Permanent magnet synchronous motors (PMSM) are widely adopted in industrial drives for their high power density, efficiency, and reliability. Model predictive control (MPC) is commonly used for inverter control because of its fast dynamics and constraint handling, but MPC performance depends strongly on model accuracy. In practice, parameter mismatches (e.g., inductance, resistance, flux linkage) and unmodeled dynamics degrade MPC robustness. Data-driven or model-free predictive control (MFPC) approaches build predictive models from measured data to mitigate sensitivity to physical parameter knowledge. Existing MFPC strategies include lookup-table mappings, ultra-local models, ARX-based schemes with recursive least squares (RLS), and neural-network methods. Each approach has trade-offs between nonlinear dynamic fidelity, computational burden, and theoretical guarantees. This study proposes a compact nonlinear ARX-based MFPC with an adaptive RLS estimator—termed NARX-ARLS-MFPC—that aims to improve nonlinear modeling of PMSM dynamics while preserving online computational feasibility and providing Lyapunov-based stability assurances.
The PMSM two-phase α-β stationary-frame dynamics are introduced by a complex-vector stator equation where stator inductance Ls, stator resistance Rs, stator voltage us, stator current is, rotor flux linkage Ψr (Ψf e jθe), and electrical speed ωe appear explicitly. A first-order discretization provides a one-step current prediction that is sensitive to parameter uncertainties in Ls, Rs, and Ψf, motivating a data-driven modeling approach.
A classical linear ARX structure is described for the α-β axes, where predicted currents are linear combinations of past currents and inverter voltage inputs. Parameter vectors ηα(k) and ηβ(k) and regression vectors φα(k) and φβ(k) are constructed from historical data. The RLS algorithm is used to estimate these parameter vectors online by minimizing a least-squares cost with a forgetting factor λ. The RLS recursion updates a covariance matrix P, gain matrix G, and computes a prediction error e(k) used in the update.
To capture PMSM nonlinearities—notably back-EMF effects tied to electrical angular velocity and rotor position—the authors introduce a compact nonlinear ARX (NARX) regression vector that augments the linear terms with specific nonlinear coupling terms. The chosen NARX formulation reduces model order to a third-order compact form with 3×1 parameter vectors for each axis (ηα(k), ηβ(k)) and corresponding 3×1 regressors (φα(k), φβ(k)).
The estimator employed is an adaptive RLS (ARLS) variant that uses a variable forgetting factor λ(k) adjusted online according to the current estimation error e(k) and a recent error fluctuation metric Δe(k). The forgetting factor is constrained between λmin and λbase (0 < λmin < λbase ≤ 1) and adapted exponentially based on thresholds δ1 and δ2. In practice, when a sudden disturbance or load change increases e(k) and Δe(k), λ(k) decreases toward λmin to accelerate adaptation by discounting older data; in steady state Δe(k) tends toward zero and λ(k) returns toward λbase to improve noise averaging and steady-state accuracy.
Control action follows a finite-control-set MPC (FCS-MPC) approach: a current-tracking cost function g, formulated from one-step-ahead NARX predictions and reference currents iα*(k+1), iβ*(k+1), is evaluated for all eight candidate inverter voltage vectors (Table 2 in the source). The vector minimizing the cost is selected and applied in each control period. The full algorithmic flow is presented in the source as a flowchart and block diagram (Figs 2 and 3).
A Lyapunov-based stability analysis is provided. A candidate Lyapunov function V(k) is defined from current tracking errors eα(k) and eβ(k). Under Assumption 1—that the online-identified NARX model yields a bounded one-step-ahead prediction error with finite bound ε—the analysis demonstrates that the forward difference ΔV(k) can be bounded and that V(k) remains bounded. Consequently, the closed-loop system is shown to be uniformly ultimately bounded for any admissible disturbance respecting the finite prediction error bound. In the idealized case of perfect one-step predictions (ε = 0), ΔV(k) ≤ 0 and the system is Lyapunov stable. The control law is designed to minimize predicted tracking error at each step, and the ARLS estimator provides rapid convergence to recover performance after transient inaccuracies.
The computational burden per control cycle splits into two parts: online parameter identification and optimal voltage-vector selection. Identification updates two independent 3×3 covariance matrices (one per axis) and the associated parameter vectors, yielding core operations of fixed complexity O(n^2) with n = 3. The control law evaluates the cost function for 8 voltage vectors; each evaluation requires a single-step current prediction using the compact NARX model and a few scalar operations. The dominant cost is the 3×3 covariance update, which is small and constant-size.
The authors report simulated execution metrics from MATLAB/Simulink: the core 3×3 matrix update requires about 0.25 μs per call in the simulation environment, and the complete algorithm executed within a 10 μs cycle in simulation. These results indicate practical real-time feasibility for typical PMSM control frequencies (10–20 kHz) and embedded motor-control DSPs (e.g., TI C2000 series). The compact third-order NARX model is emphasized as a key design choice enabling an effective balance between modeling expressiveness and computational tractability.
The source reports comprehensive simulation studies assessing robustness under parameter mismatches, sensor noise, and combined load transients with parameter perturbations. Key reported outcomes include significant reductions in current total harmonic distortion (THD) under parameter mismatch conditions: with large inductance and flux linkage mismatches the NARX-ARLS-MFPC reduced THD by 28.3% relative to a conventional FCS-MPC and by 12.3% versus a baseline control method. Under load transients combined with resistance and inductance mismatches, THD reductions of 24.1% and 15.7% were reported against the two benchmark methods, respectively. The method is noted to maintain stable performance under moderate sensor noise when properly tuned. A statistical analysis over parameter perturbations is cited in the source as confirming the overall superiority of the proposed method across key dynamic and steady-state metrics.
The authors conclude that the synergistic combination of a nonlinear regression structure (NARX) and an adaptive identification mechanism (ARLS with variable forgetting factor), together with Lyapunov-based closed-loop analysis, effectively suppresses current harmonics arising from model inaccuracies while improving dynamic response. Figures and tables in the source illustrate the algorithmic structure, switching-state definitions, and comparative performance; further simulation details and discussion are provided in Section 4 of the original article.