research article adaptive neural-sliding mode control of...

Research ArticleAdaptive Neural-Sliding Mode Control of Active SuspensionSystem for Camera Stabilization

Feng Zhao Mingming Dong Yechen Qin Liang Gu and Jifu Guan

School of Mechanical Engineering Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Liang Gu guliangbiteducn

Received 10 March 2015 Revised 29 April 2015 Accepted 30 April 2015

Academic Editor Marco Alfano

Copyright copy 2015 Feng Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The camera always suffers from image instability on the moving vehicle due to the unintentional vibrations caused by road rough-nessThis paper presents a novel adaptive neural network based on slidingmode control strategy to stabilize the image captured areaof the camera The purpose is to suppress vertical displacement of sprung mass with the application of active suspension systemSince the active suspension system has nonlinear and time varying characteristics adaptive neural network (ANN) is proposed tomake the controller robustness against systematic uncertainties which release the model-based requirement of the sliding modelcontrol and the weighting matrix is adjusted online according to Lyapunov functionThe control system consists of two loopsTheouter loop is a position controller designed with sliding mode strategy while the PID controller in the inner loop is to track thedesired force The closed loop stability and asymptotic convergence performance can be guaranteed on the basis of the Lyapunovstability theory Finally the simulation results show that the employed controller effectively suppresses the vibration of the cameraand enhances the stabilization of the entire camera where different excitations are considered to validate the system performance

1 Introduction

Camera stabilization is very important for visual guidance inautonomous vehicle and the goal of image stabilization is toremove unwanted motion from dynamic camera sequences[1] The better image-acquisition process will increase thefeasibility and reliability of the process and analysis after-ward Current efforts in the research of image stabilizationtechniques can be broadly classified as three major cate-gories optical stabilization [2] digital image stabilization[3] and mechanical stabilization [4] The ability of vibrationamplitude control in optical and digital image stabilizationis relatively limited Different from former two stabilizersmechanical stabilization involves stabilizing the entire cam-era not just the image It is suitable for large camera movingsituation In the condition of the camera fixed on the top ofthe car the vertical vibration amplitude of the camera needsto be reduced Therefore active suspension system (ASS) isa kind of mechanical stabilization that can suppress the largevertical vibration caused by road roughness

Active suspension system has been widely used in bothmilitary and civil vehicles [5ndash7] Compared with the con-ventional suspension the force actuator is added to provide

energy to the suspension system which enables the suspen-sion to control the attitude of the vehicle body [8] In previousworks such as [9 10] they ignored the actuator dynamicsHowever the nonlinear dynamic of the hydraulic actuatorshould be considered to improve the performance of thecontrol system [11] The control strategy of active suspensionsystem has attracted many researchersrsquo attention in the pastfew decades Different control approaches have been appliedsuch as Hinfin [12] LQG control [13] fuzzy control [14] andartificial neural networks [15] The road disturbances themeasurement errors and the dynamic uncertain model arethe key factors that affect the control performance

In reality the camerarsquos vibrations are inevitable andunpredictableThe vehicle body can provide a stable platformfor the camera to detection of the obstacleswith application ofactive suspension system For a nonlinear active suspensionsystem with bounded uncertainty and external disturbancethe sliding mode control (SMC) has the advantage of robust-ness to improve the antidisturbance ability of the systemIn [16] the state predictors along with SMC technique areapplied to active suspension system which consider the timedelay of the data transmission Mixed control strategies are

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 542364 8 pageshttpdxdoiorg1011552015542364

2 Shock and Vibration

proposed in [17] and an enhanced adaptive self-fuzzy slidingmode controller for a quarter-car active suspension systemis presented Furthermore an adaptive sliding controller isusedwith the function approximation approach for nonlinearsystem containing bounded unknown time-varying uncer-tainties in [18] Besides the SMC strategy neural networkapproach has received increasing attention mainly due to itsadvantages of nonlinear mapping properties and its abilityto deal with uncertainty [19] However in traditional mul-tilayer neural controller the relatively complex structure ofneuron network takesmuch time for computation andweightadjustment An adaptive neural network with learning andself-tuning ability could be used to overcome the drawbacks

This paper lays an emphasis on designing a controllerwith a quarter-car active suspension to suppress verticalvibration of the camera so that the displacement of the sprungmass changes as small as possible In this way we proposea sliding mode controller combined with adaptive neuralnetworks approach to stabilize the image captured area ofthe camera In the meantime the dynamics of the hydraulicactuator is considered and can track the desired force withPID controller in the inner loop Owing to the parametersof the model which are nonlinear conventional adaptiveschemes are not applicable Adaptive radial basis functionneural network (RBFNN) [20 21] is employed to make thecontroller robustness against system uncertainties and theupdate laws of the weighting matrix are derived on the basisof the Lyapunov stability theory The closed loop stabilityand asymptotic convergence performance can be guaranteedFinally the effectiveness of the proposed control approach isvalidated by simulations under different road profiles

This paper is organized as follows Section 2 gives sus-pension system dynamic Section 3 develops the adaptive NNcontroller In Section 4 the results of computer simulation ofthe controller are presented Section 5 concludes this paper

2 Suspension System Dynamic andProblem Formulation

This paper considers the quarter-car model [22] with acamera installed on the top of vehicle as shown in Figure 1The camera is rigidly connected with the vehicle body Thesprung mass 119898119887 represents the car body and unsprung massis 119898119908 The tire is modeled as a linear spring with stiffness119896119905 119888119901 and 119896119904 are the linear damping and stiffness of thesuspension system respectively Variables 119909119887 119909119908 and 119909119903

are the displacement of the body wheel and road profilerespectively A hydraulic actuator between the sprung andunsprung masses can exert a force 119865119886

The dynamic equations of the suspension system can beexpressed as

119898119887119887 + 119896119904 (119909119887 minus 119909119908) + 119888119901 (119887 minus 119908) minus 119865119886 = 0

119898119908119908 + 119896119905 (119909119908 minus 119909119903) + 119896119904 (119909119908 minus 119909119887) + 119888119901 (119908 minus 119887)

+ 119865119886 = 0

(1)

The electrohydraulic actuator comprises a servo-valveand a hydraulic cylinder as shown in Figure 2 Detailed

Camera

xb

xw

xr

cp ks

mw

kt

Fa

mb

Figure 1 Quarter-car model with an active suspension

Servo-valveHydraulic cylinder

xb

x

Fa

xw

Pl

PuPr

Ps

Pr

Figure 2 The electrohydraulic actuator system

introduction of such a hydraulic actuator can be found in[23] The spool valve system is formulated as

V =1

120591

(minus119909V + 119870119888119906) (2)

where 119906 is the control voltage and 120591 is the mechanical delaytime constant of the servo-valve system119870119888 is the conversiongain

The dynamic of the hydraulic cylinder equation is givenby [24]

119886 = minus120573119865119886 minus 1205721198602(119887 minus 119908) + 120574119860119909V

radic119875119904 minus

119865119886 sgn (119909V)119860

(3)

where 119860 is area of piston meanwhile 120572 120573 and 120574 are thehydraulic parameters which may be time varying Consider120572 = 4120573119890119881119905 120573 = 120572119862119905119901 and 120574 = 120572119862119889119908

radic1120588 where 120573119890is bulk modulus of hydraulic fluid 119881119905 is the total volume ofactuator cylinder chamber 119862119905119901 is the leakage coefficient 119862119889is the discharge coefficient119908 is the spool valve area gradient

Shock and Vibration 3

120588 is the hydraulic fluid density and 119875119904 is the hydraulic supplypressure

Remark 1 (see [25]) The practical hydraulic actuator hasthe properties of inherent nonlinear and time-varying char-acteristics and precise parameters in (3) are not easy toestimate which is due to the uncertainties in the actuatordynamic model Furthermore the uncertainties in the modelare bounded

Select the state variables as 119909 = [1199091 1199092 1199093 1199094]119879 where

1199091 = 119909119887 the camera (sprung mass) displacement 1199092 = 119887camera (sprung mass) velocity 1199093 = 119909119908 unsprung massdisplacement 1199094 = 119908 unsprungmass velocity 119909119903 is externaldisturbances and 119910 = [1199091 1199092 1199093 1199094]

119879 Then the equation ofquarter-car active suspension system with actuator dynamicis given as

= 119860119909 + 119861119865119886 + 119861119908119909119903 (4)

where

119860 =

[

[

[

[

[

[

[

[

[

0 1 0 0

minus

119896119904

119898119887

minus

119888119901

119898119887

119896119904

119898119887

119888119901

119898119887

0 0 0 1

119896119904

119898119908

119888119901

119898119908

minus

119896119904 + 119896119905

119898119908

minus

119888119901

119898119908

]

]

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

0

1

119898119887

0

minus

1

119898119908

]

]

]

]

]

]

]

]

]

119861119908 =

[

[

[

[

[

[

[

[

0

0

0

119896119905

119898119908

]

]

]

]

]

]

]

]

(5)

Actually the sprung mass is always time varying Thesystem dynamics can be simplified as

1 = 119891 (119909 119905) + 119892 (119905) 119865119886 (6)

where 119891(119909 119905) is an unknown bounded function and 119892(119905) isthe time-varying control gain From (4) we can obtain

119891 (119909 119905) = (minus

119896119904

119898119887

)1199091 minus (

119888119901

119898119887

)1199092 + (

119896119904

119898119887

)1199093

+ (

119888119901

119898119887

)1199094

119892 (119905) =

1

119898119887

(7)

The objective of this paper is to design a controllerto suppressing vertical vibration of the camera so that thetarget is stable in image plane Since the camera is in rigidconnection with the car body the displacement of the carbody 1199091 can reflect the vibration situation of the camera inthe following analysis

Lemma 2 (see [26]) Let119891(119909) be a continuous function whichis defined on a compact set 119863 Then there exists a neuralnetworks system 119882

119879119911(119909) which can approximate 119891(119909) with

arbitrary accuracy such that

119891 (119909) = 119882119879119911 (119909) + 120576 (119909) (8)

where 119882 = [1199081 1199082 119908119873]119879 is the ideal constant weight

vector 119911(119909) = [1199111(119909) 1199112(119909) 119911119873(119909)]119879 is the RBFs vector

119873 gt 1 is the number of the neurons and 120576(119909) is theapproximation error which is minimized by the ideal vector119882

119882= argmin119882isinR

sup119909isin119863

10038161003816100381610038161003816119891 (119909) minus 119882

119879119911 (119909)

10038161003816100381610038161003816 (9)

Assuming that 120576(119909) is bounded |120576(119909)| lt 120576lowast lt infin with 120576lowast beingan unknown constant 119911119894(119909) is Gaussian function that is

119911119894 (119909) = exp[minus (119909 minus 120583119894)

T(119909 minus 120583119894)

1205782

119894

] (10)

with 120583 = [1205831 1205832 120583119873]119879 and 120578119894 representing the centers andwidths of the Gaussian functions and 119899 is the dimension of 119909

3 Controller Design

The proposed control scheme consists of two loops as shownin Figure 3 The nonlinearity of the hydraulic actuators andtheir force generation capabilities are also considered incontrol design The outer loop performs a position controlof the vehicle body and the desired value of 119909119889 is zero Theinner loop is the PID controller for tracking purpose TheRBFNN approach is employed to approximate the unknownfunction 119891(119909 119905) and 119892(119905) This control method may be muchmore feasible in practical application

31 Designing Inner Loop In order to track the desiredforce for the inner loop a PID controller is used to controlthe electrohydraulic actuators The PID controller can bedesigned to stabilize the closed loop system and providesuitable tracking performance [27] The PID controller isgiven as follows

119906 = 119896119901119890119865 + 119896119894 int

119905

0

119890119865119889119905 + 119896119889

119889119890119865

119889119905

(11)

where 119896119901 is the proportional gain 119896119894 is the integral gain and119896119889 is the differential gain The error is defined as

119890119865 = 119865119889 minus 119865119886 (12)

where 119865119889 is the desired force and 119865119886 is the actual forcegenerated by the actuator


ActuatorSliding controller

mixed withadaptive RBFNN

Weightingupdate laws

PID Quarter car

RoaddisturbanceThe inner loop

The outer loop

xd FaFd u x

minus

Figure 3 The system control block diagram

32 Designing Outer Loop According to RFB neural networkstated previously a novel sliding mode controller for aquarter-car active suspension systems can be developed Thesliding surface is defined as

119904 = 1 + 1205821199091 (13)

where 120582 is the convergent rate of 1199091 on the sliding surfaceThe derivative of above equation (13) is given as follows

119904 = 1 + 1205821 (14)

Substituting (6) into (14)

119904 = 119891 (119909 119905) + 119892 (119905) 119865119889 + 1205821 (15)

The control force 119865119889 can be designed as

119865119889 =

1

119892

(minus119891 minus

1205821 minus 1205781119904 minus 1205782 sgn (119904)) (16)

where 120582 119891 and 119892 are the estimate of 120582 119891(119909 119905) and 119892(119905)respectively The constants 1205781 gt 0 and 1205782 gt 0 are parametersto be selected Substituting (16) into (15)

119904 = minus1205781119904 minus 1205782 sgn (119904) + (119891 minus 119891) + (119892 minus 119892) 119865119889

+ (120582 minus120582) 1

(17)

119904 = minus1205781119904 minus 1205782 sgn (119904) + 119891 + 119892119865119889 +

1205821

(18)

where 119891 = 119891minus 119891 119892 = 119892minus119892 and 120582 = 120582minus 120582 They are assumedto be unknown bounded function and satisfy the Dirichletconditions Therefore the RBFNN can represent 119891 and 119892 as

119891 =

119879

119891119911119891 (119909) + 120576119891

119892 = 119879

119892119911119892 (119909) + 120576119892

(19)

where 119879119891= 119882119879

119891minus 119879

119891and 119879

119892= 119882119879

119892minus 119879

119892 With these

representations (18) can be rewritten as

119904 = minus1205781119904 minus 1205782 sgn (119904) + 119879

119891119911119891 (119909) +

119879

119892119911119892 (119909) 119865119889

+1205821 + 120576119886

(20)

where 120576119886 = 120576119891 + 119865119889120576119892 To prove the stability of this controlsystem and to find the update laws for

119891 and 119892 the

Lyapunov function candidate is chosen as

119881 =

1

2

1199042+

1

2

1205822+

1

2

119879

119891119876119891119891 +

1

2

119879

119892119876119892119892

(21)

where 119876119891 and 119876119892 are both positive definite and symmetricmatrix Taking the time derivative of (21) yields

= 119904 119904 +120582120582 minus

119879

119891119876119891

119891 minus

119879

119892119876119892

119892

= minus12057811199042minus 1205782 |119904| + 119904120576119886 +

119879

119891(119904119911119891 minus 119876119891

119891)

+ 119879

119892(119904119911119892119865119889 minus 119876119892

119892) +

120582 (1199041 +

120582)

(22)

Therefore the update laws and parameter 120582 can be

designed as

119891 = 119876

minus1

119891119904119911119891

119892 = 119876

minus1

119892119904119911119892119865119889

120582 = minus1199041

(23)

So 119891 119892 and

120582 could be approximated by the followingequations

119891 =

119879

119891119911119891 (119909)

119892 = 119879

119892119911119892 (119909)

120582 = intminus1199041119889119905

(24)

Hence we may conclude that

= minus12057811199042minus 1205782 |119904| + 119904120576119886

(25)

In order to satisfy le 0 it must be 1205782|119904| ge 119904120576119886Then we can choose 1205782 with the condition 1205782 ge |120576119886|The

derivative of the Lyapunov function becomes

le minus12057811199042le 0 (26)

Therefore 119904 119891 119892 isin 119871infin and int

infin

01199042119889119905 le

minus120578minus1int

infin

0 119889119905 lt infin imply 119904 isin 1198712 In addition (20)

implies 119904 isin 119871infin hence asymptotic convergence of 119904 can beconcluded using Barbalatrsquos lemma [28] And this furtherimplies that the sliding surface can converge to zero at last


Table 1 Parameter values of the active suspension system

Parameter Value119898119887

290 kg119898119908 59 kg119896119904 16812Nm119896119905 190000Nm119888119901 1000Nms119860 335 times 10minus4 m2

120572 4515 times 1013 Nm5

120573 1 sminus1

120574 1545 times 109 Nm52 kg12

119875119904 10342500 Pa119879 0003 s119870119888 0001mV

4 Simulations

In this section the simulation will be given on a quarter-carmodel System parameters are shown in Table 1

In order to evaluate the performance of the designedcontroller we consider three typical cases

Case 1 As the resonance frequency of the car body isabout 1Hz an external excitation close to this frequencymight induce unwanted oscillation Then to test controlperformance near the system resonance frequency considerthat the road profile is assigned as

1199091199031 (119905) = 0025 sin 2120587119905 (27)

Case 2 Consider a sine bump in an otherwise smooth roadsurface The mathematical description of this type grounddisplacement is given by

1199091199032 (119905) =

05ℎ (1 + sin(2120587V119871

119905)) if 119905119897 le 119905 le 119905ℎ0 otherwise

(28)

where ℎ and 119871 are height and length of the bump and V isthe vehicle velocity Assume ℎ = 005m 119871 = 25m andV = 18 kmh The corresponding road excitation is shownin Figure 4

Case 3 Consider the road excitation which is consistentand typically specified as a random process with a grounddisplacement power spectral density (PSD) of

119866119902 (119899) = 119866119902 (1198990) (

119899

1198990

)

minus119882

119866119902 (119891) =

1

V119866119902 (119899)

(29)

where 119899 is the spatial frequency in119898minus1119866119902(119891) stands for PSDin time domain and 1198990 is the reference spatial frequencySelect the road roughness as 119866119902(1198990) = 64 times 10

minus6m3 1198990 = 01119882 = 2 and V = 20 kmh This PSD indicates that the roadprofile can be obtained from integrating a white noise in time

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

Road

pro

file (

m)

Figure 4 Bump road profile

0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Time (s)

Road

pro

file (

m)

Figure 5 Random road profile

domain [29] So the displacement of the road excitation intime domain is shown in Figure 5

In the simulation the method presented in this paperis represented by NN-SMC and the compared method isdenoted by LQR The matrixes 119876119891 and 119876119892 are chosen asdiagonal matrixes and all of these diagonal elements are 03and 02 respectively The initial parameter 120582 that influencesthe converging slope of the sliding surface is chosen as 100It will influence the converging rate of 1199091 on the slidingsurface We set the adaptation laws with initial conditions1205781 = 1 and 1205782 = 1 times 10

4 The initial weighting vectorsare selected to be 119879

119891(0) = [05 05 05 05 05]

119879 and119879

119892(0) = [05 05 05 05 05]

119879 respectively after a roughtest and the system initial condition is set as 119909(0) =

[01 01 01 01]

119879 The PID controllers for all the threecases are 119896119901 = 10

minus4 119896119894 = 10minus7 and 119896119889 = 10

minus7 The numberof neurons in the hidden layer is 5 and 119911119891(119909) = 119911119892(119909) =

[1199111(119909) 1199112(119909) 1199115(119909)]119879 The center of Gaussian function is

set 120575119894 = 5 (119894 = 1 2 5) and 1205831 = minus1 1205832 = minus05 1205833 = 01205834 = 05 and 1205835 = 1

According to the numerical result of sinusoidal roadexcitation in Figure 6 the displacement attenuation of thecamera (sprungmass) with the proposedNN-SMCcontrolleris dramatically improved and the maximum deflection iskept within plusmn02mm It can be observed that the maximumcamera displacement has been reduced from 60mm to01mm And the suspension deflection is almost the same


0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4

Figure 6 Body displacement (Case 1)

0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR

Figure 7 Suspension deflection (Case 1)

as that with the LQR method in Figure 7 but the maximummagnitude is about 50 smaller than the passive suspensionFrom Figure 8 the estimated value of 119891(119909) converges fast totrue value of 119891(119909) within 06 s

When a vehicle is riding on the bump terrain thedynamic responses of the camera position by using NN-SMC and LQR controllers are shown in Figure 9 It canbe observed that NN-SMC has more obvious performanceimprovement The maximum amplitude of sprung massdisplacement responses is suppressed by 65 and 30respectively compared with the passive suspension and thecurve converges to zero within 16 s by NN-SMC method InFigure 10 the maximum value of suspension deflection byNN-SMC is 35mm which is smaller than the value underLQR method In addition the trajectory of 120582 is shown inFigure 11

The dynamic responses of the sprung mass displacementwith the random excitation are shown in Figure 12 The rootmean square (RMS) value of the sprung mass displacement

0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)

Figure 8 True and estimation 119891(119909) (Case 1)

0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR


withNN-SMC is 333times 10minus4 while the RMS value is 87times 10minus3by LQR As it appears in Figure 13 the fluctuation amplitudeof the red line with NN-SMC is smaller than the blue dottedline In Figure 14 themaximum value of body acceleration byNN-SMC is 065ms2 which is smaller than the maximumvalue under LQR method But the root mean square (RMS)


0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582

Figure 11 Estimate of 120582 (Case 2)

0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR


value of the body acceleration with NN-SMC is 41 times 10minus2which is slightly larger than the RMS value 37 times 10minus2 by LQR

From the above results of the simulation it is clearlyseen from Figures 9 and 10 in Case 2 that the closed loopsystem with NN-SMC has stronger robustness FurthermoreFigure 6 indicates that the vertical vibration of the camera ismuch smaller in the NN-SMCmethod for the resonance fre-quency of the car body which improves the stabilization forcapturing images The tracking speed of the estimation valueof 119891(119909) is given in Figure 8 and the tracking performance

0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)

Figure 14 Body acceleration (Case 3)

is satisfactory Therefore the NN-SMC method possesses afaster dynamic property and has a better performance on thewhole

5 Conclusions

The vibration of the camera will directly affect the imagequality and impede the subsequent processes So the studyof the image vibration attenuation is important in applica-tions In this paper this innovative sliding mode controlbased on mixed adaptive neural networks approach hasbeen successfully employed and simulated to reduce thevertical vibration of the camera using a quarter-car activesuspension system where electrohydraulic actuators withnonlinear characteristics are considered for accurate controlThe radial basis function (RBF) network is used to overcomethe uncertainties in the dynamicmodel of ASS and bymeansof the Lyapunov theorem the weights of neural networksare regulated online without previous learning phase Thewhole control architecture contains two loopsThe inner loopis a PID controller to track the desired force whereas inthe outer loop the position controller is designed by thesliding mode strategy The proposed control scheme cansignificantly suppress the displacement oscillation amplitudeof the camera Different road conditions are used in thesimulation The results show that the presented methodcan guarantee that the active suspension system has a fastconvergence and strong robustness which can effectivelyenhance the stabilization of the camera image

Conflict of Interests

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

Acknowledgments

This research is partially supported by the National ScienceFoundation of China under Contract no 51005018 and thispaper is also funded by International Graduate ExchangeProgram of Beijing Institute of Technology


References

[1] J S Jin Z Zhu and G Xu ldquoA stable vision system for movingvehiclesrdquo IEEE Transactions on Intelligent Transportation Sys-tems vol 1 no 1 pp 32ndash39 2000

[2] P Rawat and J Singhai ldquoReview ofmotion estimation and videostabilization techniques for hand held mobile videordquo Signal ampImage Processing vol 2 no 2 p 159 2011

[3] S Erturk ldquoReal-time digital image stabilization using Kalmanfiltersrdquo Real-Time Imaging vol 8 no 4 pp 317ndash328 2002

[4] Y-C Chang and J Shaw ldquoLow-frequency vibration control ofa pantilt platform with vision feedbackrdquo Journal of Sound andVibration vol 302 no 4-5 pp 716ndash727 2007

[5] D Cao X Song and M Ahmadian ldquoEditorsrsquo perspectivesroad vehicle suspension design dynamics and controlrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 3ndash28 2011

[6] D Hrovat ldquoSurvey of advanced suspension developments andrelated optimal control applicationsrdquoAutomatica vol 33 no 10pp 1781ndash1817 1997

[7] S Ryu Y Park and M Suh ldquoRide quality analysis of atracked vehicle suspension with a preview controlrdquo Journal ofTerramechanics vol 48 no 6 pp 409ndash417 2011

[8] W Sun H Gao and B Yao ldquoAdaptive robust vibration controlof full-car active suspensions with electrohydraulic actuatorsrdquoIEEE Transactions on Control Systems Technology vol 21 no 6pp 2417ndash2422 2013

[9] H Gao W Sun and P Shi ldquoRobust sampled-data 119867infin controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 18 no 1 pp 238ndash245 2010

[10] G Verros S Natsiavas and C Papadimitriou ldquoDesign opti-mization of quarter-car models with passive and semi-activesuspensions under random road excitationrdquo Journal of Vibra-tion and Control vol 11 no 5 pp 581ndash606 2005

[11] X Shen and H Peng ldquoAnalysis of active suspension systemswith hydraulic actuatorsrdquo in The Dynamics of Vehicles onRoads and on Tracks Supplement to Vehicle System DynamicsProceedings of the 18th Iavsd Symposium Held in KanagawaJapan August 24ndash30 2003 vol 41 CRC Press 2005

[12] H Chen Z-Y Liu and P-Y Sun ldquoApplication of constrainedscript Hinfin control to active suspension systems on half-carmodelsrdquo Journal ofDynamic SystemsMeasurement andControlvol 127 no 3 pp 345ndash354 2005

[13] M P Nagarkar G J Vikhe K R Borole and V M NandedkarldquoActive control of quarter-car suspension system using linearquadratic regulatorrdquo International Journal of Automotive andMechanical Engineering vol 3 no 1 pp 364ndash372 2011

[14] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[15] S Yildirim and I Uzmay ldquoNeural network applications tovehiclersquos vibration analysisrdquo Mechanism and Machine Theoryvol 38 no 1 pp 27ndash41 2003

[16] U N L T Alves J P F Garcia M C M Teixeira S C Garciaand F B Rodrigues ldquoSlidingmode control for active suspensionsystem with data acquisition delayrdquo Mathematical Problems inEngineering vol 2014 Article ID 529293 13 pages 2014

[17] R-J Lian ldquoEnhanced adaptive self-organizing fuzzy sliding-mode controller for active suspension systemsrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 3 pp 958ndash968 2013

[18] A-C Huang and Y-S Kuo ldquoSliding control of non-linearsystems containing time-varying uncertainties with unknown

boundsrdquo International Journal of Control vol 74 no 3 pp 252ndash264 2001

[19] Y Jin and D J Yu ldquoAdaptive neuron control using an integratederror approach with application to active suspensionsrdquo Interna-tional Journal of Automotive Technology vol 9 no 3 pp 329ndash335 2008

[20] M Chen S S Ge and B V E How ldquoRobust adaptive neuralnetwork control for a class of uncertain MIMO nonlinearsystems with input nonlinearitiesrdquo IEEE Transactions on NeuralNetworks vol 21 no 5 pp 796ndash812 2010

[21] S S Ge and C Wang ldquoAdaptive neural control of uncertainMIMO nonlinear systemsrdquo IEEE Transactions on Neural Net-works vol 15 no 3 pp 674ndash692 2004

[22] M M Fateh and M M Zirkohi ldquoAdaptive impedance controlof a hydraulic suspension system using particle swarm optimi-sationrdquo Vehicle System Dynamics vol 49 no 12 pp 1951ndash19652011

[23] M-M Ma and H Chen ldquoDisturbance attenuation controlof active suspension with non-linear actuator dynamicsrdquo IETControl Theory and Applications vol 5 no 1 pp 112ndash122 2011

[24] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[25] P-C Chen and A-C Huang ldquoAdaptive sliding control ofactive suspension systems with uncertain hydraulic actuatordynamicsrdquoVehicle System Dynamics vol 44 no 5 pp 357ndash3682006

[26] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[27] M H Ab Talib and I Z Mat Darns ldquoSelf-tuning PID controllerfor active suspension system with hydraulic actuatorrdquo in Pro-ceedings of the IEEE Symposium on Computers and Informatics(ISCI rsquo13) pp 86ndash91 April 2013

[28] J-J E Slotine and W Li Applied Nonlinear Control vol 199Prentice-Hall Englewood Cliffs NJ USA 1991

[29] F Tyan Y F Hong S H Tu et al ldquoGeneration of random roadprofilesrdquo Journal of Advanced Engineering vol 4 no 2 pp 1373ndash1378 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



proposed in [17] and an enhanced adaptive self-fuzzy slidingmode controller for a quarter-car active suspension systemis presented Furthermore an adaptive sliding controller isusedwith the function approximation approach for nonlinearsystem containing bounded unknown time-varying uncer-tainties in [18] Besides the SMC strategy neural networkapproach has received increasing attention mainly due to itsadvantages of nonlinear mapping properties and its abilityto deal with uncertainty [19] However in traditional mul-tilayer neural controller the relatively complex structure ofneuron network takesmuch time for computation andweightadjustment An adaptive neural network with learning andself-tuning ability could be used to overcome the drawbacks

This paper lays an emphasis on designing a controllerwith a quarter-car active suspension to suppress verticalvibration of the camera so that the displacement of the sprungmass changes as small as possible In this way we proposea sliding mode controller combined with adaptive neuralnetworks approach to stabilize the image captured area ofthe camera In the meantime the dynamics of the hydraulicactuator is considered and can track the desired force withPID controller in the inner loop Owing to the parametersof the model which are nonlinear conventional adaptiveschemes are not applicable Adaptive radial basis functionneural network (RBFNN) [20 21] is employed to make thecontroller robustness against system uncertainties and theupdate laws of the weighting matrix are derived on the basisof the Lyapunov stability theory The closed loop stabilityand asymptotic convergence performance can be guaranteedFinally the effectiveness of the proposed control approach isvalidated by simulations under different road profiles

This paper is organized as follows Section 2 gives sus-pension system dynamic Section 3 develops the adaptive NNcontroller In Section 4 the results of computer simulation ofthe controller are presented Section 5 concludes this paper

2 Suspension System Dynamic andProblem Formulation

This paper considers the quarter-car model [22] with acamera installed on the top of vehicle as shown in Figure 1The camera is rigidly connected with the vehicle body Thesprung mass 119898119887 represents the car body and unsprung massis 119898119908 The tire is modeled as a linear spring with stiffness119896119905 119888119901 and 119896119904 are the linear damping and stiffness of thesuspension system respectively Variables 119909119887 119909119908 and 119909119903

are the displacement of the body wheel and road profilerespectively A hydraulic actuator between the sprung andunsprung masses can exert a force 119865119886

The dynamic equations of the suspension system can beexpressed as

119898119887119887 + 119896119904 (119909119887 minus 119909119908) + 119888119901 (119887 minus 119908) minus 119865119886 = 0

119898119908119908 + 119896119905 (119909119908 minus 119909119903) + 119896119904 (119909119908 minus 119909119887) + 119888119901 (119908 minus 119887)

+ 119865119886 = 0

(1)

The electrohydraulic actuator comprises a servo-valveand a hydraulic cylinder as shown in Figure 2 Detailed

Camera

xb

xw

xr

cp ks

mw

kt

Fa

mb

Figure 1 Quarter-car model with an active suspension

Servo-valveHydraulic cylinder

xb

x

Fa

xw

Pl

PuPr

Ps

Pr

Figure 2 The electrohydraulic actuator system

introduction of such a hydraulic actuator can be found in[23] The spool valve system is formulated as

V =1

120591

(minus119909V + 119870119888119906) (2)

where 119906 is the control voltage and 120591 is the mechanical delaytime constant of the servo-valve system119870119888 is the conversiongain

The dynamic of the hydraulic cylinder equation is givenby [24]

119886 = minus120573119865119886 minus 1205721198602(119887 minus 119908) + 120574119860119909V

radic119875119904 minus

119865119886 sgn (119909V)119860

(3)

where 119860 is area of piston meanwhile 120572 120573 and 120574 are thehydraulic parameters which may be time varying Consider120572 = 4120573119890119881119905 120573 = 120572119862119905119901 and 120574 = 120572119862119889119908

radic1120588 where 120573119890is bulk modulus of hydraulic fluid 119881119905 is the total volume ofactuator cylinder chamber 119862119905119901 is the leakage coefficient 119862119889is the discharge coefficient119908 is the spool valve area gradient







= 119860119909 + 119861119865119886 + 119861119908119909119903 (4)

where

119860 =

[

[

[

[

[

[

[

[

[

0 1 0 0

minus

119896119904

119898119887

minus

119888119901

119898119887

119896119904

119898119887

119888119901

119898119887

0 0 0 1

119896119904

119898119908

119888119901

119898119908

minus

119896119904 + 119896119905

119898119908

minus

119888119901

119898119908

]

]

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

0

1

119898119887

0

minus

1

119898119908

]

]

]

]

]

]

]

]

]

119861119908 =

[

[

[

[

[

[

[

[

0

0

0

119896119905

119898119908

]

]

]

]

]

]

]

]

(5)


1 = 119891 (119909 119905) + 119892 (119905) 119865119886 (6)


119891 (119909 119905) = (minus

119896119904

119898119887

)1199091 minus (

119888119901

119898119887

)1199092 + (

119896119904

119898119887

)1199093

+ (

119888119901

119898119887

)1199094

119892 (119905) =

1

119898119887

(7)





119891 (119909) = 119882119879119911 (119909) + 120576 (119909) (8)





sup119909isin119863

10038161003816100381610038161003816119891 (119909) minus 119882

119879119911 (119909)

10038161003816100381610038161003816 (9)


119911119894 (119909) = exp[minus (119909 minus 120583119894)

T(119909 minus 120583119894)

1205782

119894

] (10)


3 Controller Design



119906 = 119896119901119890119865 + 119896119894 int

119905

0

119890119865119889119905 + 119896119889

119889119890119865

119889119905

(11)


119890119865 = 119865119889 minus 119865119886 (12)






PID Quarter car


The outer loop

xd FaFd u x

minus



119904 = 1 + 1205821199091 (13)


119904 = 1 + 1205821 (14)


119904 = 119891 (119909 119905) + 119892 (119905) 119865119889 + 1205821 (15)


119865119889 =

1

119892

(minus119891 minus

1205821 minus 1205781119904 minus 1205782 sgn (119904)) (16)



+ (120582 minus120582) 1

(17)

119904 = minus1205781119904 minus 1205782 sgn (119904) + 119891 + 119892119865119889 +

1205821

(18)


119891 =

119879

119891119911119891 (119909) + 120576119891

119892 = 119879

119892119911119892 (119909) + 120576119892

(19)

where 119879119891= 119882119879

119891minus 119879

119891and 119879

119892= 119882119879

119892minus 119879

119892 With these


119904 = minus1205781119904 minus 1205782 sgn (119904) + 119879

119891119911119891 (119909) +

119879

119892119911119892 (119909) 119865119889

+1205821 + 120576119886

(20)


119891 and 119892 the


119881 =

1

2

1199042+

1

2

1205822+

1

2

119879

119891119876119891119891 +

1

2

119879

119892119876119892119892

(21)


= 119904 119904 +120582120582 minus

119879

119891119876119891

119891 minus

119879

119892119876119892

119892

= minus12057811199042minus 1205782 |119904| + 119904120576119886 +

119879

119891(119904119911119891 minus 119876119891

119891)

+ 119879

119892(119904119911119892119865119889 minus 119876119892

119892) +

120582 (1199041 +

120582)

(22)


designed as

119891 = 119876

minus1

119891119904119911119891

119892 = 119876

minus1

119892119904119911119892119865119889

120582 = minus1199041

(23)

So 119891 119892 and


119891 =

119879

119891119911119891 (119909)

119892 = 119879

119892119911119892 (119909)

120582 = intminus1199041119889119905

(24)


= minus12057811199042minus 1205782 |119904| + 119904120576119886

(25)



le minus12057811199042le 0 (26)


infin

01199042119889119905 le


infin







120572 4515 times 1013 Nm5

120573 1 sminus1

120574 1545 times 109 Nm52 kg12

119875119904 10342500 Pa119879 0003 s119870119888 0001mV

4 Simulations




1199091199031 (119905) = 0025 sin 2120587119905 (27)


1199091199032 (119905) =

05ℎ (1 + sin(2120587V119871


(28)



119866119902 (119899) = 119866119902 (1198990) (

119899

1198990

)

minus119882

119866119902 (119891) =

1

V119866119902 (119899)

(29)



0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

Road

pro

file (

m)


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Time (s)

Road

pro

file (

m)





119891(0) = [05 05 05 05 05]

119879 and119879

119892(0) = [05 05 05 05 05]


[01 01 01 01]


minus4 119896119894 = 10minus7 and 119896119889 = 10






0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4


0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR





0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)


0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















= 119860119909 + 119861119865119886 + 119861119908119909119903 (4)

where

119860 =

[

[

[

[

[

[

[

[

[

0 1 0 0

minus

119896119904

119898119887

minus

119888119901

119898119887

119896119904

119898119887

119888119901

119898119887

0 0 0 1

119896119904

119898119908

119888119901

119898119908

minus

119896119904 + 119896119905

119898119908

minus

119888119901

119898119908

]

]

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

0

1

119898119887

0

minus

1

119898119908

]

]

]

]

]

]

]

]

]

119861119908 =

[

[

[

[

[

[

[

[

0

0

0

119896119905

119898119908

]

]

]

]

]

]

]

]

(5)


1 = 119891 (119909 119905) + 119892 (119905) 119865119886 (6)


119891 (119909 119905) = (minus

119896119904

119898119887

)1199091 minus (

119888119901

119898119887

)1199092 + (

119896119904

119898119887

)1199093

+ (

119888119901

119898119887

)1199094

119892 (119905) =

1

119898119887

(7)





119891 (119909) = 119882119879119911 (119909) + 120576 (119909) (8)





sup119909isin119863

10038161003816100381610038161003816119891 (119909) minus 119882

119879119911 (119909)

10038161003816100381610038161003816 (9)


119911119894 (119909) = exp[minus (119909 minus 120583119894)

T(119909 minus 120583119894)

1205782

119894

] (10)


3 Controller Design



119906 = 119896119901119890119865 + 119896119894 int

119905

0

119890119865119889119905 + 119896119889

119889119890119865

119889119905

(11)


119890119865 = 119865119889 minus 119865119886 (12)






PID Quarter car


The outer loop

xd FaFd u x

minus



119904 = 1 + 1205821199091 (13)


119904 = 1 + 1205821 (14)


119904 = 119891 (119909 119905) + 119892 (119905) 119865119889 + 1205821 (15)


119865119889 =

1

119892

(minus119891 minus

1205821 minus 1205781119904 minus 1205782 sgn (119904)) (16)



+ (120582 minus120582) 1

(17)

119904 = minus1205781119904 minus 1205782 sgn (119904) + 119891 + 119892119865119889 +

1205821

(18)


119891 =

119879

119891119911119891 (119909) + 120576119891

119892 = 119879

119892119911119892 (119909) + 120576119892

(19)

where 119879119891= 119882119879

119891minus 119879

119891and 119879

119892= 119882119879

119892minus 119879

119892 With these


119904 = minus1205781119904 minus 1205782 sgn (119904) + 119879

119891119911119891 (119909) +

119879

119892119911119892 (119909) 119865119889

+1205821 + 120576119886

(20)


119891 and 119892 the


119881 =

1

2

1199042+

1

2

1205822+

1

2

119879

119891119876119891119891 +

1

2

119879

119892119876119892119892

(21)


= 119904 119904 +120582120582 minus

119879

119891119876119891

119891 minus

119879

119892119876119892

119892

= minus12057811199042minus 1205782 |119904| + 119904120576119886 +

119879

119891(119904119911119891 minus 119876119891

119891)

+ 119879

119892(119904119911119892119865119889 minus 119876119892

119892) +

120582 (1199041 +

120582)

(22)


designed as

119891 = 119876

minus1

119891119904119911119891

119892 = 119876

minus1

119892119904119911119892119865119889

120582 = minus1199041

(23)

So 119891 119892 and


119891 =

119879

119891119911119891 (119909)

119892 = 119879

119892119911119892 (119909)

120582 = intminus1199041119889119905

(24)


= minus12057811199042minus 1205782 |119904| + 119904120576119886

(25)



le minus12057811199042le 0 (26)


infin

01199042119889119905 le


infin







120572 4515 times 1013 Nm5

120573 1 sminus1

120574 1545 times 109 Nm52 kg12

119875119904 10342500 Pa119879 0003 s119870119888 0001mV

4 Simulations




1199091199031 (119905) = 0025 sin 2120587119905 (27)


1199091199032 (119905) =

05ℎ (1 + sin(2120587V119871


(28)



119866119902 (119899) = 119866119902 (1198990) (

119899

1198990

)

minus119882

119866119902 (119891) =

1

V119866119902 (119899)

(29)



0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

Road

pro

file (

m)


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Time (s)

Road

pro

file (

m)





119891(0) = [05 05 05 05 05]

119879 and119879

119892(0) = [05 05 05 05 05]


[01 01 01 01]


minus4 119896119894 = 10minus7 and 119896119889 = 10






0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4


0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR





0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)


0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













PID Quarter car


The outer loop

xd FaFd u x

minus



119904 = 1 + 1205821199091 (13)


119904 = 1 + 1205821 (14)


119904 = 119891 (119909 119905) + 119892 (119905) 119865119889 + 1205821 (15)


119865119889 =

1

119892

(minus119891 minus

1205821 minus 1205781119904 minus 1205782 sgn (119904)) (16)



+ (120582 minus120582) 1

(17)

119904 = minus1205781119904 minus 1205782 sgn (119904) + 119891 + 119892119865119889 +

1205821

(18)


119891 =

119879

119891119911119891 (119909) + 120576119891

119892 = 119879

119892119911119892 (119909) + 120576119892

(19)

where 119879119891= 119882119879

119891minus 119879

119891and 119879

119892= 119882119879

119892minus 119879

119892 With these


119904 = minus1205781119904 minus 1205782 sgn (119904) + 119879

119891119911119891 (119909) +

119879

119892119911119892 (119909) 119865119889

+1205821 + 120576119886

(20)


119891 and 119892 the


119881 =

1

2

1199042+

1

2

1205822+

1

2

119879

119891119876119891119891 +

1

2

119879

119892119876119892119892

(21)


= 119904 119904 +120582120582 minus

119879

119891119876119891

119891 minus

119879

119892119876119892

119892

= minus12057811199042minus 1205782 |119904| + 119904120576119886 +

119879

119891(119904119911119891 minus 119876119891

119891)

+ 119879

119892(119904119911119892119865119889 minus 119876119892

119892) +

120582 (1199041 +

120582)

(22)


designed as

119891 = 119876

minus1

119891119904119911119891

119892 = 119876

minus1

119892119904119911119892119865119889

120582 = minus1199041

(23)

So 119891 119892 and


119891 =

119879

119891119911119891 (119909)

119892 = 119879

119892119911119892 (119909)

120582 = intminus1199041119889119905

(24)


= minus12057811199042minus 1205782 |119904| + 119904120576119886

(25)



le minus12057811199042le 0 (26)


infin

01199042119889119905 le


infin







120572 4515 times 1013 Nm5

120573 1 sminus1

120574 1545 times 109 Nm52 kg12

119875119904 10342500 Pa119879 0003 s119870119888 0001mV

4 Simulations




1199091199031 (119905) = 0025 sin 2120587119905 (27)


1199091199032 (119905) =

05ℎ (1 + sin(2120587V119871


(28)



119866119902 (119899) = 119866119902 (1198990) (

119899

1198990

)

minus119882

119866119902 (119891) =

1

V119866119902 (119899)

(29)



0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

Road

pro

file (

m)


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Time (s)

Road

pro

file (

m)





119891(0) = [05 05 05 05 05]

119879 and119879

119892(0) = [05 05 05 05 05]


[01 01 01 01]


minus4 119896119894 = 10minus7 and 119896119889 = 10






0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4


0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR





0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)


0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120572 4515 times 1013 Nm5

120573 1 sminus1

120574 1545 times 109 Nm52 kg12

119875119904 10342500 Pa119879 0003 s119870119888 0001mV

4 Simulations




1199091199031 (119905) = 0025 sin 2120587119905 (27)


1199091199032 (119905) =

05ℎ (1 + sin(2120587V119871


(28)



119866119902 (119899) = 119866119902 (1198990) (

119899

1198990

)

minus119882

119866119902 (119891) =

1

V119866119902 (119899)

(29)



0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

Road

pro

file (

m)


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Time (s)

Road

pro

file (

m)





119891(0) = [05 05 05 05 05]

119879 and119879

119892(0) = [05 05 05 05 05]


[01 01 01 01]


minus4 119896119894 = 10minus7 and 119896119889 = 10






0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4


0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR





0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)


0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8 9 10

000

002

004

006

008

20 25 30 35 40 45

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR

minus002

minus004

minus006

minus008

10

05

00

minus05

minus10

times10minus4


0 1 2 3 4 5 6 7 8 9 10

minus004

minus002

0

002

004

006

Time (s)

Susp

ensio

n de

flect

ion

(m)

NN-SMCPassiveLQR





0 1 2 3 4 5 6 7 8 9 10minus6

minus4

minus2

0

2

4

6

f an

d es

timat

ed f

Estimation fTrue f

Time (s)


0 05 1 15 2 25 3 35 4 45 5minus004

minus002

0

002

004

006

008

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 05 1 15 2 25 3 35 4 45 5minus006

minus004

minus002

0

002

004

006

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 3 35 4 45 5

minus02minus01

00102030405

Time (s)

120582


0 1 2 3 4 5 6 7 8 9 10minus002

minus001

0

001

002

003

Body

disp

lace

men

t (m

)

Time (s)

NN-SMCPassiveLQR


0 1 2 3 4 5 6 7 8 9 10minus003

minus002

minus001

0

001

002

003

Susp

ensio

n de

flect

ion

(m)

Time (s)

NN-SMCPassiveLQR




0 1 2 3 4 5 6 7 8 9 10minus15

minus1

minus05

0

05

1

15

Time (s)

NN-SMCPassiveLQR

Body

acce

lera

tion

(ms2)



5 Conclusions




Acknowledgments



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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