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Temporal Regression-Based Model-Free Sensorless Control of Permanent Magnet Synchronous Motor

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

arXiv:2609.29290v1eess.SYcs.SY

TL;DR

Sensorless SPMSM position estimation is sensitive to motor-parameter variation, which can degrade estimation accuracy. TFC projects a finite-window regression to remove inductance and uses a common-amplitude cancellation to estimate position without specifying resistance, inductance, or flux linkage.

  • Problem

    Motor-parameter variation can cause back-EMF or flux estimation errors that degrade rotor-position accuracy in sensorless control.

    Source: §I · p. 1

  • Method

    TFC projects a finite-window voltage-and-current regression to eliminate inductance and uses current-flux geometry to cancel a common amplitude during position extraction.

    Source: §I · p. 2

  • Results

    The position estimate can be computed without specifying stator resistance, inductance, or flux linkage.

    Source: D. Resistance and Flux Independence · p. 5

  • Takeaways & Limitations

    The method provides a sensorless position-estimation approach that does not require online identification of the motor parameters.

    Source: §I · p. 2

Abstract

from arXiv · show

To address the widespread sensitivity of surface-mounted permanent magnet synchronous motor (SPMSM) sensorless control to motor parameters, this paper proposes a temporal regression-based model-free sensorless control (TFC) method. First, voltage integrals and current increments over consecutive short intervals are stacked to construct a finite window regression, in which the unknown stator inductance appears as a common scalar coefficient. Second, a projection operator constructed from the stacked current increments eliminates the inductance term, and a least-squares formulation is developed to reconstruct the rotor flux vector. Meanwhile, the analysis of the projected regression and current-flux geometry establishes a two-dimensional direction vector whose components share a common amplitude containing the stator resistance and flux linkage. This amplitude cancels during position extraction. By setting the resistance reference to zero, the proposed method estimates the position without specifying the stator resistance, inductance, or flux linkage. Finally, experimental results verify the effectiveness of the proposed TFC method.

I. INTRODUCTION

Sensorless position estimation is motivated by the cost, size, and reliability drawbacks of mechanical sensors and by parameter-dependent estimation errors. The paper proposes TFC to estimate position from voltage and current data without motor-parameter inputs; the SPMSM model is expressed in the stationary αβ frame.

I. INTRODUCTION

  • Mechanical position sensors add cost and size and can compromise reliability in harsh environments, motivating sensorless control.At medium and high speeds, common sensorless approaches estimate back-EMF or flux from the fundamental-frequency model and rely on motor parameters.Source: §I · p. 1
  • Motor-parameter variation can cause back-EMF or flux estimation errors that degrade rotor-position accuracy.The cited sources identify temperature and magnetic saturation as sources of parameter variation.Source: §I · p. 1
  • Existing robust observers retain motor models and add compensation or observer gains, while online identification brings computational, tuning, and convergence considerations.The introduction presents reducing parameter dependence without complicated tuning as the motivation for the proposed approach.Source: §I · pp. 1–2
  • TFC uses temporal voltage and current data to estimate rotor position without requiring resistance, inductance, or flux linkage as inputs or identifying them online.It eliminates a shared inductance term by temporal projection and cancels a common resistance- and flux-dependent amplitude during angle extraction.Source: §I · p. 2

A. SPMSM Model

  • The SPMSM is modeled as a two-dimensional vector system in the stationary αβ frame, with flux λ = L_s i + ψ_f c(θ).Here c(θ) is the unit vector along the rotor d-axis.Source: A. SPMSM Model · p. 2

B. Current-Flux Orthogonality

The method uses current-flux geometry and a finite window of relative rotor rotations to recover position. Its short-window angle approximation assumes slowly varying speed, and the direction signal weakens as speed decreases for fixed window settings.

B. Current-Flux Orthogonality

  • With id = 0, the stator current is orthogonal to the flux vector, providing direction information for rotor-position estimation.The current is expressed using the rotor d- and q-axis directions.Source: B. Current-Flux Orthogonality · p. 2

III. TEMPORAL REGRESSION-BASED MODEL-FREE SENSORLESS CONTROL METHOD

  • TFC stacks voltage and current measurements over consecutive short intervals, removes the common inductance term by projection, and recovers rotor position from projected data.The finite-window representation supplies the temporal data used by the method.Source: §III · p. 2

A. Finite Window Representation

  • The finite window uses N ≥ 2 consecutive intervals and represents each boundary flux relative to the flux at the current endpoint.The endpoint flux is the vector to be reconstructed.Source: A. Finite Window Representation · p. 2
  • For constant flux linkage, the relative-angle relation is exact even with varying speed, while its short-window approximation assumes slowly varying speed and uses estimated electrical speed in implementation.The approximation is φ_j ≃ ω̂(t_j − t_N).Source: A. Finite Window Representation · pp. 2–3

B. Short Interval Integral Regression

The short-interval regression is built from voltage and current integrals and current increments. Integrating over intervals avoids current differentiation and leaves the unknown flux linear while inductance multiplies a measured current increment.

B. Short Interval Integral Regression

  • For each interval, the method defines voltage and current integrals and a current increment from voltage and current samples.With the resistance reference set to zero, the voltage-integral term is simply the voltage integral; the current integral is needed only for resistance analysis.Source: B. Short Interval Integral Regression · p. 3
  • Integrating the motor equation over each interval produces a regression in which the endpoint flux enters linearly and inductance multiplies a measured current increment.The interval flux increment is related to the endpoint flux through the interval rotation matrix.Source: B. Short Interval Integral Regression · p. 3
  • Finite-interval integration avoids current differentiation and requires no initial stator-flux value.The regression is constructed from the interval voltage and current quantities.Source: B. Short Interval Integral Regression · p. 3

C. Temporal Projection and Inductance Elimination

Stacking interval data makes the unknown inductance a shared scalar multiplying the measured current-increment vector. Projection orthogonal to that vector removes inductance, after which least squares estimates flux for position recovery.

C. Temporal Projection and Inductance Elimination

  • The stacked regression collects interval voltage integrals, current increments, and current integrals, with the same unknown inductance multiplying the current-increment vector across intervals.The projected regression eliminates that common inductance term.Source: C. Temporal Projection and Inductance… · p. 3
  • Projecting orthogonally to the stacked current-increment vector eliminates the inductance term and yields an inductance-free regression.The projection is defined for a window with DᵀD > 0.Source: C. Temporal Projection and Inductance… · p. 3
  • The projected regression is solved by least squares to estimate the endpoint flux, whose direction then gives rotor position.The estimated flux is given by the pseudoinverse solution.Source: C. Temporal Projection and Inductance… · p. 3

D. Flux Direction Reconstruction and Angle Extraction

The projected data vector provides a direction for recovering rotor position at the window endpoint. Under the analyzed conditions, its common amplitude cancels during angle extraction.

D. Flux Direction Reconstruction and Angle Extraction

  • Under equally spaced intervals, locally constant speed, zero d-axis current, constant q-axis current, and nonzero rotational excitation, Y shares the positive amplitude χψf across its components.The rotor position is recovered from the direction of Y; the common flux-linkage amplitude cancels.Source: D. Flux Direction Reconstruction and… · pp. 3–4
  • With resistance mismatch, position extraction remains independent of flux linkage when the corresponding positive-amplitude condition holds.The algorithmic resistance reference is optional; if nonzero, it must preserve κR > 0.Source: D. Flux Direction Reconstruction and… · p. 4
  • The implementation advances a finite window, forms relative rotations, and evaluates the direction vector and angle estimate without explicitly storing the 2N × 2N projection matrix.Fig. 1 presents the method’s block diagram.Source: D. Flux Direction Reconstruction and… · p. 4, Figure 1 · p. 4

IV. THEORETICAL ANALYSIS AND PARAMETER INDEPENDENCE

The theoretical analysis explains how temporal projection preserves rotor-flux direction and avoids requiring resistance and flux-linkage information. The temporal rotation matrix has isotropic geometry, while its direction signal weakens at lower speeds.

IV. THEORETICAL ANALYSIS AND PARAMETER INDEPENDENCE

  • The analysis establishes how projection preserves rotor-flux direction and avoids requiring resistance and flux-linkage information.The algorithm updates a window of N intervals, forms the stacked regression, computes Y, and recovers endpoint position.Source: §IV · p. 4, A. Properties of the Temporal Rotation… · p. 4

A. Properties of the Temporal Rotation Matrix

  • The isotropic Gram matrix has orthogonal columns with equal squared norm χ, enabling temporal projection to be related to a spatial direction.When χ > 0, the window contains nonzero flux-rotation information; for fixed N and ∆t, this signal weakens as speed decreases.Source: A. Properties of the Temporal Rotation… · p. 4
  • The direction-recovery signal requires χ > 0; at fixed window size and interval duration, it weakens as speed decreases.The analysis considers equally spaced intervals and constant electrical speed for the temporal rotation matrix.Source: A. Properties of the Temporal Rotation… · p. 4

B. Geometry of Current Increments

Under the stated current conditions, the endpoint current has a defined relation to the rotor axes. Adjacent differences establish a geometric relation between temporally extended current data and endpoint flux direction.

B. Geometry of Current Increments

  • With id = 0 and constant nonzero iq throughout the window, the endpoint current vector q is parallel to the q-axis, with orientation set by the sign of iq.q is the current vector expressed at the window endpoint.Source: B. Geometry of Current Increments · p. 4
  • The endpoint current vector q is orthogonal to the endpoint flux vector xN.This is the stated relation qTxN = 0.Source: B. Geometry of Current Increments · p. 4
  • Taking adjacent differences establishes a geometric relation between the temporally extended current data D and the flux direction at the window endpoint.The relation follows by combining the adjacent-difference result with the preceding geometric expression.Source: B. Geometry of Current Increments · pp. 4–5

C. Preservation of the Rotor Flux Direction

The projected Gram matrix retains the rotor-flux direction in a one-dimensional subspace. Under the stated geometry, the least-squares solution recovers the physical flux and its orientation.

C. Preservation of the Rotor Flux Direction

  • The projected Gram matrix is rank one, with eigenvalue χ along xN and zero along JxN, so the physical flux lies in its retained subspace.Although it has no conventional inverse, the matrix has a pseudoinverse.Source: C. Preservation of the Rotor Flux… · p. 5
  • For matched resistance, Y = χxN and G†Y = xN, so the minimum-norm least-squares solution recovers the physical flux under the stated geometry.Because χ > 0, Y also establishes the correct orientation.Source: C. Preservation of the Rotor Flux… · p. 5
  • Preserving the flux subspace is sufficient for reconstruction; the projected regression need not have full column rank.The physical flux lies in the subspace retained by the projected Gram matrix.Source: C. Preservation of the Rotor Flux… · p. 5

D. Resistance and Flux Independence

The method removes inductance by temporal projection and extracts rotor position from a direction vector whose common resistance- and flux-dependent amplitude cancels.

D. Resistance and Flux Independence

  • Resistance and flux linkage enter the direction vector through a shared amplitude that cancels in the ratio Yβ/Yα when χ, ψf, and κR are positive.The resistance reference is optional, but a nonzero choice should preserve κR > 0.Source: D. Resistance and Flux Independence · p. 5
  • Setting the resistance reference to zero yields position estimates from voltage, current, and relative-rotation data without specifying Rs, Ls, or ψf.The actual resistance and flux linkage affect signal amplitude, not the angle computation.Source: D. Resistance and Flux Independence · p. 5
  • Temporal projection removes the stator-inductance contribution because it lies in the subspace spanned by measured current increments.The analysis considers a resistance mismatch over a window with id = 0 and constant, nonzero iq and ω.Source: D. Resistance and Flux Independence · p. 5

V. EXPERIMENTAL RESULTS AND ANALYSIS

Experiments compare TFC with two benchmarks during load transitions and parameter variations, reporting smaller load-transition position errors and robustness to the tested parameter changes.

B. Performance under Load Transition

  • After a sudden rated-load step at 1000 rpm, TFC settles near −0.15 rad, versus Method 1 at −0.23 rad and Method 2 at 0.21 rad.Before loading, the respective position errors are about −0.02, −0.19, and 0.25 rad; the proposed method also shows smoother estimation.Source: Figure 3 · p. 6, B. Performance under Load Transition · p. 6

C. Performance under Parameter Variations

  • At 1000 rpm, Method 1 is sensitive to resistance and inductance changes, whereas Method 2 shows only small position-error shifts.For Method 1, 0.5Rs changes error from about −0.22 to −0.65 rad, and 1.5Ls gives about −0.60 rad; Table II summarizes RMSE under parameter mismatch.Source: Figure 4 · p. 6, Figure 5 · p. 6, Table II · p. 6

A. Experimental Setup

  • The experimental setup uses two coupled PMSMs, with speed control at 2 kHz and current control at 10 kHz; Method 1 and an SMO-based Method 2 are benchmarks.Table I summarizes motor and control parameters, including the TFC window size N and interval duration Δt.Source: A. Experimental Setup · pp. 5–6, Table I · p. 6

C. Performance under Parameter Variations

  • Across resistance-reference settings from zero to 2Rs, TFC’s position error stays around −0.15 rad, with only slight transient fluctuations.The resistance reference is varied through 0, 0.5Rs, Rs, 1.5Rs, and 2Rs; the experiments also include 200-rpm variation tests for Methods 1 and 2.Source: Figure 6 · p. 7, C. Performance under Parameter… · p. 7, Figure 7 · p. 7

D. Performance under Low-Speed Operation

Across low-speed operation, regression-parameter variations, and speed transients, experiments report robust position estimation with bounded error. The paper also gives operating-condition guidance for choosing window size and interval duration.

D. Performance under Low-Speed Operation

  • At 200 rpm, Method 1 has errors of approximately −0.52 rad at 0.5Rs and −0.40 rad at 1.5Ls, while Method 2 reaches approximately ±1.0 rad oscillations at 0.5Ls.For TFC, resistance-reference variations have little effect on mean error but change error oscillation; Fig. 9 shows the resistance-variation experiment.Source: D. Performance under Low-Speed Operation · p. 7, Figure 9 · p. 8

E. Performance under Regression Parameter Variations

  • At 1000 rpm, small or moderate ∆t provides stable estimation, whereas large ∆t and N increase oscillation; at 200 rpm, too-small ∆t risks insufficient excitation, while overly large ∆t or N also increases oscillation.The reported tuning guidance favors smaller intervals at medium and high speeds and larger intervals at low speeds.Source: E. Performance under Regression… · p. 7, Figure 10 · p. 8, Figure 11 · p. 8

F. Performance under Speed Transients

  • From 200 to 1000 rpm, TFC maintains continuous position estimation without loss of synchronization; speed fluctuation falls from about 10 rpm to approximately 1 rpm, with position error bounded throughout.The acceleration experiment is shown in Fig. 12.Source: F. Performance under Speed Transients · p. 8, Figure 12 · p. 8

VI. CONCLUSION

  • The conclusion reports model-free position estimation without resistance, inductance, or flux-linkage inputs, alongside robustness to load and parameter variations and reliable wide-speed operation.The method is described as purely data-driven with few tuning parameters.Source: §VI · p. 8, F. Performance under Speed Transients · p. 8
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