implementation of active steering fuzzy pid controller on ...1450298/fulltext01.pdf · degree...

IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2020

Implementation of Active Steering Fuzzy PID Controller on a Two-axle Railway Vehicle

YULEI QIU

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Implementation of ActiveSteering Fuzzy PID Controlleron a Two-axle Railway Vehicle

YULEI QIU

Exchange Student in Vehicle EngineeringEmail: [email protected]: June 22, 2020Supervisor: Rocco GiossiExaminer: Carlos Casanueva PerezSchool of Engineering Science

Abstract

This thesis investigates the improvement of curving performance of a two-axlerailway vehicle using fuzzy PID controller. The control goal during curving isto maintain zero difference between front and rear yaw angle. A mathematicallinear model is built up based on a two-axle railway vehicle. A nonlinear modelis also created in SIMPACK. The controller is firstly designed and tested on thelinear model, subsequently on the nonlinear model after it works on the linearone. Finally, a comparative study of the performance of the fuzzy controlleron both linear and nonlinear model is carried out. The results show that thefuzzy PID control can improve the curving performance of the model.

ii

Contents

1 Introduction 1

2 Methodology 32.1 Research Process . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Running Cases . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Steering Control Strategy . . . . . . . . . . . . . . . . . . . . 5

3 Fuzzy Control 73.1 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Linguistic Variable . . . . . . . . . . . . . . . . . . . 93.1.2 Fuzzy Conditional Statements . . . . . . . . . . . . . 93.1.3 Fuzzy Algorithms . . . . . . . . . . . . . . . . . . . 93.1.4 Fuzzy Operators . . . . . . . . . . . . . . . . . . . . 10

3.2 Structure of a Fuzzy Controller . . . . . . . . . . . . . . . . . 103.2.1 Fuzzy Rules Base . . . . . . . . . . . . . . . . . . . . 113.2.2 Fuzzification and Defuzzification . . . . . . . . . . . 113.2.3 Mamdani Fuzzy Inference Systems . . . . . . . . . . 11

4 Linear & Nonlinear Model 144.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Equations of motion . . . . . . . . . . . . . . . . . . 154.1.2 The State-Space Model . . . . . . . . . . . . . . . . . 194.1.3 Implementation of the State-Space Model . . . . . . . 21

4.2 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Controller Design 255.1 PID Controllers . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Fuzzy Rules Base . . . . . . . . . . . . . . . . . . . . . . . . 27

iii

CONTENTS

6 Result & Discussions 306.1 Results: Linear Model . . . . . . . . . . . . . . . . . . . . . 306.2 Results: Nonlinear Model . . . . . . . . . . . . . . . . . . . . 306.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Conclusions 34

Bibliography 36

iv

Chapter 1

Introduction

With the development of science and technology, increasing attention is givento the quality of transportation. Among public transportation, railway vehiclesare of great interest since they can cover both urban (metro) and long-distancetransportation with low energy consumption. Concerning a metro system, atwo-axle vehicle can offer an alternative to standard bogie vehicles in termsof investment and maintenance cost. Nevertheless, for this kind of systemsthere are many technical difficulties in the improvement of the transportationquality, one of which is the curving performance.Conventional railway wheelset consists of two profiled wheels rigidly mountedon a common axle. Each wheel on the same axle therefore has the same angu-lar velocity and distance to the center of carbody[1]. This layout of wheelsethas some advantages such as natural curving capability and a self-centringmechanism. Between rail and wheel creep forces that depends on speed aregenerated. The configuration of the wheelset coupled with the presence ofthe creep forces can generate unstable oscillations increasing the speed. Thisphenomenon is called hunting. Key factors in this process are the suspensionelements of the railway vehicle such as springs and dampers mounted in thelongitudinal and lateral direction. They can make the vehicle stable when os-cillation occurs[2].For a two-axle vehicle with single axle running gear, high longitudinal stiffnessis required to have a satisfactory critical speed and avoid hunting. However,this can lead to a reduced curving capability by preventing the wheelset totake its natural curving position. Active steering can improve the curving per-formances of the two-axle vehicle without changing the required longitudinalstiffness.Different configurations were studied throughout history. Geuenich et. al pro-

1

CHAPTER 1. INTRODUCTION

posed a passive suspension system in which magnet powder coupling betweentwo wheels were used to improve the curving performance[3]. Here, the mag-netic field is used to adjust the coupling between left and right wheel allow-ing transfer torque from one shaft to another. Semi-active damper using aSky-Hook (SH) strategy were applied to improve both curving performanceand dynamic stability[4]. Here, the sprung mass acceleration and the relativedisplacement between the mass and road input will be measured. These twomeasurement signals are then used to determined the desired damping level.However, these passive and semi-active control approaches can only partiallyimprove wheel behavior[2]. Pérez, Busturia and Goodall in 2002 showed thebenefits of the actively steered vehicles over passively steered ones[5].Feedback control is generally applied to achieve active steering[6][7]. Amongmany controllers, the proportional–integral–derivative (PID) controller is wide-ly used in many industrial applications because it has a simple structure and iseasy to design. To improve the curving performance of a railway vehicle, thefeedback control system must be robust against a variety of working conditionsto represent a reliable solution. There are various methods to achieve robust-ness, among which the fuzzy control is a possible control strategy. Fuzzycontrol is not been widely applied to steering control in railway applicationsyet. Nevertheless, Min-Soo Kim has proved that fuzzy controller can robustlyyields good performance comparing with conventional controller[8].

The aim of this thesis it to build a steering control system for a two-axlerailway vehicle that can adapt to different running conditions. This is done bydesigning a fuzzy PID controller starting from a linear model and applying itto non-linear one. The layout of the thesis is as follows. Chapter 1 summariesthe literature surrounding the field of study. Chapter 2 introduces the appliedmethodology. Chapter 3 overviews the theory of fuzzy sets and their appli-cation to control theory. Chapter 4 describes the linear and nonlinear modelused. The design of the controller is discussed in Chapter 5. Chapter 6 showsthe results of the simulation on both the linear and the nonlinear model andprovides some discussions. Finally, a brief conclusion is given in Chapter 7.

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Chapter 2

Methodology

In this chapter, the research process of the thesis is introduced. It follows adeductive procedure by dividing a complex problem (a railway vehicle) into asequence of simpler problems. Subsequently, the running cases are introduced.There are five different running speeds ranged from 10 to 96 km/h, represent-ing low, medium and high speed running conditions. At the end of this chapter,the control strategy is introduced. The control strategy used in thesis is from apaper in 2002, which suggests controlling the difference between yaw anglesof front and rear wheelset.

2.1 Research ProcessThe research process follows the flow chart shown in Figure 2.1. A two-axlerailway vehicle unit is considered in this study. Firstly, the equations of motionfor an equivalent two-dimension linear model are derived with the Lagrangeapproach considering small displacement assumption (Chapter 4), The state-space linear model is then implemented in Simulink. At the same time, anonlinear model is built in SIMAPCK. Subsequently, due to a relatively sim-ple tuning process the PID controller is designed on the linear model underdifferent speeds condition. Different heuristic tuning methods for a PID con-troller exist. Among them, the Ziegler–Nichols method[9] is chosen. Oncethe control parameters are identified for each velocity, the fuzzy controller isdesigned to embrace the control parameters variation. The fuzzy controller isthen tested on the linear model. Once the designed fuzzy PID controller hasproven to work on the linear model for all the considered running cases, it willbe applied on the nonlinear model.

3

CHAPTER 2. METHODOLOGY

Figure 2.1: Flow chart of the research process

2.2 Running CasesTo address the feasibility of the fuzzy controller to produce stable results, therunning cases shown in Table 2.1 are introduced. The track layout has fixedturning radius (600 m) and cant (80 mm) while the vehicle speed varies from10 to 96 km/h, representing different running conditions. The track layout isshown in Figure 2.2 and it follows a sequence of tangent track (200 m), entertransition curve (100 m), circular curve (500 m), exit transition curve (100 m)and tangent track (200 m). In tangent track, the vehicle runs on a straight track(infinite turning radius). Then the vehicle enters transition curve that connectstangent track and circular curve. The turning radius increases linearly withrespect to distance in this stage, after which the vehicle goes into circular curve,keeping a constant turning radius. The track ends with another transition curveand tangent track. The turning radius in this transition curve decreases linearlywith respect to distance. Five running conditions are created: two cant excessconditions (10 and 37 km/h), one equilibrium speed (64 km/h) and two cantdeficiency (80 and 96 km/h). The control parameters of the PID controller willbe tuned under these five running conditions. The defined track and vehiclespeeds are kept the same in both linear and nonlinear model.

Table 2.1: Running cases

Radius (m) Cant (mm) Velocity (km/h)

600 80 10 37 64 80 96

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2.3 Steering Control StrategyIn railway application, a steering control strategy defines the target that thecontrol system should reach during the circular part of the curve to improvethe curving performance of a vehicle. A variety of strategies may be applied.Among them, the chosen control strategy is to control the difference betweenfront and rear attack angle of the wheelsets. In fact, Pérez et. al pointed outthat, due to the dependency of the lateral creep forces from the wheelset yawangle, equal yaw angles for front and rear wheelsets can balance the lateralcreep forces[5]. Therefore, the control objective is to maintain zero differencebetween front and rear attack angles in every working condition

ψw1 − ψw2 = 0 (2.1)

This strategy has the advantage of not relaying on track geometry conditionsor vehicle speed neither it defines a predetermined attack angle. This strategyis ideally free from unmodelled wheel and rail interactions. A scheme of thesteering control strategy is shown in Figure 2.3.

Figure 2.2: Track Layout

5


Figure 2.3: An illustration of control strategy

6

Chapter 3

Fuzzy Control

In recent years, fuzzy controllers have been applied to many control systemsincluding railway vehicle. Fuzzy control is based on fuzzy logic proposed byLotfi A. Zadeh in his 1965 paper[10]. Fuzzy logic allows imprecise and quali-tative information to be expressed in a quantitative manner. Unlike classical ordigital logic which operates on discrete values of either 0 or 1, fuzzy logic canhandle continuous logic values between 0 and 1, i.e. it can describe conceptsthat cannot be expressed as ”true” or ”false” but ”partially true” or ”partiallyfalse”[11]. Subsequently, in 1973, Zadeh developed the concept of fuzzy setwith ”three main distinguishing features”: linguistic variables, fuzzy condi-tional statements and fuzzy algorithms[12]. The following sections introducesome basic concepts of fuzzy sets and fuzzy control.

3.1 Fuzzy SetsA fuzzy set is a set whose elements have a corresponding membership valuebetween 0 to 1, mapped by a proper membership function. The concept canbe understood with the help of an example. Let X be a set of the real num-bers between 0 to 100, i.e. X = [0, 100], x an element of X and let A be afuzzy set of numbers that are much greater than 10. The membership func-tions of A can be defined as: fA(0) = 0; fA(1) = 0; fA(10) = 0; fA(11) =

0.01; fA(50) = 0.55; fA(100) = 1. Note that the nearer the value of fA(x)to 1, the higher the grade of membership of x in A, i.e. the more x matchesthe description of the fuzzy set. By the membership functions, the linguisticdescription, ”numbers that are greater than 10”, is converted to some member-ship values between 0 and 1, thus, the membership value represents to whatextent the number is greater than 10. By means of membership functions, the

7

CHAPTER 3. FUZZY CONTROL

Figure 3.1: A fuzzy set of temperature

qualitative description of a variable is converted a quantitative value whichcan be treated like other quantized signals. Membership functions can be arbi-trarily shaped. Most commonly triangular, trapezoidal and Gaussian are used.

Figure 3.1 is an example of a fuzzy set for temperature distinction. Here,trapezoidal membership functions are used. There are three descriptions oftemperature, i.e. three membership functions: hot, warm and cold. Each inputvalue (temperature) corresponds to one membership value for each member-ship function. Therefore, one input temperature has three membership valuesin this example. The corresponding membership values in Figure 3.1 are listedin Table 3.1.As shown in Table 3.1, when temperature is 4 ◦C, the corresponding mem-

bership values are: 0 for hot, 0 for warm and 0.6 for cold. In this case, atemperature of 4 ◦C does not match the description of “hot” or “warm” but it20% matches the description of “cold”. Similarly, a temperature of 15 ◦C cor-responds to membership values of: 0 for cold and hot and 1 for warm, which

Table 3.1: Membership Values

Temperature(◦C) Value

4Hot: 0

Warm: 0Cold: 0.2

15Hot: 0

Warm: 1Cold: 0

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means at this temperature it is completely warm, neither hot nor cold.

In order to operate fuzzy sets, three main distinguishing features must beintroduced: linguistic variable, fuzzy conditional statement, and fuzzy algo-rithm and operators.

3.1.1 Linguistic VariableA linguistic variable is defined as a variable whose values are sentences orwords in a natural language. A linguistic variable, temperature, can have typ-ical values such as “hot”, “warm” and “cold”. Other examples of linguisticvariable are listed in Table 3.2. There is no restriction to which extension alinguistic variable can be used.

Table 3.2: Some examples of linguistic variable

Linguistic Variable Typical ValuesTemperature Hot, Warm, Cold

Height Short, Medium, TallSpeed Slow, Creeping, Fast

3.1.2 Fuzzy Conditional StatementsFuzzy conditional statements express the logical relationship among fuzzyquantities, and they are in the form of IF A THEN B, where A and B havefuzzy meanings. For example, IF indoor temperature is low THEN heatertemperature is high. Fuzzy conditional statements are the basis of input(s)output(s) relationship.

3.1.3 Fuzzy AlgorithmsTo operate fuzzy sets, fuzzy algorithm are introduced. They are composed ofan ordered sequence of instructions which may contain fuzzy assignment andconditional statements.Example:

indoor temperature = hot,IF indoor temperature is hot THEN heater temperature is high

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3.1.4 Fuzzy OperatorsIn fuzzy algorithms, fuzzy operators may be used to express relationship be-tween variables. Logic operators like AND (union, ∪), OR (intersection, ∩)and NOT (complement) also appear in fuzzy sets. They are obvious exten-sions of the corresponding definitions for ordinary sets, although sometimesother types of definition may be applied. Assume we have two fuzzy sets, Aand B. Let C be the union of A and B, i.e. C = A OR B (C = A ∪ B). Themembership function of C can be expressed by those of A and B such that

fC(x) =Max [fA(x), fB(x)] (3.1)

Similarly, the intersection (C = A ∩ B) can be expresses as

fC(x) =Min [fA(x), fB(x)] (3.2)

The NOT operator (complement) of a fuzzy set A is denoted by A′ and isdefined by

fA′ = 1− fA (3.3)

The union and intersection of two fuzzy sets,A andB, are illustrated in Figure3.2. The membership function of the intersection is curve segments 3 and 4,while that of the union is curve segments 1 and 2.

Figure 3.2: Illustration of union and intersection of fuzzy sets[10]

3.2 Structure of a Fuzzy ControllerA fuzzy controller uses the concepts of linguistic variables, fuzzy conditionalstatements, algorithms, and operators to build an input(s) output(s) relation

10


Figure 3.3: Structure of a fuzzy controller

that can suits active systems requirements. A fuzzy controller generally con-sists of four main parts: Fuzzification, Fuzzy Inference System, Fuzzy RulesBase and Defuzzification. The input of a fuzzy controller is converted intoan input membership value by the Fuzzification part. It is then turned into anoutput membership value after being processed in the Fuzzy Inference Systemwhere the fuzzy rules are applied according to the Fuzzy Rules Base. Thefuzzy output is defuzzified in the Defuzzification part and becomes a crispoutput value. Unlike binary values that are either 0 or 1, a crisp value can beany real number between 0 and 1. A graphic illustration of fuzzy controller isgiven in Figure 3.3.

3.2.1 Fuzzy Rules BaseAs mentioned before, fuzzy conditional statement can be one fuzzy rule. Fuzzyrules base consists of a series of modus ponens expressions like the example”IF A THEN B”. Another example of fuzzy rules includes fuzzy operators,like ”IF A or B THEN C and D”. The design of fuzzy rules base will beintroduced in Chapter 5.

3.2.2 Fuzzification and DefuzzificationFuzzification and defuzzification are key instruments that allow the interpreta-tion of physical (measured) signals in terms of fuzzy quantities. Fuzzificationis the process of converting an input value to a fuzzy value by the member-ship function. Defuzzification is a reverse process of fuzzification, convertinga membership value to a crisp value. The centroid method is a very popularmethod of defuzzification[13]. It will be explained in the next subsection.

3.2.3 Mamdani Fuzzy Inference SystemsAmong different fuzzy inference systems, the Mamdani one is applied in thecontroller design of this thesis because it has advantages of being intuitive andeasy to design. The Mamdani fuzzy inference system was firstly proposed by

11


E. H. Mamdani and S. Assilian[14]. It is a method to build a control systemusing a series of linguistic rules. Once more, the functioning of the Mamdaniinference system is explained with an exampleFigure 3.4 shows the working process of Mamdani Fuzzy Inference System forthe tips determination based on the quality of the food in a restaurant. Thereare two input variables, food and service. Input service has three membershipfunctions: poor, good, excellent while food has two: rancid and delicious. Out-put variable tip has three: cheap, average and generous. There are three fuzzyrules in total. For each rule, the fuzzy values of service and food are fuzzi-fied by their membership functions and connected by a fuzzy operator such asOR. It can be useful to recall that the operator OR means the maximum oneof the two functions. Therefore, the maximum value between these two inputmembership values is accepted as the output membership value. In the outputmembership function, the area under the output membership value is takenfor the defuzzification. After the determination of all the areas from the fuzzyrules, they are added together, becoming a new combined area. Finally, thecentroid of this area is calculated, which is the output of the fuzzy controller,the crisp value.

In this chapter, some basic concepts of fuzzy sets and fuzzy control areintroduced. Besides, the structure of fuzzy controllers are included. Fuzzysets allow linguistic and qualitative description of variables compared withclassical sets. It has three main distinguishing features: linguistic variable,fuzzy conditional statements and fuzzy algorithms. Linguistic variable can bedefined by natural languages. Fuzzy control statements represent the logical re-lationship of fuzzy variables by ”IF...THEN...” expressions. Fuzzy algorithms,which are in generally a series of modus ponens, use fuzzy operators and fuzzyconditional statements to operate fuzzy variables. Fuzzy controllers consist offuzzification block, fuzzy rules base, fuzzy inference system and defuzzifica-tion block. There are two types of fuzzy inference system, where the MamdaniFuzzy Inference System is applied in this thesis for its ease of design.

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Figure 3.4: The working process of Mamdani Fuzzy Inference System[15]

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Chapter 4

Linear & Nonlinear Model

In this chapter, the process of building up a linear and nonlinear model is intro-duced. The linear model is a mathematical model derived from the equationsof motion by studying a two-axle railway vehicle unit with the Lagrange ap-proach considering small displacement assumption. The nonlinear model iscreated in SIMPACK based on this vehicle unit.

4.1 Linear ModelA scheme of a two-axle railway vehicle unit is shown in Figure 4.1, where yC ,ψC , yw1, ψw1, yw2 and ψw2 are the lateral displacement of carbody, the yawangle of carbody, the lateral displacement of front wheelset, the yaw angleof front wheelset, the lateral displacement of rear wheelset respectively theyaw angle of rear wheelset. The vehicle unit has both longitudinal and lateralsuspensions on each wheelset. The longitudinal suspensions are not shownin Figure 4.1, where a is the semi-axle distance, be is the distance from thecenter of the carbody to the actuator and b0 is the semi-wheels distance. Fouractuators are introduced, two per each wheelset. FL

F , FRF , FL

R and FRR are the

actuation force on the front left, front right, rear left respectively rear rightwheel. Besides, reaction forces due to suspension system in both longitudinaland lateral directions and the actuation forces, other external forces also act onthe front and rear wheelset. In Figure 4.2 a graphical representation is given,where the FL

ξ is the longitudinal creep force on left wheel, FLη is the lateral

creep force on left wheel, FRξ is the longitudinal creep force on right wheel,

FRη is the lateral creep force on right wheel, FQ is the force caused by self-

centering mechanism and FC is the centrifugal force. The centrifugal force isthe only force that has an effect also on the carbody while the remaining five

14

CHAPTER 4. LINEAR & NONLINEAR MODEL

forces are proper of each wheelset.The centrifugal force can be expressed as

FC = mw · ay = mwg

(h

2b0− v2

R

),

where ay is the resultant acceleration in lateral direction, h is the cant, v is thevelocity andR is the turning radius. The self-centering force can be expressedas

FQ = 2Q0κyw,

whereQ0 =

g(mC +mw1 +mw2)

4

The creep forces are derived with Kalker’s linear theory and small displace-ment assumption. Here, the longitudinal creep force can be expressed as

FLξ = −f11νξ = −f11

(λeqr0yw +

b0R

+b0vψw

)(for left wheel),

FRξ = −f11νξ = −f11

(−λeqr0yw − b0

R− b0vψw

)(for right wheel),

and the lateral creep force can be expressed as

Fη = −f22νη−f23Φ = −f22(−ψw +

ywv

− r0vϕt

)−f23

(− κ

r0yw +

1

vψw

),

where νξ is the longitudinal creepage, νξ is the lateral creepage, Φ is the spin,λeq is the equivalent conicity of the wheel, r0 is the radius of the wheel, yw isthe lateral displacement of the wheelset, ψw is the yaw angle of the wheelset,ϕt is the cant angle.

4.1.1 Equations of motionAccording to the Lagrangian approach, the equations of motion can be derivedfrom the following formula:

d

dt

(∂

∂qEk

)+

∂

∂qEk +

∂

∂qD +

∂

∂qV =

δW

δq, (4.1)

whereEk is the kinematic energy,D is the energy dissipated by viscous dampers,V is the elastic potential energy of springs, W is the work done by external

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Figure 4.1: Scheme of the railway vehicle unit

Figure 4.2: External forces acting on the wheelset

forces and q is one of the six variable, yC , ψC , yw1, ψw1, yw2 and ψw2.

According to Equation 4.1, one equation of motion can be derived for oneof the following variables: yC , ψC , yw1, ψw1, yw2 and ψw2. Hence, there aresix equations of motion in total.The left-hand side of these equations of motion can be derived as:

mcyc + 4cyyc − 2cyyw1 − 2cyyw2 + 4kyyc − 2kyyw1 − 2kyyw2 (4.2)

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Jcψc + (4a2cy + 4b2ecx)ψc − 2b2ecxψw1 − 2b2ecxψw2 − 2acyyw1 − 2acyyw2

+(4a2ky + 4b2ekx)ψc − 2b2ekxψw1 − 2b2ekxψw2 − 2akyyw1 − 2akyyw2

(4.3)

mw1yw1 + 2cyyw1 + 2kyyw1 − 2cyyc − 2kyyc − 2acyψc − 2akyψc (4.4)

Jw1ψw1 + 2b2ecxψw1 + 2b2ekxψw1 − 2b2ecxψc − 2b2ekxψc (4.5)

mw2yw2 + 2cyyw2 + 2kyyw2 − 2cyyc − 2kyyc + 2acyψc + 2akyψc (4.6)

Jw2ψw2 + 2b2ecxψw2 + 2b2ekxψw2 − 2b2ecxψc − 2b2ekxψc (4.7)

where:

subscript c: carbody related variablesubscript w1: front wheelset related variablesubscript w2: rear wheelset related variable

subscript x: longitudinal direction related variablesubscript y:lateral direction related variable

m: massJ : moment of inertiay: displacementψ: yaw angle

k: the stiffness of the springc: the damping coefficient of the damper

a: semi-axle distancebe: distance from center to the actuator

b0: semi-wheels distance

The right-hand side of these equations of motion can be written as:

mc(gh

2b0− v2

R) (4.8)

−beFLF + beF

RF − beF

RR + beFL

R (4.9)

17


[2f23r0

− gκ(mc +mw1 +mw2)

2

]yw1 +

−2f22v

yw1 + 2f22ψw1−2f23v

ψw1

+2f22r0v

ψt +mw1

(gh

2b0− v2

R

)(4.10)

−2b0f11λeqr0

yw1 +−2b20f11

vψw1 +

−b20f11R

+ beFLF − beF

RF (4.11)

[2f23r0

− gκ(mc +mw1 +mw2)

2

]yw2 +

−2f22v

yw2 + 2f22ψw2−2f23v

ψw2

+2f22r0v

ψt +mw2

(gh

2b0− v2

R

)(4.12)

−2b0f11λeqr0

yw2 +−2b20f11

vψw2 +

−b20f11R

+ beFRR − beFL

R (4.13)

where:

h: cantv: vehicle speedR: turning radiusF : actuation force

f11, f22, f34: creep coefficientsκ: gravitational stiffness

λeq: equivalent conicity of the wheelsubscript F or R: front respectively rear wheelset related variablesuperscript L or R: left respectively right wheel related variable

Therefore, the equation set consisting a series of equations of motion can bedefined as follows (Eqn. for Equation):

Eqn.4.2 = Eqn.4.8Eqn.4.3 = Eqn.4.9Eqn.4.4 = Eqn.4.10Eqn.4.5 = Eqn.4.11Eqn.4.6 = Eqn.4.12Eqn.4.7 = Eqn.4.13

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4.1.2 The State-Space ModelOnce the equations of motion are derived, the state-space model can be de-fined.The equation set above can be seen as an augmented matrix equation (Equa-tion. 4.14), where M is the mass matrix, C the damping matrix, K is thestiffness matrix and F is the external force vector.

MZ + CZ +KZ = F (4.14)

In this equation, Z is defined as:

Z = [yc ψc yw1 ψw1 yw2 ψw2]T

Correspondingly, other matrices are defined as:

M =

mc 0 0 0 0 0

0 Jc 0 0 0 0

0 0 mw1 0 0 0

0 0 0 Jw1 0 0

0 0 0 0 mw2 0

0 0 0 0 0 Jw2

C =

4cy 0 −2cy 0 −2cy 0

0 4a2cy+4b2ecx −2acy −2b2ecx −2acy −2b2ecx

−2cy −2acy 2cy+2f22v

2f23v

0 0

0 −2b2ecx 0 2b2ecx+2b0f11

v0 0

−2cy 2acy 0 0 2cy+2f22v

2f23v

0 −2b2ecx 0 0 0 2b2ecx+2b0f11

v

K =

4ky 0 −2ky 0 −2ky 0

0 (4a2ky+4b2ekx) −2aky −2b2ekx −2aky −2b2ekx

−2ky −2aky 2ky+gκ(mc+mw1+mw2)

2− 2f23κ

r0−2f22 0 0

0 −2b2ekx2b0f11λeq

r02b2ekx 0 0

−2ky 2aky 0 0 2ky+gκ(mc+mw1+mw2)

2− 2f23κ

r0−2f22

0 −2b2ekx 0 02b0f11λeq

r02b2ekx

Due to the formulation of creep forces and gravitational stiffness force, sev-eral terms of Equations 4.8 – 4.13 directly influence the damping and stiffnessmatrices. Thus, the mass, damping and stiffness matrices can be rewritten asmatrix M , C and K, leaving the system with two inputs, track input (cant andthe derivative of cant angle with respect to time) and actuation forces. The

19


right-hand side of Equation 4.13, i.e. F can be now divided into two parts, F1

and F2.F = F1 + F2 = H1u1 +H2u2

u1 =[FLF FR

F FLR FR

R

]Tu2 =

[h ψt

1

R

]T

H1 =

0 0 0 0

−be be be −be0 0 0 0

be −be 0 0

0 0 0 0

0 0 −be be

H2 =

mcg2b0

0 −mcv2

0 0 0mw1

2b0

2f22r0v

−mw1v2

0 0 −b20f11mw2

2b0

2f22r0v

−mw2v2

0 0 −b20f11

Define the state variable as:

x =

[Z

Z

],

the state-space model can be expressed as[Z

Z

]=

[O I

−M−1K −M−1C

] [Z

Z

]+

[O

M−1

]F

Denote:A =

[O I

−M−1K −M−1C

]B =

[O

M−1

]B1 = BH1

B2 = BH2

Therefore, the state-space model of this two-input system can be written incompact form as

x = Ax+B1u1 +B2u2 (4.15)

20


During situations, the delay of track input should be taken into consideration.The front wheelset enters the curve first, followed by the center of the carbodyand then the rear wheelset, i.e. there is a shift for the signal of track input tothe carbody and the rear wheelset. The value of the time delay from the frontwheelset to the center of the carbody is given by

∆t =a

v

where a is the semi-axle distance and v is the vehicle velocity. Due to thesymmetry of the vehicle the same delay is experienced between the center ofthe carbody and the rear wheelset. In this case, the input signal, u2 is modified.

u2 =

[hc ˙ψt,c

1

Rc

hF ˙ψtF

1

RF

hR ˙ψtR

1

RR

]TBesides, matrixH2 should be extended from a 6×3 to a 6×9 matrix, as below.

H2 =

mcg2b0

0 −mcv2 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 mw1g2b0

2f22r0v

−mw1v2 0 0 0

0 0 0 0 0 −b20f11 0 0 0

0 0 0 0 0 0 mw2g2b0

2f22r0v

−mw2v2

0 0 0 0 0 0 0 0 −b20f11

4.1.3 Implementation of the State-Space ModelThe linear model is implemented in Simulink now that the state-space modelis derived.As discussed above, the input signals can be divided into two parts: actua-tion forces and track inputs. Actuation forces are determined by the controller.Track inputs are determined by track layout shown in Figure 2.2. The inputsignal related to variables of track layout, u2 can therefore be defined. In Fig-ure 4.3 the track inputs are shown for the 10 km/h case. For each running casea set of track inputs is created. Considering the delay, the block diagram of thetrack input signal is shown in Figure 4.4. Here the track input signal selectoris used to select input signals for different test speed.According to Equation 4.15, the implementation of the state-space model is

shown in Figure 4.5.

21


Figure 4.3: Input signals due to track layout

Figure 4.4: Input signals due to track layout, with delay

Figure 4.5: Block diagram of the state space model

22


4.2 Nonlinear ModelThe nonlinear model of the vehicle is implemented in SIMPACK. Figure 4.6shows the visualization of the SIMPACK model. The nonlinear model is builtup based on a two-axle vehicle with single axle running gear with only onesuspension step. A composite material frame is used as structural and suspen-sion elements. The axle boxes of the running gear on the inside of the wheelsmakes the wheel axle shorter and the frame of the structure more compact.Besides, the anti-roll bar is integrated into the frame through composite ma-terial properties. To simulate the bar in SIMPACK, the required stiffness isintroduced by means of an ideal torsional stiffness[16].In Figure 4.7 the Simulink diagram for the control of the nonlinear modelis shown. The SIMPACK/Simulink co-simulation interface is used to imple-ment the nonlinear model in Simulink. Here, the outputs of the co-simulationinterface are the wheelsets yaw angles and the vehicle speed while the inputsto it are the actuation forces. A low-pass filter is applied on both inputs andoutputs as will be explained in Chapter 5. For the nonlinear model the trackinputs, and thus the running cases, are directly defined into SIMPACK. Here,the nonlinear contact interaction is calculated using the FASTSIM algorithm.

Figure 4.6: The SIMPACK model

23


Figure 4.7: Simulink implementation of the nonlinear model

24

Chapter 5

Controller Design

Fuzzy PID controllers are used to achieve good curving performance in differ-ent working conditions. The structure of a fuzzy controller is shown in Figure3.3. PID controllers are applied in the fuzzy controller. They are firstly de-termined for each running case. Subsequently, the fuzzy inference system isdesigned based on them. PID controllers are tuned for different speeds usingZiegler-Nichols rule, which gives the control parameters. Then a fuzzy infer-ence system with velocity as its input and control parameters as its output isdesigned to get a proper PID controller for each working conditions, where thefuzzy control can adjust the output, i.e. control parameters based on currentspeed to get a steady curving performance.

5.1 PID ControllersAs mentioned before, the PID controller is tuned using Ziegler-Nichols rule.Specifically, the ultimate-cycle method of Ziegler and Nichols is applied whentuning the control parameters of PID controllers. There are three control pa-rameters of a PID controller: the gains of the proportional, P , integral, I , andderivative part, D. Equation 5.1 shows the controller transfer function. Todetermine the parameters using the ultimate-cycle method, all control param-eters are set to zero except for the proportional gain P . The proportional gainP is initially set at a low value giving as reference signal is a step signal witha proper amplitude. This gain is gradually increased until the output begins toshow sustained oscillations[17]. The period of these oscillations is defined ascritical time constant and is denoted by Tc. The proportional gain at this condi-tion is called critical gain, which is denote byKc. In Figure 5.1 an illustrationof Tc is given. Ziegler-Nichols rule suggests that PID control parameters can

25

CHAPTER 5. CONTROLLER DESIGN

be calculated starting from Kc and Tc. In Table 5.1 the relation between con-trol parameters and critical gain and time constant is given. The block diagramused to tune the PID controller parameters for the linear model is shown in Fig-ure 5.2.

G(s) = P +I

s+Ds (5.1)

Table 5.1: The Ziegler-Nichols settings

Control Parameter ExpressionP 0.6Kc

I 1.2Kc/TcD 0.075KcTc

Figure 5.1: An illustration of critical time constant, Tc

Figure 5.2: A model the tuning of PID controller

26


5.2 Fuzzy Rules BaseThe PID controller is tuned under five velocities (10, 37, 64, 80 and 96 km/h).The tuning results are shown in Table 5.2. The fuzzy rules are designed basedon these reference values. By observing the data, it is clear that the controlparameters, Kc and Tc, vary with the change of the vehicle speed. Therefore,velocity is used as the input of the fuzzy controller so that it can produce aproper PID controller according to current running velocity.

Table 5.2: Results of tuning

v(km/h) 10 37 64 80 96Kc 31450 25300 18225 18225 14250Tc 1.384 0.405 0.288 0.288 0.489

Five membership functions for the input variable, velocity are chosen: Low,RelativeLow, Medium, RelativelyHigh, High; four membership func-tions forthe output variable, Kc are introduced: Big, Medium, RelativelySmall, Small;four membership functions for the other output variable, Tc: Big, RelativelySmall,Medium, Small. The membership functions are chosen to be triangular andthey are shown in Figure 5.3.

The fuzzy rules are therefore defined as

1. IF (v is Low) THEN (Kc is Big) and (Tc is Big)

2. IF (v is RelativelyLow) THEN (Kc is Medium) and (Tc is RelativlySmall)

3. IF (v is Medium) THEN (Kc is RelativeSmall) and (Tc is Small)

4. IF (v is RelativelyHigh) THEN (Kc is RelativeSmall) and (Tc is Small)

5. IF (v is High) THEN (Kc is Small) and (Tc is Medium)

Figure 5.4 shows the control parameters calculated by the fuzzy inference sys-tem for the vehicle speed equal to 37 km/h. It can be seen thatKc and Tc givenby the fuzzy controller are sufficiently close to the reference value shown inTable 5.2, 25300 and 0.405.

27


(a) Membership functions of velocity

(b) Membership functions of Kc

(c) Membership functions of Tc

Figure 5.3: Membership functions

28


Figure 5.4: Kc and Tc given by the fuzzy controller, v = 37km/h

In this chapter, the design process of PID controller and fuzzy rules areillustrated. PID controllers are tuned using Ziegler-Nichols rules under fiverunning speeds. The fuzzy controller accepts velocity as its input and calcu-lates Kc and Tc as its output. Fuzzy rules are designed based on the tuningresults of PID controllers, i.e. Kc and Tc. Finally, the fuzzy controller isshown to be useful due to the fact that the control parameters given by it areclose to the reference values.

29

Chapter 6

Result & Discussions

To investigate the curving performance of a two-axle railway vehicle unit, thechange of the difference between front and rear yaw angle is studied accordingto the chosen control strategy introduced in Chapter 2. The controller designedin Chapter 5 is applied firstly to the linear vehicle model described in Chapter 4and subsequently to the nonlinear model implemented in SIMPACK. In orderto present the simulation result in a more straightforward way, the x axis (time),is normalized according to Equation 6.1. In this way it is possible to comparedifferent running cases at the same time.

tnormalized =tcurrenttend

(6.1)

6.1 Results: Linear ModelThe simulation results obtained from the mathematical linear model are re-ported in Figure 6.1, which show a comparison among the difference betweenfront and rear yaw angle for different velocities. Note that the maximum of theabsolute value of the difference is less than 3× 10−4 radians. In the transitioncurve, the difference is kept to a relatively low level. Considering the orderof magnitude, this value is very small and therefore, the results are quite sat-isfactory, although the differences becomes big and oscillation occurs as thevelocity increases.

6.2 Results: Nonlinear ModelIn Figure 6.2 the simulation results of nonlinear model are shown. The ab-solute value of the maximum of the difference is less than 2 × 10−4 radians.

30

CHAPTER 6. RESULT & DISCUSSIONS

The difference between front and rear yaw angle is close to zero in the circularcurve for all the velocities. Considering that the difference during transitioncurve is maintained small (with the order of magnitude of 10−4 radians), thecontrol objective is considered reached. However, it takes more time for thedifference to go to zero as the speed becomes higher.

Figure 6.1: Simulation Results of Linear Model

6.3 ComparisonAs mentioned in the previous sections, the speed influences the performanceof the controller. As the speed increases, it becomes harder for the controllerto maintain a zero difference during the curving phase even if the difference iskept to a low level. Figure 6.3 and 6.4 show the performance of the controllerwhen the speed is low and high, respectively. In the low speed case, the dif-ference of nonlinear model goes to zero in a short time as for the linear one.For high speed, however, the rise time is longer. The difference of nonlinearmodel in the circular curve goes to zero very slowly. Although it takes moretime for the nonlinear model to response than the linear one. Nevertheless, thenonlinear model shows less vibration and is less sensitive to speed variationthan the linear one.

31


Figure 6.2: Simulation Results of Nonlinear Model

6.4 DiscussionsThe fuzzy PID controller proves to be able to work on the five working speedsand can improve the curving performance. The difference between front andrear yaw angle during circular curve is close to zero, which makes the resultsatisfactory. However, vibration occurs when the speed becomes high for lin-ear model. For the nonlinear model, it takes more time for the difference toreach zero during circular curve as the vehicle moves faster. The fuzzy PIDcontroller remains available at the cost of some performances when the veloc-ity of the vehicle increases, which can be considered as a trade-off betweenperformance and adaptability.Ensuring that the difference between front and rear angle is kept to zero doesnot guarantee that the absolute yaw angle for each wheelset is maintainedwithin an acceptable level. In fact, in some extreme cases it is possible that theabsolute yaw angle is excessive, putting the wheel in a flange contact condi-tion during the circular part of the curve. In this eventual condition, even if thecontrol target of zero difference between front and rear wheelset is achieved,the controlled system may result in a poor curving performance.

32


Figure 6.3: Simulations Results, Low Speed

Figure 6.4: Simulations Results, High Speed

33

Chapter 7

Conclusions

The thesis has presented the development of a fuzzy PID controller for a two-axle railway vehicle that can adapt to a series of running conditions. It has pro-vided an introduction of the fuzzy control theory including fuzzy logic, fuzzyset and structure of fuzzy controller. It has shown that fuzzy PID controllercan help improving the curving performance of the vehicle. The process ofdesigning such a controller is also included. Besides, the control strategy alsoneeds improving to overcome the deflections that the curving performance can-not be guaranteed.A drawback may be found in the choice of the control strategy. In fact, the se-lected control strategy may give rise to poor curving performance in extremeconditions when the absolute wheelset yaw angle is excessive. This conditionhas not been experienced in the current study, but it needs to be taken intoconsideration in future studies.In summary, the thesis has shown that:

1. The fuzzy controller with velocity as input and PID control parametersas output works for both linear and nonlinear model.

2. The fuzzy PID controller works in all the introduced five running con-ditions, although the performance becomes worse as the speed becomeshigher.

3. The designed controller remains available at the expenses of performanceswhile increasing the vehicle speed, which creates a compromise betweenadaptability and performances.

Although a simple two-axle railway vehicle unit has been used in the discus-sion and performance assessment, the application to a more complex railwayvehicle configurations is a feature extension.

34

Acknowledgment

I wish to express my sincere appreciation to my supervisor, Rocco Giossi, whoconvincingly guided me to carry out the project seriously and write this thesisin a professional way. He gave me much support when I tried to solve someunfamiliar problems and encouraged me to overcome challenges when I feltupset. Without his persistent help, the goal of this thesis would not have beenrealized.Besides, I would like to show my gratitude to Prof. Carlos Casanueva Perezand Prof. Susann Boij, who gave the the chance of doing my bachelor thesisin Division of Rail Vehicles of School of Science and Engineering. Boji intro-duced different project works that are suitable for bachelor students to me andhelped me contact Carlos. Carlos let Rocco supervise my work and helped mewith the bureaucratic things to finish the thesis. I would not be able to carry abachelor project work without their help.Moreover, the physical and technical contribution of my home university, Bei-jing Institute of Technology (BIT) and my exchange university, KTH RoyalInstitute of Technology is truly appreciated. BIT offered me a chance of be-ing an exchange student in one of the best institute of technology in Europe,KTH and provided allowances to me. Both institutes supported me technicallymentally during the the coronavirus COVID-19 pandemic, when I had to workfrom home. Without their support and funding, this project could not havereached its goal.Finally, I wish to acknowledge the support and great love of my family, my fa-ther, Changhong; and my mother, Wei. They kept me going on and this workwould not have been possible without their input.

35

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https://doi.org/https://doi.org/10.1016/B978-008043567-1/50016-4

https://doi.org/https://doi.org/10.1016/B978-008043567-1/50016-4

http://www.sciencedirect.com/science/article/pii/B9780080435671500164

http://www.sciencedirect.com/science/article/pii/B9780080435671500164

https://www.mathworks.com/help/fuzzy/types-of-fuzzy-inference-systems.html



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