The Fractional Order PID Method with a Fault Tolerant Module for Road Feeling Control of Steer-by-Wire System

Xu, Fei-xiang; Liu, Xin-hui; Chen, Wei; Zhou, Chen; Cao, Bing-wei

doi:https://doi.org/10.1155/2018/6386513

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 6386513 | https://doi.org/10.1155/2018/6386513

The Fractional Order PID Method with a Fault Tolerant Module for Road Feeling Control of Steer-by-Wire System

Fei-xiang Xu,¹Xin-hui Liu,¹Wei Chen,¹Chen Zhou,¹and Bing-wei Cao¹

Academic Editor: Saeed Eftekhar Azam

Received20 Aug 2018

Accepted05 Nov 2018

Published19 Nov 2018

Abstract

To improve the road feeling of the steer-by-wire (SBW) system, a fractional order PID (proportion-integral-derivative) method with a fault tolerant module is proposed in this paper. Firstly, the overall road feeling control strategy of the SBW system is introduced, and then the mathematical model of road feeling control is established. Secondly, a fractional order PID (FOPID) controller is designed to control torque of the road feeling motor. Furthermore, genetic algorithm (GA) is applied to tune the FOPID controller’s parameters. Thirdly, a fault tolerant module aiming at potential failures of the motor’s torque sensor is studied to improve the reliability of the system. Kalman Filter (KF) algorithm is utilized in the fault tolerant module so as to detect failures of the motor’s torque sensor, and then fault tolerant module reconfigures the motor’s torque estimated by KF as a substitute when the torque sensor fails. Finally, simulations based on MATLAB are performed with the proposed control strategy to identify its performance, and the results demonstrate that the proposed control method is feasible and accurate.

1. Introduction

The SBW system is playing an important role in the development of modern vehicles [1, 2], which breaks the concept of traditional steering system and replaces the mechanical joint between the steering wheel and front wheels [3]. The SBW system not only simplifies the design of cab, but also enhances the vehicle stability and driving force [4]. Drivers will not feel the road condition directly without the mechanical connection from steering wheel to the road in the SBW system [5]. Hence, it is urgent to study the control strategy to improve the road feeling of the SBW system. The road feeling control is to feed back the road condition to the driver, which controls the reaction torque that the steering wheel actuator generates [6].

Recently, some scholars have developed PID controller for the road feeling control of the SBW system [7, 8]. Compared with the PID controller, the FOPID controller adds two adjustable parameters (integral order and derivative order), which provides a larger tuning space for better control performance and robustness [9, 10]. In some practical applications, it is verified that the FOPID controller has better disturbance rejection ability and less sensitivity to the parameter variations of the control system than the traditional PID controller [11]. As a result, the common PID controller is gradually replaced by the FOPID controller with the support of the researchers [12–14]. Therefore, the FOPID controller is adopted to control the road feeling of the SBW system in this paper.

The additional varying orders of integration and differentiation of the FOPID controller are accompanied with the increase of the difficulty of parameters optimization [11]. GA is widely regarded as a global optimization method, which can only depend on the fitness function to search the best parameters with constraints, targets, and dynamic components [15, 16]. With its optimization characteristics, GA has been applied to the parameter optimization of the FOPID controller for many times [17–19]. In this way, GA is used to tune the FOPID controller’s five parameters including proportional constant, integral constant, derivative constant, integral order, and derivative order.

Most of the literatures about the FOPID controller and PID controller mentioned above only take into account the case without sensor faults in the control process. The torque sensor is so essential that it is used to measure the feedback signal of the road feeling control system. However, the faults of the torque sensor could occur in the process of road felling control, which can cause a critical effect on the road feeling control [20]. Consequently, fault tolerant control (FTC) becomes the key issue to ensure the reliability of the control system. Fault detection and diagnosis (FDD) module is essential in FTC problem and it is expected to be very important for general SBW technique [21]. Once the sensor faults occur, the FDD method can detect, diagnose, and remove the faults, thus ensuring the normal working of the control system. The most convenient FDD method for sensors is hardware redundancy, where multisensors are deployed to measure the same physical parameter and faults are detected by comparing all sensor's measurements [21]. For the hardware redundancy method, on the one hand, it is difficult to install multiple redundant sensors on the vehicle; on the other hand, it also increases the development cost. To make up the shortcomings of the hardware redundancy method, the model-based techniques have been widely used in the FDD [22–24], which provides a great fault detection effect and robustness against external disturbances and model uncertainties [25].

The primary principle of the model-based FDD method is to design an observer to evaluate a residual signal which is the difference between the measured signal and the observer output [26]. The model-based FDD method can detect and diagnose sensor faults effectively without redundant hardware. The well-known KF observer is one of the model-based FDD methods, which shows outstanding performance in dealing with the fault tolerant problem, and it has been widely applied to detect and diagnose the sensor faults [27–30]. Zhang Han et al. [5] put forward a fault tolerant module for two-way H infinity controller of the SBW system. KF is used as the FDD method to detect and diagnose front wheel sensor’s faults. The simulations and experiments were conducted to verify the proposed FTC strategy based on KF. Xiahou K. S et al. [31] presented a KF-based FTC strategy under voltage and current sensor faults. The sensor faults were detected and isolated based on the residuals calculated from observations measured by sensors and estimations provided by KF. The simulation and experimental studies validated the proposed FTC strategy. Liu Yuan et al. [32] designed a FTC method for aeroengine, where the sensor faults can be diagnosed by KF accurately. The semiphysical simulation indicated that the fault tolerant controller could ensure that the aeroengine controller worked safely and reliably in the event of sensor faults.

Inspired by the previous studies, KF is used to detect and diagnose the torque sensor’s faults of road feeling motor in this paper. KF evaluates a residual signal between the measured signal and the observer output. When the value of residual signal is greater than the defined threshold, it indicates that the torque sensor faults occur and the observer output is reconfigured as the reference value of the fault tolerant module. On the contrary, the FOPID controller works normally through obtaining the feedback torque measured by the torque sensor.

In this paper, we aim to improve the driver’s road feeling of the SBW system with the FOPID controller tune by GA and KF-based fault tolerant module. The rest of this paper is organized as follows: Section 2 briefly introduces the overall road feeling control strategy; Section 3 studies the design of the FOPID controller; Section 4 focuses on the FTC strategy based on KF; Section 5 describes the simulation results; and the final section is the conclusions.

2. The Overall Road Feeling Control Strategy of the SBW System

2.1. The Structure of the SBW System

The function of SBW system can mainly be divided into two parts: vehicle steering control and road feeling control. The overall structure of the SBW system is shown in Figure 1.

Firstly, the controller receives the angle sensor signal of the steering wheel. Secondly, the controller outputs the currents to the electromagnetic valves. Thirdly, the hydraulic cylinders are pushed by pressure energy from the electromagnetic valves. Finally, the steering of the vehicle is realized. Meanwhile, the pressure sensors installed in the hydraulic cylinders send the steering resistance signal to the controller, and then the controller sends out the control signal to the road feeling motor based on the comparison between the ideal steering resistance torque and the actual torque of the steering wheel detected by the torque sensor. Then the reducing mechanism transfers the torque to the steering wheel. In this way, the road feeling can be obtained by the drivers.

The road feeling control is discussed in this paper, and the vehicle steering control will be studied in the next step.

2.2. The Overall Road Feeling Control Strategy

The closed-loop control method is applied in the road feeling control and the overall road feeling control strategy is shown in Figure 2. The ideal steering resistance torque is calculated by the vehicle speed and the maximum pressure difference between left cavity and right cavity of four steering cylinders. The concrete formula of the ideal steering resistance torque will be drawn based on experiments conducted on the prototype vehicle in the future. The FOPID controller sends out the control signal to the road feeling motor according to the difference between the ideal steering resistance torque and the torque measured by the torque sensor. The torque sensor is so essential that it is used to measure the feedback signal of the road feeling control system.

The fault tolerant module is used to avoid the bad effect caused by the torque sensor faults of the road feeing control system. KF observer is used to observe the torque output of the road feeling motor and evaluate a residual signal between the measured value and the observer output signal. When the residual signal is greater than the defined threshold, it indicates that the torque sensor faults occur and the value measured by the torque sensor is replaced by the observer output. When the value of the residual signal is smaller than the defined threshold, the FOPID controller works normally through obtaining the feedback torque detected by the torque sensor.

2.3. System Modelling

The control system modelling is mainly to establish the mathematical model of the road feeling motor, whose moment equilibrium equation is expressed as follows.where denotes the output torque of the road feeling motor; represents the moment of inertia of the road feeling motor; stands for the damping coefficient of the road feeling motor; is the torsional stiffness of the road feeling motor; represents the reduction ratio of the reducing mechanism; is the angle of the road feeling motor; denotes the steering wheel angle.

The last subformula in formula (1) occupies a small proportion in the whole equation, which can be ignored here [33].

The electromagnetic torque and the current of the road feeling motor can be expressed as follows.where denotes the torque coefficient of the road feeling motor; represents the armature current of the road feeling motor.

The electromotive force balance equation of the road feeling motor is shown below.where denotes the armature voltage of the road feeling motor; represents the armature inductance of the road feeling motor; represents the armature resistance of the road feeling motor; is the back electromotive force coefficient of the road feeling motor.

The Laplace transform is carried out for (2)-(4) mentioned above, and the transfer function from torque of the road feeling motor to the output voltage of the FOPID controller can be shown as follows.

The actual parameters of the SBW system in this paper are shown in Table 1.

3. The FOPID Controller Design

3.1. The FOPID Controller

Compared with the conventional PID controller, the FOPID controller adds the integral and differential orders, which makes control system fine tracking accuracy, abundant dynamics, and high robustness [34]. Before describing the design of the FOPID controller, the basic knowledge of fractional calculus (FC) is introduced firstly. FC is a generalization of integration and differentiation operators, which is able to be represented as follows.where denotes the FC; is the fractional order; and represent the upper and lower limits of the operation, respectively.

There are three mainstream methods to define the fractional operators [35]. The common one is the RL (Riemann Liouville) definition, which can be expressed as follows [36].where ; is the function; is the well-known gamma function, which is described as follows.

In this paper, the FOPID controller is used to control the road feeling, whose output voltage can be expressed as follows.where are the proportional gain, integral gain, derivative gain, fractional order integrator, and fractional order differentiator, respectively.

The block diagram of road feeling control system using a FOPID controller is shown in Figure 3. The continuous transfer function of the FOPID controller is shown as follows.

Clearly, when , the FOPID controller becomes the conventional PID controller.

3.2. GA-Based Optimal Tuning of the FOPID Controller

GA is a search heuristic method that imitates the natural selection process, and it is an efficient technique to solve optimization problems [37, 38]. Hence, GA is used to tune five parameters of the FOPID controller in this paper, and its flow chart is shown in Figure 4.

Step 1. Once the approximate range and coding length of each parameter are determined, parameters are coded. In order to save the optimization time of GA, based on [11] and the optimization trials in MATLAB, the ranges of parameters are set as follows.Compared with the real coding, the binary coding has simple coding and decoding operations, and it is easier to implement genetic operations. Therefore, the binary coding is selected to code five parameters of GA. The length of the binary string is calculated based on parameter range and optimization accuracy, which is shown below.where is the length of the binary string; and are the maximum and minimum values of the optimized parameters, respectively; denotes the optimization accuracy of parameters.

Step 2. The initial population is randomly generated by the computer. For the binary coding, a random number of uniform distribution between 0~1 is created firstly, and then the random numbers between 0~0.5 are represented by 0; the random numbers between 0.5~1 are denoted by 1.

Step 3. The individual parameters of the population are decoded into corresponding values, and then the best individual is found through calculating the fitness function values. To define suitable fitness function, the objective function must be taken into account firstly. The common objective function of the FOPID controller is integral of squared error (ISE), integral of absolute value error (IAE), integral of time by absolute value error (ITAE), and integral of time by squared error (ITSE), which are defined as follows [9].The IAE criterion mentioned above is regarded as the objective function of the FOPID controller in this paper. Clearly, the smaller the value of IAE, the better the performance of the FOPID controller. GA is to search for individuals with maximum fitness function value. On that basis, the fitness function of GA is defined as follows.

Step 4. Check whether the genetic generation is reached. GA will output optimization results once the genetic generation is satisfied; otherwise, the fifth step will be executed.

Step 5. Genetic operators are implemented. The initial population can be used to obtain a new population through implementing reproduction, crossover, and mutation. The new generation is applied into Step 3, and then the fitness function value of new generation is obtained.

Step 6. Steps 3~5 are repeated until genetic generation is reached, and the best gene string is received. At last, five parameters of the FOPID controller can be obtained based on decoding operation.

All the parameters in the GA-based optimization process are defined as follows:(i)Population size: 1000(ii)Crossover probability: 0.5(iii)Mutation probability: 0.02(iv)Genetic algebra: 100

4. Fault Tolerant Module Based on Kalman Filter

By analyzing the overall road feeling control strategy mentioned above, it can be known that the torque sensor is of great significance as a feedback signal acquisition module. However, the faults of the torque sensor may appear in the long time process of the road feeling motor, which can cause a critical effect on the road feeling control. In this context, KF-based FTC method for the torque sensor is proposed in this paper.

The FTC procedure is shown in Figure 5. Firstly, the voltage output of the FOPID controller and the plant torque output signal are put into the KF observer. Secondly, the estimated torque value of the road feeling motor is obtained based on the KF observer. Thirdly, the residual signal Re is sent to a fault evaluation function . Fourthly, the torque sensor in fault condition depends on the comparison between the fault evaluation function and its threshold value. If the fault evaluation function is bigger than its threshold, the faults occur in the torque sensor; otherwise, the torque sensor works normally. Finally, the fault reconfiguration module outputs the torque signal to the FOPID controller. If the torque sensor faults occur, is the estimated torque of the KF observer; otherwise is the torque measured by the torque sensor. As mentioned above, the FTC method mainly consists of two parts: FDD module and fault reconfiguration module.

4.1. Fault Detection and Diagnosis Strategy

FDD module makes full use of KF to detect the torque sensor faults in this paper. KF is a kind of linear unscented filter, which obtains the optimal estimation of system state and system parameters under the condition of minimizing the error variance. KF can be implemented by recurrence formula in discrete domain, and the amount of computation and storage is greatly reduced, so it is easy to meet the real-time requirements. Owing to its advantages, KF is used to estimate the torque value of the road feeling motor.

4.1.1. Observer Design Based on Kalman Filter

For a dynamic system, it can be described by the following difference equation.where and denote the state matrix and the observation signal matrix of system, respectively; represents the input signal matrix; , and are the parameter matrixes of system; and denote the process noise and the measurement noise matrixes, and they need to meet the following relationships.where denotes the mean value;

For the control object’s transfer function in this paper, it can be translated into a state space, and the related parameters in (15) are shown as follows.

The continuous model described in (15) is discretized as follows.where ; denotes the system state transition matrix of the discrete description; is the system input matrix of the discrete description; is the system measurement matrix of the discrete description; T denotes the sample time.

The process of KF contains two steps: prediction module and correction module [39, 40].

(1) Prediction Module

(2) Correction Modulewhere and are the predicted state value and error covariance prediction value at the (k+1)th moment based on their values at the kth moment; and are the predicted state value and error covariance prediction value at the kth moment; is the Kalman gain matrix at the (k+1)th moment; denotes the observation matrix at the (k+1)th moment; represents the innovation information of filtering at the (k+1)th moment.

4.1.2. Residual Generation and Evaluation

By means of (20)-(24), the torque value of the road feeling motor can be estimated effectively. The residual signal is the difference between the measured and estimated values described as follows [41].where denotes the residual signal; represents the torque value measured by the torque sensor; is the torque value estimated by KF.

The fault evaluation function can be defined as follows [42].

Suppose the threshold of the evaluation function is which can be designed in experiments according to the experience of experts [42]. The torque sensor faults can be detected by the following rule.

4.2. Fault Reconfiguration Strategy

If the torque sensor faults occur, the controller cannot control the road feeling of the SBW system correctly. To minimize the potential risks of sensor faults, the FTC strategy is needed to control the road feeling, which uses the fault reconfiguration module to output the torque to the proposed FOPID controller. The key point in the FTC strategy is to reconstruct the system torque output using the fault reconfiguration module. When the torque sensor is detected with faults based on the FDD module, the system torque output is the estimated value of KF. When the torque sensor is detected with no fault, the system torque output is the torque value measured by the torque sensor. Hence, the output of the fault reconfiguration module can be described as follows.

Figure 6 shows the control reconfiguration strategy for the torque sensor faults. Based on the principle of the fault reconfiguration module, the FOPID can receive the valid torque value as a feedback signal whenever the sensor faults occur. In this way, the fault tolerant controller can ensure the normal road feeling control.

5. Simulation Results and Analyses

In this section, simulations based on MATLAB are carried out to validate the feasibility of the proposed FOPID controller and the FTC strategy.

5.1. Test of the FOPID Controller

In order to demonstrate the superiority of the FOPID controller, the conventional PID controller tuned by GA is also designed to compare the performance of the optimized FOPID controller. The code of GA is written in MATLAB to calculate the objective function mentioned in (14) for every set of parameters in the ranges described in (11). The optimization is executed for the FOPID controller and the conventional PID controller to find the optimal value through minimizing IAE of the unit step responses of the ideal steering resistance torque. Table 2 shows the optimal parameters of the FOPID controller and the conventional PID controller using GA.

From Table 2, we can see that the optimized FOPID controller has better performance because the FOPID controller decreases by 68.83% compared with the conventional PID controller in terms of IAE. The FOPID controller possessing better control effect is accompanied with the fractional integration order and differentiation order optimized by GA. Meanwhile, it verifies that the fractional integration order and differentiation order of the FOPID controller provide indeed a larger tuning space for better control performance. Moreover, Table 2 indicates that GA can tune parameters of the FOPID and the conventional PID controllers well because two controllers’ IAE are small significantly. Furthermore, the code of GA in MATLAB tuning parameters of the FOPID and PID controllers can be extended into other control applications conveniently.

The unit step responses of the ideal steering resistance torque using the optimized FOPID controller and conventional PID controller are shown in Figure 7. It can be seen that the optimized PID controller has 0.026s rise time, 0.072s settling time, and 14.823N.m IAE. Also, with the optimized FOPID controller, the rise time is 0.01s, the settling time is 0.007s, and the IAE is 4.620N.m. The rise time, settling time, and IAE of the FOPID controller are significantly smaller than the PID controller. It is concluded that the optimized FOPID controller has a better performance than the optimized PID controller in terms of transient response.

Figure 8 illustrates the following curves of the optimized FOPID controller and conventional PID controller for the sinusoidal signal. Figure 9 shows the resistance torque tracking error of the optimized FOPID and PID controllers. On the one hand, we can see that the optimized FOPID controller has smaller tracking error from Figures 8 and 9. On the other hand, compared with 30.320N.m IAE of the optimized PID controller, the optimized FOPID controller has 10.662N.m IAE. Therefore, we can safely come to the conclusion that the optimized FOPID controller has better resistance torque tracking performance than the optimized PID controller.

5.2. Test of the Fault Tolerant Module

According to the above subsection, we can draw the conclusion that the FOPID controller has better transient response and tracking performance. The FOPID controller with the fault tolerant module is demonstrated in this subsection.

Firstly, the estimation accuracy of the KF observer is validated under the condition of the fault-free torque sensor. Figure 10(a) illustrates the KF-based estimation value and the measured value of the resistance torque. Figure 10(b) shows the estimation error of the resistance torque based on KF. From Figure 10, we can see that the KF observer has great estimation accuracy and it can be used to generate residual signal and detect sensor faults in the fault tolerant strategy.

(a)

(b)

Secondly, the fault tolerant schema is demonstrated based on simulations. In order to validate the comprehensive fault tolerant effect, three kinds of sensor faults (lock fault, constant gain fault, and constant deviation fault) are taken into account, which are shown as follows.where , , and denote the measured value by the torque sensor with lock fault, constant gain fault, and constant deviation fault, respectively; represents the true value of the torque; is the constant gain; is the constant deviation; denotes the start time of sensor faults.

The specific sensor faults are simulated in the control system, which are shown below.

Figure 11 shows the fault detection result of the torque sensor with lock fault. The true value, KF-based estimation value, and value measured by the faulty sensor of the resistance torque are displayed in Figure 11(a). The fault evaluation function is shown in Figure 11(b). From Figure 11, we can clearly figure out that the sensor with the lock fault can be detected rapidly and effectively after it occurs. Furthermore, it can be seen that the estimation accuracy of KF is high and KF-based estimation value can be reconstructed as the torque feedback signal after the torque sensor’s lock fault occurs.

(a)

(b)

Similarly, Figures 12 and 13 illustrate the fault detection results of the torque sensor with constant gain fault and constant deviation fault, respectively. From Figures 12 and 13, it can be seen that the faulty sensor can be detected once the sensor faults occur. KF-based estimation value also can be reconfigured as a substitute when the torque sensor fails.

(a)

(b)

(a)

(b)

Based on the simulation results mentioned above, we can draw the conclusions: the FOPID controller tuned by GA has better control performance than the optimized PID controller in terms of transient response and tracking ability; KF observer has great estimation accuracy, which can be used to generate residual signal and detect sensor faults; the fault tolerant strategy can reconfigure the torque value rapidly and effectively when the torque sensor fails.

6. Conclusions

In this paper, the FOPID controller with a fault tolerant module for the road feeling control of the SBW system is proposed. The FOPID controller is designed to control the steering resistance torque accurately and quickly. GA is applied to tune the FOPID’s parameters including proportional constant, integral constant, derivative constant, integral order, and derivative order so as to achieve the best road feeling control performance that minimizes the IAE. In order to improve the reliability of the control system, a fault tolerant module aiming at the motor torque sensor is studied. In the FDD module, a residual signal is generated by KF, and a threshold of the fault evaluation function is used for judging sensor faults. For the fault reconfiguration module, the output torque of the control system can be reconstructed effectively when the torque sensor faults occur. Based on the fault tolerant strategy, the FOPID controller can receive the valid torque value as a feedback signal in spite of sensor faults. The results of the numerical simulations based on MATLAB demonstrate the effectiveness of the FOPID controller with a fault tolerant module. The optimized FOPID controller has better transient response and tracking performance than the traditional PID controller. The proposed FDD method can detect and diagnose the torque sensor faults (lock fault, constant gain fault, and constant deviation fault) effectively; what is more, the FTC strategy can identify and reconfigure faults within a reasonable time. Finally, the proposed FOPID controller with the FTC strategy can be extended into other industrial control applications.

Although the simulations based on MATLAB are carried out to demonstrate the feasibility of the proposed methods, the experiments conducted in practice would improve greatly the reliability and quality of this research. In the future, we will make joint efforts to complete the prototype vehicle, and then a large number of experiments with regard to the FOPID controller and the FTC strategy will be conducted and implemented on the prototype vehicle.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This research was funded by the National Key Research and Development Program of China under Grant No. 2016YFC0802904. The authors would like to acknowledge the Institute of Electrohydraulic Control Technology for Engineering Machinery in Jilin University and Xuzhou Construction Machinery Group Co., Ltd., which support this research.

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Copyright © 2018 Fei-xiang Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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