An equivalent-input-disturbance approach for current sensor faults estimation and rejection in permanent magnet synchronous motor drives

Abstract

Current sensor is commonly used in a permanent magnet synchronous motor (PMSM) drive system. Occurrence of unexpected current sensor faults may cause feedback currents deviation and system degradation, which can be extremely detrimental to the safety of the industrial system with PMSM. This paper presents an estimation and rejection strategy of current sensor faults for a PMSM drive system. Sensor faults in current measurement circuits are treated as system disturbances by constructing a new system plant. A sliding mode observer and an improved equivalent-input-disturbance (EID) estimator are designed for the plant based on the EID theory. Accurate estimates of the current sensor equivalent-input-faults are thus obtained readily. Faults rejection is performed by subtracting the equivalent-input-faults from the control input. This allows an existing controller in a PMSM system to continue to function normally even a current sensor fault occurs. An existence analysis and stability proof are also discussed in detail for the system. Finally, different faults examples and a hardware-in-the-loop experiment are given to demonstrate the efficiency of the method.

Keywords

Permanent magnet synchronous motor equivalent-input-disturbance fault estimate and reject sliding mode observer

Introduction

Permanent magnet synchronous motor (PMSM) drive systems are widely used in industry automation field and traction systems. Also, high performance closed-loop torque control is required for the above applications (Bianchi et al., 2016; Menon et al., 2016; Tian et al., 2018). However, the accuracy of current measured value usually affects the torque performance of the system (Qian et al., 2004). The current sensor, which is a measuring current component in PMSM system, with unexpected faults usually does not perform measurements in a good way due to different reasons, such as aging, vibration, and electromagnetic interference. Therefore, it is important for PMSM drive systems to study current sensor fault diagnosis (FD) or fault-tolerant control (FTC) for high performance closed loop torque control.

Many approaches of current sensor FD have been presented in recent years. Zhang et al. (2017) presented a model-based FD approach based on structural analysis to diagnose current sensor faults for a PMSM system in an electric vehicle. The approach evaluated the structural model of the system by using a mathematical model matrix form to design the structured residuals of the faults. Jlassi et al. (2017) presented a Luenberger observer method based on adaptive threshold to isolate current sensor faults. Khil et al. (2016) introduced an average normalised currents approach to diagnose current sensor faults for PMSM drives. Also, the approach did not need any information about the motor model. However, we also hope to correct distorted sensor data while carrying out FD. That is, current sensor FTC is our another main objective. But FTC can be performed only if fault estimation or reconstruction is ensured. Zhao et al. (2017) used a sliding mode observer (SMO) to estimate and reconstruct the sensors faults for $dq$ -axis currents. Beng et al. (2015) presented an observer method to estimate currents and offered sensor fault-resilient control for an interior PMSM system. Kommuri et al. (2018) discussed a Luenberger observer to estimate $α β$ -axis currents. Then, a higher-order sliding mode controller was developed to ensure robust sensor FTC for PMSM drives. Mekki et al. (2015) presented an SMO method to reconstruct current sensor faults and designed a reconstructed sliding mode controller with an integral sliding surface for an additive fault. Wu et al. (2018) developed a sensor faults detection method based on the estimation of the current amplitude. Then, different FTC strategies based on coordinate transformation are designed for signal-loss mode, gain-variation mode, and zero-offset mode.

The above FTC methods use the inverse model of a system to reconstruct fault-tolerant controller when the faults are estimated, or need to reconstruct different fault-tolerant structure for different fault modes. She et al. (2011) and Zhou et al. (2014) presented an equivalent-input-disturbance (EID) approach to reject both matched and unmatched disturbances. The approach does not require a prior information about disturbance modes or the inverse model of a system. Motivated by She et al. (2011) and Zhou et al. (2014), we use the approach to investigate faults estimation and rejection of current sensors for a PMSM drive system. Yang et al. (2017) regarded current sensor fault as an external disturbance. Therefore, the current sensor faults can be regarded as the disturbances of state equation in system model by use a output filter. A Luenberger observer, which was used to produce the estimate of the EID in She et al. (2011) and Zhou et al. (2014), played an important role in the EID approach. SMO differs from the Luenberger observer because of its robustness and low sensitivity to uncertainties (Wang et al., 2017). To improve the accuracy and efficiency of fault estimation, this study used a SMO instead of the Luenberger observer to estimate current states. The SMO and the concept of an EID were combined to construct a new EID estimator to reject the overall effect of different faults on the PMSM drive system. The stability analysis and proof of the system for faults are devised. The validity of the method is demonstrated through a hardware-in-the-loop experiment.

The contributions of the paper are as follows. First, The EID theory is first introduced to deal with sensor fault and its FTC. An EID estimator is utilized to estimate an equivalent-input-fault, and the obtained estimate is incorporated into the state-feedback control law to counteract the effect of the fault. This effectively avoids the problems of the reconfiguration of fault-tolerant controller and the influence of the reconfiguration controller on the stability of the whole system in traditional FTC. Second, an SMO, instead of a Luenberger observer in the conventional EID approach, is used to reconstruct the state for the estimation of the equivalent-input-fault. The SMO and the concept of the EID are combined to construct an improved EID estimator to reject the overall effect of sensor faults on the system. It greatly improved the robustness of the conventional EID to bounded model uncertainties and parameter variations in a sensor fault system.

System description

In this section, a PMSM model with current sensor faults is described. Then, the faults are converted to equivalent disturbances of a new system by a linear filter. Thus, it is easy and convenient to use EID approach to perform sensor faults estimation and rejection.

A PMSM $α β$ model in the stationary reference frame is described as (Zang et al., 2018)

{\begin{matrix} \frac{d i_{α}}{dt} = - \frac{R_{s}}{L_{α}} i_{α} + \frac{1}{L_{α}} u_{α} + \frac{ω}{L_{α}} ψ_{r} \sin θ, \\ \frac{d i_{β}}{dt} = - \frac{R_{s}}{L_{β}} i_{β} + \frac{1}{L_{β}} u_{β} - \frac{ω}{L_{β}} ψ_{r} \cos θ, \end{matrix}

(1)

with

{\begin{matrix} L_{α} = L_{0} + L_{1} \cos (2 θ), \\ L_{β} = L_{0} - L_{1} \cos (2 θ), \end{matrix}

(2)

and

{\begin{matrix} L_{0} = \frac{L_{d} + L_{q}}{2}, \\ L_{1} = \frac{L_{d} - L_{q}}{2}, \end{matrix}

(3)

where $i_{α} (i_{β})$ is the $α$ ( $β$ )- axis stator current; $u_{α} (u_{β})$ is the $α$ ( $β$ )- axis stator voltage; $R_{s}$ is the stator resistance; $L_{α} (L_{β})$ is the $α$ ( $β$ )- axis stator inductance; and $L_{d} (L_{q})$ is the $d$ ( $q$ )- axis stator inductance; $ψ_{r}$ is the permanent magnet flux linkage; $ω$ is the rotor speed; $θ$ is the electrical rotor position.

The electromagnetic torque is described as

T_{e} = \frac{3}{2} n_{p} [ψ_{r} + (L_{d} - L_{q}) i_{d}] i_{q},

(4)

where $i_{d} (i_{q})$ is the $d$ ( $q$ )- axis current; $n_{p}$ is the number of pole pairs.

Letting

A = [\begin{matrix} - \frac{R_{s}}{L_{α}} & 0 \\ 0 & - \frac{R_{s}}{L_{β}} \end{matrix}]; B = [\begin{matrix} \frac{1}{L_{α}} & 0 \\ 0 & \frac{1}{L_{β}} \end{matrix}];

G = [\begin{matrix} - \frac{ω}{L_{α}} & 0 \\ 0 & - \frac{ω}{L_{β}} \end{matrix}]; ψ = [\begin{matrix} ψ_{α} \\ ψ_{β} \end{matrix}] = [\begin{matrix} - ψ_{r} \sin θ \\ ψ_{r} \cos θ \end{matrix}] .

Then, the model of a PMSM system with current sensor faults can be written as

{\begin{matrix} \overset{\cdot}{x} = Ax + Bu + G ψ, \\ y = x + f_{s}, \end{matrix}

(5)

where $x$ is the state variable defined as $x = {[\begin{matrix} i_{α} & i_{β} \end{matrix}]}^{T}$ ; $y$ is the output vector; $u$ is the system input vector defined as $u = {[\begin{matrix} u_{α} & u_{β} \end{matrix}]}^{T}$ ; $f_{s}$ is the sensor fault vector for $α β$ -axis currents defined as

f_{s} = {[\begin{matrix} f_{s α} & f_{s β} \end{matrix}]}^{T} .

Define a new state variable $z$ and a filter is applied

\overset{\cdot}{z} = - z + y = - z + x + f_{s} .

(6)

Equations (5) and (6) yield a new system

{\begin{matrix} \overset{\cdot}{X} = \bar{A} X + \bar{B} u + \bar{G} ψ + F f_{s}, \\ Y = \bar{C} X, \end{matrix}

(7)

where $X$ denotes the new system state vector defined as $X = {[\begin{matrix} x & z \end{matrix}]}^{T}$ ; $Y = z$ is the new system output; $\bar{A} = [\begin{matrix} A & 0 \\ I & - I \end{matrix}]$ ; $\bar{B} = [\begin{matrix} B \\ 0 \end{matrix}]$ ; $\bar{G} = [\begin{matrix} G \\ 0 \end{matrix}]$ ; $F = [\begin{matrix} 0 \\ I \end{matrix}]$ ; $\bar{C} = [\begin{matrix} 0 & I \end{matrix}]$ .

Thus, the sensor fault problem becomes an actuator fault problem by using the filter. If the actuator fault is treated as a disturbance, the disturbance rejection method based on EID approach can be used to deal with the fault according to the system model description in She et al. (2011) and Zhou et al. (2014).

EID-based faults estimation and rejection

In this section, the configuration of the faults estimation and rejection for current sensor based on EID approach is explained. Then, the design methods of a SMO and an EID estimator to estimate the equivalent-input-fault are presented.

The permanent magnet flux linkage of (7) is assumed to be a linear constant when system bandwidth is high enough (Hasegawa and Matsui, 2009). Therefore, the EID theory can be used to estimate and reject the fault of the linear system (7). Figure 1 shows the configuration of an EID-based estimation and rejection system for current sensor faults. It contains a plant, a SMO, and an EID estimator. As explained in She et al. (2011) and Zhou et al. (2014), an EID is a signal on the control input channel that produces the same effect on the output as actual disturbances do. If a fault $f_{se}$ on the control input channel is used to describe the plant. It is given by

{\begin{matrix} \overset{\cdot}{X} = \bar{A} X + \bar{B} u + \bar{B} f_{se} + \bar{G} ψ, \\ Y = \bar{C} X . \end{matrix}

(8)

If letting the control input be $u = 0$ . Then, the output of the plant (7) for the fault $f_{s}$ is $Y$ , and the output of the plant (8) for the fault $f_{se}$ is $Y$ . The fault $f_{se}$ is called an EID of the fault $f_{s}$ . Current sensor faults in this research are the disturbances in She et al. (2011) and Zhou et al. (2014).

Figure 1.

Configuration of EID-based system.

The state-space representation of the SMO is

{\begin{matrix} \dot{\hat{X}} = \bar{A} \hat{X} + \bar{B} u_{f} + \bar{G} ψ + L (Y - \hat{Y}) + \bar{B} K_{s} s g n (Y - \hat{Y}), \\ \hat{Y} = \bar{C} \hat{X}, \end{matrix}

(9)

where $\hat{X}$ is the estimate state of $X$ ; $\hat{Y}$ is the estimate output of $Y$ ; $u_{f}$ is an input, $L$ is an observer gain to be designed; $K_{s}$ is a sliding mode gain to be designed; $sgn (\cdot)$ is the sign function.

In the EID estimator, ${\bar{B}}^{+}$ is a pseudoinverse of $\bar{B}$ and is given by

{\bar{B}}^{+} : = \frac{{\bar{B}}^{T}}{{\bar{B}}^{T} \bar{B}} .

(10)

Defining the state-estimation error to be

e = X - \hat{X} .

(11)

According to (8), (9) and (11), we have

\begin{array}{l} \dot{e} = \dot{X} - \dot{\hat{X}} \\ = (\bar{A} - L \bar{C}) e + \bar{B} (u + f_{s e} - u_{f} - K_{s} s g n (Y - \hat{Y}) . \end{array}

(12)

Substituting (9) to (12) yield

\begin{matrix} \overset{\cdot}{X} = \overset{\cdot}{e} + \overset{\cdot}{\hat{X}} \\ = (\bar{A} - L \bar{C}) e + \bar{B} [u + f_{se} - u_{f} - K_{s} sgn (Y - \hat{Y})] \\ + \bar{A} \hat{X} + \bar{B} u_{f} + \bar{G} ψ + L (Y - \hat{Y}) + \bar{B} K_{s} sgn (Y - \hat{Y}) \\ = \bar{A} e + \bar{B} u + \bar{B} f_{se} + \bar{A} \hat{X} + \bar{G} ψ \\ = \bar{A} \hat{X} + \bar{B} u + (\bar{A} e + \bar{B} f_{se}) + \bar{G} ψ . \end{matrix}

(13)

That is

\overset{\cdot}{\hat{X}} = \bar{A} \hat{X} + \bar{B} u + (- \overset{\cdot}{e} + \bar{A} e + \bar{B} f_{se}) + \bar{G} ψ .

(14)

If we introduce a variable $Δ f$ that satisfies

\bar{B} Δ f = - \overset{\cdot}{e} + \bar{A} e,

(15)

substitute (15) into (14), and define an estimation of the EID, ${\hat{f}}_{se}$ , as

{\hat{f}}_{se} = f_{se} + Δ f .

(16)

Then, we write a plant as

\overset{\cdot}{\hat{X}} = \bar{A} \hat{X} + \bar{B} u + \bar{B} {\hat{f}}_{se} + \bar{G} ψ .

(17)

Equations (16) and (17) mean that the difference between the state of the plant and that of the observer is equivalent to the difference between the exact value and the estimate of the EID. Equation (17) plays a key role in the EID estimation (She et al. 2011).

According to (9) and (17), we have

\bar{B} [{\hat{f}}_{se} - u_{f} + u - K_{s} sgn (Y - \hat{Y})] = L (Y - \hat{Y}) .

(18)

If we solve (18) for ${\hat{f}}_{se}$ , then a least square solution is

{\hat{f}}_{se} = {\bar{B}}^{+} L (Y - \hat{Y}) + (u_{f} - u) + K_{s} sgn (Y - \hat{Y}) .

(19)

The filtered fault, ${\tilde{f}}_{se}$ , is given by

{\tilde{F}}_{se} (s) = F (s) {\hat{F}}_{se} (s),

(20)

where ${\tilde{F}}_{se} (s)$ and ${\hat{F}}_{se} (s)$ are the Laplace transforms of ${\tilde{f}}_{se} (t)$ and ${\hat{f}}_{se} (t)$ . The low-pass filter $F (s)$ is introduced to select the angular frequency band for the EID estimation. It satisfies

| F (j ω_{1}) | \approx 1, \forall ω_{1} \in [0, ω_{r}],

(21)

where $ω_{r}$ is the highest angular frequency required for the EID estimation. Let $Ω_{r} : = {ω_{1}, 0 \leq ω_{1} \leq ω_{r}}$ be the angular-frequency band for fault rejection. A suitable design of the observer guarantees that ${\hat{f}}_{se}$ converges to $f_{se}$ , and ${\tilde{F}}_{se} (j ω_{1}) = {\hat{F}}_{se} (j ω_{1})$ for all $ω_{1} \in Ω_{r}$ is true for a properly designed filter $F (s)$ . Therefore, ${\tilde{f}}_{se}$ is a good approximation of $f_{se}$ (She et al., 2011).

Combining the fault estimate (20) with the original control law yields a new control law

u = u_{f} - {\tilde{f}}_{se} .

(22)

The modified control law improves the fault-rejection performance.

The stability analysis is shown as follows.

Theorem 1. The following equation

∥ f_{se} - {\tilde{f}}_{se} ∥ < γ_{1} ∥ e ∥

(23)

hold, where $γ_{1}$ is a small positive constant.

Theorem 2. A appropriate gain $L$ is designed such that

{\bar{A}}^{T} - {\bar{C}}^{T} L^{T} + \bar{A} - L \bar{C} + ξ I + \frac{1}{σ_{1}} \bar{B} {\bar{B}}^{T} + σ_{1} γ_{1}^{2} I \leq 0

(24)

hold, where $ξ$ and $σ_{1}$ are small positive constants.

Then, the error system (12) asymptotically stable, and $e$ will converge to zero in finite time.

Proof. Choose a Lyapunov functional candidate to be

V = e^{T} e .

(25)

Calculating the derivative of $V$ according to (12) and (22) yields

\begin{matrix} \overset{\cdot}{V} = {\overset{\cdot}{e}}^{T} e + e^{T} \overset{\cdot}{e} \\ = [(\bar{A} - L \bar{C}) e + \bar{B} ((f_{se} - {\tilde{f}}_{se}) - K_{s} sgn (Y - \hat{Y} {))]}^{T} e \\ + e^{T} [(\bar{A} - L \bar{C}) e + \bar{B} ((f_{se} - {\tilde{f}}_{se}) - K_{s} sgn (Y - \hat{Y}))] \\ = e^{T} {(\bar{A} - L \bar{C})}^{T} e + {[(f_{se} - {\tilde{f}}_{se}) - K_{s} sgn (Y - \hat{Y})]}^{T} {\bar{B}}^{T} e \\ + e^{T} (\bar{A} - L \bar{C}) e + e^{T} \bar{B} [(f_{se} - {\tilde{f}}_{se}) - K_{s} sgn (Y - \hat{Y})] \\ = e^{T} ({\bar{A}}^{T} - {\bar{C}}^{T} L^{T} + \bar{A} - L \bar{C}) e \\ + 2 e^{T} \bar{B} [(f_{se} - {\tilde{f}}_{se}) - K_{s} sgn (Y - \hat{Y})] . \end{matrix}

(26)

Since

2 e^{T} \bar{B} (f_{se} - {\tilde{f}}_{se}) \leq \frac{1}{σ_{1}} e^{T} \bar{B} {\bar{B}}^{T} e + σ_{1} (f_{se} - {\tilde{f}}_{se}) (f_{se} - {\tilde{f}}_{se})^{T},

(27)

and $\bar{C} = [\begin{matrix} 0 & I \end{matrix}]$ . Then

\begin{matrix} \overset{\cdot}{V} \leq e^{T} ({\bar{A}}^{T} - {\bar{C}}^{T} L^{T} + \bar{A} - L \bar{C} + \frac{1}{σ_{1}} \bar{B} {\bar{B}}^{T}) e \\ + σ_{1} ∥ f_{se} - {\tilde{f}}_{se} ∥^{2} - 2 e^{T} \bar{B} K_{s} sgn (Y - \hat{Y}) \\ \leq e^{T} ({\bar{A}}^{T} - {\bar{C}}^{T} L^{T} + \bar{A} - L \bar{C} + \frac{1}{σ_{1}} \bar{B} {\bar{B}}^{T} + σ_{1} γ_{1}^{2} I) e \\ - 2 e^{T} \bar{B} K_{s} sgn (Y - \hat{Y}) \\ \leq - e^{T} ξ e - 2 ∥ \bar{B} K_{s} ∥ ∥ e ∥ \\ = - ξ ∥ e ∥^{2} - 2 ∥ \bar{B} K_{s} ∥ ∥ e ∥ . \end{matrix}

(28)

This means that the closed-loop system is uniformly ultimately bounded for the fault.

This completes the proof.

The design algorithms for the gains $L$ and the filter $F (s)$ should not destroy the stability of the system. It is suitable to select the cutoff angular frequency of $F (s)$ being more than 5-10 times larger than $ω_{r}$ (She et al. 2011).

For the dual system of the plant

\begin{matrix} {\overset{\cdot}{X}}_{L} = {\bar{A}}^{T} X_{L} + ({\bar{C}}^{T})^{T} u_{L}, \\ Y_{L} = {\bar{B}}^{T} X_{L} . \end{matrix}

(29)

A gain $L$ can be designed according to minimizing the performance index

J = \int_{0}^{\infty} (X_{L}^{T} Q X_{L} + u_{L}^{T} R u_{L}) dt,

(30)

and satisfying theorem 2, where $Q$ and $R$ are the weighting matrices.

Figure 2 presents an illustrative architecture of the system to be designed. It mainly contains a system model block, and an EID-based fault rejection block.

Figure 2.

Architecture of the system.

Experimental verification

This section takes the example of a closed-loop torque control system for a PMSM that demonstrated the validity of the scheme. Two cases are discussed.

Case 1. Only one current sensor fault occurs, and the fault is given as

f_{α} = {\begin{matrix} 0, & t < 0.3 s, \\ 20 \sin 8 π t + \sin 30 π t, & t \geq 0.3 s, \end{matrix}

(31)

Case 2. Two current sensors faults occur simultaneously, and the faults are given as

f_{α} = {\begin{matrix} 0, & t < 0.3 s, \\ 20 \tanh (t), & t \geq 0.3 s, \end{matrix}

(32)

and

f_{β} = {\begin{matrix} 0, & t < 0.5 s, \\ 20, & 0.5 s \leq t < 1.2 s, \\ 30, & t \geq 1.2 s . \end{matrix}

(33)

Since $ω_{r}$ in (31) is $30 π$ rad/s, the low-pass filter $F (s)$ was selected to be

F (s) = \frac{1}{0.001 s + 1} .

(34)

Selecting $Q = diag {10^{6}, 10^{6}, 1, 1}$ and $R = 1$ . Then, according to the PMSM parameters (see Table 1), we found a feasible solution for (30)

L = [\begin{matrix} 778.8988 & 0 & 38.4943 & 0 \\ 0 & 778.8988 & 0 & 38.4943 \end{matrix}] .

Table 1.

Parameters of the PMSM.

Parameters	Unit	Value
Stator resistance R_s	$Ω$	0.02
$Q$ –axis inductance ( $L_{q}$ )	H	0.001500
$D$ –axis inductance ( $L_{d}$ )	H	0.003572
Inertia ( $J$ )	kg·m²	100
Magnetic flux ( $ψ_{r}$ )	Wb	0.892
Number of pole pairs ( $P$ )	pairs	4
Damping coefficient ( $B$ )	Nm·s/rad	0.0001
DC-bus voltage ( $V_{dc}$ )	V	1500

Selecting the SMO gain

K_{s} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \\ 0.01 & 0 \\ 0 & 0.01 \end{matrix}] .

An experimental system was built using a TMS320F2812 DSP controller and an RT-LAB OP5600 model (containing an inverter and a PMSM). Figure 3 and Figure 4 show the structure and the experiment setup of RT-LAB hardware-in-the-loop system, separately. The control scheme of $i_{d} = 0$ is carried out for a closed-loop torque control system of a PMSM. Set the rotor speed be 200 rad/s and the torque be 300 Nm.

Figure 3.

Structure of RT-LAB hardware-in-the-loop system.

Figure 4.

RT-LAB experiment setup.

In Case 1, $α$ -axis current sensor fault occurred as (31), and $β$ -axis current sensor had no fault. Figure 5 shows the $α β$ -axis currents. $α$ -axis current deviated from the normal value when fault occurred at $t \geq 0.3$ s. And the maximum deviation of current was 20.54, which was 36.679% of the reference current amplitude. However, the EID-based fault rejection method effectively suppressed the effect of the fault on $α$ -axis current, and the maximum deviation of current was only 0.55, which was only 0.982% of the reference current amplitude. Figure 6 shows the phase currents. The balance of the phase currents was destroyed (see Figure 6(a)) when the fault occurred. The magnitude of the phase currents is given as (Yan et al., 2018)

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

(35)

With the $α$ -axis current sensor fault was rejected, the phase currents tended to be normal in the EID-based fault rejection system (see Figure 6(b)). Figure 7 shows the $α$ -axis current sensor fault. In system (7), $\bar{B} = {[\begin{matrix} B & 0 \end{matrix}]}^{T}$ . That is, one of the state equations of system (7), equation (6), did not has input $u$ . Therefore, the EID of the fault in Figure 1, ${\tilde{f}}_{se}$ , was regarded as a true estimate of the fault. As seen in Figure 7, the estimate of the fault accurately estimated the $α$ -axis current sensor fault. The mean-square error (MSE = $\frac{1}{t} \int_{0}^{t} | e (t) |^{2} dt$ ) between the fault and its estimate of the EID method based on SMO (Figure 7 (b)) was 0.201. And that of the EID method based on Luenberger observer (LO) (Figure 7 (a)) was 0.334, which was 1.662 times bigger than the presented method one. Figure 8 shows the torque of the system. Since the torque is related to the current, the torque occurred a corresponding fluctuation due to the current sensor fault. When the designed EID-based fault rejection system was introduced, the torque returned to normal. This shows the good effect of the fault rejection.

Figure 5.

$α β$ -axis currents.

Figure 6.

Phase currents.

Figure 7.

$α$ -axis current sensor fault.

Figure 8.

Torque.

In Case 2, $α$ -axis current sensor fault occurred as (32), and $β$ -axis current sensor fault occurred as (33). Figures 9 and 10 show the $α β$ -axis currents and phase currents, separately. When fault occurred, $α$ -axis current deviated from the normal value at $t \geq 0.3$ s and increased as (32). And $β$ -axis current deviated from the normal value at $t \geq 0.5$ s, and it increased 20 A at $t = 0.5$ s and 30 A at $t = 1.2$ s as (33). At the same time, the balance of the phase currents was destroyed accordingly (see Figure 10 (a)). However, the designed fault rejection system rejected the effect of the fault on currents. It was seen from Figure 9 that the $α β$ -axis currents were normal, and the balance of phase currents was recovered (see Figure 10 (b)). Figure 11 and 12 show the current sensor faults. The maximum estimated errors of $f_{s α}$ and $f_{s β}$ with EID approach based on SMO were 0.0734 A and 0.1254 A, respectively. And the maximum estimated errors of $f_{s α}$ and $f_{s β}$ with EID approach based on LO were 0.1035 A and 0.1597 A, respectively. The MSEs of $f_{s α}$ and $f_{s β}$ with the EID approach based on SMO were 0.0116 and 0.136, respectively. And those of $f_{s α}$ and $f_{s β}$ with the EID approach based on LO were 0.0152 and 0.1762, respectively. The MSE of $f_{s α}$ with the EID approach based on LO was 1.31 times bigger than the presented method one, and the MSE of $f_{s β}$ with the EID approach based on LO was 1.296 times bigger than the presented method one. That is, the estimation accuracy of the designed method is higher than it of the EID based on LO. Figure 13 shows the torque of the system. The torque started to fluctuation due to the effect of the faults. But the effect was effectively rejected in the designed fault rejection system.

Figure 9.

$α β$ -axis currents.

Figure 10.

Phase currents.

Figure 11.

$α$ -axis current sensor fault.

Figure 12.

$β$ -axis current sensor fault.

Figure 13.

Torque.

Conclusions

This paper presented an EID approach based on SMO to estimate and reject current sensor faults for a PMSM system by supposing the faults to be external disturbances. An equivalent-input-fault was estimated by use an EID estimator and a state SMO. Then, the effect of the faults was rejected from the control input based on EID theory. The equivalent-input-fault ensures that the existing controller gives correct control input commands whether or not the faults occur. Fault-tolerant control is thus performed without control reconstruction. A robustness analysis and stability proof have also been performed. Moreover, an RT-LAB hardware-in-the-loop experiment based on different faults (periodic fault, incipient fault, and intermittent fault) scenarios in PMSM system shows the effectiveness of the presented approach. It is shown that the EID approach based on SMO not only ensure the performance for faults reject, but also make the fault estimate accuracy much better than the existing EID approach.

It would like to be pointed out that the coupling separation of sensor fault and disturbance in EID-based approach and control compensation is another interesting topic, and is under investigation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of China (No. 61773159), the program of JSPS (Japan Society for the Promotion of Science) International Research Fellows (No. 17F17807), Scientific Research Fund of Hunan Provincial Education Department (No. 17B073). Natural Science Foundation of Hunan Province (Nos. 2019JJ40072, 2018JJ2100 and 2018JJ2093).

ORCID iD

Gang Huang

References

Beng

GFH

Zhang

Vilathgamuwa

(2015) Sensor fault-resilient control of interior permanent-magnet synchronous motor drives. IEEE/ASME Transactions on Mechatronics 20 (2): 855–864.

Bianchi

Bolognani

Carraro , et al. (2016) Electric vehicle traction based on synchronous reluctance motors. IEEE Transactions on Industry Applications 52(6): 4762–4769.

Hasegawa

Matsui

(2009) Position sensorless control for interior permanent magnet synchronous motor using adaptive flux with inductance identification. IET Electric Power Applications, 3 (3): 209–217.

Jlassi

Estima

Khil

SKE

, et al. (2017) A robust observer-based method for IGBTs and current sensors fault diagnosis in voltage-source inverters of PMSM drives. IEEE Transactions on Industry Applications 53(3): 2894–2905.

Khil

SKE

Jlassi

Estima

, et al. (2016) Current sensor fault detection and isolation method for PMSM drives, using average normalised currents. IET Electronics Letters 52(17): 1434–1436.

Kommuri

Lee

Veluvolu

(2018) Robust sensors-fault-tolerance with sliding mode estimation and control for PMSM drives. IEEE/ASME Transactions on Mechatronics 23(1): 17–28.

Mekki

Benzineb

Boukhetala

, et al. (2015) Sliding mode based fault detection, reconstruction and fault tolerant control scheme for motor systems. ISA Transactions 57: 340–351.

Menon

Kadam

Azeez

, et al. (2016) A comprehensive survey on permanent magnet synchronous motor drive systems for electric transportation applications. In: IECON 2016–42nd Annual Conference of the IEEE Industrial Electronics Society, Florence, Italy, 23–26 October 2016, pp. 6627–6632.

Qian

Panda

(2004) Torque ripple minimization in PM synchronous motors using iterative learning control. IEEE Transactions on Power Electronics 19(2): 272–279.

10.

She

Xin

Pan

(2011) Equivalent-input-disturbance approach—analysis and application to disturbance rejection in dual-stage feed drive control system. IEEE/ASME Transactions on Mechatronics 16(2): 330–340.

11.

Tian

Mirzaeva

, et al. (2018) Fault-tolerant control of a five-phase permanent magnet synchronous motor for industry applications. IEEE Transactions on Industry Applications 54(4): 3943–3952.

12.

Wang

Tan

Zhou

(2017) A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions. Automatica 79: 290–295.

13.

Guo

Xie

, et al. (2018) A signal-based fault detection and tolerance control method of current sensor for PMSM drive. IEEE Transactions on Industrial Electronics 65(12): 9646–9657.

14.

Yang

Chen

, et al. (2017) Disturbance/uncertainty estimation and attenuation techniques in PMSM drives-a survey. IEEE Transactions on Industrial Electronics 64(4): 3273–3285.

15.

Zang

Yang

, et al. (2018) Improved initial rotor position estimation for PMSM drives based on HF pulsating voltage signal injection. IEEE Transactions on Industrial Electronics 65(6): 4702–4713.

16.

Zhang

Yao

Rizzoni

(2017) Fault diagnosis for electric drive systems of electrified vehicles based on structural analysis. IEEE Transactions on Vehicular Technology 66(2): 1027–1039.

17.

Zhao

Zhang

, et al. (2017) Sliding mode observer-based current sensor fault reconstruction and unknown load disturbance estimation for PMSM driven system. Sensors 17: 2833.

18.

Zhou

She

, et al. (2014) Estimation and rejection of aperiodic disturbance in a modified repetitive-control system. IET Control Theory and Applications 8(10): 882–889.

19.

Yan

Zou

, et al. (2018) A novel open-circuit fault diagnosis method for voltage source inverters with a single current sensor. IEEE Transactions on Power Electronics, 33 (10): 8775–8786.