## 3 Actuator Fault Model of EPS System  
In this paper, the actuator fault is divided into power motor fault, deceleration mechanism, gear and rack actuator fault. Because the power motor drives the deceleration mechanism and the gear and rack movement, the power motor fault is defined as active actuator fault, and the deceleration mechanism and gear and rack faults are defined as passive actuator faults. In this paper, actuator failure means that the actuator cannot complete the expected action. When the actuator fails, the EPS system cannot complete the ideal power steering. Common actuator faults are mainly divided into three categories: gain variation faults (multiplicative faults), constant deviation faults (additive faults) and stuck faults (constant output of the actuator). Stuck faults include two kinds of failure: complete failure and out of control. Complete failure refers to the motor voltage being constant zero, or the motor voltage is not zero, but it and the deceleration mechanism or the deceleration mechanism and gear rack is stuck, unable to move the failure state. Out of control refers to the motor suddenly running at a constant non-zero speed. At present, only the redundant hardware can achieve the purpose of fault tolerance, which will not be considered in this paper. In this paper, gain variation faults and constant deviation faults, namely, partial faults, are considered. Partial faults refer to the motor being in a controlled state, but the voltage loss or friction increase efficiency becomes low, or the deceleration mechanism and gear and rack wear lead to efficiency loss, that is, the actual steering power cannot reach the ideal steering power. The main factors leading to motor fault and what kinds of fault types it will produce are:  
* if the motor temperature is too high, it may lead to partial faults or stuck faults;
* if the air is damp, dusty and polluted, it may lead to partial faults or stuck faults;
* if there is no plug, the motor is broken, the motor is locked, it will lead to stuck faults;
* if the connection shorts between the wire harness end line and the ground, it will lead to stuck faults;
* if the converter is short circuited or open circuited, it will lead to stuck faults;  
The main factors that lead to the failure of the deceleration mechanism, the gear and rack and what kinds of fault types it will produce are:  
* if the clearance of the turbine worm is too large, it will lead to partial faults;
* if the tooth surface of the turbine worm is worn and the gear and rack are worn, it will lead to partial faults. The following mathematical formula of actuator fault type corresponds to the list above. According to the fault estimation, we can judge what kind of fault has occurred. In practical engineering, because different factors can lead to the same type of fault, even if we know the fault type, we need to find out what factors caused it one by one. The output of different types of fault actuators is uniformly described as follows:  
\[u_{q}=\beta u(t)+u_{0} \tag{5}\]where \(\beta\) represents the gain variation coefficient of the actuator, \(u_{0}\) represents the value of constant deviation fault or stuck fault of the actuator, and \(u(t)\) represents the output of the normal actuator. * when \(\beta=1\) and \(u_{0}=0\), it is in a normal state and the actuator has no failure;
* when \(\beta=1\) and \(u_{0}\
eq 0\), it is a constant deviation fault and belongs to partial failure;
* when \(0<\beta<1\) and \(u_{0}=0\), in this case, it is a gain variation fault and belongs to a partial failure;
* when \(0<\beta<1\) and \(u_{0}\
eq 0\), in this case, it is a mixed fault with constant deviation and gain variation and belonging to partial failure;
* when \(\beta=0\) and \(u_{0}=0\), then the actuator has no output and is in a complete failure;
* when \(\beta=0\) and \(u_{0}\
eq 0\), the output of the actuator is a constant.