Position Sensor Fault-Tolerant Control of IPMSM Drives Using Hybrid Sensorless Estimation over the Full Speed Range ^†

Abstract

This study proposes a new hybrid sensorless control estimation structure that provides a fault-tolerant control (FTC) when a position sensor fault occurs in a permanent magnet synchronous motor (PMSM). The proposed hybrid estimator combines two techniques using a speed-dependent weighting transition. High-frequency signal injection is applied at standstill and low speeds, while an extended Back-EMF observer takes over the position estimation at higher speeds. Within the transition band, the two estimators are combined by a vectorial cross-fade, resulting in a continuous position estimate. The sensor fault is detected by monitoring the angular difference between the sensor output and the complementary sensorless estimation output. The residual threshold is adjusted dynamically so that false triggers and missing triggers are avoided during acceleration and deceleration. Additionally, a decision-making delay is added to prevent noise-induced false alarms. When the fault is detected, the position feedback is smoothly switched to hybrid sensorless operation without interruption. The proposed strategy is tested across the full speed range of the PMSM via simulation, including steady-state, transient, and both fault-free and sensor fault scenarios. As a result of these scenarios, a new hybrid sensorless estimation structure capable of handling low-amplitude sensor biases in the position sensor across the full speed range is designed. Also, the Back-EMF estimation is compared with the extended Kalman filter at medium and high speeds, demonstrating a lower error detection threshold, quicker response, and reduced computational load.

1. Introduction

Interior permanent magnet synchronous motors (IPMSMs) are widely used in electric vehicle traction and industrial servo drives due to their high torque density and wide constant power region. Field-oriented control (FOC) of these machines requires continuous rotor position feedback, which is generally provided by a resolver or encoder. Sensor faults and failures such as complete loss of signal, intermittent loss or measurement error can disrupt the d-q decoupling and cause uncontrolled torque responses. Therefore, fault-tolerant control (FTC) strategies that detect faults and reconfigure the drive to sensorless operation are commonly used in safety-critical drive designs [1,2].

Sensorless position estimation methods depend on operating speed. At medium and high speeds, Back-Electromotive Force (Back-EMF)-based observers estimate the rotor angle from induced voltage. An extended Back-EMF model is used to design a simple disturbance observer for salient-pole permanent magnet synchronous motors (PMSMs) without model approximations [3]. However, these approaches fail at low and zero speeds because the Back-EMF magnitude becomes very small when the motor is near standstill. At these speeds, high-frequency (HF) signal injection techniques fill this gap by using magnetic saliency (L_d ≠ L_q) of the IPMSM [4]. A superimposed carrier voltage in high-frequency signal injection (HFSI) causes a position dependent current response and rotor angle is demodulated from this response. Pulsating injection offers higher accuracy and lower sensitivity to inverter dead-time effects than rotating injection [5]. Since neither technique alone covers full speed range, a transition mechanism is needed for continuous sensorless operation over the full speed range [6].

Two transition strategies have been reported in the literature: hysteresis-based hard switching and weighting-based soft transition. Hysteresis-based switching is computationally simple, but it causes torque and speed ripples at transition point, because two estimators produce different outputs at switching instant. On the other hand, the weighting-based transition eliminates these discontinuities by combining both estimates within speed band. This implementation reduces injection loop gain linearly to zero during the transition region [6]. In this paper, a vectorial cross-fade has been used based on sin/cos decomposition. This method also eliminates 2π wrapping effects that exist in direct angle combinations.

For fault detection, a commonly used approach in the literature monitors the deviation between the sensor angle and a sensorless estimation. The reconfiguration is triggered when this difference exceeds a threshold value [1]. A fixed threshold should be greater than worst case transient overshoot during acceleration and deceleration. This limitation causes slower detection and increases the duration of the current disturbance [7]. This trade-off is improved by adaptive threshold designs based on phase-locked loop (PLL) estimation dynamics and cumulative sum (CUSUM) algorithm [8]. However, current implementations do not cover standstill or low speeds because they mostly depend on Back-EMF estimates [9]. Statistical alternatives to geometric thresholding have also been explored, including sequential hypothesis testing [10], Mahalanobis-distance-based observation weighting [11], and residual-duration dual-parameter diagnosis for Kalman-filter-based position sensor fault detection [12].

This paper proposes a hybrid FTC architecture that performs fault detection across the entire speed range with dynamic threshold. The main contributions are as follows:

• A hybrid sensorless estimator that runs HF pulsating injection at standstill and low speeds and an extended Back-EMF disturbance observer at medium and high speeds in the background of the sensor based FOC. This estimator provides full speed range diagnostic.
• A vectorial weighted transition that combines two estimations within transition band by using sin/cos decomposition. This transition provides a smooth combined position information without wrapping.
• The residual signal is determined based on the difference between the position information from sensor and estimation from sensorless methods. Thanks to the on-delay and dynamic threshold, false alarms are prevented.
• A smooth reconfiguration is ensured, when a fault is detected. Estimation methods always run in the background for this bumpless reconfiguration.

The proposed strategy is validated on a MATLAB/Simulink (version R2023a; The MathWorks Inc., Natick, MA, USA) model of an IPMSM drive. The remainder of this paper is organized as follows. Section 2 presents the IPMSM model. Section 3 explains the hybrid estimation and transition algorithm. Section 4 describes the fault detection and reconfiguration logic. Section 5 presents the simulation results, and Section 6 concludes the paper.

2. Mathematical Model of the IPMSM

The IPMSM model used throughout this work is formulated in the rotor reference frame and then transformed into the estimated synchronous frame to support the Back-EMF observer. The model has been limited to the fundamental electrical behaviour relevant to position estimation and fault detection, cogging torque, iron losses, and magnetic saturation have been neglected.

2.1. Voltage Equations in the Rotor Reference Frame

Applying the Park transformation to three phase stator voltage equations yields the IPMSM model in rotor aligned d-q frame

$$\begin{array}{ll}{v}_d& =R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{\mathit{dt}}}-{\omega }_e{L}_q{i}_q,\\ v_q& =R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{\mathit{dt}}}+{\omega }_e\left(L_d{i}_d+{\psi }_m\right),\end{array}$$

(1)

where v_d, v_q and i_d, i_q are the d- and q-axis stator voltages and currents, R_s is the stator resistance, L_d and L_q are the d- and q-axis inductances, ψ_m is the permanent magnet flux linkage, and ω_e = p ω_m is the electrical angular velocity, with p the number of pole pairs and ω_m the mechanical angular velocity. For an IPMSM the inductances satisfy L_q > L_d, this magnetic saliency has been used on HF injection method introduced in Section 3.1.

2.2. Extended Back-EMF Formulation

The model (1) cannot be applied directly in a sensorless drive because the d-q frame is defined by the unknown rotor angle θ_e. The geometric relations between relevant reference frames have been summarized in Figure 1. Transforming (1) into the estimated synchronous frame γ-δ, whose orientation differs from d-q by the estimation error $\Delta \theta ={\theta }_e-{\widehat{\theta }}_e$, does not by itself resolve the difficulty. When L_d ≠ L_q, the rotor frame inductance matrix diag(L_d,L_q) is not invariant under this change of basis, and the transformed model contains non-diagonal terms that directly depend on Δθ.

Figure 1. Reference frames: stationary α-β, rotating d-q and estimated γ-δ.

The extended Back-EMF formulation [3,13] eliminates this coupling by an algebraic rearrangement of (1). That absorbs all saliency-related effects into a single q-axis voltage source while providing inductance matrix free of speed-dependent terms. After performing the required substitutions and simplification, the IPMSM model takes the compact matrix form

$$\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{cc}{R}_s+s{L}_d& -{\omega }_e{L}_q\\ {\omega }_e{L}_q& R_s+s{L}_d\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}0\\ E_{\mathit{ex}}\end{array}\right],$$

(2)

where s≜d/dt and the extended Back-EMF is defined as

$$E_{\mathit{ex}}={\omega }_e\left[\left(L_d-L_q\right)i_d+{\psi }_m\right]-\left(L_d-L_q\right){\displaystyle \frac{d{i}_q}{\mathit{dt}}}.$$

(3)

Two properties of (2) underpin the observer design. The inductance appears in impedance matrix as sL_d, that is independent of speed. Also, all rotor position information has been included in the single voltage vector ${\left[0,\hspace{0.17em}{E}_{\mathit{ex}}\right]}^{\mathrm{\top }}$ aligned with q-axis. Because of these two features, Equation (2) keeps the same form in any synchronous frame. Sensorless estimation therefore reduces to finding the direction of this voltage vector.

Expression (3) also exposes the principal limitation of Back-EMF. Since each term in E_ex is proportional to ω_e or di_q/dt, the rotor angle becomes unobservable during steady-state operation at standstill because the signal vanishes.

To express the model in the estimated frame, the rotation matrix

$$\mathbf{R}\left(\Delta \theta \right)=\left[\begin{array}{cc}\cos \Delta \theta & -\sin \Delta \theta \\ \sin \Delta \theta & -\cos \Delta \theta \end{array}\right]$$

(4)

is applied to the voltage and current vectors of (2) to perform the change of reference frame from d-q to γ-δ. Because the impedance matrix is independent of speed, it does not change form in the estimated frame and only the E_ex vector is rotated. Evaluating $\mathbf{R}\left(\Delta \theta \right){\left[0,\hspace{0.17em}{E}_{\mathit{ex}}\right]}^{\mathrm{\top }}$ yields the scalar projections

$$\begin{array}{rr}{e}_{\gamma }& =-E_{\mathit{ex}}\sin \left(\Delta \theta \right),\\ e_{\delta }& =-E_{\mathit{ex}}\cos \left(\Delta \theta \right).\end{array}$$

(5)

For small estimation errors sin(Δθ) ≈ Δθ, so e_γ behaves like linear function of the angular error with slope −E_ex close to Δθ ≈ 0. The Back-EMF observer described in Section 3.2 exploits this property by driving e_γ to zero through a PLL, when ${\widehat{\theta }}_e$ has converged to θ_e.

2.3. Mechanical Dynamics and Electromagnetic Torque

The electromagnetic torque of the IPMSM combines magnet alignment term and a reluctance term caused by saliency

$$T_e={\displaystyle \frac{3}{2}}\hspace{0.17em}p\hspace{0.17em}\left[{\psi }_m{i}_q+\left(L_d-L_q\right)\hspace{0.17em}{i}_d{i}_q\right].$$

(6)

The first term in (6) is conventional magnet torque, while the second is the reluctance torque exploited under maximum torque per ampere operation. The mechanical subsystem is defined by Newton’s equation

$$J\hspace{0.17em}{\displaystyle \frac{d{\omega }_m}{\mathit{dt}}}=T_e-T_L-B\hspace{0.17em}{\omega }_m,$$

(7)

together with the kinematic relation dθ_m/dt = ω_m, where J is rotor moment of inertia, B is viscous friction coefficient, and T_L is external load torque. The electrical position and speed are related to the mechanical quantities through θ_e = p θ_m and ω_e = p ω_m.

Equations (1)–(7), together with inverter and field-oriented current loop, build architecture where the hybrid sensorless estimator (Section 3) and the fault detection logic (Section 4) operate.

3. Hybrid Sensorless Position Estimation

3.1. High-Frequency Pulsating Signal Injection at Zero and Low Speeds

Since E_ex becomes zero near standstill, the HF pulsating signal injection method [4,5] is combined in this structure to restore position observability. It superimposes a high-frequency carrier voltage on the estimated d-axis of the controller and extracts rotor position from the carrier frequency current response, which is modulated by the magnetic saliency L_d ≠ L_q of the IPMSM.

A pulsating carrier $v_{\widehat{d}h}=V_h\cos \left({\omega }_{h}t\right)$ is added to the estimated d-axis voltage reference at angular frequency ω_h well above current loop bandwidth. Because the injection is defined in the γ-δ frame, its projection onto actual rotor frame is obtained through R^⊤(Δθ), giving

$$\left[\begin{array}{c}{v}_{\mathit{dh}}^r\\ v_{\mathit{qh}}^r\end{array}\right]=V_h\cos \left({\omega }_{h}t\right)\hspace{0.17em}\left[\begin{array}{c}-\cos \Delta \theta \\ -\sin \Delta \theta \end{array}\right].$$

(8)

Although pulsating carrier signal is only injected on the estimated d-axis, when Δθ ≠ 0 the same physical voltage vector projects onto both axes through cosΔθ and −sinΔθ. This q-axis projection, that becomes zero with perfect alignment, makes estimation error observable thanks to saliency L_d ≠ L_q.

At ω_h, the resistive and rotational terms in (1) are negligible when compared with inductive impedances jω_hL_d and jω_hL_q. Therefore, rotor carrier currents follow from term-by-term integration

$$\left[\begin{array}{c}{i}_{\mathit{dh}}^r\\ i_{\mathit{qh}}^r\end{array}\right]\approx {\displaystyle \frac{V_h\hspace{0.17em}\sin \left({\omega }_{h}t\right)}{{\omega }_h}}\left[\begin{array}{c}-\cos \Delta \theta /L_d\\ -\sin \Delta \theta /L_q\end{array}\right].$$

(9)

Transforming (9) back to estimated frame with R(Δθ) and applying the identity sinΔθ cosΔθ = 1/2sin(2Δθ), the carrier frequency component of the current on the estimated q-axis is

$$i_{\mathit{qh}}\approx {\displaystyle \frac{V_h\hspace{0.17em}\left(L_q-L_d\right)}{2\hspace{0.17em}{\omega }_h{L}_d{L}_q}}\hspace{0.17em}\sin \left({\omega }_{h}t\right)\hspace{0.17em}\sin \left(2\Delta \theta \right),$$

(10)

which carries the estimation error with saliency dependent term (L_q − L_d)/(L_dL_q). Critically, (10) is independent of ω_e, so the response continues at standstill.

The position information is extracted from (10) in three steps. A second order bandpass filter centred at ω_h with bandwidth B_w

$$H_{\mathit{BP}}\left(s\right)={\displaystyle \frac{B_w\hspace{0.17em}s}{s^2+B_{w}s+{\omega }_h^2}},$$

(11)

isolates the carrier frequency component of i_q from fundamental drive currents. Multiplying bandpass output by the synchronization signal sin(ω_ht) and applying sin²(ω_ht) = 1/2[1 − cos(2ω_ht)] yields

$$i_{\mathit{qh}}\sin \left({\omega }_{h}t\right)={\displaystyle \frac{V_h\left(L_q-L_d\right)}{4\hspace{0.17em}{\omega }_h{L}_d{L}_q}}\left[1-\cos \left(2{\omega }_{h}t\right)\right]\sin \left(2\Delta \theta \right),$$

(12)

in which the position information is shifted to baseband, while the modulation simultaneously generates a high-frequency image at 2ω_h. A first-order low-pass filter (LPF) with cutoff ω_LP ≪ 2ω_h,

$$H_{\mathit{LP}}\left(s\right)={\displaystyle \frac{{\omega }_{\mathit{LP}}}{s+{\omega }_{\mathit{LP}}}},$$

(13)

suppresses the high-frequency image, leaving baseband error signal

ε_hf = K_hfsin(2Δθ) ≈ 2K_hf Δθ,

(14)

where the demodulation gain is

$$K_{\mathit{hf}}={\displaystyle \frac{V_h\left(L_q-L_d\right)}{4\hspace{0.17em}{\omega }_h{L}_d{L}_q}}.$$

(15)

The error signal ε_hf behaves like linear function of the angular error while Δθ ≈ 0. The progression of the position-bearing content through this chain is shown in Figure 2. After the bandpass filter, position information is centralized at ω_h with amplitude proportional to sin(2Δθ). Then as a result of multiplication operation, position information carrier signal and high-frequency image at 2ω_h are formed. Lastly, this high-frequency image signal is filtered with LPF to obtain only position information signal, and the error signal is derived from the remaining component. The distinctive feature of pulsating injection is that position information appears as amplitude of carrier signal and is recovered after demodulation. In rotating signal injection, by contrast, the position information appears at phase of carrier signal rather than amplitude, requiring a different demodulation structure.

Figure 2. Pulsating injection demodulation: (a) bandpass output and (b) multiplication by sin(ω_ht).

A type-2 PLL is used to get ε_hf, a proportional–integral (PI) controller in this PLL drives it to zero and output of controller yields the speed estimate ${\widehat{\omega }}_e^{\mathit{hf}}$. Then that speed is also integrated to obtain the angle estimate ${\widehat{\theta }}_e^{\mathit{hf}}$. In order to clearly separate the angle dynamics from carrier residuals, the PI bandwidth is set well below ω_LP.

The reliability of the injection-based estimator degrades at higher speeds because inverter dead-time and switching effects attenuate the carrier signal. Furthermore, as speed increases, the fundamental current frequency approaches to injection frequency, degrading signal quality and demodulation. These effects set an upper bound on the speed at which the HF estimate is reliable.

3.2. Extended Back-EMF Observer at Medium and High Speeds

In this section, Back-EMF observer operates directly on the rotor frame (2) in Section 2.2. It reconstructs the extended Back-EMF components on estimated γ-δ axes and calculates θ_e from their orientation.

Rearranging (2) for current dynamic and applying same transformation to obtain (5) yields the voltage equations in estimated frame

$$\left[\begin{array}{c}{v}_{\gamma }\\ v_{\delta }\end{array}\right]=\left[\begin{array}{cc}{R}_s+s{L}_d& -{\widehat{\omega }}_e{L}_q\\ {\widehat{\omega }}_e{L}_q& R_s+s{L}_d\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right],$$

(16)

where ${\widehat{\omega }}_e$ is the speed estimation that produced by observer and (v_γ,v_δ) and (i_γ,i_δ) are measurable quantities from sensors. Solving (16) for extended Back-EMF components gives

$$\begin{array}{rr}{\widehat{e}}_{\gamma }& =v_{\gamma }-R_s{i}_{\gamma }-L_d\hspace{0.17em}s\hspace{0.17em}{i}_{\gamma }+{\widehat{\omega }}_e{L}_q{i}_{\delta },\\ {\widehat{e}}_{\delta }& =v_{\delta }-R_s{i}_{\delta }-L_d\hspace{0.17em}s\hspace{0.17em}{i}_{\delta }-{\widehat{\omega }}_e{L}_q{i}_{\gamma }.\end{array}$$

(17)

Direct implementation of (17) is not practical because pure differentiator s amplifies measurement noise on currents. Two filtering stages are necessary in there. Firstly, low-pass filter is applied on each component

$$\widetilde{s\hspace{0.17em}i}={\displaystyle \frac{{\omega }_c\hspace{0.17em}s}{s+{\omega }_c}}\hspace{0.17em}i,$$

(18)

where ω_c is chosen well above the fundamental but below the switching harmonics. That limits band of di/dt. Secondly, to suppress the slow drift and DC bias that accumulate in (17), a high-pass filter (HPF) with the same cutoff is applied on each component

$$H_{\mathit{HP}}\left(s\right)={\displaystyle \frac{s}{s+{\omega }_c}}.$$

(19)

The combined LPF and HPF structure keeps speed proportional term of E_ex as original, while rejecting current noise and steady-state offsets.

Once ${\widehat{e}}_{\gamma }$ and ${\widehat{e}}_{\delta }$ have been calculated, the estimation error is determined from their orientation with arctangent expression

$${\epsilon }_{\mathit{bemf}}=\mathrm{atan}2 \left(-{\widehat{e}}_{\gamma },\hspace{0.17em}{\widehat{e}}_{\delta }\right)\approx \Delta \theta ,$$

(20)

that takes advantage of projection structure of (5). When Δθ is small, the arctangent reduces to the linear approximation Δθ ≈ −e_γ/E_ex used in Section 2.2. However, (20) preserves the correct sign and range over the full [−π,π] interval. That is critical during transients where Δθ may briefly exceed the small angle range.

As in Section 3.1, a type-2 PLL is used to get ε_bemf. PI controller in this PLL drives it to zero and output of controller yields the speed estimate ${\widehat{\omega }}_e^{\mathit{bemf}}$. To reduce ripple and switching noise on ${\widehat{\theta }}_e^{\mathit{bemf}}$, a first-order filter is applied to the angle output with cutoff ω_pf,

$$H_{\mathit{pf}}\left(s\right)={\displaystyle \frac{{\omega }_{\mathit{pf}}}{s+{\omega }_{\mathit{pf}}}},$$

(21)

As ω_e decreases, the signal-to-noise ratio of (17) changes because ${\widehat{e}}_{\gamma }$ and ${\widehat{e}}_{\delta }$ become smaller while measurement and quantization noise remain the same. Below the threshold mechanical speed, arctangent expression in (20) begins to lose its accuracy. Therefore, observer is enabled only above this speed threshold.

3.3. Weighted Transition Between Estimation Methods

The HFSI and Back-EMF estimations only cover their operating range, so a continuous transition between methods is required for full speed range operation. The approach applies linear speed-dependent weight with vectorial combination of the two angle estimations.

Let ω₁ < ω₂ bound transition band in electrical speed. The weighting coefficient is defined as

$$w\left(\left|{\widehat{\omega }}_e\right|\right)=\left\{\begin{array}{ll}0,& \left|{\widehat{\omega }}_e\right|<{\omega }_1,\\ {\displaystyle \frac{\left|{\widehat{\omega }}_e\right|-{\omega }_1}{{\omega }_2-{\omega }_1}},& {\omega }_1\le \left|{\widehat{\omega }}_e\right|\le {\omega }_2,\\ 1,& \left|{\widehat{\omega }}_e\right|>{\omega }_2,\end{array}\right.$$

(22)

so that w = 0 selects HFSI estimation, w = 1 selects the Back-EMF estimation, and intermediate values produce combined estimation according to weighting coefficient.

A direct combination of the two angle outputs is unsuitable because of 2π wrapping. Two estimations can differ by approximately 2π even when both are accurate. Also, when the speed exits the transition region, angle estimation would cause jump on estimation. Therefore, angle combination is performed with Cosine–Sine decomposition,

$$\left[\begin{array}{c}\cos {\widehat{\theta }}_e\\ \sin {\widehat{\theta }}_e\end{array}\right]=\left(1-w\right)\left[\begin{array}{c}\cos {\widehat{\theta }}_e^{\mathit{hf}}\\ \sin {\widehat{\theta }}_e^{\mathit{hf}}\end{array}\right]+w\left[\begin{array}{c}\cos {\widehat{\theta }}_e^{\mathit{bemf}}\\ \sin {\widehat{\theta }}_e^{\mathit{bemf}}\end{array}\right],$$

(23)

from which the composite angle is recovered as

$${\widehat{\theta }}_e=\mathrm{atan}2\left(\sin {\widehat{\theta }}_e,\hspace{0.17em}\cos {\widehat{\theta }}_e\right).$$

(24)

Interpolation is performed on unit circle instead of unwrapped angle. As a result, continuity across 2π is ensured and angle jumps are avoided.

Applied low-pass filter and rate limiter further smooth the transition. That filter applied to $\left|{\widehat{\omega }}_e\right|$ suppresses high-frequency speed changes to prevent chattering near region limits. Also, rate limiter limits slew rate during abrupt speed transients. The resulting weighting profile and operating regimes are summarized in Figure 3.

Figure 3. Speed-dependent weighting profile of the hybrid sensorless estimator.

4. Fault Detection and Control Reconfiguration

The fault detection and isolation (FDI) layer runs in parallel with the sensored FOC to generate fault flag. The hybrid estimator in (Section 3) operates continuously in the background, so residual is available across the full speed range including standstill.

4.1. Residual Generation

The residual is angular difference between encoder angle θ_enc and sensorless estimation ${\widehat{\theta }}_e$ from (24). Direct subtraction is discontinuous at 2π boundary, so shortest path form

$$r=\mathrm{atan}2\left(\sin \left({\theta }_{\mathit{enc}}-{\widehat{\theta }}_e\right),\cos \left({\theta }_{\mathit{enc}}-{\widehat{\theta }}_e\right)\right)$$

(25)

maps the result into (−π, π] and prevents 2π wrapping issue. A washout high-pass filter removes accumulated bias between two estimations during normal operation.

4.2. Adaptive Threshold with Dynamic Scaling

A fixed threshold must exceed the worst-case transient excursion of r during speed ramps, which delays detection of low-amplitude faults [7]. The threshold is therefore formed as

δ_th = [(1 − w)δ_hf + wδ_be] + k_dyn|i_q|

(26)

The bracketed term is the base threshold, which is not constant but is calculated according to the same speed-dependent weight coefficient described in Section 3. Here, δ_hf and δ_be are the constant threshold values associated with the HFSI and Back-EMF methods. The base threshold is interpolated between them so that each method has its own threshold. The dynamic term k_dyn then scales the total threshold with the q-axis current. During acceleration, large |i_q| increases total threshold and suppresses false alarms. At steady state it becomes close to base threshold and sharpens sensitivity. An asymmetric rate limiter extends the decay tail so the threshold does not fade faster than the transient it tracks [8].

4.3. Fault Confirmation and Seamless Reconfiguration

When |r| ≥ δ_th, an on-delay timer requires fault to persist for T_conf continuously before raising fault flag. Thus, it detects fast position jumps by rejecting noise-induced faults [14]. Upon confirmation, SR latch sets the permanent fault flag F with no reset path. Reconfiguration is irreversible to avoid mode oscillation due to intermittent sensor faults [1]. The complete signal flow, from residual generation to reconfiguration, is shown in Figure 4: when the flag F is set, sensor feedback θ_enc is switched with hybrid sensorless estimation ${\widehat{\theta }}_e$, reducing reconfiguration to a single position source selection. Since the estimator has been running in the background during sensored operation, ${\widehat{\theta }}_e$ has already converged to θ_enc, so no transient torque is observed during transition [15].

Figure 4. Signal flow diagram of the FDI and control reconfiguration; fault flag F switches angle.

5. Simulation Results and Discussion

5.1. Simulation Platform

The motor speed reference follows step ramp profile from standstill to the maximum operating speed. This speed reference is designed to pass through the HFSI-dominant region, the transition band, and the Back-EMF-dominant region. One fault-free scenario and three fault scenarios are evaluated. Fault injection is a step angle offset applied to encoder output at specified time step. The simulation platform and system parameters are listed in Table 1.

Table 1. Simulation platform and system parameters.

Parameter	Value	Parameter	Value
Motor Parameters
Rs	0.2 Ω	ψm	0.2078 Wb
Ld	4.3 mH	p	3
Lq	8.7 mH	J	0.017 kg·m2
Simulation Platform
Vdc	400 V	Ts	1 μs
Estimator Parameters
ωh/2π	1000 Hz	HFSI PLL	Kp = 4, Ki = 100
Vh	27 V	Back-EMF PLL	Kp = 50, Ki = 2500
Limits
Transition band	40–60 rad/s	EMF Activation speed	15 rad/s

5.2. Steady-State Estimation Accuracy

Figure 5 shows hybrid estimator speed output ω_est tracking encoder speed ω_enc in the fault-free scenario, together with estimation error Δω_m between sensor speed information and speed estimation. The hybrid output follows the reference at all three steady-state speeds with the remaining error within a narrow band. This confirms that the vectorial combination introduces no bias at any operating point.

Figure 5. Fault-free scenario. (a) Encoder speed ω_enc and hybrid estimate ω_est over full speed range. (b) Estimation error Δω_m.

Figure 6 zooms into the HFSI-dominant low-speed region and the Back-EMF-dominant high-speed region, comparing each estimation method against the extended Kalman filter (EKF) benchmark [16]. The saliency-based estimation ω_est,inj closely follows ω_enc at low speed, while the Back-EMF estimation ω_est,emf converges to the encoder reference at high speed. At low speed, EKF produces larger transient deviations, because Back-EMF amplitude decreases as electrical angular velocity approaches zero, reducing observability. At high speed, both methods converge to the reference but filter group delay and saliency mismatch in EKF model cause slight steady-state bias in EKF estimation. These findings confirm that neither a single Back-EMF observer nor an EKF covers both regimes, motivating the hybrid architecture.

Figure 6. EKF benchmark comparison. (a) Low-speed region: HFSI estimate ω_est,inj vs. EKF estimate ω_est,ekf. (b) High-speed region: Back-EMF estimate ω_est,emf vs. EKF estimate ω_est,ekf.

5.3. Transient Performance During Speed Ramp

Figure 7 shows the evolution of the weighting factor w_inj and w_emf during the ramp scenario. Below the transition band w_inj = 1 and w_emf = 0, so the HFSI estimation is dominant. As the speed information enters the transition region, both weighting factors vary continuously, and the vectorial cross-fade interpolates between two estimations. Above the upper boundary w_inj = 0 and w_emf = 1, transferring full authority to Back-EMF estimation.

Figure 7. Weighting factor evolution during the speed ramp.

5.4. Fault Detection and Reconfiguration Response

Three fault scenarios inject a 0.5 rad step angle offset into the encoder output at different time steps, covering all three estimation regimes. In every case, the injected offset exceeds the base threshold for long enough to activate confirmation timer T_conf.

Scenario 1 (Figure 8) injects a fault at low speed, where HFSI estimation is dominant. The residual r(t) exceeds the adaptive threshold δ_th immediately and remains above it for the full confirmation interval, setting the fault flag F. The speed response shows a brief transient as the controller switches to sensorless operation, with steady-state tracking recovered within one settling period.

Figure 8. Scenario 1: fault injected at low speed (HFSI dominant). (a) Residual r(t), adaptive threshold δ_th and fault flag F. (b) Speed response.

Scenario 2 (Figure 9) injects a fault in the transition region, where both estimators contribute to the combined output. The base threshold is interpolated between HFSI and Back-EMF base threshold values. The residual still exceeds the threshold cleanly, the fault flag is set within one confirmation interval, and reconfiguration speed transient is of comparable magnitude to Scenario 1.

Figure 9. Scenario 2: fault injected in the transition region. (a) Residual r(t), adaptive threshold δ_th and fault flag F. (b) Speed response.

Scenario 3 (Figure 10) injects a fault at high speed, where Back-EMF estimation is in full control. The residual clearly exceeds the adaptive threshold, the fault flag is set, and the drive continues at the same speed reference using the Back-EMF estimation, with only a minor transient response. Across all three scenarios, FDI layer produces no false alarms and achieves detection within expected confirmation time.

Figure 10. Scenario 3: fault injected at high speed (Back-EMF dominant). (a) Residual r(t), adaptive threshold δ_th and fault flag F. (b) Speed response.

6. Conclusions

A position sensor fault-tolerant control architecture for IPMSM drives was developed by combining high-frequency signal injection at standstill and low speeds with an extended Back-EMF observer at medium and high speeds. These methods are combined with a vectorial weighted transition that produces a continuous rotor angle across the full speed range. A fault detection and isolation layer monitors the residual from the difference between the encoder angle and the hybrid estimation. This triggers an irreversible reconfiguration to sensorless operation with a dynamically scaled threshold and a confirmation delay at the moment of detection. Across fault scenarios, including the HFSI-dominant, transition, and Back-EMF-dominant regimes, the residual cleanly exceeds the adaptive threshold with no false alarms during speed transients and tracking recovers after reconfiguration. A comparison with the extended Kalman filter, which loses Back-EMF observability near standstill and retains a small steady-state bias at high speed, supports necessity of the hybrid structure rather than a single estimation method. Future work will focus on extending structure to multiple parallel sensor faults and performing experimental validation on a hardware test bench.