Adaptive Physics-Informed System Modeling with Control for Nonlinear Structural System Estimation

Biqi Chen; Chenyu Zhang; Jun Zhang; Ying Wang

REVIEW 4 major objections 6 minor 60 references

Adaptive Physics-Informed System Modeling with Control for Nonlinear Structural System Estimation

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read APSMC claims that online estimation of a structure's time-varying linear model from noisy data converges to the theoretically optimal solution, with the identified matrices tracking the Jacobian of the nonlinear dynamics.

desk verdict A useful online system identification recipe with real experiments, but the advertised convergence guarantee is not established and the paper's own caveats undermine it. read the letter →

arxiv 2505.06525 v1 pith:7EM4H6WY submitted 2025-05-10 nlin.AO

Biqi Chen , Chenyu Zhang , Jun Zhang , Ying Wang This is my paper · ORCID

classification nlin.AO MSC 93B3093E11

keywords nonlineardynamicalsystemsadaptivesystemmodelingstochasticsubspaceidentificationoptimalestimationstructuraldynamicsKalmanfilterproximalgradientoptimizationphysics-informed

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper seeks to establish that a nonlinear structure's dynamics can be identified online, from noisy measurements, as a time-varying linear system, and that the estimate is provably optimal once enough data has accumulated. The claim matters because real structures such as bridges and buildings are monitored continuously while their dynamic properties drift with damage, aging, and extreme loading, and existing data-driven methods are mostly offline, noise-sensitive, and yield models without physical meaning. The proposed APSMC framework couples a Kalman filter with physics-constrained proximal-gradient updates, so each new sample refines the system matrix pair $(A_k, B_k)$ while enforcing structural constraints on the model. Within the stochastic subspace identification framework, the paper asserts that the noise contribution vanishes as data accumulates and the convex updates converge to the global optimum based on the true Kalman-filtered state sequence, making the estimated $A_k$ equal to the Jacobian of the nonlinear dynamics.

What carries the argument

The mechanism is a coupled loop of two classical tools: a Kalman filter that produces the optimal state estimate $\hat{x}_k$ from the current model, and a proximal-gradient step that updates the matrix pair $(A_k, B_k)$ by minimizing $\|\hat{x}_{k+1} - A_k \hat{x}_k - B_k u_k\|^2$ plus a physical regularizer, which is effectively an online least-squares fit against filtered states. What licenses the optimality claim is a stochastic subspace identity: the noise subspace is statistically orthogonal to the signal subspace, so as data grows the residual noise vanishes and the least-squares objective is evaluated against the true Kalman-filtered sequence. Physical priors enter through the proximal mapping, with hard structural constraints such as the block form of a mechanical system's continuous-time matrix acting as projections and soft constraints as regularizers. The upshot is that the time-varying $A_k$ can be read as the Jacobian of the nonlinear dynamics, which gives the identified matrices their physical interpretability.

What would settle it

Run the algorithm on a linear time-invariant system with known $A$, initialize $A_0$ far from the truth with the sensor count below the state dimension, and monitor $\|A_k - A^*\|$: the paper itself reports that in this regime the Kalman filter fails to provide accurate predictions, so the claimed limit $\lim_{i\to\infty}\Delta A_i = 0$ would be violated. A second check targets the constrained version: because the bilinear transform used to impose continuous-time constraints makes that subproblem non-convex, exhibiting two different initial matrices that converge to two different stationary solutions would contradict the global-optimality claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that merging stochastic subspace identification with an online proximal-gradient loop turns nonlinear structural estimation into a convex problem with a provable optimum: with infinite data the subspace argument eliminates the noise component, leaving the least-squares target $A^* = \arg\min_A \sum_i \|\hat{x}_{i+1} - A\hat{x}_i\|^2$ built from the true Kalman-filtered states, and each online step solves a convex subproblem of this form, so the paper asserts $\lim_{i\to\infty}\Delta A_i = 0$ and convergence to the theoretically optimal model. Because the system matrix is locally the Jacobian of the dynamics, the identified matrices retain physical meaning, and when structural constraints are imposed through the proximal mapping the method tracks the Jacobian even where purely data-driven fits fail. Experimentally, the claim is that a single 10-second impact-test segment updates the model well enough to predict 19 unseen 10-second tests, with a best normalized mean square error (NMSE) of 0.398%.

Load-bearing premise

The argument assumes that alternating between Kalman filtering, whose state estimates depend on the current system matrix, and a convex least-squares update of that matrix drives the pair to the joint optimum; the paper never proves this coupled fixed-point convergence, and it concedes in Section 5.3 that a poor initial matrix with fewer sensors than states prevents the filter from updating the model correctly.

Editorial extensions

If this is right

Continuous monitoring becomes a live model-update loop: each new acceleration sample refines the system matrices without storing or reprocessing the full measurement history.
The identified matrices carry physical meaning, since their block structure approximates the structure's mass, stiffness, and damping, so damage and drift could be read directly from the estimates.
Extremely short calibration data may suffice: a single 10-second impact-test segment predicted 19 unseen 10-second responses with a minimum NMSE of 0.398%.
Physical constraints are what preserve accuracy under noise: on a frame structure with 30% added noise, constrained APSMC held prediction NMSE at 4.35% versus 12.56% for unconstrained APSMC and 34.54% for the global linear baseline.
The framework extends to arbitrary white-noise distributions and explicit external inputs, covering cases the earlier APSM formulation could not handle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the filter and the matrix update feed each other, a fully rigorous convergence guarantee would require an alternating-minimization or expectation-maximization analysis of the coupled problem; that analysis is not in the paper, and it may impose conditions beyond convexity of each subproblem.
The paper shows that under weak excitation the least-squares problem is not strictly convex, which suggests a testable extension: characterizing which input spectra let the constrained estimator recover the physical Jacobian rather than merely a minimum-Frobenius-norm fit.
Nothing in the argument pins the filter to the linear-Gaussian Kalman filter, so the same loop with an unscented or particle filter for non-Gaussian noise is a direct and untested extension.
Because the proximal step is generic, the loop could in principle be applied to other locally linearizable physical systems, such as identifying contact or friction laws, by changing only the constraint set.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know about this one. First, the central theoretical claim — that APSMC provably converges to the optimal state-space model as data accumulates — does not hold up. Second, the empirical work is real and the method may well be useful in practice; the paper is just oversold on the theory.

The genuinely new part is modest: the authors extend their prior APSM method by adding an explicit input term B_k u_k and allowing arbitrary white noise. That is a natural extension, and the scaled-bridge experiment with 20 impact tests is a legitimate piece of validation. They also ship code, which is appreciated. Credit where due: the paper is honest about several limitations, and the physical-constraint idea — using the known Jacobian structure to guide the update — is sensible and demonstrably helps in the Duffing and frame examples.

But the load-bearing convergence argument in Section 3.1 is a sketch, not a proof. It treats the Kalman-filtered state sequence as if it were the true one, then claims convexity of the loss implies global convergence. That ignores the coupling: the state estimates depend on the A being estimated, so the objective is not fixed even in principle. The 'optimal solution' A* is the least-squares fit to a self-generated state sequence, not to the underlying data-generating system. The paper itself concedes the point twice: in Section 5.3 it says that with a poor initial A0 the Kalman filter fails and the algorithm cannot update, and in Appendix B.2 it admits the bilinear transformation makes the constrained problem non-convex, so proximal gradient does not guarantee a global optimum. That is a direct contradiction with the abstract's claim of guaranteed convergence.

The experiments also have weaknesses. The learning rate is hand-tuned per sequence (Table 2), hyperparameters are underreported, and in sequences 15, 16, and 19 APSMC is substantially worse than the ERA baseline — not just a little, but three to four times worse. That pattern suggests the method's stability depends on getting the tuning right, which casts further doubt on the 'guaranteed' language.

Who gains from this paper? Practitioners in structural dynamics who want an online, physics-constrained identification heuristic and are willing to tune it will find the recipe useful. As a research claim about convergence to an optimal estimator, it fails. I would send it to peer review only with the expectation that the theory is removed or replaced by a much more limited statement — say, that the scheme is a heuristic for tracking time-varying linearizations. The empirical content deserves a shot, but not with the current framing.

Referee Report

4 major / 6 minor

Summary. The paper proposes the Adaptive Physics-Informed System Modeling with Control (APSMC) framework for online identification of time-varying linear state-space models from noisy input–output data. The method alternates between Kalman filtering (or another Bayesian filter) for state estimation and online proximal-gradient updates of the system matrices A and B, subject to physics-based structural constraints. The paper claims a theoretical guarantee, derived from stochastic subspace identification, that as data accumulate the estimates converge to the theoretically optimal solution (Section 3.1), and it validates the method on a Duffing oscillator, a seismic frame response simulation, and impact tests on a scaled bridge model, reporting NMSE values as low as 0.398%. An open-source Python implementation is provided.

Significance. If the central convergence and optimality claims were valid, APSMC would be a notable contribution to online nonlinear structural system identification: it combines state estimation with constrained optimization, provides physically interpretable time-varying system matrices, and is supported by code and by tests in simulation and experiment. The empirical framework is attractive and the reported prediction accuracy on the bridge data is competitive. However, the theoretical guarantee is not established and is contradicted by the paper's own statements; the experimental design involves heavy per-dataset tuning; and one of the interpretability claims rests on an algebraic error. The contribution is therefore currently an interesting heuristic with promising experiments rather than a validated theoretical method.

major comments (4)

[Sec. 3.1 (Eqs. 24-29), Sec. 5.3, App. B.2] The claimed guarantee lim_{i→∞} ΔA_i = 0 is not proven. The objective (24) is convex for a fixed state sequence, but the regressors x̂_k and targets x̂_{k+1} are outputs of a Kalman filter whose gain and innovation covariance depend on the A being estimated, so the iteration is a coupled stochastic approximation with state-dependent noise. Eq. (26) defines A* as the least-squares fit to the filtered states, so the target is self-referential rather than the data-generating A_true; no contraction, Lyapunov, or stochastic-approximation argument is supplied. The paper's own Sec. 5.3 concedes that with a poor initial A0 the Kalman filter fails and the algorithm cannot update to the correct model, and that it is theoretically unclear whether identity initialization guarantees convergence. Appendix B.2 additionally concedes that the bilinear-transform constraint is non-convex, so proximal gradient does not guarantee a global optimum. These admissions directly contradict the headline guarantee.
[Sec. 4.1 (Eqs. 46-48)] The equality asserted between the Duffing system and the Jacobian-based time-varying linear system is algebraically false. Substituting J(x) from (46) into (47) yields ẋ₂ = (α - 3βx₁²)x₁ - c x₂ + F cos(ωt) = α x₁ - 3βx₁³ - c x₂ + F cos(ωt), whereas the original system (44) has ẋ₂ = α x₁ - β x₁³ - c x₂ + F cos(ωt); the difference is -2βx₁³. Consequently x_{k+1} = J(x_k) x_k + B_k u_k is not a representation of the nonlinear system, and the numerical agreement of A_k with J(x_k) in Figs. 6-8 is not a valid ground-truth comparison for the claim that A_k tracks the Jacobian.
[Sec. 5.2 (Table 2)] The experimental evidence does not support the claimed consistent superiority of APSMC. The learning rate t_k is hand-tuned per dataset and varies by four orders of magnitude (0.004 to 100). For sequences 15 and 19, APSMC's NMSE (18.847% and 17.028%) is substantially worse than ERA's (10.065% and 11.164%). The 0.398% figure quoted in the abstract is the minimum over all sensors and datasets, not a typical result. The model order and Hankel dimensions (m = n = 3000) are also determined ad hoc, without sensitivity analysis.
[Secs. 4-5 (Algorithms 1-2)] The algorithm's reproducibility is compromised by unspecified key hyperparameters. The noise covariances Q, R, and P0 for the Kalman filter, the regularization weight λ in Eq. (37), and the step-size schedule t_k used in the numerical and experimental studies are not reported (except for selected learning rates in Table 2). Without these, the results cannot be replicated, and the claim that the method is 'theoretically optimal' cannot be assessed against the actual implementation.

minor comments (6)

[Throughout (Figs. 5-22)] Many figures contain extensive text corruption (e.g., strings of '/uni00000013/uni00000018/...'), and numerous figures lack captions; the manuscript must be regenerated cleanly before review.
[Sec. 2.1, Eq. (2)] Eq. (2) uses squared error E[(x_k - g)^2] for vector states; it should be a vector norm (or sum of squares) to define the MMSE estimator.
[Sec. 3.2] The statement that noise 'may follow arbitrary distributions of white noise' is an overclaim: Kalman filtering is optimal only under Gaussianity, and the paper neither analyzes nor tests non-Gaussian noise.
[Sec. 5.3] The analogy between the algorithm and SGD is imprecise: online proximal gradient with sequentially ordered samples is not stochastic gradient descent (which samples randomly), and the convergence properties differ.
[References] Several key references are unpublished preprints, notably [2] (SSRN) and [31] (arXiv), and results in Secs. 4-5 draw heavily on [31] and [9]; these dependencies should be updated or clarified.
[Abstract and Sec. 5.2] The abstract's claim of '19 consecutive 10-second time series using only a single initial 10-second segment for model updating' is misleading because Table 2 shows large errors for several sequences and the model was updated sequentially, not held fixed; the description should be precise.

Circularity Check

1 steps flagged · score 6.0 of 10

Section 3.1's convergence-to-optimality claim is self-referential: A* is defined as the least-squares fit to Kalman-filtered states, while the same objective over states produced by the estimated A supplies the 'optimal solution' the algorithm is said to reach.

self definitional [Section 3.1, Eq. (26) and following paragraph (Eqs. 27-29); compare Eq. (24) and Algorithm 1]
"if the true Kalman-filtered state sequence ˆXi were known, the optimal estimate of the system matrix A∗ could be defined as: A∗ = argmin A Pk i=1 ∥ˆxi+1−Aˆxi∥2 2 = ˆYk ˆX† k (26) ... Since the objective function in equation (24) is convex, and each update follows a convex optimization scheme, the algorithm is guaranteed to converge to the global optimum based on the true Kalman-filtered state sequence."

Equation (26) defines the target A* as the least-squares fit to the Kalman-filtered state sequence. Equation (24) (and its APSMC version, Eq. (38)) is the same least-squares objective, but the state matrices Xhat_k and Yhat_k entering it are produced online by a Kalman filter whose gain and state estimates depend on the very A being optimized. Thus the 'global optimum' the algorithm is guaranteed to reach is the minimizer of the algorithm's own, self-referential objective, not an independently specified system matrix. The proof that this minimizer equals A* requires that the online KF state sequence coincide with the 'true Kalman-filtered state sequence' — precisely the fixed-point convergence that is asserted rather than derived.

full rationale

The empirical core of the paper is self-contained: APSMC is benchmarked against DMDc on a frame structure and against ERA/OKID+ERA on a scaled bridge, with predictions evaluated on holdout segments via NMSE. Those comparisons do not reduce to the fitted inputs, and the self-citations to the authors' prior APSM work [2] are not load-bearing for those experiments. The circularity is concentrated in the theoretical optimality claim of Section 3.1. Eq. (26) defines A* as the least-squares fit to a Kalman-filtered state sequence, while the online algorithm (Algorithm 1/2, Eq. (24)/(38)) minimizes the same least-squares objective over state sequences produced by the Kalman filter running with the iteratively updated A. The 'optimal solution' is therefore defined internally as the minimizer of the algorithm's own loss, and the additional claim that this minimizer is the 'true Kalman-filtered state sequence' is a fixed-point assumption that is never proved. The paper's own Section 5.3 concedes that with a poor initial A0 the KF fails and 'it remains theoretically unclear whether initializing A0 as an identity matrix guarantees convergence to the optimal solution,' and Appendix B.2 admits that the bilinear-transform constraints make the optimization non-convex, so proximal gradient does not inherently guarantee global optimality. These admissions confirm that the headline convergence guarantee is an unproved assumption rather than a derived result. Because the target of the theoretical claim is defined by the same objective the algorithm minimizes, the convergence result is partially circular by construction; the numerical and experimental demonstrations retain independent value despite this theoretical gap.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method rests on a linear time-varying state-space model with white noise, a Kalman filter that is assumed optimal, and a proximal gradient step whose convergence is assumed despite the coupled state-matrix dependency. Several hyperparameters are fitted to the data or left unspecified, including per-sequence learning rates, model order, Hankel dimensions, the regularization weight, and the Kalman noise covariances. No fundamentally new physical entities are introduced.

free parameters (6)

Per-test learning rate t_k = 0.004 to 100 (listed per sequence in Table 2)
The learning rate is hand-tuned for each of the 19 test segments. The paper states that an excessively high rate causes instability and a low rate slows convergence, so each sequence uses a different value.
Model order n = 50 to 577; n=300 used for the main result
Selected from the singular value distribution; the paper tests multiple orders and reports that accuracy plateaus near n=200. No principled model-order selection criterion is applied.
Hankel matrix dimensions m and n = m = n = 3000
Chosen after 'extensive trial calculations' (Section 5.2). The paper notes that different choices significantly impact results but provides no objective selection rule.
Regularization weight λ = not reported
Appears in the objective function (Eq. 37) as the balance between data fidelity and physical constraints, but no value or selection method is given.
Kalman filter noise covariances Q, R, P0 = not reported
Algorithm 2 requires these covariance matrices for the state estimation step, but the paper does not specify how they are initialized or updated.
Iteration count = 100 to 1100 iterations
Convergence behavior depends on the number of iterations; for model order 50, 100 iterations give NMSE 9.806% and 1100 iterations give 9.858%, showing non-monotonic behavior.

assumptions (5)

domain assumption The true system is well approximated by a linear time-varying state-space model with zero-mean white process and measurement noise.
Adopted in Section 2.1; the Kalman filter, the SSI framework, and the convergence claims all rely on this linear, white-noise model.
domain assumption The Kalman filter provides the minimum-mean-square-error state estimate for the evolving model at each step.
Section 3.1 uses this to assert each updated state is more accurate than the SSI-based state sequence; this holds only under linear-Gaussian assumptions, which the paper relaxes without proof.
standard math The signal and noise subspaces in stochastic subspace identification are asymptotically orthogonal, so the projection in Eq. (13) removes all noise as j→∞.
This is a standard result of SSI theory [29], applied here without a rigorous extension to the time-varying, input-driven setting.
ad hoc to paper The objective in Eq. (24) is convex at each update, so the proximal gradient step converges to its global minimum.
Section 3.1 claims convexity, but the state sequence changes with the estimated matrix, and Appendix B.2 admits that the bilinear transformation in the constrained case makes the overall problem non-convex, so global convergence is not guaranteed.
domain assumption The physical constraint structure (e.g., Jacobian form [0 1; Unknown -0.1] or symmetric lower blocks) is known a priori.
Used to define the feasible set M in Sections 4.1 and 4.2. If the assumed structure is incorrect, the identified model will be biased regardless of data fit.

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Cite this review

Pith. "Pith review of Adaptive Physics-Informed System Modeling with Control for Nonlinear Structural System Estimation." pith.science (2026). https://pith.science/paper/7EM4H6WY

@misc{pith2026250506525,
  author       = {Pith},
  title        = {Pith review of: Adaptive Physics-Informed System Modeling with Control for Nonlinear Structural System Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EM4H6WY}},
  note         = {Machine review of arXiv:2505.06525}
}

read the original abstract

Accurately capturing the nonlinear dynamic behavior of structures remains a significant challenge in mechanics and engineering. Traditional physics-based models and data-driven approaches often struggle to simultaneously ensure model interpretability, noise robustness, and estimation optimality. To address this issue, this paper proposes an Adaptive Physics-Informed System Modeling with Control (APSMC) framework. By integrating Kalman filter-based state estimation with physics-constrained proximal gradient optimization, the framework adaptively updates time-varying state-space model parameters while processing real-time input-output data under white noise disturbances. Theoretically, this process is equivalent to real-time tracking of the Jacobian matrix of a nonlinear dynamical system. Within this framework, we leverage the theoretical foundation of stochastic subspace identification to demonstrate that, as observational data accumulates, the APSMC algorithm yields state-space model estimates that converge to the theoretically optimal solution. The effectiveness of the proposed framework is validated through numerical simulations of a Duffing oscillator and the seismic response of a frame structure, as well as experimental tests on a scaled bridge model. Experimental results show that, under noisy conditions, APSMC successfully predicts 19 consecutive 10-second time series using only a single initial 10-second segment for model updating, achieving a minimum normalized mean square error (NMSE) of 0.398%. These findings demonstrate that the APSMC framework not only offers superior online identification and denoising performance but also provides a reliable foundation for downstream applications such as structural health monitoring, real-time control, adaptive filtering, and system identification.

Figures

Figures reproduced from arXiv: 2505.06525 by the authors.

**Figure 2.** Illustration of the APSM framework for optimal system matrix estimation. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Illustration of the convex optimization formulation within the APSMC framework. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: A schematic representation of the APSMC algorithm process. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: The displacement x1(t), velocity x2(t), and phase space attractor of the Duffing oscillator. To validate the effectiveness and physical interpretability of the system matrix identified by the APSMC algorithm, we compare it with the theoretical Jacobian matrix of the Du…

**Figure 6.** Figure 6: The Frobenius norm of the difference between [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: shows the comparison of the Frobenius norm of Ak and Bk with the true values. 0 50 100 150 200 250 300 Time (s) 0.0 0.2 0.4 0.6 0.8 1.0 Frobenius Norm (a) A0 = I 0 50 100 150 200 250 300 Time (s) 0.6 0.8 1.0 1.2 1.4 Frobenius Norm (b) B0 = I 0 50 100 150 200 250 300 Ti…

**Figure 8.** Figure 8: Frobenius norm errors of Ak and Bk relative to ground truth under physical constraints With the incorporation of physical constraints, the error between Ak and J(xk) is significantly reduced. Since the constraints are applied only to Ak, the initial error in Bk remain…

**Figure 9.** Figure 9: Two-dimensional computational diagram The ground motion record from the 1994 Northridge earthquake, with Record Sequence Number 968, was selected from the PEER NGA WEST-2 database [36]. The recording station is 46.74 km from the rupture, and both the site and structure…

**Figure 10.** Figure 10: 0 10 20 30 40 50 60 Time (seconds) 0.2 0.1 0.0 0.1 0.2 Acceleration (g) Ground Motion (a) Time history 0 5 10 15 20 25 Frequency (Hz) 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 Power Spectral Density (g²/Hz) Ground Motion (b) Power spectral density [PITH_FULL_IMAGE:fig…

**Figure 11.** Figure 11: Time response of structural seismic response data with noise. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

**Figure 12.** Figure 12: System matrices identified from noise-free data [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: System matrices identified from noisy data [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]

**Figure 14.** Figure 14: Comparative analysis between true data and predicted results. [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

**Figure 15.** Figure 15: Schematic of the scaled bridge model and test equipment. [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

**Figure 16.** Figure 16: Singular values employed for determining model order and the corresponding input signal [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

**Figure 17.** Figure 17: Prediction Accuracy and Convergence Analysis [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]

**Figure 18.** Figure 18: Comparative Analysis of Monitoring and Predicted Data for Various [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]

**Figure 19.** Figure 19: Comparative Analysis of Power Spectral Density for Various Monitoring Data [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]

**Figure 20.** Figure 20: Comparative Analysis of Monitoring and Predicted Data for Various [PITH_FULL_IMAGE:figures/full_fig_p029_20.png]

**Figure 21.** Figure 21: Comparative Analysis of Power Spectral Density for Various Monitoring Data [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]

**Figure 22.** Figure 22: Convergence curves for model order 50 It is evident that the NMSE in Figure 22b exhibits oscillations, with the final NMSE reaching 9.858%. This behavior occurs because, as the model parameters approach their optimal values, the SGD algorithm requires a progressively…

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