research article rail vehicle vibrations control using

Research ArticleRail Vehicle Vibrations Control Using ParametersAdaptive PID Controller

Muzaffer Metin and Rahmi Guclu

Department of Mechanical Engineering, Yildiz Technical University, Besiktas, 34349 Istanbul, Turkey

Correspondence should be addressed to Muzaffer Metin; [email protected]

Received 26 February 2014; Accepted 7 April 2014; Published 29 April 2014

Academic Editor: Weichao Sun

Copyright © 2014 M. Metin and R. Guclu. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In this study, vertical rail vehicle vibrations are controlled by the use of conventional PID and parameters which are adaptive toPID controllers. A parameters adaptive PID controller is designed to improve the passenger comfort by intuitional usage of thismethod that renews the parameters online and sensitively under variable track inputs. Sinusoidal vertical rail misalignment andmeasured real rail irregularity are considered as two different disruptive effects of the system. Active vibration control is appliedto the system through the secondary suspension. The active suspension application of rail vehicle is examined by using 5-DOFquarter-rail vehicle model by using Manchester benchmark dynamic parameters. The new parameters of adaptive controller areoptimized bymeans of genetic algorithm toolbox ofMATLAB. Simulations are performed atmaximumurban transportation speed(90 km/h) of the rail vehicle with ±5% load changes of rail vehicle body to test the robustness of controllers. As a result, superiorperformance of parameters of adaptive controller is determined in time and frequency domain.

1. Introduction

Guided rail systems cause vibrations that affect passengercomfort significantly. With the development of technology,the expected comfort level is increased for passengers.Nowadays, while travelling by rail vehicles, passengers havehigh levels of expectations for comfort with safety. Anexcellent isolation of rail roughness caused vibration andshock from passengers cannot be provided by using onlymechanical solutions on suspension systems. To improve sus-pension performances, an additional system which can makeautonomous decisions actively is needed. In the recent past,combining active vibration control mechanism with suitablecontrol algorithms to fully meet passenger expectations is arequirement in this way.

While minimizing vibrations, there are some consider-ations to take into account: ride comfort, which is relatedto accelerations of body vibration; ride safety, which meansensuring the permanence of wheel-rail contact; suspen-sion clearance; and actuator forces. However, these perfor-mance requirements are not always compatible; for example,

required control force to achieve the desirable ride comfortcan create a negative effect on the vehicle’s stability. Regardingthis, active controlling of rail vehicle vibrations has become aserious engineering problem.

Active suspension consists of sensors, controllers, andactuators, which apply control forces to the vehicle body.The responses of a passive system to track inputs or otherdisturbances are obtained by using parameters such as inertia,spring, and damping. However, the response for an activesuspension system is developed by using an advanced controlalgorithm. Because of varying rail irregularity, effective solu-tion of this problem is using an adaptive control algorithm tosuppress rail vehicle body vibration.

In not only this but also many other studies, quartervehicle model is used due to its simplicity and convenience.In rail vehicle model, the interaction between wheel and railis not linear [1], but the interaction model is considered asa simplified linear solution using Hertz spring [2, 3]. On theother hand,Dukkipati andDong study thewheel-rail contact,which is modeled as a multispring contact [4]. Lei and Nodamodel and analyze rail vehicle and railway dynamics using

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 728946, 10 pageshttp://dx.doi.org/10.1155/2014/728946

2 Mathematical Problems in Engineering

the conventional Hertz formula in coupling vehicles andrailway tracks [5].

In the last fifteen years, there are notable studies on theactive control of vibrations on railway vehicles. Foo andGoodall apply active controller to the rail vehicle secondarysuspensions to minimize vertical accelerations of vehiclebody [6]. Detailed information about mechatronic develop-ments for railway vehicles is given by Goodall and Kortum[7]. Guclu investigates the dynamic behaviour of a nonlinear8-DOF vehicle model having active suspensions and a PIDcontrolled passenger seat [8]. Ahmadian andMohan providea methodology for improving hunting behaviour in railvehicles by semiactive control of suspension elements [9].A sliding mode control (SMC) method to improve the ridecomfort of a railway vehiclewith a flexible body is representedby Yagiz and Gursel [10]. Lin et al. present an application ofactive suspension controllers for light rail vehicles with theaim of providing superior ride comfort within the suspensionstroke limitations [11]. Skyhook damping and linear quadraticGaussian (LQG), optimal control techniqueswere consideredand the results are compared to improve ride quality of a two-axle railway vehicle by a number of authors [12]. Orvnas et al.investigate the effect of active lateral secondary suspension toimprove passenger comfort in their experimental study aboutgreen train project [13, 14]. Bruni et al. discuss how primaryand secondary suspension models can be obtained and theiruse for railway vehicle dynamics multibody simulations [15].

In recent years, significant theoretical and implementalstudies have been done on the vehicle suspension controlstrategies. Sun et al. present very significant developments onactive suspension control by using 𝐻

∞norm on a quarter

car model. To improve ride comfort, the 𝐻∞

norm fromthe disturbance to the controlled output is decreased inspecific frequency band by using the generalized Kalman-Yakubovich-Popov (KYP) lemma [16]. Also, this problem isconsidered to take into account the actuator delay by Sunet al. [17]. On the other hand, to control vehicle vibrations,semiactive suspension system and two comparable controlalgorithms are used by Lozoya-Santos et al. [18]. With theaim of extending the working time of suspension system andimproving ride comfort, an optimal control approach whichis based on the active suspension vibration control devices iscarried out by Liang and Wu [19].

Besides, there are several studies on active vibrationcontrol using fuzzy logic controller. Rao and Prahlad carryout a prestudy on rail vehicles by applying an adjustable fuzzylogic controller [20]. In another study, FLC and conventionalPID controller performances are compared using 11 degreesof freedom half-rail vehicle model with sinusoidal rail dis-turbance [21]. A fuzzy control application is achieved on aLRT system in use in Istanbul traffic by using real parameters[22]. Furthermore, secondary suspensions of rail vehicle arecontrolled actively by using a conventional PID control andFLC to improve the passenger comfort comparatively. Thevertical motions of railway vehicle body and passenger seatsare investigated at 140 km/h operating speed on random railirregularity. All parameters of both controllers are achievedby the use of genetic algorithm for the same simulationcondition and performance criteria. In conclusion, the ride

Z

Y

X

Mc

Mw1Mw2

Zc

Zb

Zw1Zw2

𝜃b

c2k2u

V

c12 c11k12 k11

kh2 kh1

Z1Z2

2La

Mb , Jb

Figure 1: The quarter rail vehicle model.

comfort is improved greatly by adding an intelligent FLCbetween the rail vehicle body and bogie [23].

The focus of this paper is to improve the PID control per-formances by using an adaptive method which regulates PIDparameters online with additional scaling factors. Geneticalgorithm can be performed to find the optimum parametersof PID controller which is used in industry widespread. Infact, the obtained parameters are optimum for just actualsituation, and the controller can lose its effectiveness undervarying rail irregularity condition. However, the adaptivecontroller system, which is used in this study, adjusts theparameters itself simultaneously according to the varying railirregularity.With these features, this study distinguishes itselffrom other prevalent studies.

In this study, active control will be implemented throughthe secondary suspension system which is located betweenvehicle body and bogie. For this purpose, rail vehicle ismodeled as 5 degrees of freedom which consists of vehi-cle body, bogie, primary and secondary suspensions, andtwo wheelsets. A quarter rail vehicle model is designedby using Manchester Benchmark dynamic parameters. Thevertical rail vehicle vibrations are controlled by the use ofconventional and parameters adaptive PID controllers. Acombination of varying loading conditions and robustness ofcontrollers are examined under different rail irregularities. Inaddition, time and frequency domain responses are investi-gated under varying loading conditions. As a result of thispaper, when control algorithms are compared under the samedynamic conditions, superior performances of adaptive PIDcontroller can be seen in time and frequency domain.

Mathematical Problems in Engineering 3

Figure 2: KRAB measurement systems.

Plant

Rail vehicle system

r = 0+

+

+

+

Kp

Kd (d/dt)

u(t)e(t) Y(Zc)

−

Ki∫t

0

C

Figure 3: Conventional feedback PID control system.

2. Dynamic Model of Rail Vehicle

The quarter rail vehicle model is used in this study (Figure 1).Uncontrolled system inputs are load changes and rail irregu-larities which can make the system unstable. System outputsare displacement and acceleration of the vehicle body. Passivesystem response depends on parameters such asmass, spring,damping ratio, and suspension geometry. Active suspensionstructure is a more complex dynamic system than passivesuspension. Performance of the active suspension depends onsensors, actuators, and hardware and software of controller.

Quarter rail vehicle model contains 𝑀𝑐which is mass

of rail vehicle body, 𝑀𝑏which is mass of bogie, 𝐽

𝑏which is

inertia moment of bogie, and 𝑀𝑤1

,𝑀𝑤2

which are massesof wheelsets that are located under bogie. 𝑘

2and 𝑐

2are

secondary suspension stiffness and damping coefficients,respectively. 𝑘

11, 𝑘12

and 𝑐11, 𝑐12

are stiffness and dampingcoefficients of primary suspension. 𝑘

ℎ1and 𝑘

ℎ2are expres-

sions of the wheel-rail linear contact form of Hertz springstiffness. The control force 𝑢 is generated by an actuatorwhich is placed between rail vehicle body and bogie.The halfdistance between front and rear wheelsets is presented by 𝐿

𝑎

and𝑉 is the velocity of rail vehicle. All these parameter valuesare given in the Appendix.

Generalized coordinates of the system𝑍𝑐,𝑍𝑏, 𝜃𝑏,𝑍𝑤1, and

𝑍𝑤2

represent the vertical displacement of vehicle body andbogie, the pitch motion of bogie, and the vertical displace-ments of front and rear wheelsets, respectively. Differentialequations of rail vehicle are obtained by using Lagrange’sequation. Differential equation of the system is

[𝑀]

..

𝑍 + [𝐶] �� + [𝐾]𝑍 = 𝐹𝑍

+ 𝐹𝑢, (1)

where 𝑍 = [𝑍𝑐, 𝑍𝑏, 𝜃𝑏, 𝑍𝑤1

, 𝑍𝑤2

]𝑇, 𝐹

𝑍=

[0 0 0 𝐾ℎ1

𝑍1

𝐾ℎ2

𝑍2]𝑇, and 𝐹

𝑢= [𝑢 −𝑢 0 0 0]

𝑇.𝐹𝑧is real rail disturbance that causes vibrations. These

disturbances act on rail vehicle via rigid wheels. 𝐹𝑢is a force

that is applied by an actuator. [M], [C], and [K] are mass,damping, and stiffnessmatrices, which are, respectively, givenin the Appendix. 𝑍

1(𝑡) and 𝑍

2(𝑡) are rail irregularity inputs

applied to wheel masses.During the interaction of rail and wheel, induced forces

are transmitted by rail-wheel contact area. Due to the geom-etry of contact area between rail and wheel, for the wheel-rail dynamic interaction, the relationship between the forceand deflection is defined by the Hertz contact spring. Hertzcontact spring is not a linear contact and depends on therelationship between contact force and deflection of wheel-rail contact surface (𝑦). This expression [2] is given in (2).In this equation, 𝐶

ℎ[𝑁𝑚−3/2

] is Hertz contact coefficientwhich depends on wheel radius, rail radius, and material; 𝐹represents static wheel load. Hertz contact spring is definedas a linear spring. Rigidity of Hertz contact linear value (𝐾

ℎ)

can be obtained from static wheel load depending on theforce-displacement relationship. This is represented by thefollowing equation (3) [3]:

𝐹 = 𝐶ℎ⋅ 𝑦3/2

[𝑁] , (2)

𝐾ℎ=

𝑑𝐹

𝑑𝑦

=3

2

(𝐶ℎ

2/3

⋅ 𝐹1/3

) [𝑁

𝑚

] . (3)

The values of the parameters static wheel load (𝐹) and Hertzcontact coefficient (𝐶

ℎ) are given in the Appendix.

3. Measurements of Rail Irregularity

In this study, in order tomeasure real rail irregularities, KRABtrack geometry measurement system, shown in Figure 2,is used. KRAB is a complete track geometry measurementsystem [24]. Istanbul Transportation Co. uses this system tomeasure deviation from the initial design geometry of railwaysuperstructure. This measurement system is a trolley thatcan be pushed or pulled by an operator. A GPS module canbe equipped on the system for easier localization of defectsand track gradient measurement. Sophisticated assessment


Rail vehicle system

Used in GA toolbox to find optimum parameters

Peak

Plant

observerParameter regulatorISE

r = 0 + ++

+

Kp

Kd d/dt

u(t)

e(t)

Y(Zc)

−

Ki

∫

∫

e2

Figure 4: Block diagram of parameters adaptive PID controller.

software of KRAB computes measured data from the railvia fast Fourier transform (FFT) technique. KRAB can beequipped with either rollers or flanges, and flanges canbe adapted for convenient tramway rail measurement. Thissystemmeasures vertical and horizontal alignment, cant, andtwist. In this study, measured vertical rail irregularities areused.

4. Controllers Design

As fit for the purpose of this study, different control algo-rithms are used on a quarter rail vehicle model as shownin Figure 1. Then, improvements of passenger comfort arecompared. To suppress the vibrations on rail vehicle, first aconventional PID controller is designed. In order to increasethe performance of this controller, the parameters adaptivemethod is used.

4.1. Conventional PID Controller. Due to the widespread usein industry and their simplicity, PID controllers are used inmore than 95% of closed loop industrial processes [25]. Also,it is pretty probable that PID control will continue to be usedin the future. Feedback has had a revolutionary influence inpractically all areas where it has been used and will continueto do so. PID controller is perhaps the most basic form offeedback [26, 27].

In Figure 3, the PID controller is used as the compensator(𝐶), reference signal (𝑟), and output of this system (𝑌,vertical displacement of rail vehicle body “𝑍

𝑐”). PID control

algorithm and its closed loop block diagram are well known.For a conventional PID controller, the control input 𝑢(𝑡) isobtained as follows:

𝑢 (𝑡) = 𝐾𝑝𝑒 (𝑡) + 𝐾

𝑖∫

𝑡

0

𝑒 (𝑡) 𝑑𝑡 + 𝐾𝑑

𝑑𝑒 (𝑡)

𝑑𝑡

, (4)

where 𝑒(𝑡) is the control error; the controller parameters𝐾𝑝, 𝐾𝑖, and 𝐾

𝑑are, respectively, proportional, integral, and

derivative gains; and𝐶(𝑠) is the transfer function of controllerin Laplace form. Consider the following:

𝐶 (𝑠) =𝑢 (𝑠)

𝑒 (𝑠)

= 𝐾𝑝+

𝐾𝑖

𝑠

+ 𝐾𝑑𝑠. (5)

0 20 40 60 80

0

0.01

0.02

0.03

Z-a

xis (

m)

Track axis (m)

System response

· · ·𝛿1

𝛿2𝛿3

𝛿n

−0.01

−0.02

−0.03

−0.04

Figure 5: Peaks of the system response.

4.2. PID Controller with Parameters Adaptive Method. Some-times using only simple PID controller does not presentthe effective response on system which has random andsufficient disturbances.Thus, some specific structured hybridcontrollers are used that involve PID. In this study, a parame-ter adaptive method is used for tuning the PID parameters(𝐾𝑝, 𝐾𝑖, and 𝐾

𝑑) in system model with different track

types. It is well known that the proportional componentof the PID has an important role on the performance ofPID system to control vertical vibrations of vehicles. If theproportional control is too weak, the response will be slow,and if the integration component is too strong, the responsewill become unstable. So it is desirable to make furtherimprovement on it which allows taking a larger value andreducing it gradually with time so as to increase the dampingof the system and make the system more stable. By thisway, an effective suppression and short settling time can beconsidered on displacement and acceleration responses of thesystem.

How to use parameters adaptive method for PID typefuzzy controller structure is described in detail by Wu andMizumoto [28]. To suppress rail vehicle vibrations on aquarter model, this method was implemented by Metin andGuclu [29]. It is known that the rail irregularities are themostimportant sources of vibrations which occur on rail vehiclebody. In this study, two different rail irregularity effects areused to obtain disturbances effect. Amplitude of sinusoidalrail irregularity is |𝑎sin| = 0.02m and maximum value of


0 20 40 60 80

0

0.005

0.01

0.015

0.02

Rail axis (m)

Z-ax

is (m

)Sinusoidal rail irregularity

−0.02

−0.015

−0.01

−0.005

(a)

0 20 40 60 80

0

0.01

0.02

0.03

Z-ax

is (m

)

Rail axis (m)

Measured rail irregularity

−0.04

−0.03

−0.02

−0.01

(b)

Figure 6: Sinusoidal (a) and measured real (b) rail irregularity.

0 1 2 3 4 5 6 7 8 9 10

00.010.02 Rail vehicle body

Time (s)

0 1 2 3 4 5 6 7 8 9 10

024

Time (s)

UncontrolledPID

Adaptive PID

−0.02

−0.01

−4

−2

Zc

(m)

d2Zc/dt2

(m/s2)

Figure 7: Uncontrolled and controlled rail vehicle body vibrationsfor sinusoidal rail irregularity (𝑀

𝑐= 8400 kg).

random rail irregularity is |𝑎ran| = 0.037m. Peak observerdetects peak values of rail vehicle vibrations and sends valuesto parameters regulator, 𝐺

𝑝and 𝐺

𝑑. 𝐺𝑝and 𝐺

𝑑are new

parameters which regulate 𝐾𝑝and 𝐾

𝑑online, respectively,

whereas increasing the value of 𝐾𝑝and 𝐾

𝑑will increase the

proportional control component; thus the reaction of thecontrol system against the errorwill be sped up and vibrationswill be suppressed effectively.

The weighted role of “𝐼” term in PID control is knownas to reduce the steady-state error of systems. A remarkablesteady-state error does not occur in this study. So, there isno need for online adjustment of the term 𝐾

𝑖by parameter

regulator. Also, this process needs additional functions whichcomplicate the controller configuration unnecessarily.

Motivated by this idea, a parameters adaptive PID con-troller is designed. This controller is composed of a PID con-troller, a peak observer, and a parameter regulator (Figure 4).

0 0.5 1 1.5 2 2.5 3 3.5

00.030.06

Rail vehicle body

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5

024

Time (s)

UncontrolledPID

Adaptive PID

−0.06

−0.03

−4

−2

Zc

(m)

d2Zc/dt2

(m/s2)

Figure 8: Uncontrolled and controlled rail vehicle body vibrationsfor real rail irregularity (𝑀

𝑐= 8400 kg).

The peak observer keeps watching on the system output,transmits a signal at each peak time, and measures the abso-lute peak value (𝛿

1–𝛿𝑛) (Figure 5). The parameter regulator

tunes the controller parameters 𝐾𝑝and 𝐾

𝑑simultaneously

at each peak time signal according to the peak value at thattime. The algorithm of tuning the scaling constants and thederivative gain are as follows:

𝐾𝑝=

𝐾𝑝𝑠

𝛿𝑘𝐺𝑝

, 𝐾𝑑=

𝐾𝑑𝑠

𝛿𝑘𝐺𝑑

for 𝛿𝑘≥ 0,001m,

𝐾𝑝= 𝐾𝑝𝑠, 𝐾

𝑑= 𝐾𝑑𝑠

for 𝛿𝑘< 0,001m,

(6)

where 𝐾𝑝𝑠

and 𝐾𝑑𝑠

are the initial values of 𝐾𝑝and 𝐾

𝑑,

respectively. 𝛿𝑘is the absolute peak value at the peak time

𝑡𝑘(𝑘 = 1, 2, 3, . . .) for 𝛿

𝑘≥ 0, 0001m. 𝐺

𝑝and 𝐺

𝑑are

additional scaling factors of peak values for parameters


Table 1: Maximum values and ranges of vibrations for sinusoidalrail irregularity.

𝑀𝑐= 8400 kg

rail vehicle body Displacement (m) Acceleration (m/s2)

Maximum valuesUncontrolled 0,01391 3,222PID 0,006705 2,357Adaptive PID 6,589𝑒 − 005 0,6163

RangesUncontrolled 0,02358 6,387PID 0,01112 4,069Adaptive PID 1,274𝑒 − 004 1,095

Table 2: Maximum values and ranges of vibrations for real railirregularity.

𝑀𝑐= 8400 kg




adaptive PID controller. Values of 𝐾𝑝, 𝐾𝑖, and 𝐾

𝑑are fixed

for both controllers. These values are established by trialand error in this study. 𝐺

𝑝and 𝐺

𝑑scaling factor values are

determined with ISE performance index by the use of geneticalgorithm toolbox in Matlab under a real rail irregularityeffect.

5. Results

The level of comfort is related to essential performancerequirements. How safe the ride is mainly depends on thedisplacement response, while the level of comfort dependson the acceleration response. Controllers are designed totake into account both two responses. New parametersderived from adaptive method are determined using geneticalgorithm toolbox according to the formula ISE (integralof square error). These parameter values of controllers aregiven in the Appendix. ISE value of controller is 2,486⋅10−11.Consider the following:

𝑆ISE = ∫

10

0

𝑒2

(𝑡) 𝑑𝑡. (7)

After obtaining the conventional PID and parametersadaptive PID controller, we will compare the two controllersto illustrate the performance of closed loop suspensionsystem in time and frequency domain. By the simulation,passive responses of rail vehicle, responses of conventional

Table 3: Maximum values and ranges of vibrations under varyingloading for sinusoidal rail irregularity.

𝑀𝑐= 7980 kg




𝑀𝑐= 8820 kg




Table 4: Maximum values and ranges of vibrations under varyingloading for real rail irregularity.

𝑀𝑐= 7980 kg




𝑀𝑐= 8820 kg




PID controller, and responses of parameters adaptive PIDcontroller are compared in Figures 7–12. In these figures, theblue dotted lines, black dotted lines, and red solid lines arethe responses of the passive system (uncontrolled), conven-tional PID controller (PID), and parameters adaptive PID


0 1 2 3 4 5 6 7 8 9 10

00.010.02

Rail vehicle body

Time (s)

0 1 2 3 4 5 6 7 8 9 10

024

Time (s)

−0.02

−0.01

−4

−2

UncontrolledPID

Adaptive PID

Zc

(m)

d2Zc/dt2

(m/s2)

(a) 𝑀𝑐= 7980 kg

0 1 2 3 4 5 6 7 8 9 10

00.010.02

Rail vehicle body

Time (s)

0 1 2 3 4 5 6 7 8 9 10

024

Time (s)

−0.02

−0.01

−4

−2

UncontrolledPID

Adaptive PID

Zc

(m)

d2Zc/dt2

(m/s2)

(b) 𝑀𝑐= 8820 kg

Figure 9: Rail vehicle body vibrations for body mass uncertainty for sinusoidal rail irregularity.

0 0.5 1 1.5 2 2.5 3 3.5

00.030.06

Rail vehicle body

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5

024

Time (s)

UncontrolledPID

Adaptive PID

Zc

(m)

d2Zc/dt2

(m/s2)

−0.06

−0.03

−4

−2

(a) 𝑀𝑐= 7980 kg

0 0.5 1 1.5 2 2.5 3 3.5

00.030.06

Rail vehicle body

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5

024

Time (s)

UncontrolledPID

Adaptive PID

Zc

(m)

d2Zc/dt2

(m/s2)

−0.06

−0.03

−4

−2

(b) 𝑀𝑐= 8820 kg

Figure 10: Rail vehicle body vibrations for body mass uncertainty for real rail irregularity.

controllers (adaptive PID), respectively. From these figures,we can see that the parameters adaptive PID controlleryields the least value of rail vehicle body displacementsand accelerations in time and frequency domain, comparedwith the passive system and the conventional PID controller,which clearly shows that an improved ride comfort has beenachieved.

In order to evaluate the rail vehicle body vibrations, withrespect to ride comfort and safety, we give two differentdisturbance signals as follows to clarify the effectiveness ofour new designed parameters adaptive controller (Figure 6).The sinusoidal rail irregularity describes the collapse of therailway substructures with time; period (𝑦) and amplitude (𝑎)of generated sinus function are 8m and 0.02m, respectively(Figure 6(a)). In Figure 6(b), measured real rail irregular-ity for a certain track (80m) is shown. Maximum urban

transportation speed (90 km/h) of rail vehicle is used for allsimulations.

Figures 7 and 8 demonstrate the rail vehicle body dis-placements and accelerations for passive, PID controlled, andparameters adaptive PID controlled systems for 8400 kg bodymass under sinusoidal and measured real rail irregularities,respectively.

Tables 1 and 2 represent the maximum values and rangesof rail vehicle (𝑀

𝑐= 8400 kg) vibrations for the same cases.

Range is the meaning of difference between maximum andminimum values.

We can clearly see that the values of the body dis-placement and acceleration with the parameters adaptivecontroller are less than that with the conventional PIDcontroller for both irregularity cases (Figures 7 and 8 andTables 1 and 2).


0

50

Frequency (Hz)

Rail vehicle body

050

100150

Frequency (Hz)

−100

−50

−100

−50

z c/z

y(d

2z c/dt2

)/z y

UncontrolledPID

Adaptive PID

10−1 100 101 102 103

10−1 100 101 102 103

Figure 11: Uncontrolled and controlled rail vehicle body responsesin frequency domain (𝑀

𝑐= 8400 kg).

Table 5: Maximum values of vibrations at first peak (1,286Hz).

𝑀𝑐= 8400 kg

rail vehicle body Displacement Acceleration

Maximum valuesUncontrolled 6,873 44,56PID −4,96 32,67Adaptive PID −43,89 −3,413

Robustness of controllers is examined under differentrail irregularities and a combination of varying loadingconditions. To test robustness of controllers for uncer-tainty of rail vehicle body mass (±5%) because of pas-sengers’ load, a set of simulations is carried out. Withinthe framework of these objectives, the frequency responsesare investigated in addition to the time domain responses(Figures 9−12).

Figures 9 and 10 represent the responses of rail vehiclebody vibrations for different body masses (7980 kg and8820 kg) under sinusoidal and real rail irregularities, respec-tively. According to these figures, superior performances ofparameters adaptive controller can be seen for both raildisturbances. These responses of the controllers indicate thestability of closed loop systems for bodymass uncertainty andvarying rail irregularity.

Table 3 demonstrates that when the rail vehicle massdecreases, the passive displacement increases for sinusoidalrail irregularity. PID controller application exhibits a parallelbehavior to passive. However, rail vehicle body vibrationamplitudes are efficiently suppressed by using parametersadaptive PID controller, while passive response increases.This event demonstrates the superiority of the parametersadaptive controller.

0

Frequency (Hz)

0

50

100

Frequency (Hz)

−100

−50

−80

−60

−40

−20

z c/z

y(d

2z c/dt2

)/z y

Mc = 7980kgMc = 8400kg

Mc = 8820 kg

Adaptive PID controlled system (Mc = 7980, 8400, 8820 kg)

10−1 100 101 102 103

10−1 100 101 102 103

Figure 12: Frequency responses of rail vehicle body for body massuncertainty.

Table 6: Maximum values of vibrations at 16,5Hz.

Rail vehicle body Displacement AccelerationMaximum values

7980 kg −9,832 71,738400 kg −9,998 71,258820 kg −10,14 70,8

Table 4 demonstrates that rail vehicle body vibrationamplitudes are suppressed efficiently by using parametersadaptive PID controller. These results reveal the superiorperformances of adaptive PID controller for both rail irreg-ularities.

Figure 11 demonstrates the uncontrolled and controlledrail vehicle body responses in frequency domain for averageloading of rail vehicle (𝑀

𝑐= 8400 kg). In this figure,

red solid line is the responses of rail vehicle body verticaldisplacement and acceleration with the parameters adaptivePID controller. We can clearly see that the value of the bodydisplacement and acceleration with the parameters adaptivePID controller is much less than that with conventionalPID controller and the passive system. These results caneffectively achieve desired ride comfort. It is well knownfrom the ISO2631 that the human body is much sensitive tovibrations between 4Hz and 8Hz in the vertical direction.Consequently, the improvement of vibration isolation of 4–8Hz is very significant.

Table 5 represents the maximum values of rail vehiclevibrations at first peak (1,286Hz) for uncontrolled andcontrolled cases.

In this study, 3 different rail vehicle body masses (𝑀𝑐

=

7980, 8400, and 8820 kg) which represent varying load-ing conditions of body are used to examine robustness


Table 7

Mass parameters Stiffness parameters Dampingparameters

𝑀𝑐= 8400 kg 𝑘

11= 12200000N/m 𝑐

11= 4000Ns/m

𝑀𝑏= 653,75 kg 𝑘

12= 12200000N/m 𝑐

12= 4000Ns/m

𝐽𝑏= 1121 kgm2

𝑘2= 320000N/m 𝑐

2= 20000Ns/m

𝑀𝑤1

= 453,25 kg 𝑘ℎ1

= 1015549000N/m𝑀𝑤2

= 453,25 kg 𝑘ℎ2

= 1015549000N/m

Other parameters PID controllerparameters

Adaptive PIDparameters

𝐿𝑎= 0,9m 𝐾

𝑝= 6,27 ⋅ 105 𝐺

𝑝= 0.55

𝐹 = 69825N 𝐾𝑖= 100 𝐺

𝑑= 52.8

𝐶ℎ= 1 ⋅ 10

11 N/m3/2𝐾𝑑= 120

of controllers. In case of 𝑀𝑐

= 8400 kg, eigenvec-tor of system is [0.98 21 30.8 239.7 239.7]

𝑇. For sec-ond case of simulations (𝑀

𝑐= 7980 kg), eigenvector

of system is [1 21 30.8 239.7 239.7]𝑇. For the last sim-

ulation case (𝑀𝑐

= 8820 kg), eigenvector of system is[0.95 21 30.8 239.7 239.7]

𝑇. Eigenvalue analysis showsthat the first eigenvalue belongs to rail vehicle body mass,predominantly. In Figure 11, the first peak of uncontrolledsystem is suppressed successfully by parameters adaptivePID controller. The use of parameters adaptive methodon PID controller demonstrates the superior performanceto control vibrations of rail vehicle body in time andfrequency domain.

Frequency responses of rail vehicle body for rail vehiclebody mass uncertainty (±5% 𝑀

𝑐) are shown in Figure 12.

This figure and Table 6 demonstrate that frequency responsesof rail vehicle body do not change significantly for varying

loading. It shows the controllers protecting their robustnessfor body mass changes (±5% 𝑀

𝑐).

6. Conclusions

In this study, a quarter rail vehicle model, which is designedas a 5 degrees of freedom system, is examined undercombinations of varying loading and rail irregularities. Tocontrol the vertical vibrations of rail vehicle body, theperformances of a conventional PID and a parametersadaptive PID controller are compared. For controlled anduncontrolled cases, displacement and acceleration responsesof rail vehicle body in time and frequency domains areshown. In the case of parameters adaptive PID controller,𝐾𝑝and 𝐾

𝑑parameters are tuned online under varying rail

irregularities. Thus, an original study is presented by theuse of a heuristic method. The controllers protect theirrobustness for varying loading conditions. The simulationresults indicate that the adaptive PID substantially improvesthe performance of the control system. It has presentedthe simulation results to demonstrate the high performanceof the proposed PID controller with parameters adaptivemethod.

For further studies, parameters adaptive PID controllerperformances can be compared with parameters adaptivefuzzy controller to improve the ride comfort of rail vehicles.On the other hand, performance of parameters adaptivePID controller can be investigated to suppress the lateralmovements of rail vehicles by using a detailed lateral dynamicmodel which contains wheel-rail contact forces and creep-ages.

Appendix

Parameters of the rail vehicle [30, 31]; see Table 7.Mass, damping, and stiffness matrices of the system:

𝑀 =

[[[[[

[

𝑀𝑐

0 0 0 0

0 𝑀𝑏

0 0 0

0 0 𝐽𝑏

0 0

0 0 0 𝑀𝑤1

0

0 0 0 0 𝑀𝑤2

]]]]]

]

,

𝐶 =

[[[[[

[

𝑐2

−𝑐2

0 0 0

−𝑐2

(𝑐2+ 𝑐11

+ 𝑐12) (𝑐11

− 𝑐12) 𝐿𝑎

−𝑐11

−𝑐12

0 (𝑐11

− 𝑐12) 𝐿𝑎

(𝑐11

+ 𝑐12) 𝐿2

𝑎−𝐿𝑎𝑐11

𝐿𝑎𝑐12

0 −𝑐11

−𝐿𝑎𝑐11

𝑐11

0

0 −𝑐12

𝐿𝑎𝑐12

0 𝑐12

]]]]]

]

,

𝐾 =

[[[[[

[

𝑘2

−𝑘2

0 0 0

−𝑘2

(𝑘2+ 𝑘11

+ 𝑘12) (𝑘11

− 𝑘12) 𝐿𝑎

−𝑘11

−𝑘12

0 (𝑘11

− 𝑘12) 𝐿𝑎

(𝑘11

+ 𝑘12) 𝐿2

𝑎−𝐿𝑎𝑘11

𝐿𝑎𝑘12

0 −𝑘11

−𝐿𝑎𝑘11

(𝑘11

+ 𝑘ℎ1

) 0

0 −𝑘12

𝐿𝑎𝑘12

0 (𝑘12

+ 𝑘ℎ2

)

]]]]]

]

.

(A.1)


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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