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Parameter-Robust Sensorless Control of IPMSM Drives With Adaptive Flux Observer

Fobao Zhou, Zhenxiao Yin, Xueyan Wang, Yang Shen, Yuanfeng Qu, Hang Zhao

arXiv:2609.39840v1eess.SYcs.SY

TL;DR

Parameter variation makes IPMSM sensorless control sensitive to errors in the motor model, with saliency causing d- and q-axis inductance mismatches to affect flux direction differently. The paper combines adaptive flux compensation with online Lq identification. Experiments report restoration of estimation error to its nominal level under several parameter mismatches.

  • Problem

    IPMSM sensorless observers are sensitive to varying motor parameters, and saliency makes d- and q-axis inductance mismatches affect the position-dependent flux vector differently.

    Source: §I · p. 1

  • Method

    The method decomposes parameter mismatch into normal and tangential equivalent-flux components, adapts flux magnitude for normal variation, and identifies Lq to address tangential sensitivity.

    Source: §I · p. 2, A. Unified Equivalent Flux Decomposition · p. 3

  • Results

    Experiments show the proposed controller restores estimation error to its nominal level under resistance, d-axis inductance, q-axis inductance, and flux mismatches.

    Source: §VII · p. 9

  • Takeaways & Limitations

    The approach avoids requiring precise offline resistance, d-axis inductance, or flux values, while supplying q-axis inductance through online identification.

    Source: §I · p. 2, §VII · p. 9

    The identified Lq value is updated only at no or light load and frozen under load to avoid saturation-induced inductance drift entering the observer.

    Source: B. Synchronous Demodulation and… · p. 5

Abstract

from arXiv · show

To address parameter sensitivity commonly in interior permanent magnet synchronous motor (IPMSM) sensorless control, this paper proposes a parameter-robust control framework by extending the adaptive flux observer from surface-mounted PMSM to salient-pole machines. First, parameter mismatches are interpreted as an equivalent flux vector with $d$-axis and $q$-axis components. Permanent magnet flux, $d$-axis inductance, and resistance mismatches are mapped to the $d$-axis flux component, whereas $q$-axis inductance mismatch is mapped to the $q$-axis component. Then, a scalar adaptive flux update is designed to compensate the $d$-axis flux variation by tracking the equivalent flux magnitude. Since the $q$-axis component rotates the observed flux direction and cannot be eliminated by scalar adaptation, a high-frequency $q$-axis voltage injection method is introduced to identify the $q$-axis inductance. Meanwhile, stability analysis proves bounded equivalent flux magnitude tracking and identifies the residual tangential flux responsible for position error. Finally, experimental results show that the proposed method does not require precise resistance, $d$-axis inductance, or flux, while the $q$-axis inductance is provided by online identification.

I. INTRODUCTION

IPMSM sensorless observers are sensitive to changing motor parameters, and saliency makes d- and q-axis inductance errors affect the flux vector differently. The paper extends adaptive flux compensation to IPMSMs and uses a nonlinear flux observer whose position estimate depends on the position-dependent inductance model.

I. INTRODUCTION

  • Motor-parameter variation distorts estimated back-EMF or flux and produces position error; IPMSM saliency makes d- and q-axis inductance mismatches affect the flux vector in different directions.Simultaneously identifying multiple parameters is constrained by identifiability and may require additional estimators and excitation conditions.Source: §I · p. 1
  • Earlier robust IPMSM approaches mainly used SMO and ESO methods, which can require tuning several gains or bandwidths and often lump parameter errors as disturbances.SMO methods also face tradeoffs among chattering suppression, phase delay, and gain tuning.Source: §I · pp. 1–2
  • The proposed method separates mismatch effects by flux direction, adapting the normal component and identifying Lq to address tangential sensitivity.The introduction reports experiments across load transitions, parameter mismatches, and a wide speed range.Source: §I · p. 2

II. IPMSM MODEL AND NONLINEAR FLUX OBSERVER

  • The IPMSM flux model uses the rotor d- and q-axis inductances and permanent-magnet flux linkage; the inductance matrix varies with rotor position.The model defines L0 as average inductance and L1 as the saliency-dependent inductance amplitude.Source: §II · p. 2
  • The observer models stator flux in the stationary αβ frame and uses the position-dependent inductance matrix L(θ), coupling its estimate to both Ld and Lq.Rotor position is obtained from the direction of an auxiliary flux vector.Source: §II · p. 2

A. Unified Equivalent Flux Decomposition

The mismatch decomposition recasts parameter errors as an equivalent flux vector. Its normal and tangential projections distinguish errors that change flux magnitude from those that rotate its direction.

A. Unified Equivalent Flux Decomposition

  • The resistance bias is treated as an equivalent flux bias, not as a new physical motor flux.The mismatches are defined as real parameter values minus their nominal controller values.Source: A. Unified Equivalent Flux Decomposition · p. 3
  • The decomposition represents parameter errors as an equivalent flux vector and expresses it through normal and tangential rotor-axis projections.The equivalent-flux magnitude is denoted Ψ.Source: A. Unified Equivalent Flux Decomposition · p. 3
  • Flux, d-axis inductance, and resistance mismatches affect the normal projection, while q-axis inductance mismatch and an id-dependent resistance term affect the tangential projection.The resistance-induced tangential term disappears under id = 0 control.Source: A. Unified Equivalent Flux Decomposition · p. 3

B. Adaptive Flux Observer Design

The adaptive observer uses an online-computable auxiliary flux to track the unknown equivalent flux caused by parameter mismatch. A scalar magnitude update adjusts the observer state to absorb mismatch effects on the flux side.

B. Adaptive Flux Observer Design

  • The adaptive update changes the equivalent flux magnitude Ψ using the computable magnitude of ηpm, with a nonlinear form that bounds update rate during large transients.The same gain is used inside and outside tanh to reduce tuning parameters.Source: B. Adaptive Flux Observer Design · p. 3
  • The computable auxiliary flux ηpm is designed to track the theoretical equivalent flux ηeq, which cannot be evaluated online because the real parameters and mismatches are unknown.The equivalent flux is used to analyze the mismatch mechanism rather than as an observer input.Source: B. Adaptive Flux Observer Design · p. 3
  • Adjusting the parameter-robust observer state absorbs flux, resistance, and inductance mismatch effects on the flux side.The observer defines ηpm as an online-computable auxiliary flux associated with its state.Source: B. Adaptive Flux Observer Design · pp. 3–4

C. Compensation Mechanism for Parameter Mismatches

The compensation mechanisms depend on whether a mismatch changes equivalent-flux radius or direction. Scalar adaptation handles radial variations, while tangential errors remain outside its independent control.

C. Compensation Mechanism for Parameter Mismatches

  • Scalar adaptation compensates equivalent-flux radius variations but cannot independently cancel a tangential component.This distinction separates the compensation mechanisms for the parameter mismatches.Source: C. Compensation Mechanism for… · p. 4

1) Flux mismatch:

  • A permanent-magnet flux mismatch changes equivalent-flux radius without a steady angular bias, so the adaptive magnitude follows the resulting flux magnitude.With only flux mismatch, the tangential component is zero.Source: 1) Flux mismatch · p. 4

2) Resistance mismatch:

  • Resistance mismatch changes the adaptive radius through its normal component; its tangential residual depends on d-axis current and decreases with electrical speed.Resistance mismatch is therefore largely compensated in the medium- and high-speed region; the dominant tangential error source remains the Lq term.Source: 2) Resistance mismatch · p. 4
  • A d-axis inductance mismatch is collinear with the d-axis and appears as a radius variation that the adaptive law can compensate.This mismatch produces no tangential component in the stated case.Source: 2) Resistance mismatch · p. 4
  • A q-axis inductance mismatch tilts the observed flux away from the rotor d-axis, and its position error grows with load current.The scalar update cannot remove the orthogonal tangential component, so Lq mismatch is not compensated by flux adaptation alone.Source: 2) Resistance mismatch · p. 4

IV. INDUCTANCE IDENTIFICATION STRATEGY

The method uses q-axis high-frequency voltage injection to identify Lq, the parameter to which the adaptive flux observer remains sensitive. Its small-signal model treats fundamental operating variables as slow during each injection period.

IV. INDUCTANCE IDENTIFICATION STRATEGY

  • High-frequency q-axis voltage injection identifies Lq, the parameter mismatch that remains sensitive after adaptive flux compensation.The observer does not require precise ψf, Ld, or Rs.Source: §IV · p. 4

A. High-Frequency Small Signal Model

  • When injection frequency exceeds the fundamental electrical frequency and current-controller bandwidth, current, speed, and saturation are treated as slow within one injection period.The local model relates high-frequency q-axis voltage and current.Source: A. High-Frequency Small Signal Model · p. 4
  • Synchronous demodulation separates the injected voltage and current into sine and cosine coefficients for estimating q-axis inductance.The coefficients are matched after substitution into the local small-signal model.Source: A. High-Frequency Small Signal Model · p. 4
  • An antisymmetric combination of the demodulated coefficients cancels effective resistance before the raw Lq estimate is formed.The passages identify resistance cancellation but do not provide the estimate's full expression.Source: A. High-Frequency Small Signal Model · p. 4

B. Synchronous Demodulation and Filtering

The identification loop demodulates high-frequency voltage and current, filters the resulting Lq estimate, and gates its use to no-load or light-load conditions. The stability analysis assumes bounded online flux magnitude and rate.

B. Synchronous Demodulation and Filtering

  • The demodulation loop computes sampled sine and cosine coefficients over windows containing an integer number of injection periods.The raw estimate is processed by a three-point median filter and a low-pass filter before reaching the controller once per window.Source: B. Synchronous Demodulation and… · p. 4
  • Only Lq identified near no load or light load is used for observer calibration, because load-dependent saturation can lower incremental inductance and increase tangential flux error if directly applied.A current threshold implements this gate.Source: B. Synchronous Demodulation and… · pp. 4–5
  • The current gate freezes the observer input under load; if gated identification reduces the difference from real Lq relative to a fixed nominal value, tangential flux and position-error bounds are reduced.The overall framework combines the adaptive flux observer with Lq identification.Source: B. Synchronous Demodulation and… · p. 5, Figure 1 · p. 5

V. STABILITY ANALYSIS

  • The stability analysis assumes the online auxiliary-flux magnitude stays positive and bounded and that its rate of change is bounded.The adaptive law uses the online flux magnitude as the equivalent-flux magnitude to track.Source: §V · p. 5

A. Equivalent Flux Magnitude Tracking

The adaptive update tracks the magnitude of the online auxiliary flux, with a uniformly bounded tracking error that enters a compact set. This scalar result does not eliminate the tangential residual, which remains tied to Lq mismatch.

A. Equivalent Flux Magnitude Tracking

  • Theorem 1 establishes that the adaptive flux magnitude is uniformly bounded and its tracking error converges to a compact set.The update law makes Ψ track the online auxiliary-flux magnitude; at a constant operating point, the tracking error converges to zero.Source: A. Equivalent Flux Magnitude Tracking · p. 5

B. Normal and Tangential Tracking Error Bounds

  • Theorem 2 bounds both d-axis and tangential residuals within compact sets under the conditions of Theorem 1.The residuals are defined by projecting the reconstructed equivalent-flux vector onto the rotor-axis directions.Source: B. Normal and Tangential Tracking… · p. 6
  • Scalar magnitude tracking does not eliminate tangential flux: as tracking error tends to zero, the d-axis residual vanishes while the tangential residual tends to ψpm,q.The tangential channel is dominated by Lq mismatch, so Lq identification is needed to reduce the remaining error.Source: B. Normal and Tangential Tracking… · pp. 5–6

VI. EXPERIMENTAL RESULTS AND ANALYSIS

The experiments use a coupled-motor IPMSM platform to test the adaptive flux observer and Lq identification loop. Parameter-mismatch comparisons are conducted at full load.

VI. EXPERIMENTAL RESULTS AND ANALYSIS

  • The experimental platform couples two PMSMs and includes a dSPACE DS1202 controller, power driver, and host computer.It is used to test the adaptive flux observer and Lq identification loop under controlled load conditions.Source: §VI · p. 6
  • The speed loop runs at 2 kHz, while the current loop and PWM switching frequency are both 10 kHz.Source: §VI · p. 6
  • All parameter-mismatch comparison tests are conducted under full-load operation, comparing the proposed controller with three sensorless controllers.The proposed controller has two independent tuning gains, Γpm and kψ, selected after practical tuning iterations.Source: §VI · p. 6

A. Performance of Inductance Identification

The experiments test inductance identification and compare sensorless estimation under load transitions and parameter mismatches. The proposed method compensates resistance, d-axis inductance, and flux variations, while q-axis inductance mismatch remains sensitive to accurate identification.

A. Performance of Inductance Identification

  • High-frequency q-axis voltage injection identifies Lq accurately at no or light load; the identified value is frozen when larger load is applied.The identified inductance fluctuates around the real Lq value under no-load operation.Source: A. Performance of Inductance… · pp. 6–7

B. Performance of Load Transition

  • During load transitions, the proposed method limits transient position-estimation error to approximately 0.2 rad and quickly returns to the steady-state range.The comparison is at 500 rpm as load changes from no load to half load and then full load.Source: B. Performance of Load Transition · p. 7, Figure 4 · p. 7

C. Performance of Rs and Ld Mismatch

D. Performance of Lq and ψf Mismatch

E. Performance of Large Range Parameter Mismatch

Tests show stable estimation under large and time-varying Rs and Ld mismatches, convergence from large initial flux errors, correction after identified Lq is applied, and tracking during acceleration. The conclusion attributes robustness to adaptive flux compensation paired with q-axis inductance identification.

E. Performance of Large Range Parameter Mismatch

  • With Rs or Ld suddenly changed to 0.2 or 1.8 times rated values, estimation error shows only limited transient deviation and returns rapidly to a stable range; sinusoidal 80% Rs and Ld variations are also handled.Sinusoidal Rs variation leaves a small slow error fluctuation because some resistance error projects onto tangential flux when id ≠ 0.Source: E. Performance of Large Range… · p. 8, Figure 10 · p. 8, Figure 11 · p. 8

F. Performance of Flux and Identified Inductance Update

  • From initial flux values of 0.5ψf and 1.5ψf, adaptive flux converges to approximately 0.124 Wb; switching from 0.8Lq or 1.2Lq to identified Lq returns error to approximately −0.1 rad.The flux mismatch cases return to a similar steady-state error range, and identified Lq rapidly reduces the original estimation bias.Source: F. Performance of Flux and Identified… · p. 8, Figure 12 · p. 8, Figure 13 · p. 9

G. Performance of Acceleration

VII. CONCLUSION

  • The method combines adaptive flux compensation for flux, Ld, and dominant resistance errors with Lq identification to reduce tangential sensitivity.The analysis proves bounded equivalent-flux magnitude tracking and clarifies the remaining tangential residual.Source: §VII · p. 9
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