Parameter-Free Model Predictive Control of Five-Phase PMSM Under Healthy and Inter-Turn Short-Circuit Fault Conditions

Abstract

Model predictive control offers high-performance regulation for multiphase drives but is critically dependent on the accuracy of mathematical models for prediction, making it vulnerable to parameter mismatches and uncertainties. To achieve parameter-independent control across both healthy and faulty operations, this paper proposes a novel dynamic mode decomposition with control (DMDc)-based model predictive current control (MPCC) scheme for five-phase permanent magnet synchronous motors. The core innovation lies in constructing discrete-time state-space models directly from operational data via the open-loop DMDc identification, completely eliminating reliance on explicit motor parameters. Furthermore, an improved fault-tolerant strategy is developed to mitigate the torque ripple induced by inter-turn short-circuit (ITSC) faults. This strategy estimates the key fault characteristic, the product of the short-circuit ratio and current, through a spectral decomposition of the AC component in the q-axis current variations, bypassing the need for complex parameter-dependent observers. The derived compensation currents are seamlessly integrated into the predictive control loop. Experimental results comprehensively validate the effectiveness of the proposed framework, demonstrating a performance comparable to a conventional MPCC under healthy conditions and a significant reduction in torque ripple under ITSC fault conditions, all achieved without any prior knowledge of motor parameters or the retuning of controller gains.

1. Introduction

Multiphase permanent magnet synchronous motors (PMSMs) have attracted significant attention across various industrial fields, particularly in high-power and high-reliability applications [1,2,3]. Compared to their three-phase counterparts, multiphase PMSM drives offer several distinct advantages, including reduced torque pulsations, decreased current stress on semiconductors and machine windings, lower DC-link current harmonics, higher power ratings, and an enhanced fault-tolerant capability [4].

Model predictive control (MPC) has emerged as a prominent strategy for the high-performance control of multiphase drives. Control objectives are realized through behavior forecasting combined with the receding optimization of predefined cost functions. However, its implementation exhibits a critical dependency on the precision of mathematical models yet demonstrates an inherent sensitivity to parametric uncertainties and exogenous disturbances of dynamic operations. Parameter variations including thermal-induced fluctuations in stator resistance, PM flux linkage variations, and load-dependent inductance saturation effects would result in a degraded performance and system instability [5].

Model-free predictive control (MFPC) is a promising technique in the field of motor drives to address the challenges caused by the conventional MPC. The implementations of the MFPC are commonly classified into two manners: prediction correction and prediction without motor parameters [6,7]. Traditionally, the former approach involves defining an aggregate disturbance term that captures discrepancies between ideal and actual models. This term is subsequently estimated and incorporated into future state predictions to minimize forecasting errors [8,9,10]. This method is dependent on the nominal parameters of motors; thus, it is not completely parameter-free control. The latter approach constructs the predictive model without relying on motor parameters, enabling control variables to be directly predicted through MFPC algorithms. These algorithms usually include two prediction methods: current difference-based prediction [11] and ultra-local model-based prediction [12,13].

Very recently, the MFPC strategy with an ultra-local model (ULM) has been extended into multiphase drive control [14,15,16,17,18,19,20]. The ULM, derived from the input-output data of systems, contains parameter uncertainties and drive nonlinearity. Future states can be easily predicted by estimating the uncertain terms [21]. In [15], a CCS-MFPC method with ULM and voltage vector correction was proposed for five-phase PMSM drives. The lumped terms of the ULM were estimated using the differences in current and voltage. The voltage stagnation was alleviated with the guidance of the power factor angle. In addition, several observers were investigated to replace the algebraic parameter identification idea above, such as the extended state observer (ESO), the disturbance observer, and the sliding mode observer (SMO). In [16], a model-free predictive current control based on ULM was proposed for asymmetrical dual three-phase PMSMs. An ESO was designed to estimate the total disturbance in the ULM. Later, a model-free predictive flux control was proposed for the fault-tolerant control of five-phase PMSM drives, where a new ESO was developed to estimate the stator inductance and the disturbances caused by open-circuit faults [17]. Based on the ULM of dual three-phase PMSM drives, a non-linear disturbance observer was designed to compensate for inverter distortion voltage and eliminate estimation spikes caused by bandwidth limitations [18]. In [19], a ULM considering back-electromotive force (EMF) harmonics was built to replace the current prediction models of the fundamental and harmonic subspaces. An SMO with improved reaching law was designed to estimate the disturbance and currents. Nevertheless, when designing observers to estimate unknown terms in ULMs, particular attention must be paid to drive system stability. Improperly tuned observer gains can potentially induce system instability, necessitating careful parameter selection during the design process [20].

In addition to the two implementation approaches of MFPC, research has also explored data-driven techniques. Dynamic mode decomposition (DMD) is a data-driven modeling method that utilizes state measurement snapshots to construct parameter-free representations of system dynamics. If the input information is available, more precise models can be obtained through DMD with control (DMDc) [22,23]. In [24], DMDc was used to identify both the accurate model of PMSMs as well as model disturbances. The identification bias caused by sensor noise was analyzed and subsequently eliminated by considering the forward and backward estimations of state matrices. Later, the identifications of the nominal system and the harmonic component were implemented for the control of PMSMs based on DMD [25]. The former was used to assign the feedback gains by matching the desired closed-loop eigenvalues and eigenvectors. The latter was to generate vectors that were multiplied by the delayed embedding of the current to predict harmonic components at the next time step. In [26], the DMD and its variants, namely online DMDc, windowed online DMDc, and weighted-online DMDc, were investigated by illustrating the estimation process for induction motor drives under normal operations, shaft misalignment faults, and speed sensor faults. The results indicate that DMDc was effective for the modeling and control of motor drives.

Inter-turn short-circuit (ITSC) faults constitute critical incipient failure modes in electrical machines, demanding immediate mitigation to prevent catastrophic damage. Among the existing approaches for ITSC fault tolerance, fault-tolerant control strategies can be broadly categorized into three types: terminal short-circuit (TSC), current injection, and motor topology design [27]. Among these, TSC and current injection are the most widely adopted methods in drive system applications. The TSC method offers straightforward implementation by cutting off the faulty phase and establishing a terminal short-circuit at the winding end. Upon detection of an ITSC fault, the healthy section current is interrupted by activating the TSC mechanism in the faulty phase. On the other hand, current injection-based fault-tolerant control serves as a prominent strategy for ITSC conditions, focusing on suppressing the torque ripple induced by short-circuit currents [28,29,30] through the introduction of specific harmonic components into the motor currents to compensate for magnetic field asymmetries and prevent fault propagation. These methods typically inject compensation currents proportional to the product of the short-circuit turns ratio and the short-circuit current [31]. However, separate estimation of the short-circuit ratio and current remains challenging, leading to conventional treatment as a combined variable observable via parameter-dependent estimators. In [32], a disturbance observer was designed to obtain the disturbance signal from the ITSC and directly utilized the fault characteristics in the disturbance signal for the fault-tolerant control of five-phase PMSMs. Nevertheless, the machine parameter dependency and observer tuning requirements critically limit the effectiveness against operational uncertainties.

To overcome these constraints, this paper proposes a parameter-free model predictive control framework for five-phase PMSMs under both healthy and ITSC fault situations. The main contributions are summarized as follows:

(1)

Construction of discrete system matrices directly from operational data via DMDc identification, eliminating dependency on explicit machine parameters.

(2)

Novel estimation of the fault characteristic product of the short-circuit ratio and current through spectral decomposition of q1-axis current variations.

(3)

Seamless integration of compensation currents into the predictive control loop, maintaining consistent operation across healthy and faulty states without retuning.

The organization of the rest of this paper is listed as follows: Section 2 establishes the mathematical foundation of model predictive current control for five-phase PMSMs. Section 3 details the DMDc-based parameter-free MPCC design. Section 4 develops the fault-tolerant strategy, including the compensation current estimation and injection mechanism. Section 5 validates the framework through comparative experiments. Section 6 concludes the paper with key findings.

2. MPCC of Five-Phase PMSM

Figure 1 demonstrates a five-phase PMSM drive fed by a two-level voltage source inverter (VSI), where A–E indicate the five phases. The VSI generates 32 basic switching states, which correspond to 32 basic voltage vectors in the fundamental subspace and 32 basic voltage vectors in the harmonic subspace, respectively [7]. With a Clarke and Park transformation, a five-phase PMSM can be mathematically modeled in the d₁-q₁-d₃-q₃ reference frame as

$$\left\{\begin{array}{l}{u}_{d1}=R_{\mathrm{s}}{i}_{d1}+L_{d1}{\displaystyle \frac{d{i}_{d1}}{dt}}-{\omega }_{\mathrm{e}}{L}_{q1}{i}_{q1}\\ \begin{array}{l}{u}_{q1}=R_{\mathrm{s}}{i}_{q1}+L_{q1}{\displaystyle \frac{d{i}_{q1}}{dt}}+{\omega }_{\mathrm{e}}(L_{d1}{i}_{d1}+{\psi }_{\mathrm{f}})\\ u_{d3}=R_{\mathrm{s}}{i}_{d3}+L_{d3}{\displaystyle \frac{d{i}_{d3}}{dt}}-{3\omega }_{\mathrm{e}}{L}_{q3}{i}_{q3}\end{array}\\ u_{q3}=R_{\mathrm{s}}{i}_{q3}+L_{q3}{\displaystyle \frac{d{i}_{q3}}{dt}}+{3\omega }_{\mathrm{e}}{L}_{d3}{i}_{d3}\end{array}\right.$$

(1)

where u, i, L, and R_s represent the phase voltage, current, inductance, and resistance, respectively. ω_e is the electrical angular velocity. ψ_f is the magnitude of the phase PM flux-linkage.

With the mathematical model and the Euler forward formula, the current predictive model considering two subspaces at the instant (k + 1) is presented as

$$\mathit{i}\left(k+1\right)=\mathit{M}\mathit{i}\left(k\right)+\mathit{N}\mathit{u}+\mathit{D}$$

(2)

where

$$\begin{gathered} \mathit{M}=\left[\begin{array}{cccc}1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{d1}}}& {\displaystyle \frac{{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{L}_{q1}}{L_{d1}}}& 0& 0\\ -{\displaystyle \frac{{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{L}_{d1}}{L_{q1}}}& 1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{q1}}}& 0& 0\\ 0& 0& 1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{d3}}}& {\displaystyle \frac{3{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{L}_{q3}}{L_{d3}}}\\ 0& 0& -{\displaystyle \frac{3{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{L}_{d3}}{L_{q3}}}& 1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{q3}}}\end{array}\right], \\ \mathit{N}=diag\left\{{\displaystyle \frac{T_{\mathrm{s}}}{L_{d1}}},{\displaystyle \frac{T_{\mathrm{s}}}{L_{q1}}},{\displaystyle \frac{T_{\mathrm{s}}}{L_{d3}}},{\displaystyle \frac{T_{\mathrm{s}}}{L_{q3}}}\right\}, \mathit{D}={[0-{\displaystyle \frac{{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{\psi }_{\mathrm{f}}}{L_{q1}}} 0 0]}^T, \\ \mathit{i}={[i_{d1} i_{q1} i_{d3} i_{q3}]}^{\mathrm{T}}, \mathit{u}={[u_{d1} u_{q1} u_{d3} u_{q3}]}^{\mathrm{T}}. \end{gathered}$$

In order to obtain the optimal voltage vector, a cost function is designed to evaluate the predicted variables. The following equation shows a cost function using the absolute errors between the referenced and predicted values of four components of phase currents:

$$\begin{array}{c}\begin{array}{c}{g}_i=\left|i_{d1}^{\mathrm{r}\mathrm{e}\mathrm{f}}-i_{d1}\left(k+1\right)\right|+\left|i_{q1}^{\mathrm{r}\mathrm{e}\mathrm{f}}-i_{q1}\left(k+1\right)\right|\end{array}\\ +\left|i_{d3}^{\mathrm{r}\mathrm{e}\mathrm{f}}-i_{d3}\left(k+1\right)\right|++\left|i_{q3}^{\mathrm{r}\mathrm{e}\mathrm{f}}-i_{q3}\left(k+1\right)\right|\end{array}$$

(3)

where T_s is the sampling time. The symbol “ref” denotes the reference command. i = {1, 2,…, 32}.

3. Parameter-Free MPCC of Five-Phase PMSM Under Healthy Conditions

3.1. Dynamic Mode Decomposition with Control

Despite the potential non-linear behavior of a system, the model derived through the application of DMDc will inherently be linear. Thus, a discrete-time dynamic system is expressed as

$${\mathit{x}}_{i+1}=\mathit{A}{\mathit{x}}_i+\mathit{B}{\mathit{u}}_i$$

(4)

where x_i is a state vector with j elements and u_i is an input vector with p elements. A and B are the state matrix and the input matrix, respectively. The purpose of DMDc is to identify state and input matrices and use them to predict the future states of systems, as shown in Figure 2. To obtain A and B, new matrices are defined as follows [22]:

$${\mathit{X}}_1=\left[{\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_{n-2},{\mathit{x}}_n\right]$$

(5)

$${\mathit{X}}_2=\left[{\mathit{x}}_2,{\mathit{x}}_3,\dots ,{\mathit{x}}_{n-1},{\mathit{x}}_{n+1}\right]$$

(6)

$$\mathit{U}=\left[{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_{n-2},{\mathit{u}}_n\right]$$

(7)

where X₁ is the state vector of n samples ranging from 1 to n, while X₂ is the state vector of n samples ranging from 2 to n + 1. U is the input vector of n samples ranging from 1 to n.

According to (4)–(7), the state vector X₂ can be derived as

$${\mathit{X}}_2=\left[\begin{array}{cc}\mathit{A}& \mathit{B}\end{array}\right]\left[\begin{array}{c}{\mathit{X}}_1\\ \mathit{U}\end{array}\right]=\mathit{G}\mathit{\Omega }$$

(8)

where G consists of the state matrix A and the input matrix B; Ω is augmented by the states X₁ and the inputs U.

The matrix G can be solved with the equation G = XΩ⁻¹. However, Ω is most probably a non-square and non-invertible matrix. Thus, the pseudo-inverse regarding the singular value decomposition (SVD) is applied. With the SVD, the pseudo-inverse of the augmented matrix is expressed as

$${\mathit{\Omega }}^{†}=\mathit{V}{\mathit{\Sigma }}^{†}{\mathit{Y}}^{\mathit{T}}$$

(9)

where

$${\mathit{\Sigma }}^{†}={\mathit{\Sigma }}^{\mathit{T}}{\left({\mathit{\Sigma }\mathit{\Sigma }}^{\mathit{T}}\right)}^{-1}.$$

The superscript “†” represents the pseudo-inverse. m is the dimension sum of states and inputs, i.e., m = j + p. Y is an m × m orthogonal matrix, Σ is an m × n diagonal matrix with non-negative real numbers, and V is an n × n orthogonal matrix. Since the sum of the state dimension and input dimension m is usually less than the samples n, Σ^† is an n × n matrix.

With (8) and (9), the following equation is derived.

$$\mathit{G}={\mathit{X}}_2{\mathit{\Omega }}^{†}$$

(10)

Considering Y = [Y₁;Y₂], Y₁ is a j × n matrix corresponding to the state matrix A, and Y₂ is a p × n matrix corresponding to the input matrix B. Accordingly, A and B are deduced as

$$\mathit{A}={\mathit{X}}_2\mathit{V}{\mathit{\Sigma }}^{†}{\mathit{Y}}_1^{\mathit{T}}$$

(11)

$$\mathit{B}={\mathit{X}}_2\mathit{V}{\mathit{\Sigma }}^{†}{\mathit{Y}}_2^{\mathit{T}}$$

(12)

The asymptotic stability of the identified discrete-time model is rigorously determined by the spectral radius of the state matrix A. According to the Lyapunov stability theory for discrete systems, a necessary and sufficient condition for stability is that all eigenvalues of A lie strictly within the unit circle of the complex plane, i.e., the spectral radius ρ(A) < 1 [33]. This eigenvalue criterion serves as the foundation for the subsequent robustness analysis against perturbations in the system matrices.

3.2. Current Prediction Based on DMDc

The identification of the matrices of the predictive current model for five-phase PMSMs is associated with states and inputs, which do not transit from one time instant to the next linearly. This causes a misidentification of the parameters if the data is stacked traditionally. To handle the non-linear transition of the aforementioned factors, the d₁-q₁-d₃-q₃ axis currents and their products with the electrical angular velocity are advanced in time in the x₂ data matrix. The inputs and the electrical angular velocity are kept at the same time step as in the x₁ data matrix. With the concept of DMDc above, the discrete-time model of five-phase PMSMs is presented as

$${\mathit{x}}_2=\mathit{A}{\mathit{x}}_1+\mathit{B}\mathit{U}$$

(13)

where

$$\begin{gathered} {\mathit{x}}_2=[\begin{array}{cccc}{i}_{d1}\left(k+1\right)& i_{q1}\left(k+1\right)& i_{d3}\left(k+1\right)& i_{q3}\left(k+1\right)\end{array} \\ \begin{array}{cc}{i}_{d1}{\left(k+1\right)\omega }_{\mathrm{e}}\left(k+1\right)& i_{q1}\left(k+1\right){\omega }_{\mathrm{e}}\left(k+1\right)\end{array} \\ \hspace{1em}{\begin{array}{cc}{i}_{d3}\left(k+1\right){\omega }_{\mathrm{e}}\left(k+1\right)& i_{q3}\left(k+1\right){\omega }_{\mathrm{e}}\left(k+1\right)\end{array}]}^{\mathrm{T}}, \\ {\mathit{x}}_1=[\begin{array}{ccccc}{i}_{d1}\left(k\right)& i_{q1}\left(k\right)& i_{d3}\left(k\right)& i_{q3}\left(k\right)& i_{d1}\left(k\right){\omega }_{\mathrm{e}}\left(k\right)\end{array} \\ {\begin{array}{ccc}{i}_{q1}\left(k\right){\omega }_{\mathrm{e}}\left(k\right)& i_{d3}\left(k\right){\omega }_{\mathrm{e}}\left(k\right)& i_{q3}\left(k\right){\omega }_{\mathrm{e}}\left(k\right)\end{array}]}^{\mathrm{T}}, \\ \hspace{1em}\hspace{1em}\mathit{U}={\left[\begin{array}{ccccc}{u}_{d1}\left(k\right)& u_{q1}\left(k\right)& u_{d3}\left(k\right)& u_{q3}\left(k\right)& {\omega }_{\mathrm{e}}\left(k\right)\end{array}\right]}^{\mathrm{T}}. \end{gathered}$$

A is an 8 × 8 state matrix and B is an 8 × 5 input matrix. Substituting the data of the inputs and outputs into (11) and (12), the state matrix A and the input matrix B are obtained.

A and B are related to the matrices M and N in the conventional predictive current model, respectively, as illustrated in (2). Given the enriched states and inputs, the constituent elements of matrices A and B demonstrate a correspondingly greater complexity compared to those in matrices M and N. The mapping relationships among A, B, M, and N are analyzed as follows.

With the same states and inputs in (13), the predictive current model in (2) can be rewritten as

$$\mathit{i}\left(k+1\right)={\mathit{M}}^{\mathit{\prime }}{\mathit{i}}^{\mathit{\prime }}\left(k\right)+{\mathit{N}}^{\mathit{\prime }}{\mathit{u}}^{\mathit{\prime }}$$

(14)

where

$$\begin{gathered} {\mathit{i}}^{\prime }={[i_{d1} i_{q1} i_{d3} i_{q3} i_{d1}{w}_e i_{q1}{w}_e i_{d3}{w}_e i_{q3}{w}_e]}^{\mathrm{T}}, {\mathit{u}}^{\prime }={[u_{d1} u_{q1} u_{d3} u_{q3} w_e]}^{\mathrm{T}}. \\ {\mathit{M}}^{\mathit{\prime }}=\begin{array}{c}\left[\begin{array}{cccccccc}{m}_{11}& 0& 0& 0& 0& m_{16}& 0& 0\\ 0& m_{22}& 0& 0& m_{25}& 0& 0& 0\\ 0& 0& m_{33}& 0& 0& 0& 0& m_{38}\\ 0& 0& 0& m_{44}& 0& 0& m_{47}& 0\end{array}\right]\end{array} \\ {m}_{11}=1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{d1}}}, m_{16}={\displaystyle \frac{T_{\mathrm{s}}{L}_{q1}}{L_{d1}}}, m_{22}=1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{q1}}}, m_{25}=-{\displaystyle \frac{T_{\mathrm{s}}{L}_{d1}}{L_{q1}}}, \\ {m}_{33}=1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{d3}}}, m_{38}={\displaystyle \frac{3T_{\mathrm{s}}{L}_{q3}}{L_{d3}}}, m_{44}=1-{\displaystyle \frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{q3}}}, m_{47}=-{\displaystyle \frac{3T_{\mathrm{s}}{L}_{d3}}{L_{q3}}}. \\ {\mathit{N}}^{\mathit{\prime }}=\left[\begin{array}{ccccc}{\displaystyle \frac{T_{\mathrm{s}}}{L_{d1}}}& 0& 0& 0& 0\\ 0& {\displaystyle \frac{T_{\mathrm{s}}}{L_{q1}}}& 0& 0& -{\displaystyle \frac{T_{\mathrm{s}}{\psi }_{\mathrm{f}}}{L_{q1}}}\\ 0& 0& {\displaystyle \frac{T_{\mathrm{s}}}{L_{d3}}}& 0& 0\\ 0& 0& 0& {\displaystyle \frac{T_{\mathrm{s}}}{L_{q3}}}& 0\end{array}\right]. \end{gathered}$$

The two subscripts of M mean the row and column number, respectively.

Most of the elements in M′ and N′ are zero due to the neglect of the non-linearity and cross-coupling. However, these factors would be reflected in A and B. Thus, the identified results can predict the future behavior of the phase current, implementing the predictive current control of five-phase PMSM drives.

Considering that the single voltage vector in MPCC would induce considerable harmonics, the SVM technique is applied in this paper. With the voltage equation in (1) and the identified parameters in (13), the voltage vector reference can be achieved.

4. Fault-Tolerant Control of Five-Phase PMSM with Inter-Turn Short-Circuit Fault

4.1. Current and Torque Analysis

The equivalent circuit model in the natural coordinate system is built to analyze the motor under ITSC faults. Figure 3 illustrates the circuit with an ITSC fault in phase A. The short-circuit turn ratio μ (0 < μ ≤ 1) is defined as the ratio of the short-circuited turns to total turns in the faulty winding, while the fault resistance R_f is designed to characterize the severity of insulation degradation. The short-circuit current i_sc is accordingly generated within the fault loop. As the ITSC fault continuously spreads, the short-circuit turn ratio increases, coupled with a declined fault resistance, both of which contribute to a substantial boost in the short-circuit current.

The ideal phase currents under healthy conditions can be expressed as

$$\left\{\begin{array}{l}{i}_{\mathrm{A}}=I_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{e}}\\ i_{\mathrm{B}}=I_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{2\pi }{5}}\right)\\ i_{\mathrm{C}}=I_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{4\pi }{5}}\right)\\ i_{\mathrm{D}}=I_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{6\pi }{5}}\right)\\ i_{\mathrm{E}}=I_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{8\pi }{5}}\right)\end{array}\right.$$

(15)

where I_m and θ_e indicate the amplitude of phase currents and the electrical position, respectively.

During an ITSC fault, an equivalent disturbance current i_sc is induced in the affected phase. Considering the star connection of stator windings, each of the other four phases will develop a compensatory current of −i_sc/4 [32]. The post-fault phase currents are expressed as

$$\left\{\begin{array}{l}{i}_{\mathrm{A}\mathrm{f}}=i_{\mathrm{A}}+{\mathrm{i}}_{\mathrm{s}\mathrm{c}}\\ i_{x\mathrm{f}}=i_x-{\displaystyle \frac{1}{4}}{\mathrm{i}}_{\mathrm{s}\mathrm{c}}\\ {\mathrm{i}}_{\mathrm{s}\mathrm{c}}=I_{\mathrm{s}\mathrm{c}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\end{array}\right.$$

(16)

where I_sc and φ_sc present the amplitude and phase angle of the short-circuit current, respectively. x = {B, C, D, E}.

The electromagnetic torque under ITSC fault conditions is calculated by

$$T_{\mathrm{e}\mathrm{f}}=-{\displaystyle \frac{e_{\mathrm{A}}{i}_{\mathrm{A}\mathrm{f}}+e_{\mathrm{B}}{i}_{\mathrm{B}\mathrm{f}}+e_{\mathrm{C}}{i}_{\mathrm{C}\mathrm{f}}+e_{\mathrm{D}}{i}_{\mathrm{D}\mathrm{f}}+e_{\mathrm{E}}{i}_{\mathrm{E}\mathrm{f}}-{\mu e}_{\mathrm{A}\mathrm{f}}{i}_{\mathrm{s}\mathrm{c}}}{{\omega }_{\mathrm{m}}}}$$

(17)

where e indicates the back EMF and ω_m represents the mechanical angular velocity.

Assuming the back EMF is given by

$$\left\{\begin{array}{l}{e}_{\mathrm{A}}={-P}_{\mathrm{r}}{\omega }_{\mathrm{m}}{\psi }_{\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{e}}\\ e_{\mathrm{B}}={-P}_{\mathrm{r}}{\omega }_{\mathrm{m}}{\psi }_{\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{2\pi }{5}}\right)\\ e_{\mathrm{C}}={-P}_{\mathrm{r}}{\omega }_{\mathrm{m}}{\psi }_{\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{4\pi }{5}}\right)\\ e_{\mathrm{D}}={-P}_{\mathrm{r}}{\omega }_{\mathrm{m}}{\psi }_{\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{6\pi }{5}}\right)\\ e_{\mathrm{E}}={-P}_{\mathrm{r}}{\omega }_{\mathrm{m}}{\psi }_{\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{\mathrm{e}}-{\displaystyle \frac{8\pi }{5}}\right)\end{array}\right.$$

(18)

where P_r represents the number of pole pairs.

Substituting (18) and (16) into (17) yields

$$T_{\mathrm{e}\mathrm{f}}={\displaystyle \frac{5}{2}}{P}_{\mathrm{r}}{\psi }_{\mathrm{f}}{I}_{\mathrm{m}}-{\displaystyle \frac{1}{2}}{P}_{\mathrm{r}}{\psi }_{\mathrm{f}}\mu I_{\mathrm{s}\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{\mathrm{s}\mathrm{c}}+{\displaystyle \frac{1}{2}}{P}_{\mathrm{r}}{\psi }_{\mathrm{f}}\mu I_{\mathrm{s}\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}(2{\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}})$$

(19)

The first term in (19) corresponds to the electromagnetic torque under healthy operating conditions, while the subsequent terms encapsulate characteristics induced by the ITSC fault. Specifically, the second term signifies a characteristic decline in steady-state torque output, and the third term introduces torque oscillations at twice the fundamental frequency.

4.2. Injection Current Construction and Fault-Tolerant Control

Electromagnetic torque oscillations induced by short-circuit current and back-EMF interactions are analytically established in (19). To suppress these oscillations, this work develops a targeted current injection strategy governed by the fault severity product μi_sc, which is directly estimated through the amplitude and phase extraction of the electromagnetic torque’s AC component during ITSC faults [30,31,32].

For the studied five-phase PMSM exhibiting minimal saliency, (20) demonstrates that the q₁-axis current variation Δi_q1 contains equivalent fault information to the torque AC component. This enables the parameter-free construction of μi_sc via the spectral decomposition of Δi_q1.

$$\begin{array}{c}\Delta i_{q1}=-{\displaystyle \frac{1}{5}}\mu I_{\mathrm{s}\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{\mathrm{s}\mathrm{c}}+{\displaystyle \frac{1}{5}}\mu I_{\mathrm{s}\mathrm{c}}\cos \left(2{\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\\ =\Delta i_{q1}^{\mathrm{D}\mathrm{C}}+\Delta i_{q1}^{\mathrm{A}\mathrm{C}} \end{array}$$

(20)

Equation (21) then constructs μi_sc using the amplitude and phase angle of the AC component in Δi_q1.

$$\begin{array}{c}\mu i_{\mathrm{s}\mathrm{c}}=5\left|\Delta i_{q1}^{\mathrm{A}\mathrm{C}}\right|\sin \left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\end{array}$$

(21)

The AC component of Δi_q1 is extracted via a real-time spectral decomposition algorithm. A Short-Time Fourier Transform (STFT) is applied to the current deviation signal, specifically tuned to isolate the magnitude and phase at twice the fundamental electrical frequency, which is the characteristic signature of the ITSC fault. Under healthy conditions, the magnitude at this frequency is negligible. Its emergence under fault conditions directly provides the amplitude |Δi_q1^AC| and phase φ_sc required to estimate the fault product μi_sc, according to (21). The detailed procedure is presented in Figure 4.

Consequently, the injected currents in the rotating coordinate system can be derived as

$$\left\{\begin{array}{l}{i}_{d1}^{\mathrm{i}\mathrm{n}\mathrm{j}}=2\left|\Delta i_{q1}^{\mathrm{A}\mathrm{C}}\right|\sin \left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{e}}\\ i_{q1}^{\mathrm{i}\mathrm{n}\mathrm{j}}=2\left|\Delta i_{q1}^{\mathrm{A}\mathrm{C}}\right|\sin \left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{e}}\\ i_{d3}^{\mathrm{i}\mathrm{n}\mathrm{j}}=2\left|\Delta i_{q1}^{\mathrm{A}\mathrm{C}}\right|\sin \left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\mathrm{s}\mathrm{i}\mathrm{n}3{\theta }_{\mathrm{e}}\\ i_{q3}^{\mathrm{i}\mathrm{n}\mathrm{j}}=2\left|\Delta i_{q1}^{\mathrm{A}\mathrm{C}}\right|\sin \left({\theta }_{\mathrm{e}}+{\phi }_{\mathrm{s}\mathrm{c}}\right)\mathrm{c}\mathrm{o}\mathrm{s}3{\theta }_{\mathrm{e}}\end{array}\right.$$

(22)

By injecting these currents into the proposed parameter-free MPCC framework, fault-tolerant control for the five-phase PMSM drive under ITSC faults is achieved. Critically, this entire tolerance mechanism remains independent of motor parameters. Figure 5 illustrates the control structure of the proposed MPCC strategy, where n* and i_q1* represent reference of speed and i_q1. In addition, the flow chart of the parameter-free modeling and fault-tolerant control is demonstrated in Figure 6.

5. Experiments

Experiments are conducted on a five-phase PMSM drive to verify the effectiveness of the proposed fault diagnosis and fault-tolerant control strategy. The experimental setup is shown in Figure 7, and the parameters of the five-phase motor are listed in

Table 1. Motor Parameters.

Parameters	Values
Pole pairs of the rotor	18
Turns per phase winding	140
Phase resistance (Ω)	0.15
PM flux-linkage (Wb)	0.07
Rated speed (rpm)	600
Rated torque (N·m)	17
Rated phase current (A)	4
d1-axis inductance (mH)	9.23
q1-axis inductance (mH)	8.92
d3-axis inductance (mH)	7.98
q3-axis inductance (mH)	8.22

. The motor is coaxially connected to a magnetic powder brake. Diagnosis and control algorithms are implemented in a DSP-TMS320C28346 control board with a sampling frequency of 10 kHz. For comparison, both the conventional MPCC with SVM and the proposed MPCC were tested. The ITSC fault was simulated by short circuiting specific turns in the phase windings using power switches.

5.1. Initialization of DMDc

To obtain the state and input matrices, the excitation signal should be properly designed. Figure 8 illustrates the experimental waveforms of excitations and speed responses. The d₁q₁-axis voltage is set as a sinusoidal signal with an amplitude of 10% of the rated voltage and a frequency of 300 Hz to excite the d₁q₁-axis inductance. Meanwhile, the q₃-axis voltage is designed as a sinusoidal signal with an amplitude of 2% of the rated voltage and a frequency of 900 Hz. The d₃-axis voltage u_d3 is similar to u_q3, but with a phase shift of π/2. The DC component of u_d1 is fixed to zero while u_q1 is adjusted collaboratively to drive the motor to reach its rated speed of 600 rpm.

The temporal evolution of the resultant i_A (k + 1) prediction error from this analysis is provided in Figure 9. The error remains bounded within a narrow band of ±0.4 A, rapidly converging as the model identification is completed. This small error validates the accuracy of the DMDc identification. Furthermore, the chosen amplitude of the excitation signals was carefully selected to ensure a strong signal-to-noise ratio for accurate identification while avoiding inverter saturation. The combination of a low prediction error and the successful implementation of the controller confirms that the excitation signal is entirely appropriate for initializing the DMDc framework.

Subsequently, the state variables i_q1, i_d1, i_d3, i_q3, i_q1ω_e, i_d1ω_e, i_d3ω_e, and i_q3ω_e and the input signals u_d1, u_q1, u_d3, and u_q3 are recorded during real implementations. These data are substituted into (11) and (12), from which the state matrix A and input matrix B are achieved. Substituting A and B into (13), the current prediction can be obtained, implementing the control of five-phase PMSM drives without any information about motor parameters.

5.2. MPCC Method Comparison

Figure 10 compares the steady-state performance of a conventional MPCC with SVM, MFPC [15], and the proposed method at 200 rpm and 17 N·m. As shown in Figure 10b, the phase currents under the MFPC method exhibit significant distortion and sparks in the q₁-axis current, especially after the ITSC fault.

A quantitative comparison is provided in

Table 2. Comparisons with different MPCC methods.

Item	Healthy Condition			ITSC Fault Condition
	MPCC with SVM	MFPC	Proposed MPCC	MPCC with SVM	MFPC	Proposed MPCC
THD (%)	11.04	18.78	8.22	32.39	29.05	23.61
Ripple of iq1 (%)	34.35	58.74	15.33	74.93	61.64	30.90

, which lists the THD of the phase-A current and the prediction error of the q₁-axis current. The results indicate that the proposed method demonstrates a superior performance. For instance, the ripple of its q₁-axis current (15.33% in the healthy state and 30.90% in the faulty state) is less than 50% of that achieved by the conventional MPCC with SVM. Moreover, the proposed method also achieves the lowest THD value of 8.22% under the healthy condition. Overall, these results confirm the superior steady-state performance of the proposed method over both the conventional MPCC and MFPC approaches.

5.3. Performance Evaluation for Short-Circuit Current Estimation

The experimental results of short-circuit current estimation by the proposed method are demonstrated in Figure 11, Figure 12, Figure 13 and Figure 14. In Figure 11, 20 turns of coils, corresponding to 14.29% of the total 140 turns, are short circuited in phase-A winding using a power switch and wires. Thus, the short-circuit resistance is neglected, i.e., R_f ≈ 0. The motor operates at 200 rpm with a load of 10 N·m. The actual short-circuit current is measured by a current clamp. As mentioned in the previous section, the integration of the short-circuit ratio and the short-circuit current is observed. Since the product of the short-circuit ratio and the measured short-circuit current cannot be directly displayed on the oscilloscope, the estimated short-circuit current is presented alongside the measured current waveform. This comparative visualization validates the estimation algorithm’s ability to accurately reconstruct the fault characteristic variable μi_sc through observable current signals.

As shown in Figure 11a, the short-circuit current is null during normal operation, and its estimated amplitude remains negligible. Upon introducing an ITSC fault in phase-A winding, an instantaneous short-circuit current with an amplitude of 3.2 A emerges. The estimated current progressively converges with the measured values in both magnitude and phase. At steady state, the estimated short-circuit current aligns precisely with the measurements, as illustrated in Figure 11b. The ITSC fault distorts phase currents, increasing the THD of the phase-A current from 10.58% to 28.39%. Meanwhile, the average torque decreases slightly from 10 N·m to 9.36 N·m, while the torque ripple increases significantly from 9.23% to 18.49%.

The experimental results of varied fault resistance R_f are illustrated in Figure 12. By comparing two subfigures under identical fault/operating conditions (200 rpm, 10 N·m), the increased short-circuit resistance reduces fault severity but preserves consistent variation trends across all variables. Crucially, the short-circuit current estimation maintains precise tracking accuracy, confirming method reliability under non-zero fault-resistance conditions.

Figure 13 presents the experimental results for short-circuit current estimation with 10 turns shorted. The reference speed is 100 rpm, and the load torque is set as 4 N·m. When an ITSC fault occurs, the real short-circuit current is generated with an amplitude of 1.1 A, being lower than that in Figure 11 due to the decrease in speed and load. The estimated current rises correspondingly and converges to match the measured current in both magnitude and phase. Concurrently, the THD value increases from 15.32% to 30.81%, and the torque ripple rises from 11.37% to 24.70%.

Experimental results for a 20-turn ITSC at 400 rpm and 11 N·m load are presented in Figure 14. A short-circuit current of 4.71 A is generated post-fault. Simultaneously, the THD value rises from 14.47% to 19.32%, while the torque ripple increases from 3.94% to 8.07%. In summary, the results in Figure 11, Figure 12, Figure 13 and Figure 14 validate the effective estimation of the fault characteristic variable μi_sc under a variety of operating and fault conditions.

5.4. Fault-Tolerant Performance

Figure 15 shows the experimental results of the short-circuit current, phase-A current, and torque under healthy, ITSC fault, and fault-tolerant conditions by the proposed control method. The speed is set to 200 rpm with a load torque of 12 N·m. A total of 20 turns of coils are shorted to simulate the ITSC fault. After the ITSC fault, the torque ripple increases significantly from 8.12% to 18.08%, as listed in

Table 3. Torque evaluation under healthy, fault, and fault-tolerant conditions.

Item	Healthy	ITSC Fault	Fault-Tolerant Operation
Average torque (N·m)	12	11.53	11.95
Torque ripple (%)	8.12	18.08	10.54

. Additionally, the average torque also declines from 12 N·m to 11.53 N·m. When the fault-tolerant algorithm is applied, the average torque recovers to the level under healthy conditions, and the torque ripple is remarkably suppressed to 10.54%. The results illustrate that the observed short-circuit current and short-circuit ratio can be successfully injected into a DMDc-based MPCC for torque performance improvement.

To further validate the robustness of the proposed method, experiments are conducted under varying load conditions at a constant speed of 200 rpm, as shown in Figure 16, which illustrates the response to a load variation from 14 N·m to 6 N·m. Figure 16a,b displays steady-state waveforms before and after the torque change, respectively. After the load variation, the amplitudes of both the phase currents and the short-circuit current decrease. Throughout this process, the estimated current maintains an accurate tracking performance. These results confirm the robustness of the proposed method and its ability to maintain a fault-tolerant operation under load variations.

Experimental studies confirm the dual efficacy of the proposed framework: DMDc-based modeling consistently delivers accurate predictive control across both healthy and fault-operating regimes without requiring motor parameters, demonstrating an inherent adaptability to winding fault conditions. Simultaneously, the q₁-axis current-variation identification approach proves highly effective in reconstructing the key fault signature, with derived compensation currents substantially mitigating torque ripples. Crucially, this unified approach maintains controller integrity during the transition from normal to fault-tolerant operations.

6. Conclusions

This paper has presented a comprehensive parameter-free predictive control framework for five-phase PMSMs, integrating data-driven system identification with advanced control theory to ensure a robust operation under both healthy and inter-turn short-circuit ITSC conditions. The primary contributions and outcomes of this work are concluded as follows:

(1)

The DMDc is successfully applied to the five-phase PMSM drive system. Using designed excitation signals, the state and input matrices of a discrete state-space model are directly identified from open-loop operational data. This process is entirely free from any prior knowledge of motor parameters, effectively decoupling the control performance from parametric uncertainties and variations, such as those caused by thermal effects or magnetic saturation.

(2)

A key innovation for fault tolerance is the novel estimation of the fault severity product μi_sc. This is achieved through a computationally efficient spectral decomposition of the q-axis current variation, extracting its AC component at twice the fundamental frequency. This method provides a precise, parameter-independent estimate of the fault signature, which is crucial for effective compensation.

(3)

The estimated fault product is used to construct optimal compensation currents, which are injected into the fundamental and harmonic subspaces. This strategy is seamlessly integrated into the MPCC loop, enabling a smooth transition from healthy to fault-tolerant operations without the need for controller reconfiguration or gain scheduling.

The proposed framework is rigorously validated on an experimental testbed. Results confirmed that: (a) The DMDc method accurately identifies system dynamics, enabling a prediction performance on par with a conventional model-based MPCC under healthy conditions; (b) the proposed estimation technique reliably tracks the fault product μi_sc across different fault severities; and (c) the injected compensation currents drastically suppress the ITSC-induced torque ripple while maintaining the average torque output, thus significantly improving operational stability and safety under fault conditions.

In summary, this work provides a promising and practical solution for high-performance, resilient motor drives in applications where accurate system parameters are difficult to obtain or subject to change. The proposed DMDc-based MPCC framework demonstrates a viable path towards truly parameter-robust and fault-resilient control in complex electromechanical systems.

This method offers significant practical value for industrial applications, such as electric vehicles, aerospace systems, and automated manufacturing, where motor reliability and minimal maintenance are critical, by enabling a continued and stable operation even under unforeseen fault conditions without reliance on precise system parameters.

Future work will focus on enhancing the noise robustness of the proposed data-driven identification and fault estimation method. Specifically, advanced signal processing techniques, such as adaptive filtering, wavelet-based denoising, or machine learning-enhanced noise suppression will be investigated to improve the accuracy and reliability of fault feature extraction under strong electromagnetic interference and sensor noise typically encountered in industrial environments. Furthermore, the integration of real-time parameter adaptation and observer-based noise compensation mechanisms will be explored to strengthen the practical applicability of the method in high-noise, real-world settings.