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Data-Driven Recursive Least Squares Estimation for Model Predictive Current Control of Permanent Magnet Synchronous Motors
Anian Brosch, Sören Hanke, Oliver Wallscheid, Joachim Böcker
TL;DR
MPC current control for PMSMs is limited by model deviations and inverter nonlinearities, especially when accurate white-box models are unavailable. The paper proposes online data-driven RLS identification with interlocking-time compensation for FCS-MPC, and experiments report significantly improved prediction and control performance over a LUT-based white-box baseline.
Problem
Fixed white-box PMSM models and offline LUTs may not capture parameter variation, inverter nonlinearities, harmonics, production deviations, and aging relevant to MPC current control.
Method
The paper combines continuously updated online RLS input-output identification with interlocking-time compensation in finite-control-set MPC.
Results
The proposed approach significantly improves prediction accuracy and FCS-MPC performance compared with a LUT-based white-box baseline across steady-state and transient operation.
Takeaways & Limitations
The RLS-based FCS-MPC approach is particularly suitable for self-commissioning drives where accurate white-box motor models are unavailable.
Abstract
from arXiv · showhide
The performance of model predictive controllers (MPC) strongly depends on the model quality. In the field of electric drive control, white-box (WB) modeling approaches derived from first-order physical principles are most common. This procedure typically does not cover parasitic effects and parameter deviations are frequent. These issues are particularly crucial in the domain of self-commissioning drives when a hand-tailored, accurate WB plant model is not available. In order to compensate for such modeling errors and, therefore, to improve the control performance during transients and steady-state, this paper proposes a data-driven, real-time capable recursive least squares (RLS) estimation method for the current control of a permanent magnet synchronous motor (PMSM). The effect of the flux linkage and voltage harmonics due to the winding scheme can also be taken into account. Moreover, a compensating scheme for the interlocking time of the inverter is proposed. The proposed identification algorithm is investigated using the well-known finite-control-set MPC (FCS-MPC) in the rotor-oriented coordinate system. The extensive experimental results show the superior performance of the presented scheme compared to a FCS-MPC-based on a state-of-the-art WB motor model using look-up tables for adressing (cross-)saturation.
I. INTRODUCTION
The paper addresses prediction errors in PMSM current-control MPC caused by model deviations and inverter nonlinearities. It proposes continuously updated data-driven identification with interlocking-time compensation for FCS-MPC.
- MPC performance depends strongly on model accuracy, while fixed-parameter fundamental-wave PMSM models can produce large prediction errors.
- Motor-parameter variation, inverter nonlinearities, flux-linkage harmonics, production deviations, and aging limit offline look-up-table compensation.
- The proposed data-driven model continuously estimates input-output behavior online using recursive least squares rather than a predefined physical-parameter structure.
- Because conventional inverter-nonlinearity compensation requires a modulator, the paper introduces an interlocking-time compensation scheme for modulator-free FCS-MPC.
- FCS-MPC with prediction horizon N = 1 offers low computational complexity, short settling time, overshoot-free response, and inherent excitation for online identification.
B. Differential Equations of PMSM
The generalized PMSM model incorporates saturation, cross-saturation, iron-loss effects, and nonlinear flux linkage in a current-based discrete-time formulation. Its resulting system matrices are generally dense.
- The paper starts from a general motor model to examine how saturation and small-angle approximations affect state-space matrix density.
- The PMSM current-based discrete-time difference equation is formulated in the stator-based dq coordinate system while accounting for cross-saturation effects.
- The differential inductance matrix depends on current through saturation and on electrical angular velocity through iron losses, while flux linkage is also nonlinear.
- The generalized model has generally dense system matrices B and E.
2) Simplified Fundamental-Wave Model:
Neglecting saturation and iron losses, equating differential and absolute inductance, and applying a small-angle approximation reduces the generalized PMSM model to the simplified fundamental-wave model.
- When saturation and iron losses are neglected, flux linkage can be expressed using the absolute inductance matrix and permanent-magnet flux.
- The simplified model assumes Ldq,∆ = Ldq and approximates cos(∆ε) ≈ 1 and sin(∆ε) ≈ ∆ε for each sample interval.
- The simplified fundamental-wave model has sparse system matrices B and E compared with the generalized model.
C. Inverter
The inverter produces finite switching-state voltages, but interlocking time creates intermediate switching vectors determined by phase-current direction. Neglecting this effect causes systematic modeling and control error in FCS-MPC.
- The finite inverter switching combinations define a finite set of voltages in the αβ coordinate system.
- The inverter switching state sabc and DC-link voltage uDC determine the applied αβ voltage vectors.
- During interlocking time Ti, both transistors in a phase leg are off and the phase current flows through one diode, creating an intermediate switching vector.
- Ignoring interlocking time in FCS-MPC causes systematic modeling and control error.
- The intermediate vector depends on current direction: positive current corresponds to an active lower and inactive upper switch, with the reverse arrangement for negative current.
D. Finite-Control-Set MPC
The FCS-MPC predicts the next-step current for each feasible switching state and applies the state that minimizes the cost based on predicted-reference current error.
- The N = 1 cost function penalizes the squared Euclidean distance between predicted next-step current and reference current.
- Intermediate and switching-vector voltages are averaged when evaluating the FCS-MPC cost function.
- The cost function is evaluated for all feasible switching states in the finite set.
- The minimum-cost switching state is applied, and the procedure repeats at the next sampling instant.
III. EXPERIMENTAL TEST SETUP
The experimental setup combines an automotive interior PMSM, a two-level IGBT inverter, a mechanically coupled induction-machine load, and rapid-control-prototyping hardware. Offline identification uses a 40000-sample dataset recorded under FCS-MPC control.
- The FCS-MPC and system-identification algorithms were implemented on a laboratory test bench.
- The test bench uses an automotive interior PMSM, a two-level IGBT inverter, a mechanically coupled induction-machine load, and a dSPACE DS1006MC system.
- The offline dataset was recorded with FCS-MPC using a prediction model whose differential-inductance matrix and flux vector were continuously updated from current-dependent LUTs.
- 40000 samples corresponded to 2 s of measurements at the rated-current operating point and 2000 min^-1.
- Offline model quality was evaluated with ordinary least squares by minimizing the quadratic deviation between measurements and model outputs.
- The system matrices A, B, and E were identified by rewriting the state-space representation as two separate least-squares problems.
A. Residual Analysis
Residual analysis validates regression models using graphical distributions and compares model structures through residual-based criteria. Fourier analysis and dense fundamental-wave modeling address harmonic and saturation-related behavior.
- Histograms are used to assess whether residuals are approximately normally distributed, with interlocking-time compensation reducing variance and improving symmetry.
- The residual-analysis section defines model structures for comparison in offline OLS and online RLS identification.
- Fourier coefficients h_d and h_q characterize residual components across electrical orders when interlocking-time compensation is active.
- The dense fundamental-wave model uses dense system matrices to represent generalized motor behavior, including cross-saturation effects.
- The model definitions include parameter vectors for the d- and q-axis regressions, with a constant-index convention simplifying later notation.
2) Sparse Fundamental-Wave Model:
The sparse fundamental-wave model removes selected coupling and offset parameters, limiting representation of cross-saturation. The augmented model adds harmonic regressors, while LASSO selects the most important terms for real-time identification.
- Sparse Fundamental-Wave Model: The sparse fundamental-wave model sets b12, b21, and e1 to zero, producing sparse B and E matrices.
- Sparse Fundamental-Wave Model: Because of its sparse structure, the model cannot represent cross-saturation effects.
- Augmented Fundamental-Wave Model: The augmented model adds sine and cosine regressors at harmonic frequencies to represent deterministic residual harmonics.
- Augmented Fundamental-Wave Model: Adding harmonics increases the number of parameters, so LASSO regression selects the most important regressors for the augmented model.
- Augmented Fundamental-Wave Model: At the evaluated operating point, LASSO set b12, b21, and e1 to zero, indicating that harmonic modeling had greater prediction relevance than cross-saturation modeling there.
- Augmented Fundamental-Wave Model: At low speeds, insufficient excitation makes the sine and cosine regressor matrix rank-deficient, preventing unique parameter identification.
A. Performance Comparison Utilizing an Offline-OLS
Offline OLS evaluation compares residual statistics across models and shows that interlocking-time compensation improves every reported criterion. With compensation, all models achieve R2 values above 0.97, including 0.993 for the DFW model in the d-axis.
- Interlocking-time compensation improves the mean, standard deviation, and coefficient of determination for every evaluated model.
- All models with interlocking-time compensation achieve a coefficient of determination larger than 0.97.This indicates high model quality according to the reported evaluation.
- R2_d = 0.993 for the DFW model with interlocking-time compensation in the d-axis.
- The RLS method updates model parameters online from each new measurement while weighting past measurements through a forgetting factor.The weighting factor λ satisfies 0 < λ < 1.
VII. EXPERIMENTAL RESULTS
Experimental evaluation across 83 operating points compares data-driven regression models with and without interlocking-time compensation against a LUT-based white-box model. The regression models reduce residual variation and improve steady-state control deviation and current distortion relative to the white-box baseline.
- Experimental setup: 83 equispaced operating points were recorded in the left id-iq half-plane, with 40000 samples collected over 2 s at 2000 min−1 per point.
- Performance criteria: Total demand distortion (TDD) is used to measure harmonic current distortion relative to nominal current.The paper identifies TDD as more suitable than THD when the fundamental current is small.
- Steady-State Behavior: All regression models have lower residual mean and standard deviation than the LUT-based white-box model.
- Steady-State Behavior: The DFW model with interlocking-time compensation is evaluated using residual-vector statistics and phase-current TDD across the left id-iq half-plane.
- Steady-State Behavior: Interlocking-time compensation produces lower standard deviation and current distortion for the compensated models.The uncompensated models can have a lower residual mean, but compensated models remain within the reported order of some 100 mA.
B. Transient Behavior
The RLS estimator remains effective as operating points change: current-step transients are short and overshoot-free, while residuals return toward steady-state values within milliseconds. During speed transients, residual magnitudes do not increase after the cutoff speed is reached, supporting the method’s reported performance across transients and operation ranges.
- Operating-point changes: The RLS system matrices are locally valid, so changing operating points are explicitly evaluated in transient operation.Parameter changes, including differential and absolute inductance variation with current, motivate the transient tests.
- Current transients: The current-step response has short settling time and overshoot-free behavior under the DFW model with interlocking-time compensation.The step response is evaluated at constant speed, with compensation applied to both DFW and WB models.
- Current transients: During the current transient, residuals rise slightly but decrease toward steady-state values within a few milliseconds.This behavior is reported for the DFW model during the step response.
- Speed transients: After a speed transient reaches constant cutoff speed, DFW residual magnitudes do not decrease, indicating no residual increase during the speed change.The paper attributes this to the mechanical system’s larger time constant and the RLS method’s fast electrical-domain adaptation.
- Overall assessment: Interlocking-time compensation combined with data-driven RLS identification improves FCS-MPC prediction accuracy and performance over a WB model using offline LUTs.The paper reports experimental validation in steady-state and transients across the electrical and speed operation range, with relevance to self-commissioning drives.