 pith:YXX5ZWTH submitted 2026-05-23 eess.SY cs.SY

Finite-Time Markov-Parameter Identification of LTI Systems Using Non-Causal FIR Models: A Unified Framework for Stable and Unstable Systems

Ahmad Al-Tawaha , Ming Jin , Khaled F. Aljanaideh This is my paper
classification eess.SY cs.SY
keywords system identificationclosed-loop identificationMarkov parametersnon-causal FIRunstable LTI systemsfinite-time analysisinstrumental variablesLaurent expansion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper develops a method to identify linear time-invariant systems that may be unstable by modeling them with a non-causal finite impulse response representation derived from a Laurent series expansion of the transfer function. Stable parts use causal coefficients while unstable parts use non-causal ones tied to reverse-time stability. This approach works on a single closed-loop input-output trajectory without needing the controller details or separating dynamics beforehand. It employs the injected excitation signal as an instrumental variable to eliminate bias from correlated noise and feedback. Under conditions on instrument strength and data concentration, it provides a non-asymptotic bound on estimation error that decays as the square root of the number of samples.

Core claim

The central claim is that the Markov parameters of the Laurent/FIR model can be estimated from closed-loop data with an error bound of order O(N^{-1/2}) (up to logs and truncation), by using the excitation as an instrumental variable that removes correlation bias, and that this bound holds uniformly for systems with both stable and unstable poles because the non-causal representation keeps all coefficient sequences decay-bounded by stable rates.

What carries the argument

The non-causal FIR model obtained from the Laurent expansion of the transfer function, which separates stable dynamics into causal Markov parameters and unstable dynamics into non-causal coefficients associated with reverse-time stable evolution.

Load-bearing premise

The system transfer function admits a Laurent expansion in which unstable dynamics are captured exactly by non-causal coefficients associated with reverse-time stable evolution, and the injected excitation acts as a valid instrumental variable that removes all correlation bias between feedback input and process noise.

What would settle it

An experiment where the estimated Markov parameters deviate from the derived O(N^{-1/2}) rate by more than logarithmic factors when the instrument-strength and closed-loop concentration conditions are satisfied.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper presents a unified finite-time framework for identifying Markov parameters of LTI systems (stable or unstable) from a single closed-loop input-output trajectory. It uses a non-causal FIR model obtained from the Laurent expansion of the transfer function, where unstable dynamics are captured by non-causal coefficients with reverse-time stable decay. External excitation serves as an instrumental variable to remove feedback-induced bias, and under explicit instrument-strength and closed-loop concentration conditions the paper derives a non-asymptotic error bound of order O(N^{-1/2}) (up to logarithmic factors and truncation) for the estimated Laurent/FIR parameters.

Significance. If the derivation of the bound holds under the stated conditions, the result supplies a controller-independent method for closed-loop identification that unifies stable and unstable plants without requiring observers or pole separation. The explicit hypotheses and finite-time rate that incorporate noise, FIR horizons, state moments, and instrument conditioning constitute a falsifiable theoretical contribution with potential utility in adaptive control and identification of unstable systems.

major comments (2)
  1. [Main theorem on the non-asymptotic error bound] The central error bound is asserted to follow directly from the instrument-strength and closed-loop concentration conditions, yet the manuscript does not make explicit (in the statement of the main theorem or its proof sketch) how these conditions control the closed-loop state moments independently of the unstable modes; this step is load-bearing for the claimed O(N^{-1/2}) rate.
  2. [Section on Laurent/FIR model and truncation analysis] The Laurent-expansion representation is claimed to keep all coefficients (input and noise) governed by stable/reverse-time-stable decay rates, but the paper should verify that the truncation error term remains controlled uniformly when the unstable poles lie outside the unit circle; this affects the overall finite-time guarantee.
minor comments (2)
  1. The abstract refers to the 'usual O(N^{-1/2}) statistical rate'; the bound statement should explicitly display the dependence on the FIR length and the number of estimated parameters.
  2. Notation for the instrumental variable and the closed-loop concentration inequalities should be introduced with a short table or list of symbols to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Main theorem on the non-asymptotic error bound] The central error bound is asserted to follow directly from the instrument-strength and closed-loop concentration conditions, yet the manuscript does not make explicit (in the statement of the main theorem or its proof sketch) how these conditions control the closed-loop state moments independently of the unstable modes; this step is load-bearing for the claimed O(N^{-1/2}) rate.

    Authors: We agree that the dependence of state-moment bounds on the closed-loop concentration conditions should be stated more explicitly. The non-causal Laurent representation ensures that all relevant coefficients (for both input and noise) decay at stable or reverse-time-stable rates, so that the closed-loop state moments remain bounded by quantities that do not grow with the unstable poles; the instrument-strength condition then guarantees that the external excitation dominates any residual effect. We will revise the theorem statement and expand the proof sketch to derive these moment bounds directly from the stated conditions. revision: partial

  2. Referee: [Section on Laurent/FIR model and truncation analysis] The Laurent-expansion representation is claimed to keep all coefficients (input and noise) governed by stable/reverse-time-stable decay rates, but the paper should verify that the truncation error term remains controlled uniformly when the unstable poles lie outside the unit circle; this affects the overall finite-time guarantee.

    Authors: The truncation error is controlled uniformly because the anti-causal coefficients associated with poles outside the unit circle decay exponentially in reverse time at a rate determined by the reciprocal of the pole magnitude. We will add an explicit lemma (or remark) in the truncation-analysis section that supplies a uniform bound on the truncation remainder in terms of the FIR horizon and the minimum distance of the poles from the unit circle, thereby confirming that the finite-time guarantee is unaffected. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained under explicit assumptions; no circularity

full rationale

The paper states explicit instrument-strength and closed-loop concentration conditions as hypotheses, then derives the non-asymptotic O(N^{-1/2}) error bound for the Laurent/FIR Markov parameters directly from those conditions plus standard concentration inequalities. The Laurent expansion is introduced as a modeling representation (stable dynamics causal, unstable captured by non-causal reverse-time terms) rather than fitted from data; the IV use of external excitation is justified by the stated conditions that eliminate feedback bias. No self-citations are load-bearing for the central bound, no parameter is fitted on a subset and then renamed as a prediction, and the bound does not reduce to its inputs by construction. The analysis is therefore independent of the target estimates themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a Laurent expansion that cleanly separates dynamics into causal stable and non-causal reverse-time stable parts, plus standard LTI assumptions on noise and closed-loop stability under feedback.

axioms (2)
  • domain assumption The transfer function of the LTI system admits a Laurent expansion separating stable and unstable dynamics into causal and non-causal FIR coefficients.
    Invoked to define the non-causal FIR model that avoids growth of unstable Markov parameters.
  • domain assumption The injected excitation signal is uncorrelated with process noise and satisfies explicit instrument-strength conditions.
    Required for the instrumental-variable bias removal and the non-asymptotic bound.

pith-pipeline@v0.9.1-grok · 5818 in / 1403 out tokens · 38538 ms · 2026-06-30T13:29:26.687122+00:00 · methodology

0 comments
read the original abstract

We present a finite-time framework for identifying stable and unstable linear time-invariant (LTI) systems from a single closed-loop input-output trajectory. The method does not require knowledge of the stabilizing controller, an intermediate observer, or prior separation of the plant into stable and unstable components. The approach uses a non-causal finite impulse response (FIR) model obtained from a Laurent expansion of the transfer function. In this representation, stable dynamics are captured by causal Markov parameters, while unstable dynamics are captured by non-causal coefficients associated with reverse-time stable evolution. This avoids the growth of causal unstable Markov parameters. A key advantage is that the coefficients multiplying both the input and the process noise remain controlled by stable and reverse-time stable decay rates, rather than by growing forward-time unstable dynamics. To handle closed-loop data, we use the injected excitation as an instrumental variable, which removes the bias caused by correlation between the feedback input and the process noise. Under explicit instrument-strength and closed-loop concentration conditions, we derive a non-asymptotic error bound for the estimated Laurent/FIR Markov parameters with the usual $\mathcal{O}(N^{-1/2})$ statistical rate, up to logarithmic factors and truncation terms. The bound captures the effects of process noise, measurement noise, FIR horizons, closed-loop state moments, and controller-dependent instrument conditioning. Numerical experiments support the finite-time analysis by showing the predicted Markov-parameter convergence rate and illustrating how controller-dependent instrument conditioning affects the sample complexity of closed-loop identification.

Figures

Figures reproduced from arXiv: 2605.24393 by Ahmad Al-Tawaha, Khaled F. Aljanaideh, Ming Jin.

Figure 1
Figure 1. Figure 1: Example 4.1: Markov-parameter estimation error [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example 4.2: Parameter estimation errors for RLS– [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 4.2: Predicted vs. measured output and [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example 4.3: Predicted and measured outputs with [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example 4.5: Test-set RMSE versus training samples [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

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    (B.1) From the truncated non-causal FIR representation, Ψ y,ℓ = θr,d Φ + Er,d,ℓ

    Define the aggregate disturbance matrix Er,d,ℓ ∆ = γr,d Φ w,r,d,ℓ + Ψ er,d ,ℓ + Ψ v,ℓ . (B.1) From the truncated non-causal FIR representation, Ψ y,ℓ = θr,d Φ + Er,d,ℓ . (B.2) Substituting (B.2) into the IV estimator gives ˆθIV r,d,ℓ = (θr,d Φ + Er,d,ℓ ) Φ ⊤ c ( ΦΦ ⊤ c )− 1 = θr,d + Er,d,ℓ Φ ⊤ c ( ΦΦ ⊤ c )− 1 . (B.3) Therefore, ˆθIV r,d,ℓ − θr,d = Er,d,ℓ Φ...

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