The Fractional Order PID Method with a Fault Tolerant ...downloads.hindawi.com/journals/mpe/2018/6386513.pdf · ResearchArticle The Fractional Order PID Method with a Fault Tolerant

Research ArticleThe Fractional Order PID Method with a Fault Tolerant Modulefor Road Feeling Control of Steer-by-Wire System

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

School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, China

Correspondence should be addressed to Wei Chen; chenwei [email protected]

Received 20 August 2018; Accepted 5 November 2018; Published 19 November 2018

Academic Editor: Saeed Eftekhar Azam

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.

To improve the road feeling of the steer-by-wire (SBW) system, a fractional order PID (proportion-integral-derivative)methodwitha 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 isdesigned to control torque of the road feelingmotor. Furthermore, genetic algorithm (GA) is applied to tune the FOPID controller’sparameters. Thirdly, a fault tolerant module aiming at potential failures of the motor’s torque sensor is studied to improve thereliability of the system. Kalman Filter (KF) algorithm is utilized in the fault tolerant module so as to detect failures of the motor’storque sensor, and then fault tolerant module reconfigures the motor’s torque estimated by KF as a substitute when the torquesensor 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 develop-ment of modern vehicles [1, 2], which breaks the concept oftraditional steering system and replaces the mechanical jointbetween the steering wheel and front wheels [3]. The SBWsystemnot only simplifies the design of cab, but also enhancesthe vehicle stability and driving force [4]. Drivers will notfeel the road condition directly without the mechanicalconnection from steering wheel to the road in the SBWsystem [5]. Hence, it is urgent to study the control strategy toimprove the road feeling of the SBW system.The road feelingcontrol is to feed back the road condition to the driver, whichcontrols the reaction torque that the steering wheel actuatorgenerates [6].

Recently, some scholars have developed PID controllerfor the road feeling control of the SBW system [7, 8].Compared with the PID controller, the FOPID controlleradds two adjustable parameters (integral order and derivativeorder), which provides a larger tuning space for bettercontrol performance and robustness [9, 10]. In some practicalapplications, it is verified that the FOPID controller hasbetter disturbance rejection ability and less sensitivity to

the parameter variations of the control system than thetraditional PID controller [11]. As a result, the common PIDcontroller is gradually replaced by the FOPID controller withthe support of the researchers [12–14]. Therefore, the FOPIDcontroller is adopted to control the road feeling of the SBWsystem in this paper.

The additional varying orders of integration and differ-entiation of the FOPID controller are accompanied with theincrease 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 searchthe best parameters with constraints, targets, and dynamiccomponents [15, 16]. With its optimization characteristics,GA has been applied to the parameter optimization of theFOPID controller for many times [17–19]. In this way, GA isused to tune the FOPID controller’s five parameters includingproportional constant, integral constant, derivative constant,integral order, and derivative order.

Most of the literatures about the FOPID controller andPID controller mentioned above only take into accountthe case without sensor faults in the control process. Thetorque sensor is so essential that it is used to measure thefeedback signal of the road feeling control system. However,

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 6386513, 12 pageshttps://doi.org/10.1155/2018/6386513

http://orcid.org/0000-0003-2734-1924http://orcid.org/0000-0001-9953-2629http://orcid.org/0000-0002-4489-6799https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/6386513

2 Mathematical Problems in Engineering

the faults of the torque sensor could occur in the processof road felling control, which can cause a critical effect onthe road feeling control [20]. Consequently, fault tolerantcontrol (FTC) becomes the key issue to ensure the reliabilityof the control system. Fault detection and diagnosis (FDD)module is essential in FTC problem and it is expected to bevery important for general SBW technique [21]. Once thesensor faults occur, the FDD method can detect, diagnose,and remove the faults, thus ensuring the normal workingof the control system. The most convenient FDD methodfor sensors is hardware redundancy, where multisensors aredeployed to measure the same physical parameter and faultsare detected by comparing all sensor's measurements [21].For the hardware redundancy method, on the one hand, it isdifficult 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 redundancymethod, the model-based techniques have been widely usedin the FDD [22–24], which provides a great fault detectioneffect and robustness against external disturbances andmodeluncertainties [25].

The primary principle of the model-based FDD methodis to design an observer to evaluate a residual signal which isthe difference between the measured signal and the observeroutput [26]. The model-based FDD method can detectand diagnose sensor faults effectively without redundanthardware. The well-known KF observer is one of the model-based FDD methods, which shows outstanding performancein dealing with the fault tolerant problem, and it has beenwidely applied to detect and diagnose the sensor faults [27–30]. Zhang Han et al. [5] put forward a fault tolerant modulefor two-way H infinity controller of the SBW system. KFis used as the FDD method to detect and diagnose frontwheel sensor’s faults. The simulations and experiments wereconducted to verify the proposed FTC strategy based on KF.Xiahou K. S et al. [31] presented a KF-based FTC strategyunder voltage and current sensor faults. The sensor faultswere detected and isolated based on the residuals calculatedfrom observations measured by sensors and estimationsprovided by KF. The simulation and experimental studiesvalidated the proposed FTC strategy. Liu Yuan et al. [32]designed a FTC method for aeroengine, where the sensorfaults can be diagnosed by KF accurately. The semiphysicalsimulation indicated that the fault tolerant controller couldensure that the aeroengine controller worked safely andreliably in the event of sensor faults.

Inspired by the previous studies, KF is used to detectand diagnose the torque sensor’s faults of road feeling motorin this paper. KF evaluates a residual signal between themeasured signal and the observer output. When the valueof residual signal is greater than the defined threshold, itindicates that the torque sensor faults occur and the observeroutput is reconfigured as the reference value of the fault tol-erant module. On the contrary, the FOPID controller worksnormally throughobtaining the feedback torquemeasured bythe torque sensor.

In this paper, we aim to improve the driver’s road feelingof the SBW system with the FOPID controller tune byGA and KF-based fault tolerant module. The rest of this

paper is organized as follows: Section 2 briefly introducesthe overall road feeling control strategy; Section 3 studiesthe design of the FOPID controller; Section 4 focuses onthe FTC strategy based on KF; Section 5 describes thesimulation results; and the final section is the conclu-sions.

2. The Overall Road Feeling Control Strategyof the SBW System

2.1. The Structure of the SBW System. The function of SBWsystem can mainly be divided into two parts: vehicle steeringcontrol and road feeling control. The overall structure of theSBW system is shown in Figure 1.

Firstly, the controller receives the angle sensor signalof the steering wheel. Secondly, the controller outputs thecurrents to the electromagnetic valves. Thirdly, the hydrauliccylinders are pushed by pressure energy from the electromag-netic valves. Finally, the steering of the vehicle is realized.Meanwhile, the pressure sensors installed in the hydrauliccylinders send the steering resistance signal to the controller,and then the controller sends out the control signal to theroad feeling motor based on the comparison between theideal steering resistance torque and the actual torque ofthe steering wheel detected by the torque sensor. Then thereducing mechanism transfers the torque to the steeringwheel. In this way, the road feeling can be obtained by thedrivers.

The road feeling control is discussed in this paper, and thevehicle 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 controland the overall road feeling control strategy is shown inFigure 2. The ideal steering resistance torque is calculatedby the vehicle speed 𝑉and the maximum pressure difference𝑃max between left cavity and right cavity of four steeringcylinders. The concrete formula of the ideal steering resis-tance torque will be drawn based on experiments con-ducted on the prototype vehicle in the future. The FOPIDcontroller sends out the control signal to the road feelingmotor according to the difference between the ideal steeringresistance torque and the torque measured by the torquesensor. The torque sensor is so essential that it is usedto measure the feedback signal of the road feeling controlsystem.

The fault tolerant module is used to avoid the bad effectcaused by the torque sensor faults of the road feeing controlsystem. KF observer is used to observe the torque output ofthe road feeling motor and evaluate a residual signal betweenthe measured value and the observer output signal. Whenthe residual signal is greater than the defined threshold, itindicates that the torque sensor faults occur and the valuemeasured by the torque sensor is replaced by the observeroutput. When the value of the residual signal is smaller thanthe defined threshold, the FOPID controller works normallythrough obtaining the feedback torque detected by the torquesensor.

Mathematical Problems in Engineering 3

Reducing mechanism

Motor

Controller

Electromagnetic valves

Steering wheel

Cylinder

Toque sensor

Pressure sensor

Angle sensor

Figure 1: The structure of the SBW system.

FOPID Road feeling motor

E(s) U(s) Ts(s)resistance

torque

Pmax

Vmechanism

Ideal

controller

Fault tolerant module basedon kalman filter observer

Reducing

Fault tolerant module basedon kalman filter observer

Fault tolerant module based on Kalman filter observer

T(s)

Controlled object ４0(s)

Road feeling motor mechanism

Reducing

Controlled object

Figure 2: The overall road feeling control strategy. (𝑇(𝑠): the ideal steering resistance torque; 𝐸(𝑠): difference between the ideal steeringresistance torque and actual steering resistance torque;𝑈(𝑠): the output voltage of the FOPID controller; 𝑇𝑠(𝑠): the output torque of the roadfeeling motor; 𝑇0(𝑠): the torque reconfigured by the fault tolerant module.)

2.3. System Modelling. The control system modelling ismainly to establish the mathematical model of the road feel-ing motor, whose moment equilibrium equation is expressedas follows.

𝑇𝑚 = 𝐽𝑚𝑑2𝜃𝑚𝑑𝑡2 + 𝐵𝑚

𝑑𝜃𝑚𝑑𝑡 + 𝐾𝑚𝜃𝑚/𝑔𝑚 − 𝜃𝑠𝑤𝑔𝑚 (1)

where𝑇𝑚 denotes the output torque of the road feelingmotor;𝐽𝑚 represents themoment of inertia of the road feelingmotor;𝐵𝑚 stands for the damping coefficient of the road feelingmotor;𝐾𝑚 is the torsional stiffness of the road feeling motor;𝑔𝑚 represents the reduction ratio of the reducing mechanism;

𝜃𝑚 is the angle of the road feeling motor; 𝜃𝑠𝑤 denotes thesteering wheel angle.

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

𝑇𝑚 = 𝐽𝑚𝑑2𝜃𝑚𝑑𝑡2 + 𝐵𝑚

𝑑𝜃𝑚𝑑𝑡 (2)The electromagnetic torque and the current of the road

feeling motor can be expressed as follows.

𝑇𝑚 = 𝐾𝑡𝐼 (3)


where 𝐾𝑡 denotes the torque coefficient of the road feelingmotor; 𝐼 represents the armature current of the road feelingmotor.

The electromotive force balance equation of the roadfeeling motor is shown below.

𝑈 = 𝐿𝑑𝐼𝑑𝑡 + 𝑅𝐼 + 𝐾𝑒𝑑𝜃𝑚𝑑𝑡 (4)

where 𝑈 denotes the armature voltage of the road feelingmotor; 𝐿 represents the armature inductance of the roadfeeling motor; 𝑅 represents the armature resistance of theroad feeling motor; 𝐾𝑒 is the back electromotive forcecoefficient of the road feeling motor.

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

𝐺 (𝑠) = 𝑇𝑠 (𝑠)𝑈 (𝑠) = 𝑔𝑚𝑇𝑚 (𝑠)𝑈 (𝑠)

= 𝑔𝑚𝐾𝑡𝐽𝑚𝑠 + 𝑔𝑚𝐾𝑡𝐵𝑚𝐽𝑚𝐿𝑠2 + (𝐵𝑚𝐿 + 𝐽𝑚𝑅) 𝑠 + 𝐵𝑚𝑅 + 𝐾𝑡𝐾𝑒(5)

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

3. The FOPID Controller Design

3.1. The FOPID Controller. Compared with the conventionalPID controller, the FOPID controller adds the integral anddifferential orders, which makes control system fine trackingaccuracy, abundant dynamics, and high robustness [34].Before describing the design of the FOPID controller, thebasic knowledge of fractional calculus (FC) is introducedfirstly. FC is a generalization of integration and differentiationoperators, which is able to be represented as follows.

𝑡0𝐷𝛼𝑡 =

{{{{{{{{{{{{{

𝑑𝛼𝑑𝑡𝛼 Re (𝛼) > 01 Re (𝛼) = 0∫𝑡𝑡0

(𝑑𝜏)−𝛼 Re (𝛼) < 0(6)

where 𝑡0𝐷𝛼𝑡 denotes the FC; 𝛼 is the fractional order; 𝑡0and 𝑡 represent the upper and lower limits of the operation,respectively.

There are three mainstream methods to define the frac-tional operators [35]. The common one is the RL (RiemannLiouville) definition, which can be expressed as follows [36].

𝑡0𝐷𝛼𝑡 𝑓 (𝑡) = 1Γ (𝑛 − 𝛼) (

𝑑𝑑𝑡)𝑛 ∫𝑡𝑡0

𝑓 (𝜏)(𝑡 − 𝜏)𝛼−𝑛+1𝑑𝜏 (7)

where (𝑛 − 1 < 𝛼 < 𝑛, 𝑛 ∈ 𝑁); 𝑓(𝑡) is the function; Γ() is thewell-known gamma function, which is described as follows.

Γ (𝑧) = lim𝑛→∞

𝑛!𝑛𝑧𝑧 (𝑧 + 1) ⋅ ⋅ ⋅ (𝑧 + 𝑛) (8)

In this paper, the FOPID controller is used to controlthe road feeling, whose output voltage can be expressed asfollows.

𝑢 (𝑡) = 𝐾𝑝𝑒 (𝑡) + 𝐾𝑖𝐷−𝜆𝑒 (𝑡) + 𝐾𝑑𝐷𝑢𝑒 (𝑡)𝑒 (𝑡) = 𝑇 (𝑡) − 𝑇𝑠 (𝑡)

(9)

where𝐾𝑝, 𝐾𝑖, 𝐾𝑑, 𝜆, 𝑢 are the proportional gain, integral gain,derivative gain, fractional order integrator, and fractionalorder differentiator, respectively.

The block diagram of road feeling control system usinga FOPID controller is shown in Figure 3. The continuoustransfer function of the FOPID controller is shown as follows.

𝐺𝐶 (𝑠) = 𝑈 (𝑠)𝐸 (𝑠) = 𝐾𝑝 + 𝐾𝑖𝑠−𝜆 + 𝐾𝑑𝑠𝑢,(0 ≤ 𝜆 ≤ 2; 0 ≤ 𝜇 ≤ 2)

(10)

Clearly, when 𝜆 = 1, 𝜇 = 1, the FOPID controllerbecomes the conventional PID controller.

3.2. GA-Based Optimal Tuning of the FOPID Controller. GAis a search heuristicmethod that imitates the natural selectionprocess, and it is an efficient technique to solve optimizationproblems [37, 38]. Hence, GA is used to tune five parametersof the FOPID controller in this paper, and its flow chart isshown in Figure 4.

Step 1. Once the approximate range and coding length of eachparameter are determined, parameters are coded. In orderto save the optimization time of GA, based on [11] and theoptimization trials in MATLAB, the ranges of parameters areset as follows.

𝐾𝑝 ∈ [0, 1] ,𝐾𝑖 ∈ [0, 20] ,𝐾𝑑 ∈ [0, 1] ,𝜆 ∈ [0, 2] ,𝜇 ∈ [0, 2]

(11)

Compared with the real coding, the binary coding hassimple coding and decoding operations, and it is easier toimplement genetic operations. Therefore, the binary codingis selected to code five parameters of GA. The length of thebinary string is calculated based on parameter range andoptimization accuracy, which is shown below.

2𝑖−1 ≤ (𝑉𝑚 − 𝑉𝑛) ∗ 10𝑎 ≤ 2𝑖 − 1 (12)where 𝑖 is the length of the binary string; 𝑉𝑚 and 𝑉𝑛are the maximum and minimum values of the optimizedparameters, respectively; 𝑎 denotes the optimization accuracyof parameters.

Step 2. The initial population is randomly generated bythe computer. For the binary coding, a random number of


Table 1: The actual parameters of the SBW system.

Symbol Value Unit𝐽𝑚 0.0068 kg.m2𝐵𝑚 0.0165 N.m.s/rad𝐿 0.00033 H𝑅 0.183 Ω𝐾𝑒 0.07 V.s/rad𝐾𝑡 0.068 N.m/A𝑔𝑚 22 \

Plant G(s)T(s) E(s) U(s)U

Proportional ＋Ｊ

Integral ＋ＣＭ-

Derivative ＋＞Ｍu

４Ｍ(s))

Proportional ＋Ｊ

Integral ＋＋ＣＭ-

Derivative ＋＋＞Ｍu

FOPID controller ＇＃(s)

Figure 3: Block diagram of road feeling control system using a FOPID controller.

uniform distribution between 0∼1 is created firstly, and thenthe random numbers between 0∼0.5 are represented by 0; therandom numbers between 0.5∼1 are denoted by 1.Step 3. The individual parameters of the population aredecoded into corresponding values, and then the best indi-vidual is found through calculating the fitness functionvalues. To define suitable fitness function, the objectivefunction must be taken into account firstly. The commonobjective function of the FOPID controller is integral ofsquared error (ISE), integral of absolute value error (IAE),integral of time by absolute value error (ITAE), and integralof time by squared error (ITSE), which are defined as follows[9].

𝐼𝐴𝐸 = ∫∞0|𝑒| 𝑑𝑡,

𝐼𝑇𝐴𝐸 = ∫∞0𝑡 |𝑒| 𝑑𝑡,

𝐼𝑆𝐸 = ∫∞0𝑒2𝑑𝑡,

𝐼𝑇𝑆𝐸 = ∫∞0𝑡𝑒2𝑑𝑡

(13)

The IAE criterion mentioned above is regarded as theobjective function of the FOPID controller in this paper.Clearly, the smaller the value of IAE, the better the per-formance of the FOPID controller. GA is to search forindividuals with maximum fitness function value. On thatbasis, the fitness function of GA is defined as follows.

𝑓 (𝑥) = 1𝐼𝐴𝐸 =1

∫∞0|𝑒| 𝑑𝑡 , 𝑥 = (𝐾𝑝, 𝐾𝑖, 𝐾𝑑, 𝜆, 𝜇) (14)

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

Step 5. Genetic operators are implemented. The initial pop-ulation can be used to obtain a new population throughimplementing reproduction, crossover, and mutation. Thenew generation is applied into Step 3, and then the fitnessfunction value of new generation is obtained.

Step 6. Steps 3∼5 are repeated until genetic generation isreached, and the best gene string is received. At last, fiveparameters of the FOPID controller can be obtained basedon decoding operation.

All the parameters in the GA-based optimization processare 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 onKalman Filter

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


Coding

Population (k)

Fitness function

Genetic operator

Initial parameters

1 Reproduction2 Crossover3 Mutation

Reaching genetic generations?

Population (k+1)

Decoding Optimization parameters

K k+1true

false

Figure 4: GA-based parameters optimization process.

FOPID controller

Plant

Kalman filter observer

J(k)

Fault alarm

Fault free

Fault reconfiguration module

T e

U

Y

+

-

+

-

Re

true

false

Kalman filter observer

J(k)

Fault alarm

Fault free

Y

++

--

ReRe

truet

lsefal

FDD module

４0

J(t)>＊ＮＢ

９

Figure 5:The fault tolerant control procedure.

The FTC procedure is shown in Figure 5. Firstly, thevoltage output of the FOPID controller and the plant torqueoutput signal are put into the KF observer. Secondly, theestimated torque value of the road feeling motor is obtained

based on the KF observer. Thirdly, the residual signal Re issent to a fault evaluation function 𝐽(𝑘). Fourthly, the torquesensor in fault condition depends on the comparison betweenthe fault evaluation function and its threshold value. If thefault evaluation function is bigger than its threshold, thefaults occur in the torque sensor; otherwise, the torque sensorworks normally. Finally, the fault reconfiguration moduleoutputs the torque signal 𝑇0 to the FOPID controller. If thetorque sensor faults occur, 𝑇0 is the estimated torque �̂� of theKF observer; otherwise 𝑇0 is the torque 𝑌 measured by thetorque sensor. As mentioned above, the FTC method mainlyconsists of two parts: FDD module and fault reconfigurationmodule.

4.1. Fault Detection and Diagnosis Strategy. FDD modulemakes full use of KF to detect the torque sensor faults inthis paper. KF is a kind of linear unscented filter, whichobtains the optimal estimation of system state and systemparameters under the condition of minimizing the errorvariance. KF can be implemented by recurrence formulain discrete domain, and the amount of computation andstorage is greatly reduced, so it is easy to meet the real-timerequirements. Owing to its advantages, KF is used to estimatethe torque value of the road feeling motor.

4.1.1. Observer Design Based on Kalman Filter. For a dynamicsystem, it can be described by the following differenceequation.

∙𝑋 (𝑡) = 𝐴1𝑋(𝑡) + 𝐵1𝑢 (𝑡) + 𝑤 (𝑡)𝑌 (𝑡) = 𝐶1𝑋(𝑡) + V (𝑡)

(15)


where 𝑋 and 𝑌 denote the state matrix and the observationsignal matrix of system, respectively; 𝑢 represents the inputsignal matrix; 𝐴1, 𝐵1, and 𝐶1 are the parameter matrixes ofsystem; 𝑤 and V denote the process noise and the measure-ment noise matrixes, and they need to meet the followingrelationships.

𝐸 [𝑤 (𝑘)] = 0,𝐸 [V (𝑘)] = 0,

𝐸 [𝑤 (𝑘) V𝑇 (𝑗)] = 0𝐸 [𝑤 (𝑘) 𝑤𝑇 (𝑗)] = 𝑄𝛿𝑘𝑗,𝐸 [V (𝑘) V𝑇 (𝑗)] = 𝑅𝛿𝑘𝑗

(16)

where 𝐸[] denotes the mean value;𝛿𝑘𝑗 = {{{

1, 𝑘 = 𝑗0, 𝑘 ̸= 𝑗. (17)

For the control object’s transfer function 𝐺(𝑠) in thispaper, it can be translated into a state space, and the relatedparameters in (15) are shown as follows.

𝑋 = [𝑥1 𝑥2]𝑇 ,𝑍 = 𝑇𝑠,𝑢 = 𝑈

𝐴1 = [[−𝐵𝑚𝑅𝑎 + 𝐾𝑡𝐾𝑒𝐽𝑚𝐿𝑎

𝐵𝑚𝐿𝑎 + 𝐽𝑚𝑅𝑎𝐽𝑚𝐿𝑎1 0 ]]

𝐵1 = [10] ,

𝐶1 = [𝑔𝑚𝐾𝑡𝐵𝑚𝐽𝑚𝐿𝑎𝑔𝑚𝐾𝑡𝐿𝑎 ]

(18)

The continuous model described in (15) is discretized asfollows.

𝑋(𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑢 (𝑘) + 𝑤 (𝑘)𝑌 (𝑘 + 1) = 𝐶𝑥 (𝑘 + 1) + V (𝑘) (19)

where 𝐴 = 𝑒𝐴1𝑇, 𝐵 = (∫𝑇0𝑒𝐴1(𝑇−𝜏)𝑑𝜏)𝐵1, 𝐶 = 𝐶1; 𝐴 denotes

the system state transition matrix of the discrete description;𝐵 is the system input matrix of the discrete description; 𝐶 isthe systemmeasurement matrix of the discrete description; Tdenotes the sample time.

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

(1) Prediction Module

State prediction: 𝑋(𝑘 + 1𝑘) = 𝐴𝑋(𝑘𝑘) + 𝐵𝑢 (𝑘) (20)Error covariance matrix prediction: �̂� (𝑘 + 1𝑘)

= 𝐴�̂� (𝑘𝑘)𝐴𝑇 + 𝑄(21)

(2) Correction Module

Kalman gain matrix: 𝐾 (𝑘 + 1) = �̂� (𝑘 + 1𝑘)𝐶𝑇 [𝐶�̂� (𝑘 +1𝑘)𝐶𝑇 + 𝑅] (22)

State update: 𝑋(𝑘 + 1𝑘 + 1) = 𝑋(𝑘 +1𝑘) + 𝐾 (𝑘 + 1) 𝜀 (𝑘 + 1)

𝜀 (𝑘 + 1) = 𝑌 (𝑘 + 1) − 𝐻�̂�(𝑘 + 1𝑘)(23)

Error covariance matrix update: �̂� (𝑘 + 1𝑘 + 1) = [𝐼 − 𝐾 (𝑘 + 1)𝐻] �̂� (𝑘 + 1𝑘) (24)

where 𝑋(𝑘 + 1/𝑘) and �̂�(𝑘 + 1/𝑘) are the predicted statevalue and error covariance prediction value at the (k+1)thmoment based on their values at the kth moment; 𝑋(𝑘/𝑘)and �̂�(𝑘/𝑘) are the predicted state value and error covarianceprediction value at the kth moment; 𝐾(𝑘 + 1) is the Kalmangain matrix at the (k+1)th moment; 𝑌(𝑘 + 1) denotes theobservation matrix at the (k+1)th moment; 𝜀(𝑘 + 1) repre-sents the innovation information of filtering at the (k+1)thmoment.

4.1.2. Residual Generation and Evaluation. By means of (20)-(24), the torque value of the road feeling motor can beestimated effectively. The residual signal is the differencebetween the measured and estimated values described asfollows [41].

𝑟 (𝑘) = 𝑌 (𝑘) − �̂� (𝑘) (25)where 𝑟(𝑘) denotes the residual signal; 𝑌(𝑘) represents thetorque value measured by the torque sensor; �̂�(𝑘) is thetorque value estimated by KF.



FOPID controller

Torque sensor


FOPID controller

Fault tolerant controller

Fault-free sensor Sensor with faults

Y

４0

Figure 6: Control reconfiguration for sensor faults.

The fault evaluation function 𝐽(𝑘) can be defined asfollows [42].

𝐽 (𝑘) = |𝑟 (𝑘)| (26)Suppose the threshold of the evaluation function is 𝐽𝑡ℎ

which can be designed in experiments according to theexperience of experts [42]. The torque sensor faults can bedetected by the following rule.

𝐽 (𝑘) < 𝐽𝑡ℎ ⇒ 𝑓𝑎𝑢𝑙𝑡 − 𝑓𝑟𝑒𝑒𝐽 (𝑘) ≥ 𝐽𝑡ℎ ⇒ 𝑓𝑎𝑢𝑙𝑡 − 𝑎𝑙𝑎𝑟𝑚 (27)

4.2. Fault Reconfiguration Strategy. If the torque sensor faultsoccur, the controller cannot control the road feeling of theSBW system correctly. To minimize the potential risks ofsensor faults, the FTC strategy is needed to control the roadfeeling, which uses the fault reconfigurationmodule to outputthe torque to the proposed FOPID controller. The key pointin the FTC strategy is to reconstruct the system torque outputusing the fault reconfiguration module. When the torquesensor is detected with faults based on the FDD module, thesystem torque output is the estimated value �̂�(𝑘) of KF.Whenthe torque sensor is detected with no fault, the system torqueoutput is the torque value𝑌(𝑘)measured by the torque sensor.Hence, the output 𝑇0 of the fault reconfiguration module canbe described as follows.

𝑇0 = 𝑌 (𝑘) (𝑓𝑎𝑢𝑙𝑡 − 𝑓𝑟𝑒𝑒)𝑇0 = �̂� (𝑘) (𝑓𝑎𝑢𝑙𝑡 − 𝑎𝑙𝑎𝑟𝑚) (28)

Figure 6 shows the control reconfiguration strategy forthe torque sensor faults. Based on the principle of the faultreconfiguration module, the FOPID can receive the validtorque value as a feedback signal whenever the sensor faultsoccur. In this way, the fault tolerant controller can ensure thenormal road feeling control.

5. Simulation Results and Analyses

In this section, simulations based onMATLAB are carried outto validate the feasibility of the proposed FOPID controllerand the FTC strategy.

5.1. Test of the FOPID Controller. In order to demonstratethe superiority of the FOPID controller, the conventionalPID controller tuned by GA is also designed to comparethe performance of the optimized FOPID controller. Thecode of GA is written in MATLAB to calculate the objectivefunction mentioned in (14) for every set of parameters inthe ranges described in (11). The optimization is executed forthe FOPID controller and the conventional PID controller tofind the optimal value (𝐾𝑝, 𝐾𝑖, 𝐾𝑑, 𝜆, 𝜇) through minimizingIAE of the unit step responses of the ideal steering resis-tance torque. Table 2 shows the optimal parameters of theFOPID controller and the conventional PID controller usingGA.

From Table 2, we can see that the optimized FOPID con-troller has better performance because the FOPID controllerdecreases by 68.83% compared with the conventional PIDcontroller in terms of IAE. The FOPID controller possessingbetter control effect is accompanied with the fractionalintegration order and differentiation order optimized by GA.Meanwhile, it verifies that the fractional integration orderand differentiation order of the FOPID controller provideindeed a larger tuning space for better control performance.Moreover, Table 2 indicates that GA can tune parameters ofthe FOPID and the conventional PID controllers well becausetwo controllers’ IAE are small significantly. Furthermore, thecode of GA in MATLAB tuning parameters of the FOPIDand PID controllers can be extended into other controlapplications conveniently.

The unit step responses of the ideal steering resistancetorque 𝑇(𝑡) = 1 (0 ≤ 𝑡 ≤ 1) using the optimized FOPIDcontroller and conventional PID controller are shown inFigure 7. It can be seen that the optimized PID controllerhas 0.026s rise time, 0.072s settling time, and 14.823N.mIAE. Also, with the optimized FOPID controller, the risetime is 0.01s, the settling time is 0.007s, and the IAE is4.620N.m.The rise time, settling time, and IAE of the FOPIDcontroller are significantly smaller than the PID controller. Itis concluded that the optimized FOPID controller has a betterperformance than the optimized PID controller in terms oftransient response.

Figure 8 illustrates the following curves of the optimizedFOPID controller and conventional PID controller for the𝑇(𝑡) = sin(2 ∗ 𝑝𝑖 ∗ 𝑡) (0 ≤ 𝑡 ≤ 1) sinusoidalsignal. Figure 9 shows the resistance torque tracking errorof the optimized FOPID and PID controllers. On the onehand, we can see that the optimized FOPID controller hassmaller tracking error from Figures 8 and 9. On the otherhand, compared with 30.320N.m IAE of the optimized PIDcontroller, the optimized FOPID controller has 10.662N.mIAE. Therefore, we can safely come to the conclusionthat the optimized FOPID controller has better resistancetorque tracking performance than the optimized PID con-troller.


Table 2: The optimal parameters of the FOPID controller and the conventional PID controller using GA.

Controller 𝐾𝑝 𝐾𝑖 𝐾𝑑 𝜆 𝜇 IAE f(x)PID 0.0684 20 0 1 1 14.823 0.067FOPID 0.118 19.521 0.155 1.067 1.875 4.620 0.216

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

time (s)

Torq

ue (N

.m)

referenceFOPIDPID

Figure 7: The unit step responses of the resistance torque using theoptimized FOPID controller and conventional PID controller.

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.4

0.6

0.8

1

time (s)

Torq

ue (N

.m)

referenceFOPIDPID

0.2

Figure 8: The sinusoidal following curve of the resistance torquebased on the optimized FOPID and PID controllers.

5.2. Test of the Fault Tolerant Module. According to theabove subsection, we can draw the conclusion that theFOPID controller has better transient response and trackingperformance. The FOPID controller with the fault tolerantmodule is demonstrated in this subsection.

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

time (s)Er

ror

FOPIDPID

Figure 9: The resistance torque tracking error using the optimizedFOPID and PID controllers.

Firstly, the estimation accuracy of the KF observer isvalidated under the condition of the fault-free torque sensor.Figure 10(a) illustrates the KF-based estimation value and themeasured value of the resistance torque. Figure 10(b) showsthe estimation error of the resistance torque based on KF.From Figure 10, we can see that the KF observer has greatestimation accuracy and it can be used to generate residualsignal and detect sensor faults in the fault tolerant strategy.

Secondly, the fault tolerant schema is demonstrated basedon simulations. In order to validate the comprehensivefault tolerant effect, three kinds of sensor faults (lock fault,constant gain fault, and constant deviation fault) are takeninto account, which are shown as follows.

𝑌1 (𝑡) = 𝑎, (|𝑎| ≥ 𝐽𝑡ℎ, 𝑡 ≥ 𝑡0)𝑌2 (𝑡) = 𝑘𝑦 (𝑡) + 𝑎, (𝑘 ̸= 0, 𝑘 ̸= 1, |𝑎| ≤ 𝐽𝑡ℎ, 𝑡 ≥ 𝑡0)𝑌3 (𝑡) = 𝑦 (𝑡) + 𝑎, (|𝑎| ≥ 𝐽𝑡ℎ, 𝑡 ≥ 𝑡0)

(29)

where𝑌1(𝑡),𝑌2(𝑡), and𝑌3(𝑡) denote themeasured value by thetorque sensorwith lock fault, constant gain fault, and constantdeviation fault, respectively; 𝑦(𝑡) represents the true value ofthe torque; 𝑘 is the constant gain; 𝑎 is the constant deviation;𝑡0 denotes the start time of sensor faults.

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


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1−0.5

00.5

11.5

time (s)

Torq

ue (N

.m)

KF−based estimation valuemeasured value

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.020.040.060.08

0.10.12

time (s)

Estim

atio

n er

ror (

N.m

)

(b)

Figure 10: KF-based estimation performance of the resistancetorque: (a) resistance torque; (b) estimation error.

𝑌1 (𝑡) = 0.5, (𝑡 ≥ 0.5)𝑌2 (𝑡) = 3 ∗ 𝑦 (𝑡) + 0.01, (𝑡 ≥ 0.4)𝑌3 (𝑡) = 𝑦 (𝑡) + 0.4, (𝑡 ≥ 0.6)

(30)

Figure 11 shows the fault detection result of the torquesensor with lock fault. The true value, KF-based estimationvalue, and value measured by the faulty sensor of the resis-tance torque are displayed in Figure 11(a).The fault evaluationfunction is shown in Figure 11(b). From Figure 11, we canclearly figure out that the sensor with the lock fault can bedetected rapidly and effectively after it occurs. Furthermore,it can be seen that the estimation accuracy of KF is high andKF-based estimation value can be reconstructed as the torquefeedback signal after the torque sensor’s lock fault occurs.

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

Based on the simulation results mentioned above, wecan draw the conclusions: (1) the FOPID controller tunedby GA has better control performance than the optimizedPID controller in terms of transient response and trackingability; (2) KF observer has great estimation accuracy, whichcan be used to generate residual signal and detect sensorfaults; (3) the fault tolerant strategy can reconfigure thetorque value rapidly and effectively when the torque sensorfails.

Torq

ue (N

.m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1−0.5

00.5

11.5

time (s)true valueKF−based estimation valuemeasured value

(a)

Resid

ual s

igna

l (N

.m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

time (s)J(k)Jth

(b)

Figure 11: The fault detection result of the torque sensor with lockfault: (a) resistance torque; (b) residual signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3−2−1

01234


Torq

ue (N

.m)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.51

1.52

2.5

time (s)J(k)Jth

Resid

ual s

igna

l (N

.m)

(b)

Figure 12: The fault detection result of the torque sensor withconstant gain fault: (a) resistance torque; (b) residual signal.

6. Conclusions

In this paper, the FOPID controller with a fault tolerant mod-ule for the road feeling control of the SBWsystem is proposed.The FOPID controller is designed to control the steeringresistance torque accurately and quickly. GA is applied to tunethe FOPID’s parameters including proportional constant,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1−0.5

00.5

11.5


Torq

ue (N

.m)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.050.1

0.150.2

0.250.3

0.350.4

time (s)J(k)Jth

Resid

ual s

igna

l (N

.m)

(b)

Figure 13: The fault detection result of the torque sensor withconstant deviation fault: (a) resistance torque; (b) residual signal.

integral constant, derivative constant, integral order, andderivative order so as to achieve the best road feeling controlperformance that minimizes the IAE. In order to improvethe reliability of the control system, a fault tolerant moduleaiming at the motor torque sensor is studied. In the FDDmodule, a residual signal is generated by KF, and a thresholdof the fault evaluation function is used for judging sensorfaults. For the fault reconfigurationmodule, the output torqueof the control system can be reconstructed effectively whenthe torque sensor faults occur. Based on the fault tolerantstrategy, the FOPID controller can receive the valid torquevalue as a feedback signal in spite of sensor faults. Theresults of the numerical simulations based on MATLABdemonstrate the effectiveness of the FOPID controller witha fault tolerant module. The optimized FOPID controllerhas better transient response and tracking performance thanthe traditional PID controller. The proposed FDD methodcan 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 reconfigurefaults within a reasonable time. Finally, the proposed FOPIDcontroller with the FTC strategy can be extended into otherindustrial control applications.

Although the simulations based on MATLAB are carriedout to demonstrate the feasibility of the proposed methods,the experiments conducted in practice would improve greatlythe reliability and quality of this research. In the future, wewill make joint efforts to complete the prototype vehicle,and then a large number of experiments with regard to theFOPID controller and the FTC strategywill be conducted andimplemented on the prototype vehicle.

Data Availability

The data used to support the findings of this study areavailable 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 Researchand Development Program of China under Grant No.2016YFC0802904. The authors would like to acknowledgethe Institute of Electrohydraulic Control Technology forEngineering Machinery in Jilin University and Xuzhou Con-struction Machinery Group Co., Ltd., which support thisresearch.

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