Current Sensor Fault-Tolerant Control Strategy for Speed-Sensorless Control of Induction Motors Based on Sequential Probability Ratio Test (2024)

1. Introduction

IMs are extensively used in various industrial manufacturing facilities due to their high reliability, low cost, and simple structure. For a long time, achieving the precise speed regulation of IMs has been a goal pursued by experts and researchers in industrial and academic circles. Through continuous improvement, variable frequency speed regulation (VFSR) has emerged as an ideal speed regulation method for IMs. In recent years, speed-sensorless vector control, a low-cost and highly reliable VFSR technique, has gained significant attention in developing modern AC speed regulation systems [1,2,3]. This technique relies on the current signal provided by a current sensor for speed estimation. However, current sensors, a vulnerable component in the driving system, are prone to various faults.

When a current sensor is experiencing a malfunction, it provides erroneous current information, leading to the failure of the speed estimation scheme. Consequently, the circuit of the current AC speed control system cannot exert stable control, affecting the reliable operation of the speed-sensorless motor driver. Therefore, it is significant to study the current sensor fault-tolerant control of IM speed-sensorless control systems [4,5,6].

In recent years, the fault detection (FD) and fault-tolerant control (FTC) of current sensors have proven to be beneficial for improving the reliability of motor control systems [7,8,9,10]. The rapid and accurate detection of current sensor faults is an important prerequisite for implementing an FTC scheme. Currently, the FD methods for current sensors mainly fall into two categories: model-based methods and signal-based methods. Signal-based methods are susceptible to environmental factors. If random signals are among the probed signals, accurately extracting the target signals will be difficult [11,12]. Therefore, the model-based approach is adopted to realize the current sensor FD.

The model-based approach in [13] is proposed to realize the current sensor FD based on the concept of coordinate translation. They used the d-axis and q-axis currents as reference currents to compare with the measured current. This scheme has the advantages of high detection speed and easy implementation. However, it relies on the speed information provided by the speed sensor. A current sensor FD scheme for speed-sensorless control was introduced in [14]. Based on the concept of coordinate translation, the scheme required the reference current and measured current to be compared in two different stationary coordinate systems, achieving fast fault diagnosis and locating the faulty current sensor without the need to obtain speed information. However, the impact of random noise is ignored in these methods, and also the issue of control loop reconfiguration is not taken into account.

The EKF has a certain level of robustness to random noise and system model parameters. It was used for the speed-sensorless vector control of IMs and elucidated the basic principle of the EKF and its application in the field of the sensorless vector control of IMs [15]. They further delved into the effects of interference and motor parameter changes on the performance of speed-sensorless control, especially the impact on the process of estimating the speed of the IM. However, they did not provide any method to estimate the speed of the IM under the condition of current sensor malfunction.

A current sensor FD method based on an open-loop speed estimator was introduced in [16]. The method made a judgment on whether a fault occurred based on the residual between the current value estimated by the EKF and the actual value provided by the current sensor. Since IM vector control systems are closed-loop systems, the current sensors in a vector control system should be able to measure the parameters accurately in both normal and abnormal working conditions to achieve accurate speed estimation. In [17], an active fault detection strategy was proposed based on the Kalman filter (KF) for doubly fed induction generators (DFIGs) to correct the faults of current and voltage sensors. The fault detection and isolation (FDI) mechanism in this strategy required residual signals obtained from the current sensor and KF. Although the experimental verification of this study was conducted in a closed-loop system, only the fault information of the current sensor was discarded. In closed-loop systems, prolonged sensor failures can lead to system divergence, thus weakening the robustness of the system.

An FTC method was proposed for a single-phase current sensor used in the drive of the main permanent magnet linear motor [18], in which the actual current was replaced by reference current and residual phase current. A phase-shifted FTC (PS-FTC) method was proposed based on the single-phase current sensor for the traction system of the main permanent magnet linear motor [19]. In this scheme, the estimated values were employed to replace the second- and third-phase currents, and the reference values were obtained using the estimated phase shift of the first-phase current. Compared with the traditional single-phase current sensor FTC method, the PS-FTC method is more robust and has better steady-state performance. However, analysis shows that in IM vector control systems, the detected current signal of the stator contains a large number of odd harmonics and DC bias due to various factors such as the nonlinearity of the inverter, the spatial harmonics in the magnetic field, and asymmetric faults in the power grid. The above residual-based FD and FTC studies mostly substitute the fault current with an ideal current and do not filter the harmonics in the detected current signal. Thus, the injection of distorted information will inevitably affect the accuracy and stability of the speed estimator.

In [20], the current space vector error projection was utilized as the correction item, and the faulty phase current was reconstructed through the SMO. In [21], a current estimation scheme based on the second-order extended state observer (SOESO) was proposed, where only one b-phase current sensor was used to estimate the d-q axis stator currents. In [22], current reconstruction is achieved using the given d-axis and q-axis currents, needing to estimate the rotor’s position information. These approaches effectively achieve the single-phase current sensor speed-sensorless control of motor drives. It is worth noting that these schemes require the involvement of estimated speed or rotor position information for current reconstruction or error reconstruction. The reconstructed current is then employed for speed and position estimation, potentially leading to undesired coupling effects.

This paper proposes a current sensor fault-tolerant control strategy based on a sequential probability ratio test (SPRT), which is suitable for the speed-sensorless control of IMs to further improve the reliability of speed-sensorless control systems for IMs. First, under the condition of no speed sensor, the EKF estimates the speed of the IM. Then, the SPRT is used to identify the fault of the current sensor, and the current signal is reconstructed using the double-cascading second-order generalized integrator (DSOGI) principle. Second, the reconstructed current signal is fed back to the speed estimation scheme based on the EKF. In this way, the odd harmonics and the DC bias in the current signal can be eliminated accurately, thus achieving the fault-tolerant control of the current sensor in the speed-sensorless vector control system for IMs. Finally, the effectiveness of the proposed strategy is validated by a series of experiments, which are conducted on a 3 kW IM drive platform, and the results are presented in Section 5.

2. Induction Motor Speed Estimation Method Based on EKF

The state-space equations for an IM whose space variables are current and the rotor magnetic fluxes can be derived as follows [15]:

$\underset{\dot{x}}{\underset{︸}{[\begin{matrix} {\dot{i}}_{s α} \\ {\dot{i}}_{s β} \\ {\dot{ψ}}_{r α} \\ {\dot{ψ}}_{r β} \\ {\dot{ω}}_{r} \end{matrix}]}} = \underset{A}{\underset{︸}{[\begin{matrix} 1 - \frac{R_{sr}}{σ L_{s}} & 0 & \frac{L_{m} R_{r}}{σ L_{s} L_{r} L_{r}} & ω_{r} \frac{L_{m}}{σ L_{s} L_{r}} & 0 \\ 0 & 1 - \frac{R_{sr}}{σ L_{s}} & - ω_{r} \frac{L_{m}}{σ L_{s} L_{r}} & \frac{L_{m} R_{r}}{σ L_{s} L_{r} L_{r}} & 0 \\ \frac{L_{m} R_{r}}{L_{r}} & 0 & 1 - \frac{R_{sr}}{σ L_{s}} & - ω_{r} & 0 \\ 0 & \frac{L_{m} R_{r}}{L_{r}} & ω_{r} & 1 - \frac{R_{sr}}{σ L_{s}} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]}} \underset{x}{\underset{︸}{[\begin{matrix} i_{s α} \\ i_{s β} \\ ψ_{r α} \\ ψ_{r β} \\ ω_{r} \end{matrix}]}} + \underset{B}{\underset{︸}{[\begin{matrix} \frac{1}{σ L_{s}} & 0 \\ 0 & \frac{1}{σ L_{s}} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}]}} \underset{u}{\underset{︸}{[\begin{matrix} u_{α} \\ u_{β} \end{matrix}]}} + w$

(1)

$\underset{z}{\underset{︸}{[\begin{matrix} i_{s α} \\ i_{s β} \end{matrix}]}} = \underset{C}{\underset{︸}{[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \end{matrix}]}} \underset{x}{\underset{︸}{[\begin{matrix} i_{s α} \\ i_{s β} \\ ψ_{r α} \\ ψ_{r β} \\ ω_{r} \end{matrix}]}} + v$

(2)

where $i_{s α}$ and $i_{s β}$ are the stator currents of the IM in the α-β coordination system, and $ψ_{r α}$ and $ψ_{r β}$ are the rotor magnetic fluxes in the α-β coordinate system, respectively. In addition, $ω_{r}$ represents the angular velocity of the rotor in $rad / s$ , and $R_{s}$ and $L_{s}$ indicate the equivalent resistance and self-inductance of the stator, respectively. $R_{r}$ and $L_{r}$ are the equivalent resistance and self-inductance of the rotor, respectively. $L_{m}$ denotes the mutual inductance between the stator and rotor, $L_{σ} = σ L_{s}$ shows the transient inductance of the stator, $σ = 1 - L_{m}^{2} / (L_{r} L_{s})$ represents the magnetic leakage factor, and $u_{α}$ and $u_{β}$ are the voltage components on the α and β axes, respectively. $w$ and $v$ symbolize the additive state noise and measurement noise, respectively. In addition, $R_{sr} = R_{s} + {(L_{m} + L_{r})}^{2} R_{r}$ .

As can be seen from (1) and (2), the stator currents $i_{s α}$ and $i_{s β}$ , rotor fluxes $ψ_{r α}$ and $ψ_{r β}$ , and rotation speed $ω_{r}$ constitute the state variable array $x$ , $A$ denotes the system matrix, $B$ indicates the input matrix, $C$ represents the output matrix, stator voltages $u_{s α}$ and $u_{s β}$ constitute the input of the equation, and the stator currents constitute the output array $z$ of the state equation, thus forming the state equation and measurement equation of the system model. Since matrix $A$ contains state variable $ω_{r}$ , the state equation of the system model is nonlinear. Nevertheless, the EKF can be used to perform state estimation for nonlinear systems.

Formulas (1) and (2) have been re-expressed as:

$\{\begin{array}{l} x_{k} = f (x_{k - 1}) + B u_{k} + w_{k} \\ z_{k} = h (x_{k}) + v_{k} \end{array}$

(3)

where $f (x_{k - 1}) = A x_{k - 1}$ , it is non-linear, and $h (x_{k}) = C x_{k}$ , the state noise $w_{k}$ and measurement noise $v_{k}$ are uncorrelated Gaussian white noises, whose statistical characteristics satisfy the following:

$\{\begin{cases} E [w_{k}] = 0 \\ E [v_{k}] = 0 \\ cov (w_{k}, v_{k}) = 0 \\ cov (w_{k}, w_{j}) = Q_{k} δ_{k j} \\ cov (v_{k}, v_{j}) = R_{k} δ_{k j} \end{cases}$

(4)

where $Q_{k}$ represents the system process noise, which is a non-negative definite symmetric matrix, $R_{k}$ indicates the system measurement noise, which is a positive definite symmetric matrix, and $δ_{k j}$ denotes a $K r o n e C k e r - δ$ function. The EKF filtering can be expressed as

$\{\begin{cases} {\hat{x}}_{k / k - 1} = f_{k - 1} ({\hat{x}}_{k - 1}) + B u_{k - 1} \\ P_{k / k - 1} = Φ_{k / k - 1} P_{k - 1} Φ_{k / k - 1}^{T} + Q_{k - 1} \\ K_{k} = P_{k / k - 1} H_{k}^{T} {(H_{k} P_{k / k - 1} H_{k}^{T} + R_{k})}^{- 1} \\ {\hat{x}}_{k} = {\hat{x}}_{k / k - 1} + K_{k} (z_{k} - h ({\hat{x}}_{k / k - 1})) \\ P_{k} = (I - K_{k} H_{k}) P_{k / k - 1} \end{cases}$

(5)

where $P_{k / k - 1}$ and $P_{k}$ are the prior covariance matrix and posterior covariance matrix, respectively. In addition, $K_{k}$ represents the Kalman gain, $Φ_{k / k - 1} = J (f ({\hat{x}}_{k - 1}))$ indicates the $n \times n$ -dimensional one-step state transition matrix of the system, $J (\cdot)$ denotes the Jacobian matrix for solving the nonlinear state equation, ${\hat{x}}_{k}$ signifies the state estimation value, and $H_{k} = C$ symbolizes the $m \times n$ -dimensional measurement matrix. The initial state $x_{0}$ , state noise $w_{k}$ , and measurement noise $v_{k}$ are independent of each other and conform with Gaussian normal distribution. The initial conditions can be calculated as

$\{\begin{cases} {\hat{x}}_{0} = E [x_{0}] \\ P_{0} = E [(x_{0} - {\hat{x}}_{0}) {(x_{0} - {\hat{x}}_{0})}^{T}] \end{cases}$

(6)

where ${\hat{x}}_{0}$ and $P_{0}$ are the initial state. As can be seen in the above EKF calculation steps, the covariance matrix $P_{k}$ of the state variables $x_{k} = \{i_{s α}, i_{s β}, ψ_{s α}, ψ_{s β}, ω_{r}\}$ of the IM is a diagonal matrix, which has a significant impact on the convergence rate of the EKF algorithm and the transient amplitude and has only a slight effect on the steady state.

Usually, the initial value of the covariance matrix $P_{0}$ is set to a unit matrix. The system noise covariance matrix $Q_{k}$ can include external disturbances, parameter perturbations of the IM, and rounding errors in actual discrete processes. The transient response speed of the system filter can be changed by adjusting the value of $Q_{k}$ . The measurement noise covariance matrix $R_{k}$ can include sensor measurement accuracy and A/D conversion accuracy, etc. The accuracy of the current measurement value $z_{k} = \{i_{s α}, i_{s β}\}$ can be changed by adjusting the value of $R_{k}$ .

Under the ideal model conditions of the IM, the accurate real-time estimation of the motor speed can be accomplished using the EKF algorithm and properly set parameters. However, when the current sensor goes wrong, both the state value $x_{k}$ and the measurement value $z_{k}$ change simultaneously. Under such conditions, it is difficult for the EKF algorithm to estimate the motor speed accurately, even though the values of $Q_{k}$ and $R_{k}$ can be adjusted. Therefore, in the event of a current sensor malfunction, it is necessary to accurately determine the occurrence time of the sensor fault and perform fault tolerance on the current fault information to ensure the normal operation of the IM vector control system.

It can be known from the EKF filtering formulas shown in (5) that the state estimation ${\hat{x}}_{k}$ contains measurement innovation $z_{k} - h ({\hat{x}}_{k / k - 1})$ . When deviations in $i_{s α}$ and $i_{s β}$ caused by a current sensor fault occur, the measurement value changes accordingly. Therefore, if this innovation is used as the eigenvalue of the fault, whether the current sensor has malfunctioned can be conveniently determined. Since this innovation contains a current residual value and measurement noise $v_{k}$ , it is necessary to analyze the eigenvalue of this measurement value using probability and statistical tools. The SPRT is a simple and effective outlier diagnosis method based on probability and statistics. By combining this method with the Kalman filter, a model-based diagnostic method is constructed. In this study, a method containing the SPRT and EKF is used to perform the real-time diagnosis of a current sensor fault, called the sequential probability extended Kalman filter (SPEKF).

3. Diagnosis of Current Sensor Fault

3.1. Theoretic Basis for Diagnosing Current Sensor Fault Based on SPRT

It is very important to quickly identify and locate the faulty current sensor of the faulty phase after a current sensor goes wrong, and this is the key to the successful implementation of fault-tolerant control. This paper proposes to use the SPRT method to determine whether a current sensor has malfunctioned. According to (2), the measurement equation of the IM vector control system can be expressed as

$z_{k} = C_{k} x_{k} + v_{k}$

(7)

where $C_{k} = C$ . If a fault occurs in the vector control system of the IM, a certain amount of fault information is added to the measurement equation. The measurement equation of the IM vector control system can be established as

$z_{k} = C_{k} x_{k} + v_{k} + ρ_{k}$

(8)

where $ρ_{k}$ represents the current residual, and the current innovation of the IM vector control system is expressed as

$υ (k) = z_{k} - C_{k} {\hat{x}}_{k / k - 1} = v_{k} + ρ_{k}$

(9)

where $υ (k)$ is the current innovation. All current information quantities at the moment of $k$ are collected to form an innovation sequence $\{υ (j) |j = 1, 2 \dots k\}$ .

When there is no fault in the speed-sensorless vector control system of the IM, it is assumed that the elements of the current innovation sequence are independent of each other and conform with a normal distribution $N (0, Ξ)$ . At this point, the expected value and variance of the current innovation are expressed as

$E [υ (j)] = 0$

(10)

$E [υ (j) υ {(j)}^{T}] = Ξ$

(11)

where $Ξ$ is the variance of the current innovation. When there is a current fault in the system, the expected value and variance of the current innovation can be calculated as follows:

$E [υ (j)] = {\hat{φ}}_{j} \neq 0$

(12)

$E [(υ (j) - {\hat{φ}}_{j}) {(υ (j) - {\hat{φ}}_{j})}^{T}] = Ξ$

(13)

where ${\hat{φ}}_{j}$ is the expected of the current innovation. Therefore, the following hypothetical testing conditions can be set:

$H_{0} : {\hat{φ}}_{j} = 0$ , the system is normal and stable.

$H_{1} : {\hat{φ}}_{j} \neq 0$ , the system is experiencing malfunction.

Under two hypothetical conditions, the probability density of the current innovation samples is expressed as

$ρ (υ (j) |H_{0}) = \frac{1}{2 π^{n / 2} {(Ξ)}^{1 / 2}} e^{- \frac{1}{2} {(υ (j))}^{T} {(Ξ)}^{- 1} υ (j)}$

(14)

$ρ (υ (j) |H_{1}) = \frac{1}{2 π^{n / 2} {(Ξ)}^{1 / 2}} e^{- \frac{1}{2} (υ (j) - {\hat{φ}}_{j}) {(Ξ)}^{- 1} {(υ (j) - {\hat{φ}}_{j})}^{T}}$

(15)

By adopting the mean value method, the result can be obtained as

${\hat{φ}}_{j} \approx \bar{υ} (j) = \frac{1}{k} \sum_{j = 1}^{k} υ (j)$

3.2. SPRT Enhanced Judgment Based on Sliding Window Function

In IM vector control systems, single-phase current sensors are prone to faults such as open circuits, DC bias, and harmonics. The magnitude of the fault information $ρ_{k}$ in (9) differs when different types of current sensor faults occur. Therefore, when a current sensor experiences an open-circuit or harmonics fault, the abrupt change in $ρ_{k}$ is large, and the statistical quantity $χ (k)$ satisfies $χ (k) > \ln (Τ (H_{1}))$ in a short time. When a DC bias or gain change fault occurs, the accumulation of the statistical quantity $χ (k)$ in a certain period will satisfy $χ (k) > \ln (Τ (H_{1}))$ . Now, the above two scenarios together are considered. Suppose (16) to (19) are used to calculate the statistical quantity $χ (k)$ , and the current innovation $υ (k)$ is greatly influenced by historical information. In that case, the ability of the SPRT to judge abrupt anomalies is weakened. Hence, there will be a relatively long judgment delay when the SPRT judges the start of a slowly progressing anomaly. Inspired by the idea of the sliding window function, this paper provides a new expression for the estimated value of current innovation, improving the fault diagnosis efficiency of the SPRT. The estimated value of the current innovation obtained based on the sliding window function is represented as

$\hat{υ} (k) = \frac{1}{N} \sum_{j = 1}^{N} υ (k - j + 1)$

(24)

According to (24), we can rewrite the current information sequence $υ (j)$ in (18) and (19) as $\{υ (j) |j = k - N + 1, 2, \dots, k\}$ . After a sliding window function is added, the SPRT is no longer affected by historical information other than the number of accounted objects N, and the statistical quantity changes faster when faults such as DC bias and gain change occur. In this way, the effectiveness of the SPRT in diagnosing fault current is improved.

At present, the number of current sensors in most IM vector control systems has been reduced from three to two for cost-saving reasons [20,21,22]. This means that under normal circ*mstances, only the currents of two phases are sampled by current sensors. However, if one of the current sensors malfunctions, the measured $i_{s α}$ and $i_{s β}$ of the IM provided to the EKF by the remaining current sensor are inaccurate, leading to the inability of the EKF to estimate the IM speed accurately [23,24,25,26,27,28,29,30,31]. Based on the idea of coordinate translation, this paper proposes a double-cascading second-order generalized integrator (DSOGI) to correct the fault phase current and obtain the required measurement information for high-accuracy motor speed estimation in the α-β coordinate system.

4. Current Sensor Fault-Tolerant Control Algorithm

4.1. Principle of Current Sensor Fault-Tolerant Control in α-β Coordinate System

If $i_{a}$ and $i_{b}$ are normal, they can be determined according to the fault detection result of $i_{s α}$ and $i_{s β}$ yielded by the SPEKF algorithm and the correspondence relationship between the α-β two-phase coordinate system and the a-b-c three-phase coordinate system. On this basis, a current sensor fault tolerance algorithm is devised. The details are as follows:

Assume that the two current sensors are installed in A-phase and B-phase. If A-phase coincides with the $α$ axis of the $α - β$ coordinate system, then

$[\begin{matrix} i_{s α} \\ i_{s β} \end{matrix}] = \sqrt{\frac{2}{3}} [\begin{matrix} 1 & 0 \\ \frac{\sqrt{3}}{2} & \sqrt{3} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \end{matrix}]$

(25)

if B-phase coincides with the α axis of the α-β coordinate system, then

$[\begin{matrix} i_{{s α}^{'}} \\ i_{{s β}^{'}} \end{matrix}] = \sqrt{\frac{2}{3}} [\begin{matrix} 0 & 1 \\ - \sqrt{3} & - \frac{\sqrt{3}}{2} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \end{matrix}]$

(26)

According to (25) and (26), the value of $i_{sa}$ is determined by the measured value of A-phase current sensor, and the value of $i_{{sa}^{'}}$ is determined by the measured value of B-phase current. Therefore, the current sensor of A-phase is faulty when the SPEKF detects an anomaly of the $i_{sa}$ signal. Similarly, detecting an anomaly of the $i_{{sa}^{'}}$ signal by the SPEKF indicates that the current sensor of B-phase is faulty. If the current sensor of A-phase goes wrong, a fault-free signal $i_{{sa}^{'}}$ can be obtained using (26), and the flag bit switches to 1: Flag A = 1. If the current sensor of B-phase goes wrong, a fault-free signal $i_{sa}$ can be obtained using (25), and the flag bit switches to 1: Flag B = 1. If normal signal $i_{{sa}^{'}}$ is utilized to construct fault signal $i_{{s β}^{'}}$ or normal signal $i_{sa}$ is used to construct fault signal $i_{s β}$ , the accurate estimation of the IM speed by the EKF algorithm can be guaranteed.

In a two-phase stationary coordinate system, the phase difference between $i_{sa}$ and $i_{s β}$ is 90°. Considering that a second-order generalized integrator (SOGI) changes the phase of the input signal by 90° without changing the amplitude of the signal, using the SOGI to construct fault current information is feasible. This study uses the current value of the α (or α′) axis as the input signal of the SOGI to construct the current value of the β (or β′) axis, thereby forming a fault tolerance strategy for current sensors. The scheme for constructing the current value of the β (or β′) axis is shown in Figure 2.

In Figure 2, $ω$ denotes the resonant frequency of the SOGI, which is the same as the operating angular frequency of the motor, $λ$ signifies the damping coefficient, whose value is set to $\sqrt{2}$ , $i_{s β_est}$ represents the current value of the β axis reconstructed according to $i_{sa}$ , $i_{s β_est}$ is orthogonal to $i_{sa}$ , so $i_{s β_est}$ is called the orthogonal output signal, and $i_{sa_est}$ is synchronous with $i_{sa}$ , so $i_{sa_est}$ is called synchronized output signal. Similarly, this strategy can be employed to reconstruct $i_{s β'_est}$ using $i_{{sa}^{'}}$ . Subsequently, the reconstructed current value of the β (or β′) axis is used instead of the fault current value as the input signal of the EKF, ensuring the accuracy of motor speed estimation. In addition, the estimated current value is used as a feedback signal for closed-loop control. When the faulty sensor is placed in the α-β coordinate system, the phase angle of the rotor magnetic flux is $θ_{1}$ , while if the faulty sensor is placed in the α’-β’ coordinate system, the phase angle of the rotor magnetic flux is $θ_{2}$ , and it satisfies $θ_{2} = θ_{1} - 120^{\circ}$ . The transfer functions $G_{1} (s)$ and $G_{2} (s)$ of the current signal reconstruction can be inferred from the input and output signals in Figure 3, which are expressed as follows,

$G_{1} (s) = \frac{i_{s β_est}}{i_{sa}} = \frac{λ ω^{2}}{s^{2} + λ ω s + ω^{2}}$

(27)

$G_{2} (s) = \frac{i_{sa_est}}{i_{sa}} = \frac{λ ω s}{s^{2} + λ ω s + ω^{2}}$

(28)

4.2. Suppressing Effect of Current Fault-Tolerant Control on DC Bias and Odd Harmonics

In an IM vector control system, current sensors often encounter problems related to DC bias and odd harmonics during the measurement process, attributed to factors such as mechanical vibration. Figure 3a shows the Bode plot of $G_{1} (s)$ plotted according to (27). As can be seen, the SOGI-reconstructed current signal $i_{s β_est}$ of the $β$ axis is very sensitive to the DC bias and odd harmonics contained in the input signal $i_{sa}$ . $G_{2} (s)$ can be regarded as a second-order bandpass filter, as shown in the Bode plot presented in Figure 3b. The bandwidth center frequency is the resonant frequency $ω$ of the SOGI, and the phase delay is 0°. As the value of $ω$ varies, the bandwidth center frequency of its amplitude–frequency characteristic curve changes, and $G_{2} (s)$ attenuates the DC component.

In [22], the authors analyzed the ability of the SOGI to eliminate high-order harmonics. When the input signal $i_{sa}$ contains n-th harmonic, i.e., the resonance frequency $ω = n ω_{0}$ , where $ω_{0}$ is the fundamental angular frequency of the input signal, the amplitude of the synchronous output signal $i_{sa_est}$ of the SOGI is approximately equal to $\sqrt{2} / n$ . This indicates that after processing the input signal containing the n-th harmonic by the SOGI, the amplitude of the output signal attenuates dramatically. The greater the value of n, the greater the amplitude attenuation and vice versa. Therefore, the SOGI has the function of a high-frequency filter, so its ability to suppress low-order harmonics is limited. Therefore, it is necessary to develop a strategy to attenuate both higher and low-order harmonics to handle the current signals containing low-order harmonics.

In addition, when the input signal $i_{sa}$ contains the n-th harmonic, the amplitude of the fundamental wave in the orthogonal output signal $i_{s β_est}$ of the SOGI no changed, but its phase is delayed by 90°. However, the phase change of the harmonic, the amplitude also changes to $\frac{\sqrt{2}}{n^{2}}$ of the original. If the orthogonal output signal $i_{s β_est}$ is processed by the SOGI once again, the amplitude of the fundamental wave remains unchanged, but the amplitudes of the harmonics attenuate significantly. Table 1 lists the attenuation gains of cascaded SOGI filtering systems of m stages (m = 1, 2, 3, 4).

It can be seen from Table 1 that for every increase of 1 in the number of SOGI stages, the attenuation gains of each harmonic increase proportionally. Because the SOGI is affected by the characteristics of the filter, the increase in the number of SOGI stages will elongate the steady-state output adjustment time, indicating that a large m will affect the dynamic performance of the SOGI cascade system. To strike a balance between suppressing harmonics and maintaining the high dynamic performance of the system, this paper proposes a two-stage cascaded SOGI called the DSOGI to reconstruct the current signal of the β axis. The DSOGI can effectively suppress the DC component and harmonics in the current signal. The structure of the DSOGI is depicted in Figure 4.

The transfer functions of the improved system can be inferred from Figure 4, which are as follows,

$G_{3} (s) = \frac{i_{s β_est}}{i_{sa}} = \frac{λ_{1} λ_{2} ω^{3} s}{(s^{2} + λ_{2} ω s + ω^{2}) (ω^{2} + s^{2}) + λ_{1} λ_{2} ω^{2} s^{2}}$

(29)

$G_{4} (s) = \frac{i_{sa_est}}{i_{sa}} = \frac{λ_{1} λ_{2} ω^{2} s^{2}}{(s^{2} + λ_{2} ω s + ω^{2}) (ω^{2} + s^{2}) + λ_{1} λ_{2} ω^{2} s^{2}}$

(30)

It can be known from (30) that the gain in $G_{4} (s)$ is lower than $G_{2} (s)$ in both the low- and high-frequency bands, meaning that its bandpass filtering performance is better than $G_{2} (s)$ . It can be known from (29) that the gain in $G_{3} (s)$ is negative in the low-frequency band, and its gain is lower than that of $G_{1} (s)$ in the high-frequency range, indicating that it can eliminate the harmonics of the input signal. Therefore, the DSOGI has a good suppressing effect on DC bias and harmonics. In Figure 5, the performance of the proposed DSOGI scheme is shown under the conditions of $ω = 50 π$ (rad/s) and $λ_{1} = λ_{2} = 0.02$ .

As revealed by the above analysis, in an IM speed-sensorless control system, the SPEKF can determine the occurrence time and location of a fault when an anomaly of $i_{s α}$ or $i_{s β}$ occurs. In addition, the faulty current signal can be reconstructed using the DSOGI algorithm, and the DC bias and harmonics in $i_{s α}$ and $i_{s β}$ can be effectively suppressed by the DSOGI. Compared with the scheme of the SOGI based on the SPEKF (SPEKF-SOGI), the proposed scheme of the DSOGI based on the SPEKF (SPEKF-DSOGI) can achieve a higher accuracy of motor speed estimation when the current sensor of one phase experiences a fault such as open circuit, DC bias, or harmonics.

Figure 6 shows the overall block diagram of the proposed current sensor fault-tolerant control scheme for the speed-sensorless control system of an IM. In Figure 6, superscript “*” indicates that the variables are reference value of the system, Flag A is the flag of the A-phase current sensor and Flag B is the flag of the B-phase current sensor.

5. Analysis of Experimental Results

To verify the feasibility of the proposed scheme, the proposed scheme is tested on an IM experimental platform shown in Figure 7. The hardware components of the experimental platform include an IM, driver, loading system, oscilloscope, etc. The actual motor speed is measured by a photoelectric encoder using the M/T method. The system software includes an initialization program, main loop program, PWM interruption program, current measurement program, etc. Among them, PWM is mainly responsible for coordinating translation, implementing speed estimation algorithms, generating SVPWM, and dead zone compensation, with a terminal cycle of 250 μs. The parameters of the IM used in the experiment are shown in Table 2.

5.1. Test of Motor Speed Estimation Performance when Current Sensors Are Functioning Normally

To verify the effectiveness of SPEKF-DSOGI, an IM speed-sensorless control method is tested based on the SPEKF alongside the SPEKF-DSOGI method for comparison. To ensure a fair comparison, both methods use the same noise covariance matrix parameters as other parts of the sensorless control system. The value of the $Q_{k}$ and $R_{k}$ matrices for the EKF are as follows:

$Q_{k} = diag ([0.002, 0.002, 0.002, 0.002, 1]), R_{k} = diag ([0.01, 0.01])$

The effectiveness of the speed-sensorless control method for the IM based on SPEKF-DSOGI is verified when the current sensors are functioning normally. Figure 8 compares the speed ${\hat{n}}_{r}$ of the IM yielded by the SPEKF-DSOGI method and the speed $n_{r}$ measured by the photoelectric encoder. The initial speed of the IM is 400 r/min, and the speed increases to 1000 r/min. After a period, the speed decreases to 700 r/min. As seen from the waveforms of ${\hat{n}}_{r}$ and $n_{r}$ , the speed yielded by the SPEKF-DSOGI method is consistent with the speed measured by the photoelectric encoder, exhibiting a maximum angle error of 12.6°.

Figure 9 shows the dynamic tracking performance test results of the two methods during sudden increases or decreases in load. At first, 65% of the rated load is applied to the motor at a speed of 1000 r/min, Then, the load is abruptly raised to 100% of the rated load. After a period, the load is reduced to 65% of the rated load. As seen in Figure 8, the speed-sensorless control system based on the SPEKF-DSOGI algorithm can operate stably at a speed of 1000 r/min under the rated load condition. Throughout the load increase and reduction processes, the maximum speed estimation error is only 8 r/min, and the convergence speed is up to 0.1 s, indicating that the convergence speed of the error is quite fast.

5.2. Test of Motor Speed Estimation Performance under Condition of Current Sensor Malfunction

Figure 10 shows the experimental results obtained from the proposed fault-tolerant strategy after the current sensor of A-phase experiences an open-circuit fault, demonstrating the signal reconstruction performance of SPEKF-DSOGI compared to SPEKF-SOGI. Figure 10a presents the current waveforms of A-phase and B-phase when an open-circuit fault occurs in the A-phase, the current waveform of $i_{{s β}^{'}_est}$ reconstructed by the SPEKF-DSOGI algorithm based on current $i_{{sa}^{'}}$ , and the waveform of FLAG A (the SPRT fault diagnosis flag). When the A-phase current is normal, FLAG A = 0, while when the A-phase current is abnormal, FLAG A = 1. Figure 10b illustrates the waveforms of stator currents $i_{{sa}^{'}}$ and $i_{{s β}^{'}_est}$ extracted by the SPEKF-SOGI algorithm during the fault period, as well as the waveforms of the actual position $θ_{r}$ and estimated position ${\hat{θ}}_{r}$ of the rotor. The maximum and average values of position estimation error $Δ {\hat{θ}}_{r}$ are 12.2° and 9.23°, respectively. Figure 10c shows the waveforms of stator currents $i_{{sa}^{'}}$ and $i_{{s β}^{'}_est}$ extracted by the SPEKF-SOGI algorithm, along with the waveforms of the actual position $θ_{r}$ and estimated position ${\hat{θ}}_{r}$ of the rotor. Under the same conditions, the maximum value of the position estimation error $Δ {\hat{θ}}_{r}$ is 5.2°, with an average value of 2.01°, representing a reduction of 78.2% compared to the average error of the SPEKF-SOGI algorithm. This demonstrates the excellent performance of the proposed algorithm when a current sensor is experiencing an open-circuit fault.

To further verify the performance of the proposed current sensor fault-tolerant control strategy under different operational conditions, the suppressing effects of SPEKF-SOGI and SPEKF-DSOGI on DC bias are tested when the IM operates at a speed of 900 r/min. The experimental results are shown in Figure 11. Figure 11a shows the waveforms of A-phase current $i_{a}$ and B-phase current $i_{b}$ during a DC bias fault in A-phase current and the waveform of the current $i_{{s β}^{'}_est}$ reconstructed by the SPEKF-DSOGI algorithm based on current $i_{{sa}^{'}}$ . When a 1A DC bias is added to the given current $i_{a}$ , the flag bit Flag A switches from 0 to 1. Figure 11b shows the waveform of FLAG A (the fault flag bit) obtained by the SPEKF-SOGI algorithm during the fault period, as well as the waveforms of the actual position $θ_{r}$ and estimated position ${\hat{θ}}_{r}$ of the rotor. The maximum error between the estimated position ${\hat{θ}}_{r}$ obtained by the SPEKF-SOGI algorithm and the actual position $θ_{r}$ is 18.2°, and the maximum error decreases to 14.6° after 0.08 s. The orange line represents the error $Δ θ_{r}$ which between actual position and estimated position. Figure 11c shows the waveform of FLAG A (the fault flag bit) obtained by the SPEKF-DSOGI algorithm during the fault period, as well as the waveforms of the actual position $θ_{r}$ and estimated position ${\hat{θ}}_{r}$ of the rotor. The maximum error between the estimated position ${\hat{θ}}_{r}$ obtained by the SPEKF-DSOGI algorithm and the actual position $θ_{r}$ is 4.2°, and the maximum error decreases to 1.6° after 0.02 s. The orange line represents the error $Δ θ_{r}$ which between actual position and estimated position. These results highlight that the SPEKF-DSOGI algorithm can reconstruct the faulty current signal and effectively suppress DC bias, enabling accurate motor speed estimation.

In addition, the suppressing effects of different fault tolerance strategies are tested on the 2k + 1 harmonics. The experimental results are shown in Figure 12. Figure 12a shows the waveform change in the given current $i_{a}$ when a harmonic (3rd, 5th, or 7th) is added to it, along with the waveforms of the current $i_{{sa}^{'}}$ and the reconstructed current $i_{{s β}^{'}_est}$ . Figure 12b displays the waveforms of the actual position $θ_{r}$ and estimated position ${\hat{θ}}_{r}$ of the rotor extracted using the SPEKF-SOGI algorithm. When the flag bit Flag A switches from 0 to 1, the maximum error and average error between the estimated position ${\hat{θ}}_{r}$ and actual position $θ_{r}$ are 22.2° and 16.1°, respectively. $Δ θ_{r}$ is the error between the actual position and the estimated position. Figure 12c shows the waveforms of the estimated position ${\hat{θ}}_{r}$ and actual position $θ_{r}$ extracted using the SPEKF-DSOGI algorithm. The maximum error and average error are 5.6° and 3.5°, respectively. $Δ θ_{r}$ is the error between the actual position and the estimated position. A comparison of Figure 12b,c reveals that when there are odd harmonics in the current, the harmonics can be effectively suppressed using the DSOGI. Consequently, the SPEKF-DSOGI algorithm can accurately estimate the speed of the IM.

5.3. Comparison of Experimental Results between SPEKF-DSOGI and SEPLL

A current sensor FTC scheme based on the SEPLL current reconstruction for the speed-sensorless control of IM drives was proposed in [13], which is an extension of [1]. In this scheme, the SEPLL is used to reconstruct current, without the use of estimated speed or rotor position information; after the introduction of the SMO-based speed estimation scheme, and the FD scheme based on the coordinate transformation was developed to target fast fault detection and location. The results of SPEKF-DSOGI and SEPLL were compared to demonstrate the superiority of SPEKF-DSOGI.

(a): The open-circuit fault of the A-phase current sensor of the IM drives

First, taking the open-circuit fault of the A-phase current sensor of the IM drives as an example, when there is no fault in the A-phase current sensor, the motor speed is 800 r/min. Then, the performance comparison between the proposed SPEKF-DSOGI, SEPLL, and the conventional current reconstruction scheme in [1] is experimentally made, and the results are shown in Figure 13.

Figure 13a shows the open-circuit fault of the A-phase current sensor of the IM drives. The conventional current reconstruction scheme was used to obtain $i_{s β^{'}_est_2}$ , SEPLL was used to obtain $i_{s β^{'}_est_1}$ , and SPEKF-DSOGI was used to obtain $i_{s β^{'}_est}$ . Figure 13b shows the estimated rotational speed for the three schemes and speed error value. In the conventional current reconstruction scheme, obvious harmonics appear in the reconstructed current due to the use of the estimated speed ${\hat{ω}}_{r_2}$ . However, due to SPEKF-DSOGI’s and SEPLL’s use of the proposed current reconstruction scheme, the reconstructed α-axis and β-axis currents are hardly affected by the A-phase current sensor fault. The actual speed of the motor $ω_{r}$ , the estimated speed ( ${\hat{ω}}_{r}$ , ${\hat{ω}}_{r_1}$ ), and the phase current of the motor are also not greatly affected by the A-phase current sensor failure. The errors between the estimated speed of the three schemes and the actual speed are $Δ {\hat{ω}}_{r}$ , $Δ {\hat{ω}}_{r_1}$ , and $Δ {\hat{ω}}_{r_2}$ , respectively.

(b) The DC bias fault in the A-phase current sensor of the IM drives

To further evaluate the performance of the proposed FTC scheme with the DC bias fault in the A-phase current sensor, the experimental test is conducted, as shown in Figure 14. As can be seen from the figure, when 2A DC bias is added to the A-phase current sensor, the conventional current reconstruction scheme will cause bias in the estimated β-axis current, thus affecting the performance of the system control. However, with SPEKF-DSOGI and SEPLL, the effect of DC bias is reduced. Their reconstructed β-axis current returns to normal after a small fluctuation, achieving an accurate estimation of motor speed.

Figure 15 shows the physical experimental results of the diagnosis algorithm in the case of the variable gain fault of the A-phase current sensor. At 0.5 s, the output value of the A-phase current sensor is forced to become four-thirds of the original through software setting. According to the experimental data, it can be seen that the bias between the measured value and the estimated value of the A-phase current changes significantly during the simulation of fault occurrence. Because the A-phase current output of the motor is a sine wave during normal operation, the amplitude of the sine wave increases obviously after a fault. According to Figure 15a, the β-axis current values obtained by the three schemes are the same before the gain fault occurs. After the gain fault occurs, the conventional current reconstruction scheme greatly affected, resulting in an increase in the estimated speed value. Both SPEKF-SOGI and SEPLL can accurately estimate the motor speed value, as shown in Figure 15b.

(d) The harmonic fault of the A-phase current sensor of the IM drives

Figure 16 shows the simulation of 2k + 1 harmonics with different fault-tolerant strategies. When the given current $i_{a}$ is increased by 3, 5, and 9 harmonics, as shown in Figure 16a, the waveforms of the current $i_{{sa}^{'}}$ and the reconstructed current $i_{{s β}^{'}_est}$ are plotted. Figure 11b shows the waveforms of the actual motor speed $ω_{r}$ extracted and the estimated motor speed ${\hat{ω}}_{r}$ using the three schemes. According to Figure 16a, the β-axis current values obtained by the three schemes are the same before the gain fault occurs. After the gain fault occurs, SEPLL and the conventional current reconstruction schemes are greatly affected, resulting in a large number of harmonics in the reconstructed current values. However, the SPEKF-DSOGI-reconstructed β-axis current does not contain harmonics and can accurately estimate the motor speed. According to Figure 16b, using SPEKF-DSOGI, harmonics can be suppressed well, and the speed of the induction motor can be estimated accurately.

To sum up, in order to compare the advantages of SPEKF-SOGI and SEPLL schemes, A-phase faults such as open circuit, DC bias, gain, and odd harmonics are verified on the induction motor experiment platform shown in Figure 7. The experimental results show that when the first three faults occur, both SPEKF-DSOGI and SEPLL solutions accurately reconstruct β-axis current and estimate motor speed. However, when an odd harmonic fault occurs in A-phase, the β-axis current reconstructed by SEPLL contains harmonic components, and the motor speed estimation is not accurate. The test results suggest that satisfactory performance and robustness against disturbances are achieved by using the proposed current sensor FTC scheme.

6. Conclusions

This paper proposed a current sensor fault-tolerant control strategy suitable for speed-sensorless vector control systems of IMs, addressing the accuracy of motor speed estimation posed by any current sensor fault in a speed-sensorless IM vector control system. The conclusions were drawn as follows:

(1): The SPRT method was used to determine whether a current sensor had gone wrong. Then, the expression of the estimated value of the current innovation was modified based on the principle of the sliding window function, improving the accuracy of the SPRT in diagnosing current sensor faults.
(2): A fault-tolerant strategy for inductive motors installed with two current sensors in two phases was proposed. A DSOGI was used to reconstruct faulty current information based on the concept of coordinate translation, providing accurate current information for the speed estimation scheme based on the EKF. When the current sensor of a certain phase experiences a fault, such as an open circuit, DC bias, or odd harmonic, the high-precision current sensor fault-tolerant control scheme for the IM speed-sensorless control system handles the fault and ensures the stable operation of the motor.
(3): The results of a series of experiments conducted on a 3 kW induction motor experimental platform demonstrated that the proposed algorithm can ensure the accurate estimation of motor speed in various working conditions of inductive motors, including normal operation, an open circuit of the current sensor in a single phase, DC bias, gain, odd harmonics, etc.

Author Contributions

Conceptualization, F.Z.; methodology, S.G. and F.Z.; software, F.Z.; validation, W.Z.; investigation, C.Z.; writing—original draft preparation, F.Z. writing—review and editing, F.Z. and W.Z.; visualization, F.Z.; supervision, G.L. and S.G.; project administration, F.Z. and W.Z.; funding acquisition, F.Z. and W.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by industrial Science and Technology Research of Shaanxi Province (2023-YBGY-060), the Research Program of Education Department of Shaanxi Province (23JC005).

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to laboratory regulations.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The flowchart for diagnosing a current sensor fault based on the SPRT.

Figure 2. Scheme of reconstructing current signal of $β$ axis.

Figure 3. Bode plots of (a) $G_{1} (s)$ and (b) $G_{2} (s)$ .

Figure 4. Improved scheme of reconstructing current signal of β axis.

Figure 5. Bode plots of $G_{3} (s)$ (red) and $G_{4} (s)$ (blue).

Figure 6. Block diagram of proposed current sensor fault-tolerant control strategy for speed-sensorless IM drive system based on EKF.

Figure 7. Induction motor experiment platform.

Figure 8. Rotor speed estimation of SPEKF-DSOGI under no load.

Figure 9. Results of dynamic tracking performance test during sudden increase or decrease in load.

Figure 10. Experimental results of fault-tolerant strategy obtained under open-circuit fault in A-phase: (a) open-circuit fault occurred in A-phase, (b) SPEKF-SOGI estimated motor speed, and (c) SPEKF-DSOGI estimated motor speed.

Figure 11. Experimental results of fault-tolerant strategy obtained under DC bias fault in A-phase: (a) DC bias fault occurs in A-phase, (b) SPEKF-SOGI estimated motor speed, and (c) SPEKF-DSOGI estimated motor speed.

Figure 12. Experimental results of fault tolerance strategy obtained when odd harmonics were added to A-phase current: (a) odd harmonics were added to A-phase current, (b) SPEKF-SOGI estimated motor speed, and (c) SPEKF-DSOGI estimated motor speed.

Figure 13. The open-circuit fault in the A-phase current sensor of the IM drives: (a) performance comparison between SPEKF-DSOGI, SEPLL, and the conventional current reconstruction scheme, and (b) estimated motor speed for the three schemes.

Figure 14. The DC bias fault in the A-phase current sensor of the IM drives: (a) performance comparison between SPEKF-DSOGI, SEPLL, and the conventional current reconstruction scheme and (b) estimated motor speed for the three schemes.

Figure 15. The gain fault in the A-phase current sensor of the IM drives: (a) performance comparison between SPEKF-DSOGI, SEPLL, and the conventional current reconstruction scheme and (b) estimated motor speed for the three schemes.

Figure 16. The harmonic fault in the A-phase current sensor of the IM drives: (a) performance comparison between SPEKF-SOGI, SEPLL, and the conventional current reconstruction scheme and (b) estimated motor speed for the three schemes.

Table 1. Attenuation gains of cascaded SOGI filtering systems of m stages (m = 1, 2, 3, 4).

m	Harmonic
m	5th	7th	11th	13th
1	25	31	39	42
2	50	62	78	84
3	75	93	116	129
4	Inf	Inf	Inf	Inf

Table 2. Parameters of induction motor.

Name	Value	Name	Value
$P_{N}$	3 kW	$R_{s}$	3.127 Ω
$U_{N}$	380 V	$R_{r}$	3.55 Ω
$I_{N}$	4.55 A	$L_{m}$	0.165 H
$f_{N}$	50 Hz	$L_{s}$	0.171 H
$L_{r}$	0.173 H	$p$	3
$L_{ls}$	0.006 H	$L_{lr}$	0.008 H

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