Uncertainty-Aware Parameter Estimation for Condition Monitoring of Power Converters

Tomas Monopoli; Jiahong Liu; Shuai Zhao

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Uncertainty-Aware Parameter Estimation for Condition Monitoring of Power Converters

Tomas Monopoli, Jiahong Liu, Shuai Zhao

arXiv:2609.00218v1Aug 31, 2026eess.SY

TL;DR

Existing converter parameter-estimation methods often provide point estimates without quantifying whether changes are statistically significant. This paper develops a Bayesian framework using a differentiable physical model and Laplace posterior, and validates calibrated uncertainty, efficient estimation, cross-condition comparison, pooling, and change detection on synthetic and hardware Buck-converter data.

Problem
Point-estimation methods for converter condition monitoring do not quantify uncertainty or whether observed parameter changes are statistically significant.
Method
The framework combines sequential Adam and L-BFGS optimization, FIR whitening, spectral covariance regularization, and a Laplace approximation on a differentiable converter model.
Results
The method provides calibrated uncertainty and roughly 20× speedup against a typical PINN benchmark, while hardware validation recovers well-identified parameters and detects statistically significant component-level changes.
Takeaways & Limitations
Posterior uncertainty supports estimator-resolution analysis, precision-weighted pooling, cross-operating-point comparison, and principled change detection within the tested converter setting.

Abstract

from arXiv · show

Parameter estimation is widely used for condition monitoring of power converters, but most existing methods provide only point estimates and therefore cannot quantify whether an observed parameter change is statistically significant. This paper proposes an uncertainty-aware parameter estimation framework based on Bayesian maximum a posteriori optimization and a differentiable converter model. A Laplace approximation is used to obtain a local Gaussian posterior, enabling uncertainty quantification, consistency testing, estimator-resolution analysis, and precision-weighted pooling across data windows. The method is validated on synthetic and hardware Buck converter. It demonstrates accurate estimation of well-identified parameters and reveal the weak practical identifiability of parameters such as MOSFET on-resistance under the available sensing configuration.

I. INTRODUCTION

Existing converter-identification methods generally return point estimates, leaving uncertainty, prior knowledge, and statistically grounded change detection unresolved. This paper develops a Bayesian framework that produces calibrated parameter posteriors and supports comparison, pooling, and resolution analysis.

Research gap: Existing methods commonly produce point estimates without quantifying uncertainty or estimation reliability.Reliability can vary with noise level, data quantity, and data quality.
Research gap: Point estimates also lack a principled way to incorporate prior knowledge or distinguish estimation noise from genuine parameter drift.This makes comparison with earlier readings or healthy references ill-posed.
Proposed framework: The framework combines Bayesian MAP estimation with a differentiable physical model and a Laplace approximation of the posterior.Prior knowledge enters through the prior distribution, while posterior curvature supplies the local approximation.
Proposed framework: FIR whitening addresses correlated residuals and periodic disturbances, while spectral covariance regularization makes posteriors from different operating points comparable.These steps target calibrated uncertainty and stable inverse-covariance-weighted comparison or pooling.
Capabilities: The posterior supports estimator resolution, uncertainty-based quality assessment, precision-weighted pooling, and statistically rigorous change detection.Resolution quantifies the smallest detectable parameter change at a selected significance level.
Validation: Validation covers simulated data and hardware Buck-converter measurements, including a roughly 20× speedup against a PINN benchmark and quantified component-level changes.The hardware study reports a current-dependent inductance shift and emulated capacitor aging.

B. Laplace Posterior Approximation

The Laplace approximation replaces the posterior near the MAP estimate with a Gaussian whose covariance is determined by local loss curvature. Its statistical interpretation depends on a correctly specified likelihood, and misspecification can miscalibrate uncertainty.

Approximation: A second-order expansion of the negative log-posterior around the MAP produces a local Gaussian posterior with covariance H−1.The approximation is local and is most accurate when a single well-defined mode dominates.
Interpretation: H−1 encodes parameter sensitivity and correlations: high curvature implies low variance, whereas low curvature indicates weak identification and high uncertainty.Off-diagonal terms reveal coupled directions that the data cannot resolve separately.
Computation: Because L-BFGS already approximates the Hessian during optimization, uncertainty quantification follows without additional sampling at marginal computational cost.This avoids the expense of fully characterizing a nonlinear posterior with methods such as MCMC.
Statistical meaning: Under regularity conditions and a correctly specified model, Laplace covariance, inverse Fisher information, and the Cramér–Rao bound coincide asymptotically.The covariance then represents the asymptotic sampling variability of the estimator.
Limitation: If the loss is not the true negative log-likelihood, H−1 may yield confidence intervals that are over- or under-confident.Comparing Laplace and empirical covariance across repeated experiments can diagnose this failure.

III. METHODOLOGY

The methodology models a Buck converter with measured inductor current and output voltage, then differentiably integrates its dynamics to enable gradient-based estimation of eight unknown parameters.

Buck converter model: The benchmark is a Buck DC–DC converter whose topology identifies parameters of interest and measured signals.The measured signals are inductor current and output voltage.
Buck converter model: The model estimates eight unknowns, including inductance, capacitor and resistor quantities, MOSFET on-resistance, load resistance, input voltage, and diode forward drop.The parameter vector includes L, RL, C, RC, Rdson, R, Vin, and VF.
Differentiable forward model: The forward model integrates the switching dynamics with a fourth-order Runge–Kutta scheme to form a one-step predictor.Within each switching regime, the predictor is affine in the state.
Differentiable forward model: Analytic dependence of the predictor on the parameters makes the full computation graph differentiable and supports automatic-differentiation-based optimization.The predictor uses the current state, switch state, integration step, and parameter vector.

B. Loss Function

The loss function measures disagreement between predicted and observed next states using a residual covariance derived from the measurement-noise model. This yields a generalized least-squares objective that accounts for residual weighting and correlation structure.

Residual construction: Parameter estimation minimizes the mismatch between the differentiable one-step prediction and the observed next state.The mismatch is represented by a residual vector over the data sequence.
Loss formulation: The residual loss is quadratic with a weighting matrix W, reducing to ordinary least squares when W = I.The weighting determines how residual components contribute to the objective.
Probabilistic formulation: Assuming zero-mean Gaussian residuals at the true parameters, the negative log-likelihood defines the statistically motivated loss.A wrong parameter vector produces systematic residual bias that the quadratic form penalizes.
Covariance construction: The residual covariance is derived from additive white measurement noise through a first-order expansion of the residual around the true state.The forward-model Jacobian determines how state-measurement noise propagates into residual uncertainty.
Covariance construction: Using the per-step covariance blocks on the diagonal produces the generalized least-squares formulation.This specifies the covariance structure used to weight the residual sequence.

2) Time-Correlated Likelihood Form:

The method models sequentially correlated residuals with an exact block-tridiagonal covariance, then uses a differentiable optimization pipeline to obtain and regularize a Laplace posterior over converter parameters.

2) Time-Correlated Likelihood Form:: Residuals are sequentially dependent even when measurement noise is white because neighboring residuals share predictor-input noise.The residual sequence therefore has a vector moving-average structure of order one.
2) Time-Correlated Likelihood Form:: The exact block-tridiagonal covariance gives the correct negative log-likelihood, but direct evaluation scales as O(N^3C^3) and is impractical inside iterative optimization.Here C = 2 measurement channels.
C. Two-Phase Optimizer: The optimizer combines Adam for robust initial convergence with L-BFGS for curvature-aware final refinement near the MAP solution.Adam uses adaptive gradient moments, while L-BFGS approximates the inverse Hessian from recent gradient pairs.
D. Overall Optimization Approach: The Bayesian pipeline converts measured inductor-current and output-voltage sequences into a MAP estimate and a Laplace covariance.The resulting posterior is represented as N(θ̂, H^-1) before covariance regularization.
D. Overall Optimization Approach: The pipeline initializes parameters and covariance from nominal specifications, performs an initial MSE optimization, estimates unknown data noise, and then minimizes the correlated likelihood with Adam and L-BFGS.Noise covariance and FIR filters are refreshed during the main optimization, while the residual covariance is dynamically updated in the final L-BFGS phase.
D. Overall Optimization Approach: The final posterior is spectrally regularized by capping covariance eccentricity so posteriors from different operating points become mutually comparable.This replaces H^-1 with a regularized covariance while preserving the posterior's operational comparability.

E. FIR Whitening of Correlated Residuals

FIR whitening approximates the exact correlated-residual likelihood with a differentiable, linear-time transformation that produces approximately uncorrelated, unit-variance innovations for optimization.

E. FIR Whitening of Correlated Residuals: The exact likelihood requires dense covariance inversion, motivating a banded operator T that approximates Σr^-1 through T^⊤T.This reformulation avoids explicitly forming the dense precision matrix.
E. FIR Whitening of Correlated Residuals: The pipeline combines the RK4 forward model, covariance-aware FIR whitening, and an Adam-then-L-BFGS MAP update to produce the posterior.The forward model generates predictions, whitening transforms residuals into innovations, and the optimizer updates θ.
E. FIR Whitening of Correlated Residuals: Instantaneous whitening uses the lag-zero covariance to decorrelate the two measured channels, inductor current and output voltage, and normalize their variance.A Cholesky factorization is used for this channel-wise transformation.
E. FIR Whitening of Correlated Residuals: Temporal whitening fits finite-order autoregressive predictors to the instantaneously whitened sequence and applies a banded FIR operator.This removes predictable temporal correlation before evaluating the innovation loss.
E. FIR Whitening of Correlated Residuals: The filter order is selected by Bayesian information criterion to capture genuine correlation without overfitting.Each channel filter is rescaled to unit innovation variance.
E. FIR Whitening of Correlated Residuals: The FIR coefficients are periodically refitted and held fixed between updates so the loss remains differentiable in θ without extra gradient cost.This refresh strategy keeps the whitening operator compatible with iterative optimization.
E. FIR Whitening of Correlated Residuals: The whitening loss is equivalent to the time-correlated likelihood while reducing evaluation to O(Nm) operations.The whitened residual norm can therefore be used inside iterative optimization.

F. Mitigating System Model Misspecification with FIR Filters

Hardware model misspecification creates persistent periodic residuals that can bias estimates, while operating-point-dependent posterior geometry can destabilize comparison and pooling; FIR filtering and spectral regularization address these effects.

F. Mitigating System Model Misspecification with FIR Filters: Hardware residuals persist at the true parameters because idealized switching omits real MOSFET behavior, producing periodic error at switching harmonics.This systematic mismatch is termed System Model Misspecification.
F. Mitigating System Model Misspecification with FIR Filters: Persistent periodic residual energy accumulates with sample count and can bias optimization toward weakly identifiable parameter directions.The optimizer may favor parameters that absorb systematic mismatch rather than represent physical values.
F. Mitigating System Model Misspecification with FIR Filters: The FIR filter mitigates this bias by predicting and removing periodic and slowly varying residual components from the innovation sequence.In this role, the filter acts as a high-pass preconditioner that also suppresses coloured noise.
G. Spectral Regularization of the Posterior Covariance: Weak identifiability makes the Laplace covariance highly elongated, so inverse-covariance weighting becomes fragile for comparing or pooling estimates.Small changes in posterior orientation can dominate downstream statistical separation.
G. Spectral Regularization of the Posterior Covariance: In the {Vin, RL} subspace, nearly constant inductor current makes increases in Vin difficult to distinguish from compensating increases in RL.The resulting curvature block is nearly rank-deficient and produces elongated probability contours.
G. Spectral Regularization of the Posterior Covariance: Different duty cycles or load currents rotate otherwise similarly elongated posteriors because their geometry depends on switch state and inductor current.This rotation is structural rather than evidence of parameter change.
G. Spectral Regularization of the Posterior Covariance: A small MAP displacement can appear statistically large when posterior rotation projects it onto a heavily weighted stiff direction.The effect follows from the large inverse-variance weight assigned to the smallest-variance axis.
G. Spectral Regularization of the Posterior Covariance: Spectral regularization floors small variances relative to the largest, limiting eccentricity and bounding the influence of near-degenerate directions.The uncertain directions remain unchanged while overconfident directions are relaxed.

IV. COMPARISON AND FUSION OF POSTERIOR ESTIMATES

The framework compares posterior estimates statistically, derives a resolution threshold from their uncertainty, and pools mutually consistent estimates while screening outliers.

Consistency testing: Posterior compatibility is tested with a Mahalanobis statistic whose p-value distinguishes significant differences from separation explained by estimation noise.The test follows a chi-squared distribution under the null hypothesis that both estimates target the same parameter vector.
Consistency testing: Statistically significant parameter changes are flagged when estimated drift exceeds the joint uncertainty between independent posteriors.The same statistic can be restricted to parameter subsets for per-parameter consistency checks.
Estimator resolution: Estimator resolution is the smallest parameter change reliably distinguishable from estimation noise at a specified significance level.For equal marginal standard deviation σ, the uncertainty of the difference is √2 σ, linking posterior covariance to a detection threshold.
Posterior fusion: Mutually consistent posteriors can be fused by precision-weighted pooling into a single, more robust posterior.When each window uses a shared prior, the corrected pooling expression subtracts K −1 copies of the prior precision and information vector.
Posterior fusion: Splitting one record into shorter windows supports approximately constant operating conditions and enables consistency-based outlier rejection before pooling.The trade-off is additional computation from independently optimizing each sub-window.

1) Optimizer Comparison:

The simulated benchmark compares a PINN baseline with explicit-model optimizers across noise levels, then evaluates posterior calibration and transfers the framework to hardware validation.

Optimizer Comparison: Three estimators are compared: the PINN-based baseline, the introduced optimizer with GLS loss, and the optimizer with FIR-filtered time-correlated loss.The benchmark evaluates all three methods at noise levels of 1 LSB, 5 LSB, and 10 LSB.
Optimizer Comparison: Replacing the neural-network branch with an explicit RK4 integrator improves estimation accuracy while reducing optimization time from ∼15 m to ∼45 s.The change reduces the optimization problem to pure parameter fitting rather than a coupled network-and-physics problem.
Optimizer Comparison: FIR filtering increases computation time to ∼80 s but further improves estimation accuracy by accounting for noise covariance.The full pipeline retains the efficiency benefit of the explicit-model optimizer.
Posterior calibration: Across N = 30 repeated noisy datasets, FIR whitening restores agreement between the Laplace posterior spread and the empirical spread of MAP estimates.Without FIR filtering, the predicted covariance fails to match empirical variability for several parameters because residuals are correlated.
Hardware validation: The hardware validation uses a laboratory Buck converter to assess estimation accuracy under non-ideal model and noise conditions.The evaluation examines component-reference agreement, prior effects, and sub-window pooling.

A. Experimental Setup and Dataset

The hardware study uses a closed-loop Buck prototype with controlled sensing and excitation, then establishes component references while accounting for operating-condition effects and weakly identified parameters.

Experimental setup: A 130 Hz, 1 V sinusoidal reference perturbation around a 6 V setpoint provides non-intrusive excitation for 30 ms.This avoids physically switching resistor banks and the associated abrupt operating-point change.
Component references: The output filter has a nominal capacitance of 156.6 µF, while frequency-dependent measurements give an effective reference range of C ∈[124.7, 143.5] µF and RC ∈[293, 360] mΩ.The ranges account for harmonic content beyond the switching fundamental.
Component references: Waveform slope estimates bracket the operating inductance as Loff < L < Lon, with the LCR measurement below the off-time lower bound.The on-time estimate overestimates inductance and the off-time estimate underestimates it when parasitic drops are neglected.
Component references: The MOSFET reference range for Rdson is 45 mΩ to 65 mΩ because direct measurement is impractical at the converter’s approximately 1.5 A operating current.The range is based on datasheet behavior below the 20 A test condition.

C. Priors on Parameters

Lognormal priors are used to incorporate component knowledge, while six-window pooling separates parameters that the data identify well from those that remain weakly identified. The pooled estimates preserve well-identified values, improve some weakly identified estimates, and support consistency checks before pooling.

Priors on Parameters: Lognormal priors encode datasheet ratings, operating conditions, tolerances, temperature dependence, and frequency-dependent behavior, with wider uncertainty for difficult parameters such as Rdson, RL, and RC.
Sub-window pooling: The hardware record is partitioned into six consecutive 5 ms sub-windows, each fitted with and without the lognormal priors before posterior pooling.
Well-identified parameters: L, C, RC, and R are well identified: their estimates remain within reference ranges, their medians barely move, and prior-free uncertainty intervals are only a few percent wide.
Weakly identified parameters: RL, Rdson, and VF remain weakly identified and correlated because the model primarily constrains their joint resistive or voltage-drop contribution.
Weakly identified parameters: Pooling improves weakly identified estimates in some cases, recovering VF at 0.98 V inside its 0.9 V to 1.1 V reference band, while priors provide additional external information.
Consistency and pooling: The six sub-windows pass consistency testing with p = 91.1 % without priors and p = 58.2 % with priors, both above the 5 % rejection level.Without regularization, both p-values fall below 0.001%, falsely indicating inconsistency.

E. Change Detection

The framework detects statistically significant hardware changes while localizing them to the affected parameters. It identifies current-dependent inductance and progressive capacitance reduction, and its posterior uncertainty predicts the smallest detectable drift.

Detection framework: The change-detection experiments compare each hardware configuration with a baseline using pairwise posterior consistency tests to identify shifted parameters while checking that others remain stable.
Current-dependent inductance: As DC current increases across loads of 3.3, 2.6, and 1.9 Ω, estimated L decreases monotonically, while L, RL, and R show p-values below 5 % against the 3.3 Ω baseline.
Progressive capacitance reduction: Removing two or four parallel capacitors makes C decrease and RC increase; the four-capacitor case is detected with p ≪1 %, while other parameters remain statistically stable.
Estimator resolution: At α = 5 %, the baseline and two-capacitor posteriors give a resolution threshold of ∆C_res ≈5.1 µF, about 3.7 % of baseline capacitance.
Estimator resolution: The two-capacitor removal shifts C by 5.3 µF and is borderline significant at p ≈6.8 %, whereas four-capacitor removal shifts it by 9.9 µF and is detected at p ≈0.01 %.
Validation scope: Hardware validation shows accurate estimation for well-identified parameters, consistent pooled sub-window posteriors, and correct localization of inductance and capacitor changes.