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Equivalent Flux Compensation for SPMSM Sensorless Control under Parameter Mismatch

Fobao Zhou, Jiaqiao Liang, Zhenxiao Yin, Xueyan Wang, Yang Shen, Zhongyu Shi, Hang Zhao

arXiv:2609.29313v1eess.SYcs.SY

TL;DR

Parameter mismatch in resistance, inductance, and flux can degrade sensorless rotor-position estimation. The paper proposes equivalent flux compensation, deriving mismatch effects from a nonlinear flux observer and updating equivalent flux using geometric error. Experiments report full steady-state compensation for resistance and flux mismatches and partial mitigation of inductance variation.

  • Problem

    Resistance, inductance, and flux mismatches degrade observer accuracy and rotor-position estimation in sensorless PMSM control.

    Source: §I · p. 1

  • Method

    The method maps parameter-mismatch effects into an equivalent flux variable and adaptively updates it using a nonlinear flux observer.

    Source: §I · p. 2, §VIII · p. 9

  • Results

    The method achieves complete steady-state compensation for resistance and flux mismatches and partially mitigates inductance variation.

    Source: §I · p. 2

  • Takeaways & Limitations

    The reported compensation offers robustness to resistance and flux variations while inductance sensitivity remains partly unresolved.

    Source: §I · p. 2, §VIII · p. 9

    The method’s decoupling against inductance variations remains limited, and extension to IPMSMs requires experimental validation.

    Source: §VII · p. 9

Abstract

from arXiv · show

Parameter mismatch is the main source of rotor position estimation error in sensorless control of surface-permanent magnet synchronous motors (SPMSMs). To this end, this paper proposes a simple yet efficient equivalent flux compensation (EFC) method that directly estimates the equivalent flux disturbance caused by parameter mismatches in real time. First, the equivalent flux disturbance caused by parameter mismatches is derived from a nonlinear flux observer. Second, a flux update law is proposed to minimize both magnitude and directional errors by leveraging geometric error together with the derived equivalent flux disturbance. To enhance numerical stability, a saturation function is introduced to improve gradient continuity in the update process. Additionally, Lyapunov analysis is employed to ensure the stability of the proposed update law, from which the corresponding error bounds and convergence properties are derived. Finally, experimental results validate that the proposed method fully compensates for the steady-state effects of resistance and flux mismatches, and partially mitigates the influence of inductance variation, effectively constraining the position estimation error within a relatively small range.

I. INTRODUCTION

Parameter mismatch in resistance, inductance, and flux degrades sensorless rotor-position estimation, while multi-parameter identification and robust observers can entail complexity. The paper proposes equivalent flux compensation (EFC), mapping mismatch effects into the flux domain for adaptive compensation.

II. PRELIMINARIES

  • PMSM sensorless control estimates rotor speed and position from motor signals, avoiding mechanical sensors that add cost and size and can reduce reliability in harsh environments.The observer model uses measured current and voltage in the α–β frame; the back-EMF depends on electrical speed, flux linkage, and rotor position.Source: §I · p. 1, A. Mathematical Model of SPMSM · p. 2

I. INTRODUCTION

  • Parameter-identification methods can estimate several motor quantities online, while robust observers address parameter variation; the cited approaches may add computational complexity, tuning demands, or chattering risk.The paper frames EFC as an alternative that combines identification and observer concepts.Source: §I · pp. 1–2
  • EFC absorbs mismatch-related offset components into an equivalent flux variable, aiming for a simple, decoupled implementation without high-order robust structures or multi-parameter identification.The paper reports complete steady-state compensation for resistance and flux mismatches and partial mitigation of inductance variation.Source: §I · p. 2

B. Nonlinear Flux Observer

  • The nonlinear flux observer defines an auxiliary flux vector whose norm is constrained by the permanent-magnet flux, then uses an observer gain to estimate state and rotor position.Position is obtained from the geometric relationship between the estimated state and flux.Source: B. Nonlinear Flux Observer · p. 2

III. ERROR DYNAMICS WITH PARAMETER MISMATCH

This section formulates the nonlinear flux observer under parameter mismatch and establishes definitions and relationships used to analyze its error dynamics. It distinguishes actual from nominal motor parameters and relates state error to current and flux deviations.

III. ERROR DYNAMICS WITH PARAMETER MISMATCH

A. Nonlinear Flux Observer of Parameter Mismatch

  • The mismatch observer is described using actual and nominal motor parameters, with estimated states and rotor position distinguished from their mismatch-free counterparts.Parameter deviations are defined for resistance, inductance, and permanent-magnet flux linkage.Source: A. Nonlinear Flux Observer of… · p. 3

B. Lemma

  • A Park-frame relationship links the state error projected onto rotor direction to d-axis current deviation and flux mismatch.The lemma defines actual and estimated d-axis currents using the actual and estimated rotor positions.Source: B. Lemma · p. 3

C. Error Dynamics of Parameter Mismatch

Parameter mismatches add an observation-error term whose influence varies with rotor angle and current. The proposed compensation strategy maps these effects into the flux domain and estimates an equivalent flux to counteract them.

C. Error Dynamics of Parameter Mismatch

  • Parameter mismatch introduces an additional observation-error term, and its influence varies with rotor angle and current, compromising angle-estimation accuracy.The section also notes that long-term parameter changes can increase oscillations and reduce angle-estimation accuracy.Source: C. Error Dynamics of Parameter Mismatch · p. 3

D. Compensation Strategy for Parameter Mismatch

  • The strategy transfers mismatch-related angle-estimation degradation to the flux domain and compensates it through an equivalent flux.The equivalent flux is intended to compensate for the added observation-error term.Source: D. Compensation Strategy for Parameter… · p. 3
  • The equivalent flux combines nominal flux with mismatch compensation, and its practical estimate is related to η(x) through a scalar projection.The text states that positive estimated flux ensures the scalar projection is positive and bounded.Source: D. Compensation Strategy for Parameter… · pp. 3–4
  • Under a ±50% resistance step at 1000 rpm and full load, the residual ∥η(x) −η(ˆx∗)∥ remains relatively small compared with fundamental harmonics, supporting the approximation used for compensation.Fig. 1 presents the dynamic response of this residual under the stated operating condition.Source: Figure 1 · p. 4, D. Compensation Strategy for Parameter… · p. 4

IV. EQUIVALENT FLUX UPDATE LAW

The equivalent-flux update law uses a geometric loss that accounts for both flux-magnitude and direction errors. A nonlinear saturation form is introduced to improve update behavior, and the paper analyzes stability and convergence using Lyapunov methods.

IV. EQUIVALENT FLUX UPDATE LAW

  • The loss function is constructed from geometric error to account for both flux-magnitude and direction errors, and the resulting update law regulates both quantities.The linear update form is described as having linear asymptotic convergence.Source: §IV · p. 4
  • A saturation-based nonlinear update is introduced to improve stability and convergence; its tanh function is near-linear for small errors and limits the update rate for large errors.The text says this form can prevent oscillations and overshoot during startup or transients, while gradient continuity improves numerical stability and avoids saturation dead zones.Source: §IV · p. 4

V. STABILITY ANALYSIS

  • The paper uses direct Lyapunov analysis to examine observer stability under mismatch and the stability and convergence of the equivalent-flux update law.The update gains are described as accelerating equivalent-flux convergence.Source: §V · p. 4, §IV · p. 4

A. Stability Analysis Under Nominal Parameter Conditions

Under nominal parameters, the analysis establishes boundedness and convergence of the observer state; under parameter mismatch, stability depends on mismatch conditions, and excessive deviations can increase errors or cause divergence.

A. Stability Analysis Under Nominal Parameter Conditions

B. Stability Analysis Under Parameter Mismatch

  • Under parameter mismatch, observer stability is parameter-governed; excessive deviations in resistance, inductance, or flux can violate Lyapunov criteria, increasing steady-state error or causing divergence.Source: B. Stability Analysis Under Parameter… · p. 5
  • Mismatch is not always detrimental: under certain operating conditions, changing nominal resistance or inductance can reduce convergence error, depending on load and current conditions.Source: B. Stability Analysis Under Parameter… · p. 5

C. Stability Analysis of the Equivalent Flux Update Law

The equivalent flux update law is shown to have bounded convergence under parameter mismatch. Experiments report full steady-state compensation for resistance and flux mismatch, partial mitigation of inductance sensitivity, and low position error across load transitions.

C. Stability Analysis of the Equivalent Flux Update Law

B. Performance Under Variable Load

A. Experimental Setup

  • Experiments used a two-PMSM/dSPACE DS1202 platform with a 2 kHz speed loop and 10 kHz current loop and PWM, comparing three controllers under ±50% step and sinusoidal mismatches.Source: §VI · p. 6, A. Experimental Setup · p. 6

C. Performance Under Step Parameter Mismatch

D. Performance Under Sinusoidal Parameter Mismatch

Experiments show EFC compensates for sinusoidal resistance and flux mismatches, while inductance variation remains a source of error; tests also examine low-speed operation, noise, initialization, and observer gains.

D. Performance Under Sinusoidal Parameter Mismatch

  • Under sinusoidal resistance variation, EFC holds angle error near −0.13 rad; under inductance variation, its error remains periodic and correlated with the mismatch.Methods 1–3 show periodic errors synchronized with resistance variation; under inductance variation, Method 2 shows an inverse trend.Source: D. Performance Under Sinusoidal… · p. 7, Figure 8 · p. 7, Figure 9 · p. 7
  • Under sinusoidal flux variation, EFC converges to about −0.13 rad from either tested initial flux, although the higher initial flux causes larger transient fluctuations.Methods 1–3 produce periodic errors with the flux variation.Source: D. Performance Under Sinusoidal… · pp. 7–8, Figure 10 · p. 8

E. Performance Under Low-Speed and Noise Immunity

  • At 200 rpm, mismatch error trends remain consistent with those at 1000 rpm, but all methods have larger estimation errors as back-EMF is reduced.With Gaussian noise N(0, 0.5) injected into α-β currents, all methods retain bounded errors and stable steady-state bias despite high-frequency fluctuations.Source: E. Performance Under Low-Speed and… · p. 8, Figure 11 · p. 8, Figure 12 · p. 8

F. Analysis of Initial Sensitivity and Convergence Dynamics

  • With resistance variation increased to 0.8Rz, error remains around −0.13 rad; larger inductance variation slightly increases error, while initial flux values from 0.2ψz to 1.8ψz converge to the same steady-state value.Larger observer gains accelerate convergence, but excessively large gains introduce transient oscillations.Source: Figure 13 · p. 8, F. Analysis of Initial Sensitivity and… · p. 9
  • During acceleration from 200 to 1000 rpm, estimation error remains bounded, and different observer gains yield nearly identical steady-state performance after full-load application.Table III reports slightly higher current THD but shorter DSP computation time for the proposed method.Source: Figure 14 · p. 8, F. Analysis of Initial Sensitivity and… · p. 9, Table III · p. 9

VIII. CONCLUSION

  • Against three methods, EFC restores estimation error to its original baseline under severe resistance and flux mismatches, but its decoupling against inductance variation remains limited.The discussion notes that extending the method to IPMSMs requires an inductance matrix and experimental validation.Source: §VIII · p. 9, §VII · p. 9

VII. DISCUSSION

  • The conclusion characterizes EFC as robust across diverse operating conditions, while the discussion identifies limited decoupling from inductance variation as a remaining boundary.The discussion also says IPMSM extension requires an inductance matrix and experimental validation.Source: §VIII · p. 9, §VII · p. 9

APPENDIX A PROOF OF LEMMA 1

The appendix proves a scalar inequality, derives error-related quantities, and uses Lyapunov analysis to characterize convergence of the observer error.

APPENDIX A PROOF OF LEMMA 1

  • The Lemma 1 proof reduces the claim to tanh(a) ≥ a/(1+a) for a ≥ 0 and applies the fundamental inequality e^x ≥ 1 + x.Equality holds only at a = 0.Source: Appendix A · p. 9

APPENDIX B PROOF OF LEMMA 2

  • The Lemma 2 proof expresses Q1 and Q2 using state-estimation errors, currents, inductance mismatch, and flux terms.The displayed expressions relate Q1 to projected state error and Q2 to current-weighted error terms.Source: Appendix B · p. 9

APPENDIX C PROOF OF THEOREM 1

  • The Lyapunov analysis shows the squared state-error norm converges to the larger root; zero convergence is stated for initial states satisfying V1(0) < ψmQ1.The latter initial-state condition is omitted from the subsequent analysis because observers are typically initialized with large values.Source: Appendix C · p. 10

APPENDIX D PROOF OF THEOREM 2

The stability analysis considers mismatch effects separately, then derives flux-mismatch steady-state error boundaries that grow with the mismatch from nominal flux.

APPENDIX D PROOF OF THEOREM 2

  • When the discriminant is nonpositive, the Lyapunov derivative is nonpositive, ensuring stability and convergence of the state error norm to zero.The subsequent mismatch analysis considers the positive-discriminant case.Source: Appendix D · p. 10

A. Effect of Flux Mismatch

  • Under flux mismatch, the steady-state error norm converges to a mismatch-dependent boundary, and larger deviation of ψz from ψm increases the estimation-error boundary.The analysis treats flux mismatch alone by setting resistance and inductance mismatches to zero; Fig. 15 depicts the boundaries for two flux ranges.Source: A. Effect of Flux Mismatch · p. 10, Figure 15 · p. 10

B. Effect of Inductance Mismatch

The analysis derives steady-state error boundaries for inductance and resistance mismatch, showing that their effects depend on mismatch direction and magnitude.

B. Effect of Inductance Mismatch

  • The inductance-mismatch error boundary is parabolic with a minimum at its symmetry axis; a properly sized negative mismatch reduces error when id > î*d.A positive mismatch is beneficial when id < î*d, while excessive deviations in either direction can degrade stability and lead to observer divergence.Source: B. Effect of Inductance Mismatch · p. 10

C. Effect of Resistance Mismatch

  • For resistance mismatch, a negative mismatch reduces the convergence radius when id < î*d but amplifies error when id > î*d; excessive deviations can increase error or remove a real solution.The resistance mismatch vertically shifts the quadratic used to obtain the modified roots, and the analysis sets flux and inductance mismatch to zero.Source: C. Effect of Resistance Mismatch · pp. 10–11

APPENDIX E PROOF OF THEOREM 3

The proof bounds the mismatched-system auxiliary error using a Lyapunov derivative inequality and a scalar comparison function. It establishes a unique stable equilibrium that bounds the error’s eventual behavior.

APPENDIX E PROOF OF THEOREM 3

  • The Lyapunov derivative is bounded using the disturbance-rate limit Ψ and a lower bound on the saturation term, yielding a scalar comparison inequality.The proof sets γ = |ψ̃z| to express the bound in terms of the error magnitude.Source: Appendix E · p. 11
  • The comparison function f(γ) is Lipschitz continuous and strictly decreasing for γ ≥ 0, so setting f(γ) = 0 yields a unique positive equilibrium γ*.Source: Appendix E · p. 11
  • The comparison theorem makes γ* stable, so all solutions converge to the region it bounds, establishing transient and ultimate steady-state error bounds.Source: Appendix E · p. 11
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