Hybrid Type-2 Fuzzy Logic–Extended Kalman Filter Approach for ITSC Fault Detection in PMSM Drives for Electric Vehicles

The mathematical representation of PMSM in the dq reference frame is a required phase for the implementation of the vector control for model simulation through MATLAB/SIMULINK. The projection of the stator variables onto the orthogonal axis leads to separating the flux control d-axis from the electromagnetic torque q-axis. In this frame, the motor equations are expressed as follows: (1) Vd=Rsid+Lddiddt−ωeLqiqVq=Rsiq+Lqdiqdt+ωeLdid+ψf \left\{ {\matrix{ {{V_d} = {R_s}{i_d} + {L_d}{{d{i_d}} \over {dt}} - {\omega _e}{L_q}{i_q}} \hfill \cr {{V_q} = {R_s}{i_q} + {L_q}{{d{i_q}} \over {dt}} + {\omega _e}\left( {{L_d}{i_d} + {\psi _f}} \right)} \hfill \cr } } \right.

• V_d and V_q: Stator voltages;

• i_d and i_q: Stator currents;

• R_s: Stator resistance (0.44 Ω);

• L_d and L_q: d-q axis inductances (0.0031 H);

• ω_e: Electrical angular speed;

• ψ_f : Flux linkage produced by the permanent magnets (0.124 Wb).

The PMSM’s generated electromagnetic torque is determined by the following expression: (2) Te=32ρψfiq+Ld−Lqidiq {T_e} = {3 \over 2}\rho \left( {{\psi _f}{i_q} + \left( {{L_d} - {L_q}} \right.} \right){i_d}{i_q}

In which ρ denotes the four pole pairs of the PMSM.

The mechanical dynamics of the rotor are governed by: (3) Jdωdt+Bωm=Te+Tl J{{d\omega } \over {dt}} + B{\omega _m} = {T_e} + {T_l}

Where, J, B, ω_m and T_l are, respectively, J: Rotor’s moment of inertia (0.0002 kg/m²), B: Viscous friction coefficient, ω_m: Mechanical angular speed, T_l: Load torque

The machine analysed in this study is characterised by the following:

• Surface permanent magnet (SPM) motor;

• Integral-slot concentrated winding;

• Three-wire Y configuration.

The conventional way to model an ITSC fault is by introducing a resistance R_f that represents the fault as it is illustrated in Figure 1 (Romdhane et al., 2023).

Figure 1.

Faulty phase equivalent circuit.

The stator phase is divided into a healthy portion a₁ and a faulty portion a₂, while the fault resistance R_f is connected in parallel with a₂ (Liu et al., 2023; Yu et al., 2025).

The electrical representation of the faulty PMSM is described by the following equation (Amiri Ahouee and Mola, 2020): (4) VITSC=RITSC⋅IITSC+LITSC⋅dIITSCdt+eITSC {V_{{\rm{ITSC}}}} = {R_{{\rm{ITSC}}}} \cdot {I_{{\rm{ITSC}}}} + {L_{{\rm{ITSC}}}} \cdot {{d{I_{{\rm{ITSC}}}}} \over {dt}} + {e_{{\rm{ITSC}}}}

The following parameters are expressed in the (a, b, c) three-phase reference frame, where:

V_{a_ITSC}, V_{b_ITSC}, V_{c_ITSC} : Stator voltages, I_{a_ITSC}, I_{b_ITSC}, I_{c_ITSC} : Stator currents; I_{f_ITSC} : Fault current, L_ITSC: Mutual inductance and e_{a_ITSC}, e_{b_ITSC}, e_{c_ITSC}: back electromotive EMF.

Where: (5) VITSC=Va_ITSCVb_ITSCVc_ITSC0RITSC=Rs00−Ra20Rs0000Rs0−Ra200Ra2+RfIITSC=Ia_ITSCIb_ITSCIc_ITSCIf_ITSCLITSC=Ls00−Las−Ma1a20Ls0−Ma2b00Ls−Ma2b−Las+Ma1a2−Ma2b00La2 \left\{ {\matrix{ {{V_{{\rm{ITSC}}}} = \left[ {\matrix{ {{V_{a\_{\rm{ITSC}}}}} & {{V_{b\_{\rm{ITSC}}}}} & {{V_{c\_{\rm{ITSC}}}}} & 0 \cr } } \right]} \hfill \cr {{R_{ITSC}} = \left[ {\matrix{ {{R_s}} & 0 & 0 & { - {R_{{a_2}}}} \cr 0 & {{R_s}} & 0 & 0 \cr 0 & 0 & {{R_s}} & 0 \cr { - {R_{{a_2}}}} & 0 & 0 & {{R_{{a_2}}} + {R_f}} \cr } } \right]} \hfill \cr {{I_{ITSC}} = \left[ {\matrix{ {{I_{a\_ITSC}}} & {{I_{b\_ITSC}}} & {{I_{c\_ITSC}}} & {{I_{f\_ITSC}}} \cr } } \right]} \hfill \cr {{L_{ITSC}} = \left[ {\matrix{ {{L_s}} & 0 & 0 & { - {L_{{a_s}}} - {M_{{a_1}{a_2}}}} \cr 0 & {{L_s}} & 0 & { - {M_{{a_2}b}}} \cr 0 & 0 & {{L_s}} & { - {M_{{a_2}b}}} \cr { - \left( {{L_{{a_s}}} + {M_{{a_1}{a_2}}} - {M_{{a_2}b}}} \right)} & 0 & 0 & {{L_{{a_2}}}} \cr } } \right]} \hfill \cr } } \right.

With: (6) eITSC=ea_ITSCeb_ITSCec_ITSC−ea2Ra1=1−TfaultTtotal⋅RsRa2=TfaultTtotal⋅Rs \left\{ {\matrix{ {{e_{{\rm{ITSC}}}} = \left[ {\matrix{ {{e_{a\_{\rm{ITSC}}}}} \cr {{e_{b\_{\rm{ITSC}}}}} \cr {{e_{c\_{\rm{ITSC}}}}} \cr { - {e_{{a_2}}}} \cr } } \right]} \hfill \cr {{R_{{a_1}}} = \left( {1 - {{{T_{{\rm{fault}}}}} \over {{T_{{\rm{total}}}}}}} \right) \cdot {R_s}} \hfill \cr {{R_{{a_2}}} = {{{T_{{\rm{fault}}}}} \over {{T_{{\rm{total}}}}}} \cdot {R_s}} \hfill \cr } } \right.

The resistance of the healthy portion of the phase is presented by R_a1 while the resistance of the inter-turn circuited portion is represented by R_a2 and T_fault and T_total are, respectively, the number of shorted turns in the sub-winding and the total number of turns per phase.

Using the ratio of these parameters, we can define the ITSC coefficient as follows: (7) μ=TfaultTtotal \mu = {{{T_{{\rm{fault}}}}} \over {{T_{{\rm{total}}}}}}

The formulation of the torque and its associated mechanical dynamics in the dq-axis frame is as follows: (8) Te=32pψdiq−ψqidJdωrdt+frωr+Tload=Te \left\{ {\matrix{ {{T_e} = {3 \over 2}p\left( {{\psi _d}{i_q} - {\psi _q}{i_d}} \right)} \hfill \cr {J{{d{\omega _r}} \over {dt}} + {f_r}{\omega _r} + {T_{{\rm{load}}}} = {T_e}} \hfill \cr } } \right.

Here, ω_r: Rotor angular velocity, f_r: Friction coefficient (0.0812 N.m.s./rad).

The following equation is the voltage representation vαvβ \left[ {\matrix{ {{v_\alpha }} \cr {{v_\beta }} \cr } } \right] in the αβ-frame reference prevailed by the application of Clark transformation to Eq. (1): (9) vαvβ=Rsiαiβ+ddtψαψβ+eαeβ \left[ {\matrix{ {{v_\alpha }} \cr {{v_\beta }} \cr } } \right] = {R_s}\left[ {\matrix{ {{i_\alpha }} \cr {{i_\beta }} \cr } } \right] + {d \over {dt}}\left[ {\matrix{ {{\psi _\alpha }} \cr {{\psi _\beta }} \cr } } \right] + \left[ {\matrix{ {{e_\alpha }} \cr {{e_\beta }} \cr } } \right] with (10) ψα=Ldiα−ψfsinθ, ψβ=Lqiβ−ψfcosθ {\psi _\alpha } = {L_d}{i_\alpha } - {\psi _f}\sin \theta ,\;\;{\psi _\beta } = {L_q}{i_\beta } - {\psi _f}\cos \theta

In addition to the voltage representation, the PMSM mechanical dynamics are defined as: (11) Jdωdt+frωr =Te−Tl J{{d\omega } \over {dt}} + {f_r}{\omega _r}\; = {T_e} - {T_l}

The EKF adapts the conventional Kalman filter for use with non-linear systems. It provides recursive, real-time estimates of system states in the presence of measurement and process noise. The application of the first-order Taylor expansion in the approximation of the linearity function at the current state estimation helps the EKF to transform non-linear models into locally linearised forms described by their Jacobian matrices (Khodarahmi and Maihami, 2023).

Each update cycle consists of two fundamental stages: prediction, where the next state is forecasted using the system model, and correction, where this prediction is refined using newly acquired measurements (Kaniewski, 2020). The process iterates every sampling period T_s. The notion k signifies the moment after measurement updates, while k-1 indicates the moment before these updates occurred (Naoui et al., 2023).

The prediction process and update of the EKF are illustrated in Figure 3.

Figure 3.

EKF flow diagram. EKF, extended Kalman filter.

The initialisation stage is a crucial step where the state-space formulation of the EKF is expressed as follows: (12) xk+1=gxk,uk+ωkzk=hxk+vk \left\{ {\matrix{ {x\left( {k + 1} \right) = g\left( {x\left( k \right),u\left( k \right)} \right) + \omega \left( k \right)} \hfill \cr {z\left( k \right) = h\left( {x\left( k \right)} \right) + v\left( k \right)} \hfill \cr } } \right. Where:

• x(k): State matrix;

• g(·): Non-linear state-transition model;

• z(k): Measurement matrix;

• h(·): Measurement function;

• w(k), v(k): The process and measurement zero-mean Gaussian noise vectors.

The second step is the prediction of the state, and the covariance of the predicted estimate is computed as follows: (13) P(k+1|k)=GkpkG(k)T+Q P(k + 1|k) = G\left( k \right)p\left( k \right)G{(k)^T} + Q

Where:

Q is the covariance matrix of the process noise, while G(k) is the Jacobian matrix represented by the following equation: (14) Gk=∂gxk,uk∂xTkxk=x^k G\left( k \right) = {\left. {{{\partial g\left( {x\left( k \right),u\left( k \right)} \right)} \over {\partial {x^T}\left( k \right)}}} \right|_{x\left( k \right) = \hat x\left( k \right)}}

When the measurement becomes available, the correction phase includes the updates beginning from the gain parameter K(k + 1) followed by the updated error covariance matrix update P(k + 1 | k + 1), which are calculated as follows: (15) Kk+1=P(k+1|k)HTk+1P(k+1|k)HTk+1+R−1 K\left( {k + 1} \right) = P(k + 1|k){\left[ {{H^T}\left( {k + 1} \right)P(k + 1|k){H^T}\left( {k + 1} \right) + R} \right]^{ - 1}} (16) P(k+1|k+1)=I−Kk+1Hk+1P(k+1|k) P(k + 1|k + 1) = \left[ {I - K\left( {k + 1} \right)H\left( {k + 1} \right)} \right]P(k + 1|k)

Where h (k + 1) is the Jacobian of h(·), and R represents the measurement noise covariance.

Finally, the estimated state vector is corrected according to: (17) x^k+1|k+1=x^k+1|k+Kk+1zk+1−hx^k+1|k \hat x\left( {k + 1|k + 1} \right) = \hat x\left( {k + 1|k} \right) + K\left( {k + 1} \right)\left[ {z\left( {k + 1} \right) - h\left( {\hat x\left( {k + 1|k} \right)} \right)} \right]

A PMSM model formulation in the stationary α–β reference frame is preferred to ensure independence from position sensors. This choice helps to guard against rotor position initialisation errors, the impact of which could be detrimental to the accuracy of the EKF.

The state equation of the PMSM is given by the following equation: (18) x˙=Ax,u+Bu \dot x = A\left( {x,u} \right) + B\left( u \right) where: x = [iα, iβ, ω, θ]^T: State matrix, u = [vα, vβ]: Input matrix, and y = [iα, iβ]^T: Output matrix.

For transposition into the discrete frame time, Eq. (19) can be linearised as follows: (19) xk+1=fxk,uk+ωk {x_{k + 1}} = f\left( {{x_k},{u_k}} \right) + {\omega _k} where ω_k characterises the discrete noise vector.

As we mentioned previously, this proposed method is an enhancement of EKF’s sensitivity for ITSC faults, by upgrading the stator resistance of the studied PMSM from FL to T2-FL, consistently with the same objectives of maintaining the best performance of the detection algorithm, in addition to conservation of the simplified implementation.

Figure 4 depicts the configuration of the type-2 fuzzy-based stator-resistance estimator (Dilmi et al., 2021).

Figure 4.

T2-FL estimator configuration. T2-FL, type-2 fuzzy logic.

3.3.1.

FL principle

The inputs of the T2-FL estimator are E and E′, processed by a Mamdani inference pattern composed of 25 rules. The Karnik–Mendel algorithm, which is the crucial component, ensures the reduction used to estimate uncertainties before proceeding to the defuzzification phase using the centroid method. This principle is illustrated in Figure 5.

Figure 5.

T2-FL estimator structure. T2-FL, type-2 fuzzy logic.

3.3.2.

Fuzzy inputs and outputs

The estimator measures the stator current matrix error, characterised as the deviation of I*, the estimated current and I relative to the reference I. It is expressed as follows: (20) e=I*−II*=(iα*)2+(iβ*)2I=iα2+iβ2 \left\{ {\matrix{ {e = {I^*} - I} \hfill \cr {{I^*} = \sqrt {{{(i_\alpha ^*)}^2} + {{(i_\beta ^*)}^2}} } \hfill \cr {I = \sqrt {i_\alpha ^2 + i_\beta ^2} } \hfill \cr } } \right.

The inputs to the T2-FL system are the current error E and its derivative E′, both normalised to lie within the interval [−1, 1]. The type-2 fuzzy output is the incremental adjustment amount in stator resistance (ΔR_s). The final stator resistance is then obtained through the recursive addition: (21) Rsk+1=Rsk+ΔRsk {R_s}\left( {k + 1} \right) = {R_s}\left( k \right) + \Delta {R_s}\left( k \right)

As illustrated in Figure 6, the application of triangular membership functions enabled fuzzification of the exact input and output data into their respective T2-FL variables. In this investigation, the variables of the T2-FL estimator are resolved into three constituent levels.

Figure 6.

T2-FL membership functions configuration. T2-FL, type-2 fuzzy logic. (a) Input variables and (b) Output variable.

During the simulation evaluation, the faulty condition is represented by R_f = 1 Ω beside a variation of T_fault. The number of shorted turns in the sub-winding indicates the severity of the ITSC fault.

An application of a load torque of 1 N.m takes place in both the healthy and faulty states of the PMSM.

The two main objectives to be achieved in this simulation are the performance of the PMSM in terms of speed and control, and the sensitivity of ITSC detection under various conditions. Hence, the evaluation includes two tasks to examine:

• Robustness evaluation

• Sensitivity evaluation

In order to evaluate the proposed approach, we opt to implement a reduced driving cycle starting with acceleration at t = 0.1 s to a speed of 160 rpm. Then, starting at 0.4 s, braking occurs, resulting in a deceleration until reaching 80 rpm. At 0.5 s, the engine rotation stops due to a second braking action, followed by acceleration to 120 rpm at t = 0.8 s. Finally, the rotational speed stabilises at 80 rpm until the end of the simulation.

This scenario represents a real-world driving experience with its various braking, acceleration and stopping states, which are typical in an urban area.

Figure 7 illustrates the temporal evolution of the PMSM speed obtained using the T2-FL–EKF method, compared with the actual measured speed. The results show excellent agreement between the two signals, demonstrating the high accuracy of the proposed speed observer. The small temporary differences observed during rapidly changing acceleration and deceleration conditions can be explained by the estimation model’s response time and the effects of mechanical inertia. Under steady-state conditions, it was found to coincide with the measured speed, demonstrating the high accuracy and robustness of the T2-FL–EKF control strategy applied to the PMSM system.

Figure 7.

Dynamic speed regression.

In parallel with the development of speed, Figure 8 illustrates the reference driving cycle and the estimated angular velocity of the rotor (rad/s), giving a clearer view of how the estimator performs dynamically as expected. The comparison shows differences during sudden changes in the system due to the low delays of the observer in detecting rapid speed variations.

Figure 8.

Rotor angular velocity.

As shown in Figure 9, the estimated and measured angular positions are in acceptable agreement between the two signals. The curves follow the same model, showing that the estimation model is well synchronised with the actual motor.

Figure 9.

Angular position.

These results demonstrate that the FL–EKF type II control provides an accurate estimate of the rotor position, which is required for effective vector control and sufficient synchronisation of stator and rotor.

This part concerns evaluation of the sensitivity of our upgraded approach to the detection of ITSC defects even in severe cases.

In fact, for this task R_f the fault resistance is set to 1 Ω, while we applied a variation in T_fault which implicates a variation in the ITSC fault severity. For the simulation, the ITSC fault appears at 0.8 s and its parameter variations are as indicated in Table 2.

The impact of a 50% ITSC fault, triggered at 0.8 s, is immediately identifiable by the asymmetry and instability of the current signals in Figure 10. Beyond transient fluctuations, ITSC faults can lead to mechanical vibrations and speed instabilities that are detrimental to ride quality. The persistence of such anomalies exacerbates thermal stresses on the windings, thus accelerating the motor’s ageing process and compromising its operational reliability.

Figure 10.

I_abc stator currents.

In Figure 11, the vector control currents evolution is carried out. At 0.8 s, increased ripples, rapid peaks and decaying oscillations can be observed in I_d and I_q currents. This reflects the perturbation of the electromagnetic torque. These anomalies in I_q directly translate into torque excursion and velocity variations, while alterations in I_d affect magnetic saturation and flux control accuracy.

Figure 11.

Vector control currents.

Figure 12 illustrates the estimated and measured stator resistance, respectively, for µ = 25% and µ = 50%.

Figure 12.

Stator resistance measurements. (a) μ = 25% and (b) μ = 50%.

By calculating the root mean square error (RMSE), the T2-FL–EKF method shows a significant reduction in RMSE value as the fault coefficient µ decreases: the RMSE drops from 0.0076 when µ = 50% to 0.0052 when µ = 25%. This result reflects an increased ability of the proposed method to converge towards the real value despite the level of the disturbances induced by the ITSC fault.

Figure 13 shows the evolution of the estimated and measured fault current. Here, we proceeded to quantitatively evaluate the precision and accuracy of the T2-FL–EKF method.

Figure 13.

Fault current measurements. (a) μ = 25% and (b) μ = 50%.

Table 3 summarises the fault current value (I_{f moy}) determined at different fault levels, where the calculation of the RMSE.

The comparison of the estimated and measured average values I_{f moy}, shows that the two signals generally follow the same evaluation with a limited difference presented by RMSE.

For μ = 50%, the RMSE value is 0.05513, while for μ = 25% it is 0.2303. This significant reduction confirms that the proposed method effectively extracts fault signatures, regardless of its severity level, and thus offers more precise and sensitive detection in situations where the fault remains weakly visible and thus hardest to detect.

To evaluate the proposed methodology (T2-FL–EKF) for ITSC fault detection, a comparative study was conducted with the classical EKF (Maanani and Menacer, 2019) and the hybrid EKF based on FL (Romdhane et al., 2023). In this study, and to ensure the validity of comparison, the three algorithms were evaluated within a unified framework. Each simulation, with a total time of 0.75 s, involved the application of an ITSC fault at time t = 0.2 s. The comparative evaluation of the algorithms was intentionally conducted under low fault visibility conditions (µ = 25%). After initial tests showed that ITSC fault detection was comparable for all algorithms at a 50% visibility level, it became necessary to focus on a more challenging scenario. The study at µ = 25% highlights the limitations of each algorithm and allows for a better assessment of its effectiveness under degraded conditions.

The comparison of the stator resistance R_s estimation precision of the three algorithms mentioned is presented in Figure 14. The EKF estimator observation in Figure 14a, shows large-amplitude fluctuations, although it is capable of tracking the stator resistance value. Without an intelligent compensation mechanism, the estimation remains subject to stochastic uncertainties. The integration of FL, as illustrated in Figure 14b, smooths the estimation. By acting as a smart filter, the fuzzy module absorbs some of the measurement noise. However, the fixed membership functions of Type 1 do not fully account for variable uncertainty. From Figure 14c, we conclude that the proposed approach T2-FL–EKF offers the most stable signal with a significant reduction in fluctuation amplitude. This performance is explained by the FOU, which allows the system to better tolerate inaccuracies in the EKF residuals, ensuring smooth convergence and increased accuracy in the stator resistance estimation.

Figure 14.

R_s estimation precision: (a) EKF approach, (b) FL–EKF approach and (c) T2-FL–EKF approach. T2-FL–EKF, type-2 fuzzy logic and extended Kalman filter; FL–EKF, fuzzy logic–extended Kalman filter.

The comparative analysis of the three approaches in Table 4, based on the RMSE and mean absolute percentage error (MAPE) indices, highlights the superiority of the T2-FL–EKF method.

The RMSE results reveal that the upgrade to the T2FL–EKF approach meets a minimum error of 0.0066416, representing a 36% improvement over the basic EKF method. This reduction demonstrates the system’s ability to effectively stabilise the stator resistance estimation under high uncertainties. The MAPE analysis confirms this efficiency with a relative error of only 1.6101%.

Although the difference between the FL–EKF estimator and the T2-FL–EKF estimator appears small at the high-visibility level, the upgraded T2-FL–EKF approach offers superior robustness to non-linear variations, ensuring reliable diagnosis, especially in weakly visible ITSC faults where conventional methods might diverge.

Nevertheless, the high sensitivity of the proposed approach in ITSC fault detection for PMSM, even under low visibility of the fault, T2-FL–EKF has some limits. Mainly, the algorithm complexity compared with FL–EKF, regardless, is manageable by the main Karnik–Mendel algorithm. We can also find that the sensibility can be affected through footprint of uncertainty (FOU), and an inappropriate calibration can lead to an increase in the response time.