An Active Fault-Tolerant Control Strategy for Current Sensors Failure for Induction Motor Drives Using a Single Observer for Currents Estimation and Axes Transformation

Fault Tolerant Control (FTC) methods are crucial in variable speed drives used in critical and high-sensitivity applications, such as, space exploration, nuclear power plants, electric vehicles, air and rail transportation; in order to ensure system security, and uninterrupted operation, even with degraded performance, when an unexpected failure occurs in the variable speed drive. Consequently, a fault tolerant control strategy must be implemented to guarantee that faults are handled in such a way that there will be no damage.

Generally, variable speed Induction Motor (IM) drives need feedback information from a speed/position transducer, two or three current sensors and a dc-link voltage sensor at least [1], [2]. Unfortunately, sensors are very prone to failures, and for this reason several research results have been published regarding sensors fault-tolerant control, as well as Fault Detection, Isolation, and Reconfiguration (FDIR) methods [3-10].

The study [3] presents a sensor FTC of an induction motor drive, based on the extended Kalman filter, and a reduced number of adaptive observers, to keep the operation of the system under the occurrence of sensors failures. In [4], a fault-tolerant vector control is proposed. A logic circuit is used to identify the faulty current sensor. The missing current information is replaced by an appropriate stator current, which value is calculated from algebraic equations and from the other measured healthy currents. However, this approach is not practical in the event of a successive failure in the different sensors. A fault-tolerant control scheme is proposed in Ref. [5], in which an adaptive fuzzy logic-based observer is used to generate residuals, which analysis allows the detection, and isolation of the faulty sensor. However, the dynamic response of this scheme for isolating defective sensors is too slow. In Ref. [6], it is proposed a sensor fault detection and isolation scheme for an IM drive, based on currents estimation and rotor resistance variations, which are then used in a decision unit for the identification and isolation of the faulty sensor. According to Yu et al. [7], a FTC technique against speed and current sensors failures in IM drives is proposed, and based on three adaptive observers with different inputs. However, this method is not suitable to achieve an efficient FTC in case of failures in two or three sensors. The results obtained by Yu et al. [8] offer a solution to the succession of faults in the different sensors but the number of the used adaptive observers is still three. Another type of FTC methods has been also presented in the literature, in which the sensors FTC algorithm changes the control according to the type of the faulty sensor, as proposed in Refs. [11-14]. A smooth FTC of IM drives with sensor failures is presented in the works [11, 12]. In the healthy state, the IM drive's operation is accomplished using direct torque control; indirect field-oriented control is then used when the dc-link voltage sensor fails, and v/f control is utilized during current sensors failure. The major drawback of this method, as reported by Tabbache et al. [15], is that these techniques are experimentally difficult to implement.

This paper proposes a direct torque fault-tolerant control strategy for the current sensor’s failure in induction motor drives. Some features are distinct from the other proposed methods. First, the estimation of stator currents is performed using a single observer. Then, the method proposed here allows the detection of faulty sensors even when several sensors fail successively, case in which a decision unit, based on a logic circuit, performs the task of reconfiguring and isolating the effects corresponding to the faulty sensor. Finally, the proposed technique can be reorganized in an adaptive way in the event of the loss of one or two sensors. The proposed scheme is simulated in MATLAB/Simulink environment, and experimentally implemented in a laboratory test bench under different operating conditions.

The remainder of the paper is organized into 5 sections. Section 2 analyses the direct torque control. Afterwards, section 3 describes the concept of the proposed FTC. Simulation and experimental results are presented and discussed in section 4. Finally, section 5 presents the main conclusion of the paper.

In the proposed fault-tolerant direct torque control, two current sensors in phase-a, and phase-b are used in addition to a dc-link voltage sensor, and an incremental encoder. Furthermore, using the control signals S_A, S_B and S_C, a voltage synthesizer constructs the three-phase stator voltages applied to the machine from the voltage measured in the dc-link.

3.1 Stator currents estimation

In order to estimate the line currents, a modified adaptive observer is used, which gives high-performance estimation stator currents, based on the IM model, which is described by the following state equation:

$\left\{\begin{array}{c} \dot{X}=A\left(\omega_{r}\right) X+B U \\ Y=C X \end{array}\right.$         (4)

here:

$\begin{gathered} X=\left[\begin{array}{lll} i_{s \alpha} & i_{s \beta} & \psi_{r \alpha} \psi_{r \beta} \end{array}\right]^{T} ; Y=\left[\begin{array}{ll} i_{s \alpha} & i_{s \beta} \end{array}\right]^{T} \\ U=\left[\begin{array}{l} V_{s \alpha} V_{s \beta} \end{array}\right]^{T} \end{gathered}$

and:

$\begin{gathered} A=\left[\begin{array}{cccc} a_{1} & 0 & a_{2} & \omega_{r} a_{3} \\ 0 & a_{1} & -\omega_{r} a_{3} & a_{2} \\ a_{4} & 0 & a_{5} & -\omega_{r} \\ 0 & a_{4} & \omega_{r} & a_{5} \end{array}\right] ; B=\left[\begin{array}{cc} \frac{1}{\sigma L_{s}} & 0 \\ 0 & \frac{1}{\sigma L_{s}} \\ 0 & 0 \\ 0 & 0 \end{array}\right] ; \\ C=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]  \end{gathered}$

as well as:

$\begin{gathered} a_{1}=-\left(\frac{1}{\tau_{s} \sigma}+\frac{(1-\sigma)}{\tau_{r} \sigma}\right) ; \\ a_{2}=\frac{M}{\sigma L_{s} L_{r} \tau_{r}} ; a_{3}=\frac{M}{\sigma L_{s} L_{r}} ; a_{4}=\frac{M}{\tau_{r}} ; \\ a_{5}=\frac{1}{\tau_{r}} ; \tau_{s}=\frac{L_{s}}{R_{s}} ; \tau_{r}=\frac{L_{r}}{R_{r}} ; \end{gathered}$

The idea of the proposed stator currents estimator is based on the general theory of Luenberger observer [16], which allows the estimation of variable or unknown parameters of a system. The equation of the Luenberger observer is expressed in (5), where the symbol “^” denotes the estimated values:

$\left\{\begin{array}{c} \dot{\hat{X}}=A\left(\omega_{r}\right) \hat{X}+B U+K \xi \\ \hat{Y}=C \hat{X} \end{array}\right.$         (5)

where:

$\begin{gathered} \hat{X}=\left[\widehat{l_{s \alpha}} \widehat{l_{s \beta}} \widehat{\psi_{r \alpha}} \widehat{\psi_{r \beta}}\right]^{T} ; \hat{Y}=\left[\widehat{l_{s \alpha}} \widehat{l_{s \beta}}\right]^{T} ; \\ \xi=\left[i_{s \alpha}-\widehat{l_{s \alpha}} i_{s \beta}-\widehat{l_{s \beta}}\right]^{T} \end{gathered}$

Since there is no information about the measured currents, the vector ξ becomes:

$\xi=\left[-\widehat{l_{s \alpha}}-\widehat{l_{s \beta}}\right]^{T}$

The estimation is performed throw the conservation of the state model (4) of the IM with the gain matrix K. This matrix ensures the convergence of the observer, and has been determined by the classical pole placement procedure. Then, replacing the adaptive mechanism by the measured rotational speed, as well as, feeding the observer with the stator voltages provided by the voltage synthesizer. This gives the stator currents estimation, as illustrated in Figure 1.

1.png

Figure 1. Proposed stator currents estimator

Considering the following system of equations, that present the current observer model in the complex format, the gain matrix K is defined as follows:

$\left\{\begin{array}{c} \dot{\bar{X}}=\bar{A} \hat{X}+B \bar{V}_{s}+\bar{K} \xi \\ \hat{Y}=C \hat{\bar{X}} \end{array}\right.$        (6)

The determination of the gain matrix K uses the conventional pole placement procedure, through the imposition of the observer's poles and consequently of its dynamics [17-20].

The characteristic equation of the observer is as follow:

$G(s)=\operatorname{det}(s I-(\bar{A}-\bar{K} C))$         (7)

Developing the different matrices A, K and C knowing that $\bar{K}=\left[\begin{array}{l} K^{\prime} \\ K^{\prime \prime} \end{array}\right]$ the following equation is obtained:

$\begin{aligned} G(s)=s^{2}+\left(\frac{1}{\sigma T_{s}}\right.&\left.+\frac{1}{\sigma T_{r}}-j \omega_{r}+K^{\prime}\right) s \\ &+\left(\frac{1}{T_{r}}-j \omega_{r}\right)\left\{\left(\frac{1}{\sigma T_{s}}+\frac{1}{\sigma T_{r}}\right)+K^{\prime}\right\} \\ &+\left(\frac{M}{T_{r}}-K^{\prime \prime}\right)\left(\frac{M}{\sigma L_{s} L_{r}}\right)\left(\frac{1}{T_{r}}-j \omega_{r}\right) \end{aligned}$         (8)

where, K' and K'' are complex gains.

The dynamic equation of the observer is defined as:

$\begin{aligned} H(s)=s^{2}+l\left(\frac{1}{\sigma T_{s}}\right.&\left.+\frac{1}{\sigma T_{r}}-j \omega_{r}\right) s \\ &+l^{2}\left(\frac{1}{T_{r}}-j \omega_{r}\right)\left\{\left(\frac{1}{\sigma T_{s}}+\frac{1}{\sigma T_{r}}\right)\right\} \\ &+\left(\frac{M}{T_{r}}\right)\left(\frac{M}{\sigma L_{s} L_{r}}\right)\left(\frac{1}{T_{r}}-j \omega_{r}\right) \end{aligned}$             (9)

where, $l$ is the proportionality constant, slightly bigger than the unit ($l$=1.004).

Eqns. (8) and (9) are in the form of the following equation: 

$H P(s)=s^{2}+2 \varepsilon \omega_{n} s+\omega_{n}^{2}$          (10)

The comparison of Eqns. (8) and (9) gives for the expressions of K' and K'':

$\left\{\begin{array}{c} K^{\prime}=(l-1)\left(\frac{1}{\sigma T_{s}}+\frac{1}{\sigma T_{r}}-j \omega_{r}\right) \\ K^{\prime \prime}=(l-1)\left[\begin{array}{c} \left\{\left[\frac{1}{\sigma T_{s}}+\frac{1}{\sigma T_{r}}\right] \frac{\sigma L_{s} M}{L_{r}}-\frac{M}{T_{r}}\right\} \\ (l+1)-\frac{\sigma L_{s} M}{L_{r}}\left[\frac{1}{\sigma T_{s}}+\frac{1}{\sigma T_{r}}\right] \\ +j \omega_{r} \frac{\sigma L_{s} M}{L_{r}} \end{array}\right] \end{array}\right.$           (11)

To get the gain matrix K it is supposed that:

$\left\{\begin{array}{l} K^{\prime}=K_{1}+j K_{2} \\ K^{\prime \prime}=K_{3}+j K_{4} \end{array}\right.$                 (12)

So, 

$\begin{gathered} K_{1}=(l-1)\left(\frac{1}{\sigma \tau_{s}}+\frac{1}{\sigma \tau_{r}}\right) \\ K_{2}=-(l-1) \omega_{r} \\ K_{3}=\left(l^{2}-1\right)\left[\left(\frac{1}{\sigma \tau_{s}}+\frac{1}{\sigma \tau_{r}}\right) \frac{\sigma L_{s} M}{L_{r}}-\frac{M}{\tau_{r}}\right] \\ +\frac{\sigma L_{s} M}{L_{r}}\left(\frac{1}{\sigma \tau_{s}}+\frac{1}{\sigma \tau_{r}}\right)(l-1) \\ K_{4}=-(l-1) \frac{\sigma L_{s} M}{L_{r}} \omega_{r} \end{gathered}$

According to the anti-symmetry of matrix A, the gain matrix K is set as follows:

$K=\left[\begin{array}{cccc} K_{1} & K_{2} & K_{3} & K_{4} \\ -K_{2} & K_{1} & -K_{4} & K_{3} \end{array}\right]^{T}$

3.1.1 Currents estimator stability

Considering the following equations:

$\text { Induction motor model: } \dot{X}=A X+B U$          (13)

and

$\text { Observer model: } \dot{\hat{X}}=A \hat{X}+B U+K \xi$                (14)

The estimated stator currents error $\left(e=I_{S}-\widehat{I}_{S}\right)$  is the deference between the observer and the induction motor models, so:

$\dot{e}=(A-K C) e$          (15)

Considering the Lyapunov function below:

$S=e^{T} e+\frac{\left(\Delta \omega_{\mathrm{r}}\right)^{2}}{\lambda}$          (16)

Its time derivative is:

$\frac{d S}{d t}=\left(\frac{d e^{T}}{d t}\right) e+\left(\frac{d e}{d t}\right) e^{T}+\frac{1}{\lambda} \frac{d}{d t}\left(\Delta \omega_{r}\right)^{2}$         (17)

Knowing that $\Delta \omega_{r}=0$ , then:

$\frac{d S}{d t}=\left(\frac{d e^{T}}{d t}\right) e+\left(\frac{d e}{d t}\right) e^{T}$          (18)

Replacing Eq. (15) in Eq. (18) gives:

$\frac{d S}{d t}=e\left\{(A-K C)^{T}+(A-K C)\right\} e^{T}$            (19)

A sufficient condition for this estimator stability is that Eq. (19) is equal to zero, and indeed, this equation is always equal to zero.

3.2 Fault detection, isolation and system reconfiguration

The logic-circuit presented in Figure 2 is intended to ensure the fault detection, isolation, and system reconfiguration. The detection of failures is performed by analyzing the residual signal between the measured and estimated quantities, passing through a Low Pass Filter (LFP) to extract the useful signal that will be compared to a well-defined threshold (Th=0.8), then a proper estimation is used for the isolation and reconfiguration of the system.

2.png

Figure 2. Fault detection, isolation and system reconfiguration scheme

As mentioned previously, two current sensors (in phase-a and phase-b) were used in the proposed method for measuring a and b line currents. The axis transformation presented in Figure 3 developed by Chakraborty and Verma [21] and used in both [22, 23], which is also used in the proposed FTC, in order to have the component currents $(\alpha, \beta)$ using only the measurements of (a, b) components.

3.png

Figure 3. Axis transformation model

As shown in Figure 3, the β-axis is aligned with the b-axis. Considering the equation of transformation from (a, b) to $(\alpha, \beta)$, which is given in (20), it is clear that $i_{s \alpha}$ depends on the two-phase currents $I_{s a} \text { and } I_{s b}$, whereas the current $i_{s \beta}$ depends only on the current $I_{s b}$. Therefore, if the phase-a current sensor becomes faulty, only $i_{S \alpha}$ that will be affected. But, if the phase-b current sensor becomes faulty, both currents $i_{s \alpha} \text { and } i_{s \beta}$ will be affected.

$\left[\begin{array}{l} i_{s \alpha} \\ i_{s \beta} \end{array}\right]=\left[\begin{array}{cc} 2 / \sqrt{3} & 1 / \sqrt{3} \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} I_{s a} \\ I_{s b} \end{array}\right]$          (20)

So, the faulty measured current components $(\alpha, \beta)$ must be replaced by the corresponding estimated currents. As a consequence, Table 1 shows the identification of the faulty sensors and the correct components of the estimated currents that will replace the erroneous components.

Table 1. Faulty sensor identification and correct component selection

As presented in Figure 2 and defined in Table I, it appears that in the case of sensor-a failure, fault detection part of the logic circuit generates an impulse Za=1, whereas, in the case of sensor-b failure, fault detection logic circuit generates an impulse Zb=1. Once the faulty sensor is identified, it is isolated, and the system is reconfigured, using the fault isolation and system reconfiguration part of the logic circuit. The logic gate OR output is 1 when any one of its inputs Za or Zb or both is 1 due to the effect of the failure of sensor-b on $i_{s \alpha}$ current. As a consequence, in the case of sensor-b failure, $i_{s \alpha \text { mes }}$ is isolated and replaced by i_{sa est}.

This paper presents a direct torque fault-tolerant control diagram for current sensors failures in the induction motor drives. For the identification of a sensor failure, a logic circuit is used in the phases where the current sensors are located, which is able to detect the failures of the used current sensors. A current estimator is used to estimate the stator currents. Immediately after the failure of the current sensor, a decision logic circuit automatically selects the correct current signal to maintain the operation of the system. Simulation and experimental results show the effectiveness of the proposed method, where the fault detection is accomplished with fast dynamics, and the drive is performing successfully, without compromising the control system.