Data-Driven Recursive Least Squares Estimation for Model Predictive Current Control of Permanent Magnet Synchronous Motors

• MPC performance depends strongly on model accuracy, while fixed-parameter fundamental-wave PMSM models can produce large prediction errors.Source: §I · p. 1
• Motor-parameter variation, inverter nonlinearities, flux-linkage harmonics, production deviations, and aging limit offline look-up-table compensation.Source: §I · p. 1
• The proposed data-driven model continuously estimates input-output behavior online using recursive least squares rather than a predefined physical-parameter structure.Source: §I · p. 1
• Because conventional inverter-nonlinearity compensation requires a modulator, the paper introduces an interlocking-time compensation scheme for modulator-free FCS-MPC.Source: §I · p. 1
• FCS-MPC with prediction horizon N = 1 offers low computational complexity, short settling time, overshoot-free response, and inherent excitation for online identification.Source: §I · p. 2

• The paper starts from a general motor model to examine how saturation and small-angle approximations affect state-space matrix density.Source: B. Differential Equations of PMSM · p. 2
• The PMSM current-based discrete-time difference equation is formulated in the stator-based dq coordinate system while accounting for cross-saturation effects.Source: 1) Generalized Fundamental-Wave Model · p. 2
• The differential inductance matrix depends on current through saturation and on electrical angular velocity through iron losses, while flux linkage is also nonlinear.Source: 1) Generalized Fundamental-Wave Model · p. 2
• The generalized model has generally dense system matrices B and E.Source: 1) Generalized Fundamental-Wave Model · p. 2

• When saturation and iron losses are neglected, flux linkage can be expressed using the absolute inductance matrix and permanent-magnet flux.Source: 2) Simplified Fundamental-Wave Model · p. 3
• The simplified model assumes Ldq,∆ = Ldq and approximates cos(∆ε) ≈ 1 and sin(∆ε) ≈ ∆ε for each sample interval.Source: 2) Simplified Fundamental-Wave Model · p. 3
• The simplified fundamental-wave model has sparse system matrices B and E compared with the generalized model.Source: 2) Simplified Fundamental-Wave Model · p. 3

• The finite inverter switching combinations define a finite set of voltages in the αβ coordinate system.Source: C. Inverter · p. 3
• The inverter switching state sabc and DC-link voltage uDC determine the applied αβ voltage vectors.Source: C. Inverter · p. 3
• 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.Source: C. Inverter · p. 3
• Ignoring interlocking time in FCS-MPC causes systematic modeling and control error.Source: C. Inverter · p. 3
• 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.Source: Figure 2 · p. 3

• The N = 1 cost function penalizes the squared Euclidean distance between predicted next-step current and reference current.Source: D. Finite-Control-Set MPC · p. 3
• Intermediate and switching-vector voltages are averaged when evaluating the FCS-MPC cost function.Source: D. Finite-Control-Set MPC · p. 3
• The cost function is evaluated for all feasible switching states in the finite set.Source: D. Finite-Control-Set MPC · p. 3
• The minimum-cost switching state is applied, and the procedure repeats at the next sampling instant.Source: D. Finite-Control-Set MPC · p. 3

• The FCS-MPC and system-identification algorithms were implemented on a laboratory test bench.Source: §III · p. 3
• The test bench uses an automotive interior PMSM, a two-level IGBT inverter, a mechanically coupled induction-machine load, and a dSPACE DS1006MC system.Source: §III · pp. 3–4
• 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.Source: §IV · p. 4
• 40000 samples corresponded to 2 s of measurements at the rated-current operating point and 2000 min^-1.Source: §IV · p. 4
• Offline model quality was evaluated with ordinary least squares by minimizing the quadratic deviation between measurements and model outputs.Source: §IV · p. 4
• The system matrices A, B, and E were identified by rewriting the state-space representation as two separate least-squares problems.Source: §IV · p. 4

• Histograms are used to assess whether residuals are approximately normally distributed, with interlocking-time compensation reducing variance and improving symmetry.Source: A. Residual Analysis · p. 4
• The residual-analysis section defines model structures for comparison in offline OLS and online RLS identification.Source: §V · p. 4
• Fourier coefficients h_d and h_q characterize residual components across electrical orders when interlocking-time compensation is active.Source: Figure 4 · p. 5
• The dense fundamental-wave model uses dense system matrices to represent generalized motor behavior, including cross-saturation effects.Source: 1) Dense Fundamental-Wave Model · p. 5
• The model definitions include parameter vectors for the d- and q-axis regressions, with a constant-index convention simplifying later notation.Source: 1) Dense Fundamental-Wave Model · p. 5

Sparse Fundamental-Wave Model

• The sparse fundamental-wave model sets b12, b21, and e1 to zero, producing sparse B and E matrices.Source: 2) Sparse Fundamental-Wave Model · p. 5
• Because of its sparse structure, the model cannot represent cross-saturation effects.Source: 2) Sparse Fundamental-Wave Model · p. 5

Augmented Fundamental-Wave Model

• The augmented model adds sine and cosine regressors at harmonic frequencies to represent deterministic residual harmonics.Source: 3) Augmented Fundamental-Wave Model · p. 5
• Adding harmonics increases the number of parameters, so LASSO regression selects the most important regressors for the augmented model.Source: 3) Augmented Fundamental-Wave Model · p. 5
• 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.Source: 3) Augmented Fundamental-Wave Model · p. 5
• At low speeds, insufficient excitation makes the sine and cosine regressor matrix rank-deficient, preventing unique parameter identification.Source: 3) Augmented Fundamental-Wave Model · p. 5

• Interlocking-time compensation improves the mean, standard deviation, and coefficient of determination for every evaluated model.Source: A. Performance Comparison Utilizing an… · p. 5
• 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.Source: A. Performance Comparison Utilizing an… · p. 6
• R2_d = 0.993 for the DFW model with interlocking-time compensation in the d-axis.Source: A. Performance Comparison Utilizing an… · p. 6
• 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.Source: §VI · p. 6

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.Source: A. Steady-State Behavior · p. 6

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.Source: A. Steady-State Behavior · p. 6

Steady-State Behavior

• All regression models have lower residual mean and standard deviation than the LUT-based white-box model.Source: A. Steady-State Behavior · pp. 6–7
• The DFW model with interlocking-time compensation is evaluated using residual-vector statistics and phase-current TDD across the left id-iq half-plane.Source: Figure 5 · p. 7, Figure 6 · p. 7, A. Steady-State Behavior · p. 7
• 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.Source: A. Steady-State Behavior · p. 7

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.Source: B. Transient Behavior · p. 7

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.Source: B. Transient Behavior · p. 7, Figure 7 · p. 7
• 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.Source: B. Transient Behavior · p. 7

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.Source: B. Transient Behavior · p. 7, Figure 8 · p. 8

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.Source: §VIII · pp. 7–8