Sistema ABS Com Fuzzy-genetico Tunning

Engineering Applications of Artificial Intelligence 23 (2010) 1041–1052

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence

0952-19

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/engappai

Genetic fuzzy self-tuning PID controllers for antilock braking systems

Abdel Badie Sharkawy

Mechanical Engineering Department, Faculty of Engineering, Assiut University, Assiut 715 16, Egypt

a r t i c l e i n f o

Article history:

Received 16 June 2008

Received in revised form

18 June 2010

Accepted 23 June 2010Available online 14 July 2010

Keywords:

Antilock braking system (ABS)

Braking basics

Slip ratio

Decoupled rule bases

Takagi–Sugeno (T–S) fuzzy systems

Self-tuning

Genetic algorithm (GA) and Ziegler–Nichols

(Z–N) method

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016/j.engappai.2010.06.011

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a b s t r a c t

Since the emergence of PID controllers, control system engineers are in pursuit of more and more

sophisticated versions of these controllers to achieve better performance, particularly in situations

where providing a control action to even a minimal degree of satisfaction is a problem. This work is an

attempt to contribute in this field. Variations in the values of weight, the friction coefficient of the road,

road inclination and other nonlinear dynamics may highly affect the performance of antilock braking

systems (ABS). A self-tuning scheme seems necessary to overcome these effects. Addition of automatic

tuning-tool can track changes in system operation and compensate for drift, due to aging and parameter

uncertainties. The paper develops a self-tuning PID control scheme with an application to ABS via

combinations of fuzzy and genetic algorithms (GAs). The control objective is to minimize the stopping

distance, while keeping the slip ratio of the tires within desired range. Computer simulations are

performed to verify the proposed control scheme. Results are reported and discussed.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Difficulties in designing an ABS may be classified as follows.First, the vehicle braking-dynamics are nonlinear. Second,there are many unknown environmental parameters, like roadcoefficient of friction which may be wet, snowy or dry. Finally,parameter changes due to mechanical wear and aging. Recently, agreat deal of research has been performed on the antilock brakesystem (ABS) (Assadin, 2001; Nouilllant et al., 2002; von AltrockNov., 1997; Choi et al., 2002). Currently, most commercial ABSsare based on look-up tabular approach (Chih-Min Lin and HsuMarch, 2003). These tables are calibrated through iterativelaboratory experiments and engineering field tests.

The conventional PID controller for automated machines iswidely accepted by industry. According to a survey reported in Yu(1999), more than 90% of control loops used in an industry usePID. This is because PID controllers are easy to understand (hasclear physical meanings i.e. present, past and predictive), easy toexplain to others, and easy to implement. Unfortunately, many ofthe PID loops that are in operation are in continual need ofmonitoring and adjustment since they can easily becomeimproperly tuned (Passino and Yurkovich, 1998). Generallyspeaking, in order to meet the demands of real time operation,self-tuning is necessary.

Motivated by the success of fuzzy controllers in controllingnonlinear, complex, time-varying dynamic processes in real

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world, there has been steep increase in the research work onthe theoretical aspects of fuzzy logic controller (FLC). The mainreason is that FLCs essentially incorporate human expertise in thecontrol strategy, exploiting easier understanding of linguisticinterpretation.

Among the different types of FLC structures, PI-type andPD-type FLCs are very common (Passino and Yurkovich, 1998;Sharkawy et al., 2003). Other related works have employeddifferent adaptation policies to improve one or more performanceindices (Sharkawy, 2005; Mudi. and Pal, 1999; Xu et al., 1998).However, development of PID-type FLCs was not popular becausethey need the construction of three dimensional rule-base, whichcomplicates the design. Moreover to make PID-type FLC adaptivein nature, the number of free adaptable parameters increases andtheir interaction and interdependence further complicates thesituation (Mann et al., 1999).

The area of auto-tuning of PID controller using fuzzy systemshas attracted many authors (Xu et al., 1998; Mann et al., 1999;S.-Z. et al., 1993; Visioli January, 2001; Bhattacharya et al., 2003;Visioli, 1999; Macvicar-Whelan, 1976; Tzafestas and Papanikolo-poulos, 1990; Marsh, 1998). A fuzzy self-tuning incremental PIDcontroller has been proposed by He et al. S.-Z. et al. (1993). Thecontroller implements a conventional PID structure, which startsits operation with values of proportional gain, integral time, andderivative time, obtained from the well-known Ziegler–Nichls(Z–N) tuning formula. This scheme implements a supervisoryfuzzy system, which adaptively changes the control parameterafter each sampling instant to improve the control performance.In another study, a survey for tuning PID controllers with fuzzy

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Nomenclature

ax the vehicle body acceleration, m/s2

ai,j,k i¼1,2,3, j¼1,2, y, 9, k¼1,2,3 are the coefficients ofthe T–S first order output-model

A areaABS antilock braking systemB1,B2 fuzzy setsc center of Gaussian membership functionc1, c2 scaling factorsC1 the maximum value of friction curveC2 the friction curve shapeC3 the friction curve difference between the maximum

value and the value at l¼1C4 the wetness characteristic valuee(t) error of the closed-loop systemF objective functionFIAE objective function that uses the stopping distance and

IAE as the performance measuresFITAE objective function that uses the stopping distance and

ITAE as the performance measuresFN the normal force, N

FSW PID controller using fuzzy set-point weightingG number of generationsGA genetic algorithmG-PID PID controller in which the three parameters are

determined using GAsH number of iterationsIFE incremental fuzzy expert PID controlIAE integrated absolute errorITAE integrated time multiplied by the absolute errorJITAE performance indexJIAE performance index which uses IAE and the stopping

distance as the performance measureJITAE performance index which uses ITAE and the stopping

distance as the performance measureJw the wheel inertia, kg m2

k1,k2,k3 constant parametersKd the derivative gainKi the integral gainKp is the proportional gainKu ultimate gainm the quarter vehicle mass, kg

n number of rulesN negativeNB negative bigNS negative smallNVB negative very bigP positivePB positive bigPS positive smallPVB positive very bigPID-Fuzzy PID controller tuned using initially guessed fuzzy

systemsPID-IAE PID controller tuned by GA using FIAE as the objective

functionPID-ITAE PID controller tuned by GA using FITAE as the objective

functionR the wheel radius, mRe low reproduction rateRh high reproduction rateSx stop distance, mSSP PID controller using fuzzy self-tuning of a single

parametertu oscillation periodT–S Takagi–Sugeno fuzzy systemsTd the derivative time constantTi the integral time constantu the braking torque, N mV population sizeVx the speed of the vehicle, m/sw positive constantwi firing strength of rule i

Z zeroZ–N PID controller tuned by the Ziegler–Nichols tuning

methoda fuzzy tuning parameteraw angular acceleration of the wheel, rad/s2

g positive constantZ1,Z2 weighting factorsl the slip ratiold desired value of the slip ratiom membership grademr road coefficient of frictions slope of Gaussian membership functiono angular velocity of the wheel, rad/s

A.B. Sharkawy / Engineering Applications of Artificial Intelligence 23 (2010) 1041–10521042

logic has been made by Visioli January (2001). In his work, severalPID control schemes have been simulated for linear systems.Although in most cases, the controllers showed superior perfor-mance for linear processes, they could only reduce the peakovershoot at the expense of increasing rise time, and thedegradation becomes more and more significant with increasingtime delay. This restricted the overall acceptance as a goodcontroller mechanism (Bhattacharya et al., 2003).

Referring to aforementioned works, one may conclude thatZ–N method has been widely accepted as a base for tuning PIDcontrollers off- and on-line. This may be referred to the ability ofthe method to preserve good load disturbance attenuation. Most ofthese studies however have transformed the tuning problem fromusing the Z–N tuning parameters to other author-defined para-meters and use fuzzy logic as the tuning-tool for the newparameters; (S.-Z. et al., 1993; Visioli January, 2001; Bhattacharyaet al., 2003; Visioli, 1999; Macvicar-Whelan, 1976), [others]. Thisindicates the lack of a generalized approach which can be followedfor linear and nonlinear plants. Furthermore, the suitability and

application range of the Z–N method are very limited (Marsh,1998; Zhuang and Atherton, 1993). For example, it is not suitablefor plants with a delay to time-constant ratio smaller than 0.15 orlarger than 0.6. This method also yields poor damping and highsensitivity and does not achieve robustness of the closed-loopwhen considerable parameter variations take place.

In this study, a self-tuning PID controller is proposed for theABS. Controlling the braking torque of an ABS is necessary to avoidlocking of the wheels, so that the driver can keep control on thevehicle’s motion. Parameter variations and uncertainties implythe need for an auto-tuning operation to achieve the performanceconsistency. The article describes a generalized procedure for thedevelopment of a simple, model free fuzzy PID-type structure asan effective combination of three independent fuzzy systems.Each PID parameter is tuned via first order Takagi–Sugeno (T–S)fuzzy system, whose parameters are optimally determined off-line using a modified genetic algorithm (GA). The control goal is tokeep the slipping ratio of the tires within the desired range, whilemaintaining minimal stopping distance when braking is

A.B. Sharkawy / Engineering Applications of Artificial Intelligence 23 (2010) 1041–1052 1043

requested by the driver. With this approach, the PID controllercan be automatically adjusted to meet the system uncertaintiesand achieve satisfactory response. Robustness against roadconditions is examined via numerical tests and results arecompared with previous works.

This paper is organized as follows. Section 2 presents themathematical model of an ABS based on a quarter car model.Section 3 introduces previous works in the area of tuning PIDcontrollers. Section 4 demonstrates the proposed PID self-tuningscheme based on first order T–S fuzzy system. Learning of thefuzzy systems’ database is achieved via optimal performanceindices (fitness functions), using a modified genetic algorithm.Simulation results and discussions are given in Section 5. Section6 offers our concluding remarks.

2. Modeling of an antilock braking system

2.1. The quarter vehicle dynamic model

To verify the control performance, this section demonstrates asimplified model for the quarter vehicle dynamic motion. It wasderived by Assadin and Nouilllant (Assadin, 2001; Nouilllant et al.,2002); Fig. 1. The nonlinear dynamics can be described as follows.The force balance in the longitudinal direction

max ¼ mrFN ð1Þ

The slip ratio is defined by

l¼Vx�oR

Vxð2Þ

Summing torques about the wheel center

Jwaw ¼�uþmrRFN ð3Þ

using Eqs. (1) and (2) and rearranging for _l yields

_l ¼�mrFN

Vx

1�lmþ

R2

Jw

� �þ

R

JwVxu ð4Þ

Eq. (4) clearly shows that, during braking, the slip ratio isdependent on the input torque u and the vehicle velocity Vx. Instate space, the system state variables are: x1¼Sx, x2¼Vx, x3¼l,where Sx is the stopping distance. The state space equations are

_x1 ¼ x2

_x2 ¼mrFN

m

_x3 ¼�mrFN

x2

1�x3

mþ

R2

Jw

� �þ

R

Jwx2u ð5Þ

Vehicle Body

NF

Nr Fμ

uJwαw

Rmax

xV

Fig. 1. The quarter vehicle dynamic model.

This state space model in Eq. (5) has been used in thesimulation tests to evaluate the performance of the ABS, using
different PID control strategies.
During braking, it is assumed that the wheel radius is constant.Also, the vehicle speed Vx and the wheel angular velocity o areavailable signals through transducers mounted on suitable places.So that the slip ratio l, is an available parameter for the ABSclosed-loop system.

2.2. Braking basics and problem definition

The ability of the ABS to maintain vehicle stability andsteerability, and still produce shorter stopping distances thanthose from a locked wheel stop, comes from the relation of theadhesion coefficient mr versus wheel slip ratio l. The frictioncoefficient can vary in a very wide range, depending on factorslike:

(a)

TablFrict

Su

Dr

W

Dr

Sn

Ice

road surface conditions (dry or wet),
(b) tire side-slip angle, (c) tire brand (summer tire, winter tire), (d) vehicle speed, and (e) the slip ratio between the tire and the road.
In this paper, the tire friction model introduced by Burckhardt(1993) and adopted in Harifi et al. (2008) has been used. Itprovides the tire-road coefficient of friction mr as a function of thewheel slip l and the vehicle velocity Vx.

mrðl,VxÞ ¼ bC1ð1�e�C2lÞ�C3lce�C4lVx ð6Þ

The parameters in Eq. (6) denote the following: C1 is themaximum value of friction curve; C2 the friction curve shape; C3

the friction curve difference between the maximum value and thevalue at l¼1; and C4 is the wetness characteristic value and in therange 0.02–0.04 s/m. Table 1 shows the friction model parametersfor different road conditions.

Dependence of the road friction coefficient on surface condi-tions and slip ratio is shown in Fig. 2, (von Altrock Nov., 1997).The lateral force is essential to the steering of vehicle. It is obviouswhen slipping is equal to one; this force is equal to zero, whichexplains why the steering ability is lost during wheel lockup. Theeffective coefficient of friction between the tire and the road hasan optimum value which differs according to the road type andthe worst performance occurs at l¼1 (locked wheel). Mostmanufacturers use a set point for the slipping ratio ld equal to 0.2,which is a good compromise for all road conditions, (Chih-Min Linand Hsu March, 2003). So that the control problem can bedescribed as a set-point control system that may implement PIDcontroller; Fig. 3. Next section describes several PID strategies forthis closed-loop system.

e 1ion model parameters (Burckhardt (Burckhardt, 1993)).

rface conditions C1 C2 C3

y asphalt 1.2801 23.99 0.52

et asphalt 0.857 33.822 0.347

y concrete 1.1973 25.168 0.5373

ow 0.1946 94.129 0.0646

0.05 306.39 0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

λ

Wet Road

Icy Road

Design Region

Lateral Force

Dry Road

� r

Fig. 2. Coefficient of road friction versus wheel slip ratio.

dλ λu WheelBrakes(ABS)

PID

Fig. 3. The closed-loop control of ABS.


3. PID tuning with fuzzy logic

In the early 1940s, after extensive manual experimentation bythe way of trial and error, Ziegler and Nichols invented thewell-known Z–N formula for off-line tuning (Ziegler and Nichols,1942). In time domain for example, the method yielded PIDcoefficients directly from the three important parameters of astable plant to be controlled, namely the plant gain, time-constantand transport delay. These parameters can be easily obtainedgraphically from a step response of the open loop plant. Theconcept of fuzzy systems instead may be used to emulate humanexpertise and on-line tune the control gains using Z–N formula.The coming sub-sections demonstrate some of the previous worksin this area. They are used in later sections for comparisonpurposes.

3.1. Standard and nonlinear PID controllers

There are many types of PID controllers, e.g., PID plus gravitycompensator, PID plus friction compensator, PID plus disturbanceobserver, etc. (Astrom, 1996). Here, we shall consider the basicforms for PID controller which is placed in a unity feedbackcontrol system. A typical PID control law in its standard form is

uðtÞ ¼ Kp eðtÞþTddeðtÞ

dtþ

1

Ti

Z t

0eðtÞdt

� �ð7Þ

where u(t) is the control variable, e(t)¼ld(t)�l(t) is the systemerror (difference between the demand input ld and the systemoutput, l), Kp is the proportional gain, Td the derivative timeconstant and Ti is the integral time constant. Eq. (7) can berewritten as

uðtÞ ¼ KpeðtÞþKi

ZeðtÞdtþKd

deðtÞ

dtð8Þ

Kd¼KpTd the derivative gain and Ki¼Kp/Ti the integral gain.In the design and tuning of a PID controller, the P, I and D

actions need to be coordinated. This is not a trivial process, sincecoefficients of these three actions interact mutually and cannot be

simply tuned individually in a de-coupled manner. An experimentis carried out on the process using Z–N tuning method, can bestated as follows (Yu, 1999; Ziegler and Nichols, 1942). First, theprocess is controlled using the proportional gain Kp. The value ofKp is slowly increased until continuous oscillations are happened.At the time of oscillation, the values of the gain Ku and theoscillation period tu are noted. The method assumes that Kp is 60%of the gain at the time of oscillation. The integral time constant Ti

is 50% of the oscillation period tu and Td the derivative timeconstant is 12.5% of the oscillation period. The Z–N method isdevised for off-line tuning of continuous systems and can also beused on discrete cases for a fast sampling time.

Nonlinear PID controllers have been widely considered inliterature, (Yu, 1999; Mann et al., 1999; Visioli January, 2001;Tzafestas and Papanikolopoulos, 1990). A large class of nonlinearPID controllers can be generalized as follows:

uðtÞ ¼ Kpf ðe, _eÞþKddeðtÞ

dtþKi

Z t

0eðtÞdt ð9Þ

where f ðe, _eÞ is a nonlinear function that depends on the closed-loop error and the process delay (Mann et al., 1999). The f ðe, _eÞmay be represented by a fuzzy logic system, in which the inputsare e(t) and _eðtÞ (Tzafestas and Papanikolopoulos, 1990). It comesout that the proportional gain depends on the current error andother parameters contained on f ðe, _eÞ. Effectiveness of this non-linear PID has been discussed in Mann et al. (1999) for linearsystems.

3.2. Fuzzy self-tuning of a single parameter (SSP)

Because of its simplicity, this scheme has been widelyconsidered in literature (S.-Z. et al., 1993; Visioli January, 2001;Bhattacharya et al., 2003). The method devised by He et al. S.-Z.et al. (1993), consists of parameterizing the Ziegler–Nicholsformula by means of a single parameter a, then using an onlinefuzzy inference system to self-tune the parameter. In this method,the three PID parameters can be expressed as

Kp ¼ 1:2aðtÞKu

Ti ¼ 0:751

1þaðtÞtu

Td ¼ 0:25Ti ð10Þ

where Ku and tu are the ultimate gain and ultimate period,respectively. The value of a(t) is determined recursively with thefollowing equation:

aðtþ1Þ ¼aðtÞþghðtÞð1�aðtÞÞ for aðtÞ40:5

aðtÞþghðtÞaðtÞ for aðtÞr0:5

(ð11Þ

where g is a positive constant that has to be chosen in the range[0.2,0.6] and h(t) is the output of the fuzzy inference system. Thefuzzy system has seven membership functions for each of the twoinput (e and _e) and seven for the output, i.e. the rule-base consistsof 49 rules. The initial value of a(t) is set equal to 0.5, whichcorresponds to the Ziegler–Nichols formula. With respect to themethod however, the tuning of the scaling coefficient of the fuzzymodules and of the parameter g is left to the user and no rule ofthumb is given for this task.

3.3. Fuzzy set-point weighting (FSW)

This approach proposed by Visioli (1999) consists of fuzzifyingthe set-point weight, leaving fixed the other three parameters(again, determined with the Ziegler–Nichols method). In this way,

Table 2Rule table of the FSW (Macvicar-Whelan, 1976).

f(t) Change in error, _e

NB NS Z PS PL

Error, e NB NVB NB NM NS Z

NS NB NM NS Z PS

Z NM NS Z PS PM

PS NS Z PS PM PB

PB Z PS PM PB PVB


the control law can be written as

uðtÞ ¼ Kp bðtÞldðtÞ�lðtÞ� �

þKddeðtÞ

dðtÞþKi

Z t

0eðtÞdt

bðtÞ ¼wþ f ðtÞ ð12Þ

where w is a positive constant less than or equal to 1, and f(t) isthe output of the fuzzy inference system, which consists of fivetriangular membership functions for each of the two inputs e(t)and _eðtÞ and nine triangular membership functions for the output.The fuzzy rules are based on the Macvicar-Whelan matrix(Macvicar-Whelan, 1976), as shown in Table 2. The linguisticvariables are negative small NS, negative big NB, negative very bigNVB, zero Z, positive small PS, positive big PB, and positive verybig, PVB.

The method however, is not a straightforward one and largenumber of arithmetic operations is needed for on-line tuning.

Err

or, e

(t)

∫=τ

0

)( dtteA

0

A

Time, sec

Fig. 4. A typical closed-loop error time-history.

3.4. Incremental fuzzy expert PID control (IFE)

Tzafestas and Papanikolopoulos (Tzafestas and Papanikolo-poulos, 1990), introduced a procedure for scaling the valuesof the three control parameters (initially determined by theZiegler–Nichols formula) during the transient period based onthe system error and its rate. In other words, the current values ofthe proportional, integral and derivative gains are increased ordecreased by means of a fuzzy inference system according to thefollowing relations:

Kp ¼ KpþFS½c1eðtÞ,c2 _eðtÞ� � k1

Ki ¼ KiþFS½c1eðtÞ,c2 _eðtÞ� � k2

Kd ¼ KdþFS½c1eðtÞ,c2 _eðtÞ� � k3 ð13Þ

where the basic tuning method is that of the Ziegler–Nichols;FS½c1eðtÞ,c2 _eðtÞ� is the output of fuzzy inference system, based onMacvicar-Whelan fuzzy rule matrix (Macvicar-Whelan, 1976)(see Table 2), which reflects the typical action of a human control.For example, the integral action has to be increased at thebeginning of the transient response to decrease the rise time, andthen has to be decreased when the system error is negative toreduce the overshoot. The range of the input membershipfunctions is normalized between 1 and �1. Finally, c1, c2 arescaling factors and ki, i¼1,2,3 are constant parameters thatdetermine the range of variation of each term. The whole fuzzysystem involve 14 quantization levels for both e and _e. Similarapproach has been followed in Bhattacharya et al. (2003), wherethe authors have to rely on trial and error procedure in order toidentify parameters of the fuzzy systems.

However, tuning the three parameters ki and the two scalingfactors (c1, c2) that multiply the inputs e and _e, is left to theuser, and it is not clear how these parameters influence theperformance of the overall controller.

4. The proposed fuzzy self-tuning PID control scheme

The proposed control scheme does not rely on Z–N method noron a previous rule-base, especially designed for some plants.Instead, it uses first order T–S fuzzy systems as the tuning-tool foreach of the PID control modules. A modified genetic algorithm isused to optimally select parameters of the fuzzy systems.

The main control objectives of an ABS are:

�
minimization of the stopping distance Sx, and � maintaining the fastest possible response with no/low over-
shoot and close to zero steady state error.

The first objective is a key issue related to ABS, while thesecond is a common objective for any closed-loop control system.

In this work, optimality of the second objective is based onconventional cost functions; i.e. integrated absolute error (IAE)and the integrated time multiplied by the absolute error (ITAE).For example, in an IAE, the hatched area in Fig. 4 represents thecost function. A minimum area is achieved at the fastestphysically possible response, no/small overshoot and close tozero steady state error. It means that decreasing overshoot willnot result in an increase in the rise time or vise versa, as it is thecase with Z–N method. The expected tuning result is, simply, thebest physically possible response.

In short, the objective of an ABS design is to minimize aperformance index that contains the stopping distance and an IAEor ITAE.

4.1. The control system architecture

Motivated by the work of Bhattacharya et al. (2003); Tzafestasand Papanikolopoulos (1990), three decoupled fuzzy systemsconstitute the proposed self-tuning system; each for one para-meter of the PID controller, i.e. Kp, Ki and Ki; Eq. (8). The error e

and change in error _e are used as behavior-recognizers of theclosed-loop performance. They are available signals in the closed-loop system of the ABS and do not require extra hardware. Theself-tuner can be expressed as

Kp ¼ FS1½eðtÞ, _eðtÞ�

Ki ¼ FS2½eðtÞ, _eðtÞ�Kd ¼ FS3½eðtÞ, _eðtÞ� ð14Þ

where a fuzzy P controller, a fuzzy I controller and a fuzzy Dcontroller are connected in parallel to give the resultant controllersignal. With this structure, independent control actions can begenerated, which should necessarily eliminate the problemsassociated with most practical two-terms or three-terms FLCs.The basic approach is summarized in Fig. 5, where each fuzzysystem is trying to recognize when the corresponding parameter

dλ λ

e.

dtd

∫dt

dtd

Update Kp, Ki and Kd

dK

pK

iK

e

e Fuzzy System1

Fuzzy System2

Fuzzy System3

ABS+ ++

+–

Fig. 5. The self-tuning fuzzy PID controller.

Product inference

PN

WeightedAverage

......

......

Fig. 6. Takagi–Sugeno fuzzy system for a module.


is not properly tuned and then seeks to adjust it to obtainimproved performance. In such a way, each fuzzy system can belooked at as gain scheduler (module). T–S type fuzzy systems areused to synthesize each of the self-tuning systems.

The T–S fuzzy system (also known as functional fuzzy system(Passino and Yurkovich, 1998)) was proposed in an effort todevelop a systematic approach to generating fuzzy rules from agiven input–output data set; (Takagi and Sugeno, 1985; Jang et al.,1997). A typical rule has the following form

IF x1 IS B1 AND x2 IS B2 THEN z¼ f ðx1,x2Þ

where B1 and B2 are fuzzy sets in the antecedent, while z¼ f(x1,x2)is a crisp function in the consequent. With this form, the fuzzysystem can be characterized as two input one output fuzzysystems.

Usually f(x1,x2) is a polynomial in the input variables x1 and x2,but it can be any function as long as it can appropriately describethe output of the model within the fuzzy region specified bythe antecedent of the rule. Although there are no restrictions onthe form of the input membership functions, Gaussians are usedin the premise through out this work. A Gaussian membershipfunction is specified by two parameters (c,s)

mBljðxjÞ ¼ gaussianðxj;c,sÞ ¼ exp �

1

2

xj�c

s

� �2" #

ð15Þ

where m is the membership grade, c represents the membershipfunction’s center, s determines its spread; B is the membershipwhich represents a linguistic variable, i¼1,2, y, n is the rulenumber, j¼1,2 is subscript of the input variables.

In the proposed self-tuner, the inputs e and _e are normalizedusing three Gaussian membership functions; negative N, zero Z,and positive P. So that nine rules constitute the rule-base for eachmodule. For simplicity, the consequent part has been chosen to bea first order function of e and _e. So that the rules have thefollowing form:

Rulej : IF e IS B1 AND e IS B2 THEN Ki ¼ ai,j,1eþai,j,2 _eþai,j,3 ð16Þ

where both B1 and B2 are positive (P), zero (Z) or negative (N),Ki ¼ f ðe, _eÞ is the gain to be tuned, i.e. Kp, Ki or Kd. ai,1,j, ai,2,j are theconstants, and j¼1,2, y, 9 is the rule number.

Fig. 6 shows the fuzzy reasoning procedure for a first order T–Smodel. The fuzzy part is only in its antecedent. Each rule has acrisp output and the overall output is obtained via weightedaverage. This fuzzy procedure avoids the time-consuming processof defuzzification required in a Mamdani fuzzy model. Theweighted sum has been also used in Jang et al. (1997) to furtherreduce computations.

With the above structure, the data-base of each moduleconsists of 27 free parameters (coefficients of the first order

polynomial) and 12 free parameters for the inputs membershipfunctions. So that, the total number of free parameters is 117. If anidentical input membership is chosen for the three modules, thetotal number of parameters is reduced to 93; i.e. parameters ofthe input membership functions cN1, sN1, cZ1, sZ1 cP1, sP1 for e, cN2,sN2, cZ2, sZ2, cP2, sP2 for _e, and aijk where i¼1,2,3 are thecoefficients of the first order polynomial, j¼ 1,2, � � � , 9 the rulenumber for each tuner and k¼1,2,3 the number of modules.Determination of optimal values for these parameters is thesubject of the next sub-section.

4.2. Genetic algorithm-based parameter learning

GAs are optimization stochastic technique mimicking thenatural selection, which consists of three operations, namely,reproduction, crossover, and mutation (Jang et al., 1997). Themost general considerations about GA can be stated as follows:

(i)
The searching procedure of the GA starts from multiple initialstates simultaneously and proceeds in all of the parametersubspaces simultaneously.
(ii)
GA requires almost no prior knowledge of the concernedsystem, which enables it to deal with completely unknownsystems that other optimization methods may fail.
(iii)
GA cannot evaluate the performance of a system properly atone step. For this reason, generally, it cannot be used as anon-line optimization strategy and is more suitable for fuzzymodeling.
In practice, training data can be obtained by experimentationor by the establishment of an ideal model. In this work, the ABSmodel in Section 2 is used to emulate the behavior of the ABS, inorder to collect training data. Fig. 7 shows the training process foreach fuzzy module involved in the self-tuning system.

Fuzzy Expert

ABSPID

Genetic – FuzzyOptimizer

Modify membership functions

Modify coefficients of

the consequent parts

Update the gains

dλλ

Search for maximum fitness (performance)

PerformanceIndex

ee ,

IAE/ITAE

Fuzzy System 1

Fuzzy system 3

Fuzzy System 2

pK

iK

dK

Fig. 8. Genetic training of the overall fuzzy PID self-tuning system.

GA

Fuzzy system

Fig. 7. Genetic learning of the data-base for each module.

Table 3Genetic parameter settings.

Genetic parameter value

Number of generations, G 150

Population size, V 100

Crossover probability Rh,Re 0.90, 0.60

Mutation Probability 0.01

Bit number for each variable 32


The following two closed-loop performance indices have beenexamined. The first uses the stopping distance and the integratedabsolute error (IAE). The second uses the stopping distance andthe integrated time multiplied by the absolute error (ITAE). Thesetwo performance indices are defined as follows:

JIAE ¼ Z1SxþZ2IAE¼ Z1SxþZ2

Z T

0eðtÞ�� dt

¼ Z1SxþZ2

XMk ¼ 0

9ld�lðkÞ9Dt ð17Þ

JITAE ¼ Z1SxþZ2ITAE¼ Z1SxþZ2

Z T

0t9eðtÞ9dt

¼ Z1SxþZ2

XMk ¼ 0

k9ld�lðkÞ9Dt2 ð18Þ

where e¼ld�l is the closed-loop error, ld is the desired slip ratio,l(k) is the current slip ratio and Dt is the time step. M is thenumber of training samples. In addition, the coefficients Z1 and Z2

are the weighting factors to emphasize the relative importance ofthe associated terms.

Because GA endeavors to maximize the fitness function, thefitness function of each gene is calculated as follows:

F ¼1

1þ Jð19Þ

where J is the performance index and 1 is introduced at thedenominator to prevent the fitness function from becominginfinitely large. The overall training procedure is shown in Fig. 8.

To simplify the presentation, let us denote F in Eq. (19) as FIAE

when J¼ JIAE and FITAE when J¼ JITAE. With this notation, thecontroller is called PID-IAE when the performance index iscalculated by Eq. (17) and PID-ITAE when the performance indexis given by Eq. (18).

Coding of the parameters to be adjusted can be stated asfollows:

cN1,cN2, � � � ,sN1,sN2, � � � , a111,a121, � � � , a339 ð20Þ

where a certain number of binary bits stands for each parameter.The combined string composes a gene (possible solution) in a

population. Evaluation of each possible solution is performed viaIAE/ITAE and genes of best solutions are allowed to reproduce.

Although, genetic algorithms were developed a few decadesago, concrete theoretical analysis of the algorithm have not beenprovided until recent years (Golberg, 1989; Rudolph, 1994).Reference (Rudolph, 1994) concludes that the canonical GAcannot always find the optimal solution within the definite time.Furthermore, the paper pointed out that if the chromosome withthe best performance in each generation is reserved for the nextgeneration, the algorithm will globally converge. Inspired by theseconclusions, the following two measures have been considered inthis work:

(i)
In reproduction, we stochastically introduce a randomlygenerated gene at a probability of Rh to replace one of thetwo parents selected for reproduction.
(ii)
Select the best performed genes in the current population at arate of Re and place them directly in the next generation.
If the reproduction is carried out in the traditional way, thebest gene will globally be lost, and thus convergence cannotbe guaranteed. However, if we only adopt the second measure,the procedure of the evolution is no longer a Markov process, andthus it does not satisfy the assumption of the convergence theory.Despite our successful application of these measures, mathema-tical analysis of them still lacks. The parameters Rh and Re areadjusted such that at the beginning of the learning, Rh is relativelylarge and Re is relatively small, and later, vice versa.

Fig. 9. Membership functions of the antecedent part before tuning.

Fig. 10. Membership functions of the antecedent part after tuning using FITAE.

Fig. 11. Membership functions of the antecedent part after tuning using FIAE.

Fig. 12. The output surfaces of the fuzzy gain-tuners.


Table 4Performance of PID tuning methods.

Controller type Performance measure

Sx, m IAE ITAE Maximum O.S%

Z–N 51.61 0.1171 0.0909 73.24

G-PID 48.77 0.0435 0.0210 13.52

SSP 47.28 0.0526 0.0309 31.61

FSW 48.06 0.0998 0.0494 0.05

PID-fuzzy 45.97 0.0482 0.0297 13.16

PID-ITAE 41.82 0.0442 0.0240 15.35

PID-IAE 39.92 0.0433 0.0291 12.84


5. Numerical tests

5.1. Simulation data

Several numerical tests have been performed using theexample data presented in Chih-Min Lin and Hsu March (2003).They are: R¼0.33 m, m¼342 kg, Jw¼1.13 kg m2, g¼9.81 m/s2 andthe desired slip ratio ld¼0.2. Due to the fact that when the wheeland vehicle velocity are nearly zero at the end of braking time, themagnitude of slip tends to infinity. Therefore, simulations areconducted up to the point when the vehicle is slowed to 0.5 m/s.

The following case study deals with braking on dry asphalt, thenafter 1 s, the road changed to be a snowy one. The initial speedVx¼27.78 m/s, i.e. 100 km/h. These conditions have been considered,in order to examine robustness against road conditions.

To verify the full potentialities of the investigated PIDcontrollers, during genetic training, it is assumed that no saturationlevels are imposed to the control signal. After training, a saturationlevel us¼74000 N m has been imposed to test the controllers inthe presence of typical process nonlinearity. Genetic training hasbeen performed using the parameter settings listed in Table 3.

The tuning procedure has been initialized using the membershipfunctions shown in Fig. 9. After training, the obtained membershipfunctions using FITAE and FIAE are depicted in Fig. 10 and 11,respectively. The corresponding controllers are denoted PID-Fuzzy(before training), PID-IAE and PID-ITAE, respectively. The outputsurfaces of the gain self-tuners, before and after learning, are depictedin Fig. 12. It should be noted that the shown universe of discourse of e

(from E�2.5 to E2.5) is not fully used by the controllers. The aim ofdisplaying the membership functions within this range is to fullydemonstrate their diversities. This also applies to the shown universeof discourse of _e.

Referring to Fig. 10 and 11, it is shown that the negative (N)membership functions of both e and _e are widely spread throughthe whole universe of discourses. It means that they have thelargest impact in the computed control signal. This may explainwhy the PID-IAE and PID-IAE controllers exhibit faster conver-gence relative to PID-fuzzy controller, as will be shown in Section5.3. To some extent, this can be attributed to that the closed-loopcycle (braking) starts with a positive error e (e¼ld�l) equal to0.2 leading to a negative overshoot at the transient period.This negative overshoot (e) invokes large control signal whichcounter-attacks this error, resulting in a faster convergence of theerror relative to the PID-fuzzy controller.

For the other investigated methodologies (SSP and FSWdescribed in Section 3), the control parameters have been

Fig. 13. Slip ratio under d

determined using the GA with FIAE as the objective function. Forthe sake of brevity, only the membership functions of theproposed self-tuning scheme are reported here. Furthermore,the genetic algorithm with the parameters listed in Table 3 hasbeen used to retune the PID parameters. It is intended for theresulted PID to use the best possible fixed parameters. This PIDcontroller is denoted by G-PID.

5.2. Computational considerations

The initial fuzzy modules were determined after small number oftrials. It is our point of view that the initially guessed modules shouldwork properly for successful learning process. Dealing with geneticlearning, the total number of iterations H should be chosen so as thesystem has enough chance to converge. A suitable choice ensurescorrect training and saves computation time since each gene is apossible solution which has to be evaluated according to Eq. (19). Inliterature, two approaches are generally used for selecting thesuitable number of population and generation for optimizationproblems similar to the problem considered in this work. The firstrelies on a relatively small number of populations (e.g. 20-30) andhigh number of generations (e.g. 700-3000) (Hwang, 1999;Rahmoun and Benmohamed, 1998; Skarmeta and Jimenez, 1999).The second uses high number of populations (e.g. 100-200) and lownumber of generations (Attia and Horacek, 2001). We have chosenthe former to assign the suitable genetic parameters. Adaptive changeof the crossover probability helps more in speeding up theconvergence. The coming results have been obtained using geneticparameters listed in Table 3.

With a population of V¼100 individuals for G¼150 generations,the fitness function in Eq. (19) is evaluated 15,000 times. Indeed,

ifferent control laws.

Fig. 14. The input torques.

Fig. 15. Performance of ABS under three PID strategies.

Fig. 16. Time-history of the PID-IAE gains (when input saturation is imposed).

Fig. 17. Slip ratio under different control schemes when saturation on the input is

not imposed.


this number (H¼V�G) represents the number of evaluated pointsinside the search space, which may be used as a reference forsimilar optimization problems. The wining gene (optimal solution)is the best of stochastically competitive 15,000 genes. Referring toEq. (20), with 32 bits for each variable, each possible solution(gene) has the length of 2976 bits. Performing the learning processusing Matlab-7, M-file under Windows XP on a PC Pentium IV,2800 Hz speed, requires about one and half hour. Due to the

stochastic nature of GAs, at least two or three trials should beperformed, in order to be sure that convergence has taken place.

5.3. Results and discussion

Fig. 13 shows the ABS response under seven investigatedcontrol schemes. As it can be noticed, PID-IAE gives the bestresponse compared with other schemes. To provide a moredetailed insight of the results, Table 4 gives the values of theperformance measures Sx, IAE, ITAE, and maximum attainedovershoot for the different PID controllers. G-PID and PID-fuzzyshow similar performances, while PID-fuzzy exhibits shorterstopping distance. With respect to constant gain schemes, G-PIDshows better results than Z–N, which exhibits the highestovershoot, longest settling time and stopping distance; i.e. theworst performance. Nevertheless, the shortest stopping distancehas been taken place by the PID-IAE controller.

The barking torques have significantly changed theirmagnitude after one second to meet the new road condition(snowy road); Figs. 14(a) and (b). Low braking torque is requiredfor the vehicle to move on icy road, in order to avoid locking.Fig. 15 shows different stopping times have been achieved by the

Fig. 18. Time history of the PID-IAE gains when saturation is not imposed.

Table 5Performance of PID tuning methods when saturation is not imposed.

Controller type Performance measure

Sx, m IAE ITAE Maximum O.S.%

Z–N 50.16 0.1142 0.0918 69.60

G-PID 46.62 0.0282 0.0188 18.59

SSP 45.55 0.0477 0.0318 31.34

FSW 46.14 0.0837 0.0588 no O. S.

PID-fuzzy 44.41 0.0661 0.0336 122.82

PID-ITAE 41.85 0.1246 0.0471 265.60

PID-IAE 38.83 0.0784 0.0303 193.50

Fig. 19. Percent overshoot with and without input saturation.


investigated controllers. Larger overshoot can be noticed by Z–Nand an SSP with respect to PID-IAE.

Time-history of the PID-IAE modules is depicted in Fig. 16.Unlike constant gain schemes (e.g. Z–N), their values arecontinuously changing during the braking period, which isimposed by the fuzzy modules, in order to counter attack theerror and change in error. After one second, smooth transition ontheir values has taken place, in order to meet the new roaddemands. This behavior resulted in a relatively smoother torque,faster convergence of the slip ratio to the desired value andminimum stopping distance.

Because the input saturation level is significantly high, it maybe interesting for the study to investigate the response when nosaturation is imposed. Fig. 17 shows the response of 4 PIDschemes. It comes out that fuzzy logic is more useful if saturationsare significant in the process. This remark strengthens the factthat the performance of the classic PID can be improved usingtime-varying parameters (Astrom 1996). Fig. 18 depicts thePID-IAE gains when saturation is not imposed. Higher variationscan be noticed at the beginning of the braking. Table 5 gives theperformance measures of the investigated seven controllers.PID-IAE exhibits the shortest stopping distance although itexhibits much higher overshoot than the case when saturationis imposed. As a performance measure, the percent overshootwhen input saturation is and is not imposed is depicted in Fig. 19.It is clear that saturation has little or no effect on the fixed gainschemes, i.e. Z–N and G-PID. On the other hand, PID-IAE andPID-Fuzzy have been greatly influenced.

Finally, one should consider the case of manual tuning of thecontrollers i.e. the fuzzy module’s parameters, since in practicalindustrial applications; the use of genetically trained controllers ispossible only if a good process model (or experimental data) isavailable. It emerges that for the fuzzy-logic-based tuningmethods for which a technique for the selection of the parametersof the fuzzy logic has not been considered, it might be difficult toperform this task effectively. For example, for the IFE method(Sub-Section 3.4), the choice of parameter k3 is critical since itsvalue has to be kept very low, otherwise the overall controlsystem is destabilized. Also, the SSP control structure is verydifficult to set. Otherwise, it is not always easy to improve the Z–Nresponse and, when it succeeds the selected rules is very difficult

to interpret because other tuning parameters interfere (g and a).Manual tuning of the proposed scheme is straightforward since itdepends on three decoupled modules, which is advantageous overmost of the investigated schemes.

6. Conclusions

In this article, a fuzzy self-tuning scheme has been proposedfor PID controllers. The proposed scheme utilizes three decoupledmodules, each for one of the PID parameters. Each module is two-input one-output T–S type fuzzy system. Optimal selection of thefuzzy modules has been obtained using a modified geneticalgorithm. The performance has been verified using the auto-mobiles’ ABS. In the presented case study, robustness against roadfriction characteristics has been considered. The control goal is tokeep the slip ratio to an assigned value (0.2) despite road frictioncharacteristics, while keeping the shortest possible stoppingdistance. Comparison with previous works shows the competi-tiveness of the proposed scheme.

The salient features can be summarized as follows:

1.
The proposed control scheme presents a generalized proce-dure, which can be followed for linear and nonlinear systems.
2.
Improved control action has been obtained by replacing theZ–N tuned PID controllers with fuzzy self-tuning systems. This isthe case in SSP and FSW proposed by earlier investigators, andfurther improvement can be obtained by using the proposedcontrol scheme.
3.
The proposed PID-IAE controller resulted in relatively fastresponse with low overshoot and short stopping distance. As aconsequence, there is a remarkable improvement with regardto the examined performance measures.
4.
The proposed scheme has the ability to switch on/off any of thecontrol actions, due to its basic decoupled nature. Thisprocedure is needed for practical implementation becauseof the drift which usually exists between theoreticalestablishment and actual experimentation.
5.
A probable area of future work is to achieve adaptivity of each/some of the three control actions, while marinating its simplicity.


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