Sliding Mode Observer-Based Current Sensor Fault Reconstruction and Unknown Load Disturbance Estimation for PMSM Driven System

The permanent magnet synchronous motor (PMSM) has been widely used in high-speed trains and electric vehicles, due to its good dynamic response, high torque-to-current ratio, high power density and excellent tracking precision [1]. The reliable operation of PMSM is the basic requirement for the application of high-speed trains and electric vehicles. However, current sensor failure often occurs in the complex PMSM-driven system. The current sensor is located in the feedback channel of the PMSM control system. Even a tiny fault of the current sensor may lead to the misoperation of the PMSM system and destroy the stability of the system. Moreover, the nonlinearities and system disturbances can cause a misleading alarm. Therefore, it is necessary to study the fault diagnosis and fault reconstruction of the PMSM current sensor and improve the overall performance of the PMSM system. This paper proposes a new scheme of reconstructing current sensor faults and estimating unknown load disturbance for the PMSM-driven system.

In recent years, many control methods have been proposed to diagnose the current sensors fault of PMSM. An approach of current sensor fault detection and isolation (FDI) for PMSM drives is presented based on signal analysis instead of currents’ residual generation through observers in [2]. A new approach is proposed for current sensor fault diagnosis of permanent magnetic synchronous generator (PMSG) drives in wind energy conversion systems (WECSs) using the measured phase currents in [3]. In [2,3], it can only detect current fault, but not reconstruct and locate current fault. The method of current sensor fault reconstruction for PMSM drives is presented on the $\alpha$-$\beta$ axis in [4]. A current sensor fault reconstruction algorithm for the interior PMSM (IPMSM) system based on active flux linkage by the sliding mode observer is presented in [5,6]. An on-line software fault detection, localization and system reconfiguration method is proposed for PMSM based on the monitoring signals of “abc” currents, DC-link voltage and rotor speed/position in [7]. The current sensor fault diagnosis is presented for the PMSM drive system in [8] based on the differential algebraic method. However, the unknown load disturbance is not considered for the PMSM drive systems in [4,5,6,7,8].

Almost all real dynamic systems can be represented as fully-Lipschitz nonlinear systems, at least locally [9]. The problem of incipient fault diagnosis for a class of Lipschitz nonlinear systems with sensor biases is dealt with by sliding mode observers in [10]. An observer-based sensor fault estimation method for generalized Lipschitz nonlinear systems in the presence of input disturbance and measurement noise is developed in [11]. A sensor fault estimation method is presented for a general class of uncertain Lipschitz nonlinear augmented systems in [12]. A model-based FDI scheme for robot manipulators with actuator and sensor faults is investigated in [13]. State and fault estimations for a class of uncertain Lipschitz nonlinear systems is proposed using a descriptor from the observer and an adaptive sliding mode observer in [14]. The problems of simultaneous actuator and sensor fault detection for a class of uncertain Lipschitz nonlinear systems are considered when the observer matching condition is not satisfied in [15]. Fault reconstruction problems for a class of uncertain Lipschitz nonlinear systems with actuator faults, sensor faults and external disturbances are considered in [16]. An observer-based fault reconstruction method for polymer electrolyte membrane (PEM) fuel cells is presented by the adaptive-gain second-order sliding mode (SOSM) observer in [17]. A high gain observer with multiple sliding modes for simultaneous state and fault estimations for MIMO nonlinear systems is developed in [18]. A higher-order sliding mode observer based on the super-twisting algorithm for state and unknown input estimations is developed for estimating the road adhesion coefficient in [19]. A new scheme for estimating the actuator and sensor fault for Lipschitz nonlinear systems with unstructured uncertainties is proposed using the sliding mode observer technique in [20,21].

Inspired by the above surveys, the PMSM is taken as a Lipschitz nonlinear system. This paper proposes a new scheme of reconstructing current sensor faults and estimating unknown load disturbance for the PMSM-driven system. Two sliding mode observers (SMOs) are designed: the unknown load disturbance is estimated by the first SMO in subsystem, which has unknown load disturbance, and the sensor faults are reconstructed using the second SMO in augmented subsystem which have sensor faults. The gains of the proposed SMOs and their stability analysis are developed via the solution of linear matrix inequality (LMI). The adoption of the LMI algorithm makes it easier to obtain the key parameters of the design of SMOs and relaxes the selection criteria for the PMSM-driven system. The scheme can be applicable for incipient fault, intermittent fault, high frequency and low frequency fault or any other types of faults. This makes the theory of the sliding mode observer applicable to the engineering of PMSM current sensor fault reconstruction.

The remainder of this paper is organized as follows: The system description of PMSM is presented in Section 2, and the model of PMSM is converted into two subsystems. Section 3 designs two SMOs such that the unknown load disturbance is estimated, and the current sensor faults are reconstructed. Stability of the system is proven using Lyapunov analysis. The sufficient conditions for the stability of the scheme are derived and expressed as linear matrix inequalities (LMI). The proposed method is applied to the PMSM systems in Section 4. The overall architecture is tested in simulation and experiment, providing good results. The simulation and experiment results are shown in Section 5. Finally, conclusions are given in Section 6.

The dynamic mathematical model of PMSM can be defined in the d-q reference frame as follows [22]: where $R_s$ is the stator resistance ($\mathsf{\Omega })$; $u_d$, $u_q$, $i_d$, $i_q$, $L_d$ and $L_q$ represent the d-q axis stator voltages (V), currents (A) and inductances (Wb), respectively; ${\psi }_r$ is the amplitude of the permanent magnet flux linkage (Wb); ${\omega }_e$ and ${\theta }_e$ are the electrical angular velocity (rad/s) and the electrical angle (rad); $n_p$ is the number of pole pairs; $T_L$ is load torque (N·m); J and B are the total moment of inertia (kg·m${}^2$) and the viscous friction coefficient (Nm·s/rad).

The dynamic mathematical model of IPMSM with current sensors fault can be described as follows: where $f_q$ and $f_d$ denote the current sensor fault of the q-axis and d-axis.

Equation (2) can be transformed into the following nonlinear system: where $\mathit{x}\left(t\right)\in {\mathit{R}}^n$, $\mathit{x}={\left[\begin{array}{cccc}{\omega }_e& {\theta }_e& i_q& i_d\end{array}\right]}^T$ is the state vector; $\mathit{u}\left(t\right)\in {\mathit{R}}^m$, $\mathit{u}={\left[\begin{array}{cc}{u}_q& u_d\end{array}\right]}^T$ is the measurable input vector; $\mathit{y}\left(t\right)\in {\mathit{R}}^p$, $\mathit{y}={\left[\begin{array}{ccc}{\omega }_e& i_q& i_d\end{array}\right]}^T$ is the measurable output vector; $\mathit{d}\left(t\right)\in {\mathit{R}}^r$, $\mathit{d}=T_L$ is unknown input disturbances; ${\mathit{f}}_s\left(t\right)\in {\mathit{R}}^q$, ${\mathit{f}}_s\left(t\right)=\left[\begin{array}{cc}{f}_q& f_d\end{array}\right]$ denote the sensor faults, which is unknown bounded; $\mathit{A}\in {\mathit{R}}^{n\times n}$, $\mathit{B}\in {\mathit{R}}^{n\times m}$ and $\mathit{C}\in {\mathit{R}}^{p\times n}$ are known matrices; $\mathit{D}\in {\mathit{R}}^{n\times r}$ is the known corresponding disturbance distribution matrix; $\mathit{E}\in {\mathit{R}}^{p\times q}$ is the known distribution matrix of sensor faults; $\mathit{f}(x,t)\in {\mathit{R}}^n$ is the known nonlinear function. For System (2), $n=4$, $m=2$, $p=3$, $q=2$, $r=1$ and $p\ge q+r$.

If Assumption 3 holds, there are two transformation nonsingular matrices T and S [23]. The original System (3) can be transformed into the following two subsystems.

where $\mathit{x}={\left[\begin{array}{cc}{\mathit{x}}_1& {\mathit{x}}_2\end{array}\right]}^T$, ${\mathit{x}}_1\in {\mathit{R}}^r$, ${\mathit{x}}_2\in {\mathit{R}}^{n-r}$; $\mathit{y}={\left[\begin{array}{cc}{\mathit{y}}_1& {\mathit{y}}_2\end{array}\right]}^T$, ${\mathit{y}}_1\in {\mathit{R}}^r$, ${\mathit{y}}_2\in {\mathit{R}}^{p-r}$. ${\mathit{A}}_{11}\in {\mathit{R}}^{r\times r}$, ${\mathit{A}}_{12}\in {\mathit{R}}^{r\times (n-r)}$, ${\mathit{A}}_{21}\in R^{(n-r)\times r}$, ${\mathit{A}}_{22}\in {\mathit{R}}^{(n-r)\times (n-r)}$, ${\mathit{B}}_1\in {\mathit{R}}^{r\times m}$, ${\mathit{B}}_2\in {\mathit{R}}^{(n-r)\times m}$, ${\mathit{C}}_{11}\in {\mathit{R}}^{r\times r}$, ${\mathit{C}}_{22}\in {\mathit{R}}^{(p-r)\times (n-r)}$, ${\mathit{D}}_1\in {\mathit{R}}^{r\times r}$, ${\mathit{E}}_2\in {\mathit{R}}^{(p-r)\times q}$, ${\mathit{f}}_1\in {\mathit{R}}^r$,${\mathit{f}}_2\in {\mathit{R}}^{n-r}$.

The matrix transformation in Systems (4) and (5) is as follows: where ${\mathit{C}}_{11}$ and ${\mathit{D}}_1$ are invertible matrices, and $rank\left({\mathit{D}}_1\right)=r$.

For Subsystem (5), define a new state ${\mathit{x}}_3={\int }_0^t{y}_2\left(\tau \right)d\tau$, so that: where ${\mathit{x}}_3\in {\mathit{R}}^{p-r}$ is the new state vector; ${\mathit{A}}_s$, ${\mathit{B}}_s$ are the filter matrices to be designed, ${\mathit{A}}_s\in {\mathit{R}}^{(p-r)\times (p-r)}$, ${\mathit{A}}_s$ is an arbitrary filter matrix; ${\mathit{B}}_s\in {\mathit{R}}^{(p-r)\times (p-r)}$, ${\mathit{B}}_s$ is a full rank constant matrix.

If ${\mathit{A}}_{\mathbf{s}}$ is chosen as the zero matrix and ${\mathit{B}}_{\mathbf{s}}$ is chosen as the identity matrix, substituting Equation (5) into Equation (6), Equation (6) can now be expressed as follows:

Based on Equation (5) and (7), the augmented new subsystem can be expressed as follows:

The augmented System (8) can then be rewritten in a more compact form as: where ${\mathit{x}}_0={\left[\begin{array}{cc}{\mathit{x}}_2& {\mathit{x}}_3\end{array}\right]}^T$, ${\mathit{y}}_0={\mathit{x}}_3$; ${\mathit{x}}_0\in R^{n+p-2r}$, ${\mathit{y}}_0\in {\mathit{R}}^{p-r}$, ${\mathit{A}}_0=\left[\begin{array}{cc}{\mathit{A}}_{22}& \mathbf{0}\\ {\mathit{C}}_{22}& \mathbf{0}\end{array}\right]$, ${\mathit{A}}_1=\left[\begin{array}{c}{\mathit{A}}_{21}\\ \mathbf{0}\end{array}\right]$, ${\mathit{F}}_2=\left[\begin{array}{c}{\mathit{f}}_2(\mathit{x},t)\\ \mathbf{0}\end{array}\right]$, ${\mathit{B}}_0=\left[\begin{array}{c}{\mathit{B}}_2\\ \mathbf{0}\end{array}\right]$, ${\mathit{E}}_0=\left[\begin{array}{c}\mathbf{0}\\ {\mathit{E}}_2\end{array}\right]$, ${\mathit{C}}_0=\left[\begin{array}{cc}\mathbf{0}& {\mathit{I}}_{p-r}\end{array}\right]$. ${\mathit{A}}_0\in {\mathit{R}}^{(n+p-2r)\times (n+p-2r)}$, ${\mathit{A}}_1\in {\mathit{R}}^{(n+p-2r)\times r}$, ${\mathit{B}}_0\in {\mathit{R}}^{(n+p-2r)\times m}$, ${\mathit{C}}_0\in {\mathit{R}}^{(p-r)\times (n+p-2r)}$, ${\mathit{E}}_0\in {\mathit{R}}^{(n+p-2r)\times q}$, ${\mathit{F}}_2\in {\mathit{R}}^{n+p-2r}$.

Accordingly, Subsystem (4) is rewritten as: where ${\mathit{A}}_2=\left[\begin{array}{cc}{\mathit{A}}_{12}& 0_{r\times (p-r)}\end{array}\right]$, ${\mathit{A}}_{11}\in {\mathit{R}}^{r\times r}$, ${\mathit{A}}_{12}\in {\mathit{R}}^{r\times (n-r)}$, ${\mathit{A}}_2\in {\mathit{R}}^{r\times (n+p-2r)}$, ${\mathit{C}}_{11}\in {\mathit{R}}^{r\times r}$, ${\mathit{B}}_1\in {\mathit{R}}^{r\times m}$, ${\mathit{D}}_1\in {\mathit{R}}^{r\times r}$, ${\mathit{f}}_1\in {\mathit{R}}^r$, ${\mathit{y}}_1\in {\mathit{R}}^r$.

In this section, two sliding mode observers are designed for Subsystem (9) and Subsystem (10) respectively; one of which is to reconstruct the sensor fault, while the other one is to estimate the unknown disturbance. The simultaneous reconstruction of the sensor faults and estimation of the unknown disturbance are studied.

In this section, taking the PMSM drive system as an example, the effectiveness of the scheme in sensor fault reconstruction and unknown load estimation is demonstrated. The nonsingular transformation matrices T and S are chosen as: then, the PMSM System (2) can be converted into the following two subsystems:

For Subsystem (47), a new state ${\mathit{x}}_3={\int }_0^t{y}_2\left(\tau \right)d\tau$ is defined, and ${\mathit{A}}_{\mathit{s}}=0$ and ${\mathit{B}}_{\mathit{s}}=\mathit{I}$ are chosen; it can be obtained from Equation (7):