Fault Estimation of Rack-Driving Motor in Electrical Power Steering System Using an Artificial Neural Network Observer

Abstract

In this paper, we present the fault estimation of a motor in rack-type electrical power steering (R-EPS) system using an artificial neural network (ANN) observer and the comparison study of estimation performance between ANN observer and model-based ones. For various amplitudes and frequencies of fault, it is not easy to obtain the accurate fault estimation using the model-based observers inherently possessing the accumulated errors and disturbance effect. Such model-based methods often result in undesired consequences; hence, this study employed the “model-free” ANN observer, without using any dynamics and parameters of a motor, to accomplish the better outcomes. Furthermore, the advantages of ANN observer for the fault estimation of the motor have been clearly investigated under several control/fault scenarios, and the effectiveness of proposed work has been validated through an actual experimental study. It is found that the performance of model-based approaches is degraded when the frequencies and amplitudes of fault and control scenarios are changed, but the ANN observer guaranteed performance that was almost the same regardless of fault and control scenarios. Notably, ANN observers showed 84% to 95% of estimation accuracy with almost no delay between estimates and actual faults while model-based approaches did 68% to 86% accuracy along with noticeable delay.

1. Introduction

Once a fault occurred in the actuator/motor of an electrical steering system (EPS), the behavior of system may be gradually or dramatically decreased, ranging from performance degradation to instability. Furthermore, the vehicle is also subject to the critically dangerous situation or unavoidable accidents due to malfunctioned steering condition. Therefore, the immediate and accurate fault estimation/detection is an important step for further remedies and fault accommodations in EPS. Therefore, the related many studies using model-based approaches and ANN techniques have been proposed to resolve this concern. In addition, most of the fault estimation strategies are also incorporated with a fault-tolerant or fail-operation controls (FTC/FOC).

In [1], the reduced-order observer method and switched Lyapunov function technique are used for fault estimation, and the discrete-time switched system with actuator fault and state delay has been addressed by the proposed technique. However, it involves a complicated design process compared to other works, and it is limited to the simulation study. To accurately describe the fault in a nonlinear system, many studies designed observers based on an LPV (linear parameter varying) system [2,3,4,5]. In [2], the actuator fault is denoted by multiplying the control input by efficiency coefficient λ∈[0,1], and the fault and the system state are estimated using the switched linear parameter varying (LPV) extended observer. However, the fault is assumed to be slowly varying or constant nature over time. Ref. [3] compared three observers for the damper fault estimation: fast adaptive fault estimation (FAFE) using an additive fault term, adaptive observer (AO) and LPV observer (LPVO) using multiplying an effectiveness coefficient. The effectiveness of proposed work was also investigated for only limited slowly varying fault scenarios. Ref. [4] presented an LPV method for Fault Estimation for Electro-Rheological dampers of automotive suspension systems. The Faults of the dampers are modeled as Loss of Effectiveness multiplicative factors and are estimated using the LPV extended-state observer, whose gain is derived from the mixed H2/H∞ norm minimization. However, the estimation results show the slow convergence to the assumed fault. Ref. [5] showed a novel and robust algorithm for discrete-time LPV fault estimation using inversion-based concepts and described how to attenuate an additive disturbance signal on the fault reconstruction error. The methods in [4,5] required delicate design procedures for observer and addressed the estimation performance for the step-like fault scenarios.

Furthermore, various fault estimation techniques are presented in [6,7,8,9,10,11,12,13,14]. The authors of [6] proposed two observers; a fixed detection observer that could estimate system state and detect actuator fault and an adaptive diagnostic observer that could accurately estimate the degree of actuator fault. However, the fault was assumed to be constant, and the performance of proposed techniques has not been validated for time-varying fault scenarios. In [7], an extended state space equation containing parameters for fault of rotor and stator was developed, and based on that equation, motor electrical fault was estimated using the UKF algorithm. In [8], the authors proposed an adaptive observer and estimated the active steering system of vehicle state, loss-of-effectiveness coefficient and additive fault coefficient that can classify three types of the fault (i.e., Loss-of-effectiveness fault, Additive fault, Stuck-at-a-fixed-level fault). However, depending on the threshold setting, the fault may not be detected, and there may be a delay in detecting the fault by the time window length. Ref. [9] presented the fuzzy logic interference system to monitor the health of motor through vibration analysis. Ref. [10] proposed an intermediate estimator designed by the intermediate variable that consists of fault signal and system state for nonlinear systems with Lipschitz nonlinearities. Ref. [11] proposed adaptive sliding mode FTC using fault estimated by nonlinear unknown input observer (NUIO). However, Refs. [9,10,11] conducted only simulation studies to exhibit the effectiveness of proposed works. Ref. [12] proposed two observers: FFE (Frequency-based Fault Estimator) and RPS (Robust Parity Space) method. The first approach is based on the frequency response for damper fault of the suspension system, which is simple to be implemented but accurate over the specific frequency range and, the second one is designed with consideration of model uncertainties and obtained best fault estimation results, but its implementation is quite challengeable. In [13], the adaptive observer is designed by an augmented model and the LMI (Linear Matrix Inequality) approach for component fault for the three-phase inverter of high-speed trains. In [14], the fault of actuator was estimated using the augmented state observer, and the active FTC approach is designed to compensate for the online actuator fault estimation. In [15], actuator-integrated fault estimation (FE) and fault tolerant control (FTC) for the electric power steering (EPS) system of a forklift has been addressed. Specifically, the nonlinear unknown input observer (NUIO) was used to estimate the system states and actuator faults, and an adaptive sliding mode FTC system was proposed based on it. The gain of the observer and controller is computed by H∞ optimization and one-step linear matrix inequality (LMI) formula operation. However, Refs. [13,14,15] need to solve the LMI condition and the fault scenarios used are also limited.

Contrary to the model-based approaches [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], the ANN/RNN techniques have been applied to overcome the difficulties of the fault diagnosis/estimation in nonlinear system, Refs. [16,17] proposed adaptive fault estimation observer designed based on RBF (Radial Basis Function) Neural Network. The authors of [18] presented a neural-network-based robust state and fault estimation method for the satellite attitude control system with actuator and sensor fault. Here, they trained the recurrent neural network (RNN) with the inputs and outputs of the current attitude control system. The works [16,17] are performed for limited fault scenarios and only simulation studies. Furthermore, the approaches used in [16,17,18] are designed based on a partial-model-based approach; thus, they are not completely described as model-free method since ANN technique is only used for a fault estimation. Ref. [19] proposed two-stage machine learning techniques that can accurately capture the motor fault modes only by using motor vibration time-domain signals without any complex preprocessing. However, the two-step method imposes a heavy burden on the computational load. Ref. [21] approximated the unknown nonlinearity of vehicle system using ANN techniques, which can be useful for the identification of critical parameters of system. The theoretical frame in this work can be applied to the fault estimation.

Taking into consideration of all the previous ideas, in this paper, we focus on the fault estimation of a motor in R-EPS using the “model-free” ANN observer, which does not require any dynamics and parameters of system, and the performance comparison study between the ANN observer and the model-based approaches is clearly presented. On the contrary to the previous work only containing the simulation studies, the estimation performance for the proposed observer is entirely based on an actual experimental study.

Three major contributions clearly distinguish our endeavor from other existing studies:

(1)

The ANN observer to estimate a fault in a motor in R-EPS is designed using the minimum sensor signal, the rotational angle of the motor measured by a rotational encoder. Only a chirp-like fault signal containing various frequencies has been used for training the proposed ANN observer, resulting in saving training time and obtaining the ANN observer describing the relation between input data and output reference. Additionally, the suitable number of nodes in ANN observer has been explored to reduce the computational burden in a practical implementation while guaranteeing the accurate estimation performance. The hybrid learning rule has been employed for training instead of standard gradient descendent.

(2)

In this study, the performance of the ANN observer is compared with the model-based approaches under the various control and fault scenarios, and the advantage of the ANN observer is clearly addressed. Most of the previous works only focus on fault estimation using a specific control scenario (or without mentioning the control one), which implies that they neglect that the estimation performance and accuracy of a fault are slightly varied by the control scenarios. Therefore, not only different fault scenarios but also several control ones have been employed for this study to address the superiority of ANN observer.

(3)

Most previous works of fault estimation are limited to the simulation-based studies. Therefore, in this paper, the investigation of the estimation performance for the proposed observer is entirely based on an actual experimental study under several control/fault scenarios. The useful estimation results are contained and have been thoroughly investigated.

The rest of the paper is followed. In Section 2, we present problem formulation, and Section 3 contains the design of the control system for the motor to track the desired trajectory. In Section 4, we describe the design of the ANN observer and introduce two representative model-based approaches, and the experimental results are presented in Section 5. Finally, the conclusions have been made.

2. Problem Formulation

In this section, we present the problem formulation for a fault estimation of a motor in R-EPS. A fault is defined here as a disturbing torque to the desired control torque of the motor due to a malfunction of the controller or the motor itself.

To adopt this problem, the mathematical model of motor with a fault is formulized by,

$$J_R{\ddot{\theta }}_R\left(t\right)+B_R{\dot{\theta }}_R\left(t\right)=\left({\tau }_{m,R}\left(t\right)+f_a\left(t\right)\right)+d_R$$

(1)

where $J_R\in ℝ$ and $B_R\in ℝ$ are a moment inertia and a viscous damping of motor, respectively. ${\theta }_R\in ℝ$ is the rotational angle of motor. ${\tau }_{m,R}\in ℝ$ and $d_R\in ℝ$ indicate the control torque of motor and the external disturbance including a friction from the mechanical components of EPS and the equivalent self-alignment torque from tire/road, respectively, and $f_a\left(t\right)\in ℝ$ describes a general unknown nonlinear additive fault which should be identified. Hence, the motor will run according to the amount of torque $\left({\tau }_{m,R}\left(t\right)+f_a\left(t\right)\right)$ under an unhealthy condition. Since we chose the additive fault rather than the loss-of-effectiveness fault, it can be more conveniently used for direct torque compensation in fault tolerant control, which is our next study.

Furthermore, the simple block diagram of estimation and control process for entire system is displayed in Figure 1. According to Figure 1, three major components of the system are also mentioned below.

(i)

The sensor used here is only the rotational encoder measuring the rotational angle of a motor.

(ii)

The selected controller is a Sliding Mode Controller for tracking the desired trajectories and dealing with an external disturbance/model uncertainty and will be designed in the next Section.

(iii)

The ANN observer for estimating a fault, $f_a\left(t\right)$, in a motor is proposed in Section 4. In addition, two representative model-based approaches, an adaptive observer and a Kalman filter, are presented to compare the estimation performance with the one via proposed ANN observer.

3. Design of Robust Controller for Motor in R-EPS

In this section, we present the designed robust controller to track the desired trajectory and manage both an external disturbance and model uncertainty. Specifically, the continuous sliding mode controller has been selected to do its given task. However, it does not have any function related to a fault-tolerant control strategy since this study purely focuses on the estimation of fault in a motor under this proposed controller.

Therefore, considering no fault in a motor, Equation (1) becomes

$$J_R{\ddot{\theta }}_R\left(t\right)+B_R{\dot{\theta }}_R\left(t\right)={\tau }_{m,R}\left(t\right)+d_R$$

(2)

Based on (2) and the idea in [22], the following control law is proposed,

$$\begin{array}{cc}\hfill {\tau }_{m,R}=B_{R.0}{\dot{\theta }}_R+J_R(F_R+{\ddot{\theta }}_{R,d}-& \left(3\lambda \dot{\epsilon }+3{\lambda }^2\epsilon +{\lambda }^3\int \epsilon dt\right)+u_R)\hfill \\ \begin{array}{c}\\ \begin{array}{cc}{F}_R=\{\begin{array}{c}-\frac{\left|S_R\right|}{S_R}\frac{\rho }{J_R}\left|{\dot{\theta }}_R\right| if \left|S_R\right|>N_R\\ 0 if \left|S_R\right|\le N_R\end{array}& \end{array}\end{array}& u_R=\{\begin{array}{c}-\frac{H_R}{J_R}sign\left(S_R\right) if \left|S_R\right|>N_R\\ -\frac{H_R}{J_R}\frac{S_R}{N_R} if \left|S_R\right|\le N_R\end{array}\end{array}$$

(3)

where $S_R\left(t\right)=\dot{\epsilon }+2\lambda \epsilon +{\lambda }^2\int \epsilon dt\in ℝ$ is the designed sliding surface and $\epsilon ={\theta }_R-{\theta }_{R,d}\in ℝ$ is the error between the desired trajectory (${\theta }_{R,d}$) and an actual angle (${\theta }_R$). $N_R$ is the thickness of layer, and $B_{R.0}$ is the nominal value of $B_R$ and is assumed to be $\left|B_{R.0}-B_R\right|\le \rho \in ℝ$. $\rho$ is the positive upper bound value of $B_R$.

Furthermore, substituting (3) into (4) yields the closed-loop system as follows,

$$\ddot{\epsilon }=-\left(3\lambda \dot{\epsilon }+3{\lambda }^2\epsilon +{\lambda }^3\int \epsilon dt\right)+\frac{\mathsf{\Delta }{B}_R}{J_R} {\dot{\theta }}_R+F_R+u_R+\frac{1}{J_R}{d}_R$$

(4)

where $\mathsf{\Delta }{B}_R=B_{R.0}-B_R\in ℝ$. Consequently, Equation (4) can be turned into

$${\dot{S}}_R\left(t\right)=-\lambda S_R\left(t\right)+\frac{\mathsf{\Delta }{B}_R}{J_R}{\dot{\theta }}_R+F_R+u_R+\frac{1}{J_R}{d}_R$$

(5)

Selecting the following Lyapunov candidate function,

$$V_R\left(t\right)=\frac{1}{2}{S}_R^2\left(t\right)>0$$

(6)

The derivative of (6) with respect to a time yields

$${\dot{V}}_R\left(t\right)=S_R\left(t\right){\dot{S}}_R\left(t\right)$$

(7)

For $\left|S_R\right|>N_R$, substituting (5) into (7),

$${\dot{V}}_R\left(t\right)=\frac{\mathsf{\Delta }{B}_R}{J_R}{\dot{\theta }}_R{S}_R-\frac{\rho }{J_R}\left|{\dot{\theta }}_R\right|\left|S_R\right|-\lambda S_R^2+S_R\frac{d_R}{J_R}-S_R\frac{H_R}{J_R}sign\left(S_R\right)$$

(8)

Furthermore, if we select $H_R$ such as $H_R$>${\mathsf{\delta }}_R\ge \left|d_R\right|$,

$$\begin{array}{c}{\dot{V}}_R\left(t\right)\le -\lambda S_R^2+S_R\frac{d_R}{J_R}-\frac{H_R}{J_R}\frac{{\left|S_R\right|}^2}{\left|S_R\right|}\\ \le \frac{\left|d_R\right|}{J_R}\left|S_R\right|-\frac{H_R}{J_R}\frac{{\left|S_R\right|}^2}{\left|S_R\right|}\le \frac{\left|S_R\right|}{J_R}\left({\mathsf{\delta }}_{d.L}-H_R\right)<0\end{array}$$

(9)

Therefore, by control law in (3), the system is ultimately bounded within $\left|S_R\left(t\right)\right|\le N_R$. The controller (3) designed here will be used to track the desired trajectory ${\theta }_{R,d}$ and the observer will estimate the fault of motor (if any fault) at the same time.

4. Design of ANN Observer for a Fault Estimation

For various amplitudes and frequencies of fault, it is not easy to obtain the accurate fault estimation using the model-based observer [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] inherently possessing the accumulated errors and unknown disturbance effect. Therefore, such methods often result in undesired outcomes. Recognizing this issue, in this paper, we employed the ANN-based observer achieving the high order approximation capability for the fault estimation of motor even under various control and fault scenarios. This “model-free” ANN observer does not require any structure and parameters of system; thus, it can be generally used for any estimation problems. Additionally, the fast adaptive and the Kalman Filter-based observers, which are the well-known symbolic brand of adaptive model-based approaches, are introduced to compare with the performance of the ANN observer.

4.1. Design of ANN Observer for a Fault Estimation

The design of proposed ANN observer includes two distinctive phases. In the first phase, according to a given representative fault scenario, collecting and preprocessing the related data are conducted to secure the input data for the reference output. In the second one, an ANN observer shown in Figure 2 is trained based on the data prepared in the first stage. Finally, through two consecutive phases, the best-fitted ANN observer between input and output can be achieved. Here, the proposed ANN observer consists of the input layer (4 nodes), 3 hidden layers (containing n nodes, n nodes and m nodes, respectively) and one output layer (1 node). The number of nodes n and m in the hidden layers has been selected based on the trial-error approach to capture accurate high-order approximation, but the suitable number of nodes should be determined to reduce the computational load and implement it in the cost-effective ECU.

Furthermore, the input vector of the ANN observer is specifically chosen by x$={\left[{\theta }_R {\dot{\theta }}_R {\tau }_{m,R} \left|{\theta }_{R.d}-{\theta }_R\right|\right]}^T\in {\Re }^{4\times 1}$, the angular position of motor, the angular rate and the control torque as well as the absolute error between desire trajectory and actual angular position of motor.

Furthermore, the total signals in the forward path can be calculated as the following equations

$${\varphi }_1=f_1\left(w_{1}x+b_1\right)\in {\Re }^{n\times 1}$$

(10)

$${\varphi }_2=f_2\left(w_2{\varphi }_1+b_2\right)\in {\Re }^{n\times 1}$$

(11)

$${\varphi }_3=f_3\left(w_3{\varphi }_2+b_3\right)\in {\Re }^{m\times 1}$$

(12)

$${\widehat{f}}_a=f_4\left(w_4{\varphi }_3+{\mathrm{b}}_4\right)\in \Re$$

(13)

where x$\in {\Re }^{4\times 1}$ is the input of the proposed ANN classifier. $w_1\in {\Re }^{n\times 4}$, $w_2\in {\Re }^{n\times n}$, $w_3\in {\Re }^{m\times n}$ and $w_4\in {\Re }^{1\times m}$ represent the weight matrices (or vectors) of the first hidden layer, the second hidden layer, the third hidden layer and the output layer, respectively. b₁$\in {\Re }^{n\times 1}$, b₂$\in {\Re }^{n\times 1}$, b₃$\in {\Re }^{m\times 1}$ and b₄$\in \Re$ indicate the bias vectors of the three hidden layers and output layer. The activation function of the first hidden layer is the hyperbolic tangent function, and the sigmoid function is chosen as the activation functions of the 2nd and 3rd layers, and those are followed by

$$f_1\left(\mathsf{\varsigma }\right)=\frac{2}{\left(1+exp\left(-2\varsigma \right)\right)}-1$$

(14)

$$f_i\left(\mathsf{\varsigma }\right)=\frac{1}{\left(1+exp\left(-\varsigma \right)\right)} \mathrm{for} i=2,3$$

(15)

where $\varsigma$ indicates the input vector of above functions. The 4th activation function is simply a linear function to scale the final output signal as follows:

$$f_4\left(\varsigma \right)=\varsigma \in \Re$$

(16)

This proposed ANN observer utilizes a BP (Backpropagation) algorithm to search for the best-fitted weights and biases representing the best mapping between the inputs and the output. Each weight and bias in (10) through (13) are updated in each iteration (specified as step k) via the following learning rules:

$$W_{i|k+1}=W_{i|k}+{\eta }_W{\mathsf{\Phi }}_{i|k} \mathrm{for} i=1,2,3,4$$

(17)

$$b_{i|k+1}=b_{i|k}+{\eta }_b{\mathsf{\Psi }}_{i|k} \mathrm{for} i=1,2,3,4$$

(18)

where ${\eta }_W$ and ${\eta }_b$ are the update rates of weights and biases, and ${\mathsf{\Phi }}_{i|k}$ and ${\mathsf{\Psi }}_{i|k}$ in (17) and (18) are defined by

$${\mathsf{\Phi }}_{i|k}=\{\begin{array}{c}-\left(\frac{\partial J}{\partial W_{i|k}}\right) if k=0\\ -\left(\frac{\partial J}{\partial W_{i|k}}\right)+{\alpha }_k{\Phi }_{i|k-1} if k=1,2,\cdots \end{array}$$

(19)

$${\mathsf{\Psi }}_{i|k}=\{\begin{array}{c}-\left(\frac{\partial J}{\partial b_{i|k}}\right) if k=0\\ -\left(\frac{\partial J}{\partial b_{i|k}}\right)+{\beta }_k{\Psi }_{i|k-1} if k=1,2,\cdots \end{array}$$

(20)

where ${\Phi }_{i|k-1}$ and ${\Psi }_{i|k-1}$ are the previous of ${\Phi }_{i|k}$ and ${\Psi }_{i|k}$, and ${\alpha }_k$ and ${\beta }_k$ are the conjugate parameters and computed based on the method proposed by Rivaie–Mustafa–Ismail–Leong (RMIL) in [24]. As seen from (17) through (20), instead of the standard gradient descendent approach, we used the hybrid method for fast learning.

The cross-entropy function is selected due to the fact that it is more sensitive to the error [23], and the loss function $J$ is specifically given by

$$J={\displaystyle \sum }_{n=1}^{\mathsf{{\rm Y}}}\left[-{\mathrm{E}}_n\ln \left({\mathrm{E}}_n\right)-\left(1-{\mathrm{E}}_n\right)\ln \left(1-{\mathrm{E}}_n\right)\right]$$

(21)

Here, $\mathsf{{\rm Y}}$ is the total number of training data and the error ${\mathrm{E}}_n=f_{a.e}=f_a-{\widehat{f}}_a$.

Remarks. Before proceeding further, the challenge of design for the “model-free” ANN observer for direct fault estimation and the corresponding remedies are elaborated below.

First, the clever selection of input data for ANN observer is one of difficulties in this study. Choosing proper input data capturing the essence of system behavior in the event of a fault is a crucial element to guarantee the accurate estimation results with the minimum delay. The particular input vector x $={\left[{\theta }_R {\dot{\theta }}_R {\tau }_{m,R} \left|{\theta }_{R.d}-{\theta }_R\right|\right]}^T\in {\Re }^{4\times 1}$ well represents the characteristics of the R-EPS system under an abnormal condition.

Second, an inappropriate selection of activation function in each layer often results in undesired consequence according to the previous other studies. To overcome these issues, we selected the activation function as a linear function for the output layer to scale the output signal and the other nonlinear activation functions in hidden layer to capture the nonlinearity of system.

Third, the choice of training scenario is also important, as training time and cost do not allow training of all possible scenarios. Other existing literature studies addressed this training problem related to the scenarios thus the chirp-like fault scenario containing various frequencies is chosen to overcome this issue.

The above remarks will be addressed along with the ANN training results in Section 5.

4.2. Fast Adaptive Observer

In this subsection, we briefly introduce the main results of fast adaptive observer presented in [20] that is capable of estimating a time-varying fault signal. Although the Linear Matrix Inequality (LMI) technique is required to compute the design parameters of this observer, the structure of the observer is relatively simple, and it is easy to be implemented.

Equation (1) can be rewritten into the following state-space equation

$$\{\begin{array}{c}\dot{X}=AX+Bu+E{f}_a\\ y=CX\end{array}$$

(22)

where $X={\left[\begin{array}{cc}{\theta }_R& {\dot{\theta }}_R\end{array}\right]}^T\in {ℝ}^{2\times 1}$ is the state of a given system and $u={\tau }_{m,R}\in ℝ$. The matrices/vectors of the system are $A=\left[\begin{array}{cc}0& 1\\ 0& -\frac{B_R}{J_R}\end{array}\right]\in {ℝ}^{2\times 2}$, $B={\left[\begin{array}{cc}0& \frac{1}{J_R}\end{array}\right]}^T\in {ℝ}^{2\times 1},$ $C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\in {ℝ}^{2\times 2}$, and $E={\left[\begin{array}{cc}0& \frac{1}{J_R}\end{array}\right]}^T\in {ℝ}^{2\times 1}$, respectively.

Based on (22), the observer is proposed by

$$\begin{array}{c}\dot{\widehat{X}}=A\widehat{X}+L\left(y-\widehat{y}\right)+Bu+E{\widehat{f}}_a\\ \widehat{y}=C\widehat{X}\end{array}$$

(23)

with an adaptive law for ${\widehat{f}}_a$,

$$\dot{{\widehat{f}}_a}=-\mathsf{\Gamma }F\left(\dot{e_y}\left(t\right)+\sigma e_y\left(t\right)\right)$$

(24)

where $e_y=\widehat{y}-y\in {ℝ}^{2\times 1}$. $\mathsf{\Gamma }\in ℝ$ and $\mathsf{\sigma }\in ℝ$ are constant update parameters and $F\in {ℝ}^{1\times 2}$ is a constant vector to be chosen. With (24), the error dynamics between (22) and (23) are bounded. The specific proof is omitted, and see [20] for the detail. The estimation results of the fast adaptive observer are presented in Section 5.

4.3. Observer Using Kalman Filter

In this subsection, we briefly present KF-based observer for a fault estimation in a motor. To obtain the prediction model of KF, (1) can be expressed the following extended state-space equation including a new state $f_a$,

$$\{\begin{array}{c}\dot{X}=AX+Bu\\ y=HX\end{array}$$

(25)

where $X={\left[\begin{array}{ccc}{\theta }_R& {\dot{\theta }}_R& f_a\end{array}\right]}^T\in {ℝ}^{3\times 1}$ is the extended state of a given system and the control input $u={\tau }_{m,R}\in ℝ$. The matrices/vectors of the system are $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& -\frac{B_R}{J_R}& \frac{1}{J_R}\\ 0& 0& 0\end{array}\right]\in {ℝ}^{3\times 3}$, $B={\left[\begin{array}{ccc}0& \frac{1}{J_R}& 0\end{array}\right]}^T\in {ℝ}^{3\times 1}$ and $H=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\in {ℝ}^{2\times 3}$, respectively.

Using the proper sampling time $∆t$, the discretized version of (25) is given by,

$$X_{K|K-1}=A_c{X}_{K-1|K-1}+∆tB{\tau }_{m,R}$$

(26)

where $A_c=I+A∆t$ and the subnotations K and K − 1 represent the current and previous time steps.

Based on (26), the predictions of the state vector and the covariance matrix are given by

$$X_{K|K-1}=A_c{X}_{K-1|K-1}+B{\tau }_{m,R}\in {ℝ}^{3\times 1}$$

(27)

$$P_{K|K-1}=A_c{P}_{K-1|K-1}{A}_c{}^T+Q_K\in {ℝ}^{3\times 3}$$

(28)

where $X_{K-1|K-1}\in {ℝ}^{3\times 1}$ and $X_{K|K-1}\in {ℝ}^{3\times 1}$ represent a previous state and a predicted one at time step K. $P_{K-1|K-1}\in {ℝ}^{3\times 3}$ and $P_{K|K-1}\in {ℝ}^{3\times 3}$ are a previous covariance matrix and a predicted one. $Q_K~N\left(0,w_k\right)\in {ℝ}^{3\times 3}$ is a process noise.

The update of Kalman gain, state and covariance matrix are provided by

$$K_K=P_{K|K-1}{H}_K^T{S}_K^{-1}\in {ℝ}^{3\times 2}$$

(29)

$$X_{K|K}=X_{K|K-1}+K_K(Z_K-H_K{X}_{K|K-1})\in {ℝ}^{3\times 1}$$

(30)

$$P_{K|K}=\left(I-K_K{H}_K\right)P_{K|K-1}\in {ℝ}^{3\times 3}$$

(31)

where $S_K=H_K{P}_{K|K-1}{H}_K^T+R_K\in {ℝ}^{2\times 2}$. $Z_K\in {ℝ}^{2\times 1}$ and $H_K=H\in {ℝ}^{2\times 3}$ are the measurement of true state and the observation model. And, $P_{K|K}$ indicate the update state and covariance matrix. The estimation outcomes of KF-based observer are presented in Section 5.

5. Experimental Setup and Results

In this section, we present the experimental setup for fault estimation of the motor in R-EPS and the discussion of estimation results. Figure 3 describes the actual R-EPS connected to DAQ(Q-PID)/PC. MATLAB/Simulink is used to implement the motion tracking control (presented in Section 3) and the fault estimation (via three observers in Section 4) of the motor, and, to mimic the rack force generated by road/tire, the spring has been attached to the rod of R-EPS as shown in Figure 3.

An AC motors (Mitsubishi HG-KR73) driven by a servo driver (Mitsubishi MR-J4-70A) was employed to run the motor in R-EPS. It should be noted that the assumed additive fault $f_a$ is generated in the MATLAB/Simulink code, not physically implemented. In other words, the motor will run via the servo driver according to the amount of torque ${\tau }_{m,R}\left(t\right)+f_a\left(t\right)$ in the abnormal condition (i.e., $f_a\left(t\right)\ne 0$). The data collection for the training of ANN observer is also specified in Figure 3. The rotational angle of motor and the desired control torque are gathered by DAQ(Q-PID)/PC with MATLAB/Simulink and then used as the primary data for training of ANN observer.

In the rest of this section, the training of ANN observer has been discussed and the fault estimation results of three observers have been presented according to the different tracking control and fault scenarios. The strength of the proposed ANN observer has been compared with the outcomes generated by the two model-based approaches.

5.1. Training of ANN Observer

Figure 4 includes the input data for the training of ANN observer and a true fault reference. Figure 4a–d represent the input data and Figure 4e does the true reference. In other words, the 4-input data in Figure 4a–d have been obtained by a fault scenario shown in Figure 4e, which is the chirp-like fault scenario owning different frequencies varying from 0.3 to 4 rad/s. The chirp-like fault scenario is the perfect fault one for training the ANN observer due to the fact that it contains the signal with various frequencies and the nature of a fault can be any type of combined signal. Here, it should be noted that the scenario of motion tracking control is a sine-wavelike trajectory (as shown in Figure 4a) and, due to the fault scenario in Figure 4e, the absolute error between ${\theta }_R$ and ${\theta }_{R,d}$ is gradually increased after a fault is injected at 3000 data-point (as shown in Figure 4d,e). Specifically, the error has been maintained below 0.015 rad under the normal operation of the motor (i.e., before 3000 data-point) but increased up to 0.06 rad after the fault is injected. The control torque of the motor was also contaminated by a fault and becomes irregular as seen in Figure 4c. Based on these input and an output shown in Figure 4, the training of the ANN observer has been initiated using the technique in Section 4.1. Furthermore, Figure 5 includes the training results. Specifically, Figure 5a contains the training epochs of the ANN observer using the different numbers of nodes, and Figure 5b does the comparisons between the true reference and the outputs of the trained ANN observers. Here, the five cases of node selection are considered as n = 5 and m = 4, n = 6 and m = 5, n = 8 and m = 6, n = 10 and m = 8 as well as n = 12 and m = 10. As seen in Figure 5, each error $\left|E\right|=\left|f_{a.e}\right|=\left|f_a-{\widehat{f}}_a\right|$ is reduced as the epoch increases regardless of the number of nodes and the acceptable agreement between the true reference and the outputs of the trained ANN observers has been made. Especially, $\left|E\right|$ for all cases reach below 0.01 after 400 epochs but the final steady-state value of $\left|E\right|$ slightly increases with fewer nodes. Nevertheless, the hybrid learning rule enable us to achieve fast learning. Meanwhile, for the efficient implementation of ANN observer in ECU, it is more advantageous to choose the minimum number of nodes, resulting in relieving the computational load. Therefore, we selected the case with the smallest node (n = 5 and m = 4) that still guarantees an accurate result with little delay between the actual reference and the output, as shown in Figure 5c.

5.2. Estimation Performance Comparison between ANN Observer and Model-Based Approaches

Figure 6 shows the control performance of sine-wave trajectory tracking under a sine-wavelike fault of the motor. Figure 6a,b indicate the actual angular position of motor ${\theta }_R$ along with the desired trajectory ${\theta }_{R.d}$ and the tracking error between ${\theta }_{R.d}$ and ${\theta }_R$, respectively. Figure 6c describes the control torque comparison between the desired (healthy) torque, ${\tau }_{m,R}\left(t\right)$, and contaminated torque, ${\tau }_{m,R}\left(t\right)+f_a\left(t\right)$, and Figure 6d represents the actual fault scenario, $f_a\left(t\right)$. As shown in Figure 6a,d, it is clear that the control is initiated at 5 s and the fault is injected at 20 s. Therefore, after 20 s, the tracking performance has been significantly degraded thus the error is increased up to almost 50% compared to the healthy condition. Consequently, the control torque became abnormal and irregular.

With the same control scenario (i.e., sine-wave trajectory tracking) shown in Figure 6, the fault estimation results via three observers (in Section 4) are presented in Figure 7. Here, we used three different sine-wavelike fault scenarios with the frequencies 0.5 rad/s, 1 rad/s, and 1.5 rad/s (with the same amplitude 0.15 Nm). Figure 7a–c indicate the estimation results for a fault signal with a driving frequency 0.5 rad/s, and Figure 7d–f do the estimation outcomes for a given fault with a driving frequency 1 rad/s. Finally, Figure 7g–i describe the estimates for the fault with a driving frequency 1.5 rad/s. As shown in Figure 7c,f,i, the fault estimation errors for the first two observers (fast adaptive observer and KF-based observer) increase as the frequency of fault signal does but the error generated by ANN observer is smallest and not heavily perturbed by the change of frequency in a fault. Specifically, the error magnitude produced by the ANN observer is almost 2~3 times smaller than the one via the model-based approaches. This constant performance is the strength of ANN observer since it has high order approximation capability for a nonlinear smooth signal containing all different frequencies. Even though the model-based approaches include an adaptive capability, the accuracy of estimation still relies on the frequency of input signals. Another merit of ANN observer can be found from Figure 8 zooming the estimates (via the three observers) from 4 s to 12 s (i.e., healthy condition, $f_a\left(t\right)=0$). The fault estimates obtained by first two observers are characterized by the signals with certain level of amplitude and frequency. This estimate might be misunderstood as the abnormal condition (even though it is indicated as the healthy one), and definitively affected by the given control scenario (sine-wave trajectory). However, the results via ANN observer is described as a noise featured signal with small amplitude, which is almost negligible.

Furthermore, for the same control scenario shown in Figure 6 and Figure 9 indicates the control performance under a square-wavelike fault of the motor and Figure 10 does the corresponding fault estimation results of three observers. Here, we used three different fault scenarios with the maximum steady-state amplitudes 0.5 Nm, 1.0 Nm, as well as 1.5 Nm. It is found that the tracking performance is not heavily degraded by a fault, except the errors (with intensive peaks) observed at 40 and 60 s, and the estimation error via ANN observer is smallest among them (almost 20% smaller than the others). The ANN observer creates little fluctuation of estimate for the steady-state fault condition (as seen in Figure 10b,e,f), but the results from two other observers exhibit a certain level of fluctuation, which is again associated with the control scenario (i.e., sine-wave trajectory tracking control).

Figure 11 displays the fault estimation results of three observers for chirp-like fault signal owning frequencies ranging from 0.5 to 3.5 rad/s. It is clear that the performance of ANN observer is superior to those with other approaches. The errors via KF and Adaptive observers significantly increase as the frequency of fault signal does, and the results are exactly matched with the trend shown in Figure 7. Especially, under the fault signal with higher frequencies, the estimated results of both model-based approaches are significantly degraded along with noticeable delays and show the errors that are nearly 2.5 to 3 times greater than those with the ANN observer.

Furthermore, Figure 12 indicates the estimation performance of the three observers for the square-wavelike fault scenarios under trapezoidal trajectory tracking control scenario. On the other hand, Figure 13 presents the estimation performance of three observers for sine-wavelike fault scenario under trapezoidal trajectory tracking control scenario. Specifically, Figure 12a and Figure 13a respectively describe the tracking performance of control system for two different fault scenarios, and Figure 12b and Figure 13b do the comparison between the desired torque, ${\tau }_{m,R}\left(t\right)$, and the contaminated one, ${\tau }_{m,R}\left(t\right)+f_a\left(t\right)$. Figure 12c and Figure 13c include the fault estimation performance of three observers, and Figure 12d and Figure 13d do the absolute errors between an actual fault and the estimates.

Although we can observe the non-smooth spike-nature signal from the estimates generated by the ANN observer (see Figure 12c and Figure 13c), the estimation error via ANN observer is relatively small to the those via two other observers. The estimation delay through the ANN observer is also insignificant compared to the results through the model-based approaches. Especially, it can be seen that the estimation results in Figure 12 and Figure 13 are also influenced by the control scenario due to the fact that the non-smooth spike characteristic signal generated by the control scenario can be found in the fault estimation.

In

Table 1. Average estimation errors of three observers.

Control Scenario	Fault Scenario	Fast Adaptive Observer	Kalman Filter	ANN Observer
Sine-wave trajectory tracking	Sine-wavelike fault (Figure 7/Figure 8)	0.02~0.04	0.02~0.04	0.007
	Square-wavelike fault (Figure 10)	0.012~0.014	0.012~0.014	0.008
	Chirp-like fault (Figure 11)	0.055	0.056	0.021
Trapezoidal trajectory tracking	Sine-wavelike fault (Figure 12)	0.12	0.125	0.071
	Square-wavelike fault (Figure 13)	0.17	0.17	0.085

, the average estimation performances of three observers are summarized and, it is found that the average errors via the model-based approaches are almost identical for each other and the average error via the ANN observer is less than the half of the others. According to the control scenario, the estimation performances of the observers are slightly varied but the ANN observer shows the minimum error and least perturbation among them regardless of control scenarios.

Table 2. Correlation coefficient between true fault signals and the estimates of three observers.

Control Scenario	Fault Scenario	Fast Adaptive Observer	Kalman Filter	ANN Observer
Sine-wave trajectory tracking	Sine-wavelike fault (Figure 7/Figure 8)	76%	73%	92%
	Square-wavelike fault (Figure 10)	88%	86%	95%
	Chirp-like fault (Figure 11)	68%	67%	91%
Trapezoidal trajectory tracking	Sine-wavelike fault (Figure 12)	72%	73%	88%
	Square-wavelike fault (Figure 13)	79%	78%	84%

lists the correlation coefficients between actual fault scenarios and the estimation results of three observers, and the ANN observer exhibit at least greater than 84% matching for the given fault signals. The model-based approaches relatively show a weak correlation for several cases. Note that the correlation coefficient is computed by,

$$C.C=\frac{\mathrm{n}\left(\sum ab\right)-\left(\sum a\right)\left(\sum b\right)}{\sqrt{\left[\mathrm{n}\left(\sum a^2\right)-{\left(\sum a\right)}^2\right]\left[\mathrm{n}\left(\sum b^2\right)-{\left(\sum b\right)}^2\right]}}$$

(32)

where a and b are the objective data of correlation and n is the total number of data for a and b.

The estimation performance of model-based approaches is degraded when the frequency and amplitude of the fault signal and control scenario are changed but the ANN observer still guarantees the unperturbed estimation performance regardless of fault and control scenarios. Through the actual experimental study and corresponding interpretations, the superiority of ANN observer has been clearly addressed here.

6. Conclusions

In this paper, we present the fault estimation performance of the ANN observer for the rack-driving motor in R-EPS system. The estimation performances between the ANN observer and the model-based approaches are compared for one another according to the different control and fault scenarios. Finally, the effectiveness of the ANN observer clearly addressed based on actual experimental results. It is found that the performance of model-based approaches is degraded when the frequency and amplitude of fault signal and control scenario are changed but the ANN observer guaranteed the almost same performance regardless of fault and control scenarios. Specifically, according to the correlation coefficient, the accuracy of estimation results via ANN observer ranges 84% to 95% for the actual fault signals, while the model-based approach shows 68% to 88% accuracy for actual ones. It was obvious that the proposed ANN observer was more robust than the model-based ones for various fault and control scenarios as long as the training data are accurate and representative. This work provides the insight and effectiveness of ANN observer for the fault estimation in a motor in R-EPS and will be a valuable asset for those who wish to implement ANN observer for the fault estimation. In our future study, we will use this estimation technique for the FTC/FOC of EPS systems including the multi-driving motor configuration system. Since it directly estimates the disturbing torque for unhealthy motor, it can be used for direct torque compensation strategy of FTC/FOC.