Research Article Adaptive Vibration Control System for MR …downloads.hindawi.com/journals/sv/2015/163694.pdf · 2019. 7. 31. · frequency switched ,andmeasurement-based ), type

Research ArticleAdaptive Vibration Control System for MR Damper Faults

Juan C Tudoacuten-Martiacutenez and Ruben Morales-Menendez

Tecnologico de Monterrey Escuela de Ingenierıa y Ciencias Avenida Garza Sada No 2501 64849 Monterrey NL Mexico

Correspondence should be addressed to Juan C Tudon-Martınez jctudonphdmtyitesmmx

Received 30 January 2015 Revised 29 April 2015 Accepted 4 May 2015

Academic Editor Miao Yu

Copyright copy 2015 J C Tudon-Martınez and R Morales-Menendez This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Several methods have been proposed to estimate the force of a semiactive damper particularly of a magnetorheological damperbecause of its importance in automotive and civil engineering Usually all models have been proposed assuming experimental datain nominal operating conditions and some of them are estimated for control purposes Because dampers are prone to fail faultestimation is useful to design adaptive vibration controllers to accommodate the malfunction in the suspension systemThis paperdeals with the diagnosis and estimation of faults in an automotivemagnetorheological damper A robust LPV observer is proposedto estimate the lack of force caused by a damper leakage in a vehicle corner Once the faulty damper is isolated in the vehicle and thefault is estimated an Adaptive Vibration Control System is proposed to reduce the fault effect using compensation forces from theremaining healthy dampers To fulfill the semiactive damper constraints in the fault adaptation an LPV controller is designed forvehicle comfort and road holding Simulation results show that the fault observer has good performance with robustness to noiseand road disturbances and the proposed AVCS improves the comfort up to 24 with respect to a controlled suspension withoutfault tolerance features

1 Introduction

In general a damper is characterized by [1] (a) the dimen-sional data (b) operating factors and (c) force characteristicsDimensional data include the stroke the length betweenmountings diameters and weight Operating factors includethe limitations on temperature power dissipation and cool-ing requirements while themost fundamental characteristicsare the magnitude and shape of the force that is forcecharacteristics indicate how the force varies with respect tothe velocity production tolerances on these forces frictioneffects saturation and so forth When the damper fails theforce characteristics are modified

According to the capability to adjust the force char-acteristics the automobile dampers can be classified aspassive active or semiactive Passive dampers are only ableto dissipate the energy and their damping characteristics aretime invariant while active ones are able to store dissipateand generate energy through a variable damping coefficientbut they are very expensive to apply because they require anexternal power supply Semiactive dampers represent a good

solution to control the vertical vehicle dynamics using lowerenergy consumption and lower cost [2] Different semiactivedamper technologies have been examined whose bandwidthoscillates from 0 to 30Hz the main technologies in the auto-motive industry are electrohydraulic dampers (EH) pneu-matic actuators (PA) andmagnetoelectrorheological dampers(MRER)

In an automotive suspension control system the dampingforce on a controllable shock absorber is adjusted to improvethe comfort andor safety of the passengers in a road vehicleovercoming the weaknesses of passive suspensions Becausethe cost-benefit ratio is better in semiactive suspensions thanin active ones the major interest of the control commu-nity related to automotive suspensions is focused on thesemiactive suspension control An extensive classification ofdifferent control strategies for semiactive dampers accordingto the type of manipulation (continuous or on-off ) controlgoal (comfort road holding or both) type of control lawto include the semiactiveness (clipped frequency adaptivefrequency switched and measurement-based) type of controldesign (model-based or free of model) and so forth is

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 163694 17 pageshttpdxdoiorg1011552015163694

2 Shock and Vibration

presented in [3] Indeed without considering the natureof the control strategy the interesting challenges of thesemiactive suspension control problem are as follows [4]

(1) Realistic design according to the nonlinearities of thesemiactive damper

(2) Constrained control according to the dissipativityand saturation of the damper to avoid impracticalsolutions even if this actuator is partially faulty

(3) Multiobjective control design that is capable ofimproving the vehicle comfort with no compromise(and perhaps enhancement) for the road holding

(4) Feasibility of implementation with adaptation to dif-ferent damper technologies small number of controlinputs and low computing cost to be programmed inmicrocontrollers

An extensive work on semiactive suspension systemsemphasizes that MR dampers are one of the best solutionsto manage the compromise between the objectives of anautomotive suspension control comfort and road holdingMR dampers are as fast as the EH actuators with less powerrequirement than ER dampers and higher robustness totemperature variations they are easier to manufacture andimplement than EH dampers and to control than PA Indeedmost of the existing work on semiactive damper modelingand control is based on the MR damping technology [1]

Several mathematical models are available to representthe nonlinear behavior of MR dampers In general theycan be grouped as parametric and nonparametric modelsParametric models explain the physical phenomena in math-ematical expressions for instance the classical viscoelastic-plastic models [5 6] the phenomenological evolutive Bouc-Wenmodels [7ndash12] and the algebraic models [13ndash15] On theother hand coefficients of nonparametric structures do nothave a physical meaning for instance fuzzy models [16 17]statistical models [18] polynomial models [19ndash22] or modelsbased on artificial neural networks (ANN) [23ndash29] Usuallyall aforementioned structures have presented good modelingresults andor good properties to design control modules butwhen the damper is faulty nothing can be said

The most common fault in MR dampers is oil leakage[1] There are different aspects that cause a damper leakage[30] which are related to bad installation excessive usedamage by external devices or bad design of control systemthat saturates the manipulation The direct consequence ofa faulty shock absorber is the loss of damping force thatcauses a performance reduction in the suspension capabilities(comfort and safety for passengers) at any driving situationRecently the Linear Parameter-Varying (LPV) control theoryhas been extended to fault tolerant control (FTC) systems tomaintain the desired control goals as much as possible whena damper fault occurs A scheduling parameter based on theload transfer is used to keep the equilibrium (comfort) whena semiactive damper fails in [31] in [32] a supervisory systemcomputes the compensation force that each healthy dampermust add to reduce the fault effect considering the semiactiveconstraints

Recently Tudon-Martınez [33] studied the effect of the oilleakage degree in the damping force using a commercial auto-motive shock absorber Several degrees of oil leakage wereimplemented to analyze how the dynamic behavior of theMRdamper is modified through the characteristic curves force-velocity (FV) and force-displacement (FD) Figure 1 showsthat the friction effect is reduced proportionally to the oilleakage degree the linear stiffness also is reduced when theoil leakage increases the nonlinearity in the viscous dampingis modified and evidently the saturation force decreasesinversely proportional to the damper leakage

Experimental data of a Design of Experiments (DoE)with central composite design [34] illustrate that the damperleakage can be represented as a fault model of multiplicativenature [33] By using an extension of a parametric modelstructure for instance the algebraic model of Guo et al[13] a faulty MR damper can be modeled by a parameteridentification algorithm as a static equation In this modelrepresentation the nominal parameters do not change onlythe fault parameter modifies the force characteristics

Because the parameter estimation approaches are sensi-tive to noise measurements and their online adaptation canrequiremany computational resources a robust estimation ofthe faulty force is essential Indeed the accuracy of the faultestimation not only determines the damper condition thisinformation can be used also to design Adaptive VibrationControl Systems to accommodate the lack of force caused bythe faulty MR damper [32 35ndash37]

Most of the work in Fault Detection and Isolation (FDI)modules for automobile suspensions is based on analyticalredundancy such as the parity space theory [38 39] robustfiltering [40] and nonlinear-robust observers [35 36 41]The above approaches perform well for sensor faults or evenfor actuator faults considering active dampers that do nothave any constraint on the force characteristics in contrastto an MR damper whose semiactiveness friction and forcesaturation constraint the actuator properties

An Adaptive Vibration Control System (AVCS) based on arobust LPV observer used to estimate faults in anMRdamperis proposed in this paper By using an 119867

infindesign the LPV-

AVCS system is robust to the road disturbances and noisymeasurements The LPV approach enables embedding thenonlinearities of the vehicle dynamics into varying param-eters such that the observer becomes linearly parameterdependent with internal stability and robust performanceThe inclusion of the faulty damper constraints as schedulingparameters into the LPV119867

infinobserver makes the fault

estimation fulfill the semiactiveness of the shock absorber ina practical implementationTheAVCS based also on the LPVframework uses the fault estimation to compute the neededcompensation force in the healthyMR shock absorbers of thevehicle to reduce the effect of lack of force caused by the faultydamper (comfort deterioration)

The outline of this paper is as follows in the next sectionthe problem statement is formulated Section 3 details thedesign of the FDI system used to diagnose damper faults andSection 4 describes the AVCS Section 5 illustrates the casestudy to validate the proposed FDI strategy under differenttests and discusses the results in the AVCS Concluding

Shock and Vibration 3

Displacement (m)

Forc

e (N

)

0 0005 001 0015

0

200

400

600

800

Friction

StiffnessStiffness

minus200

minus400minus0015 minus001 minus0005

(a) FD characteristic curve

Velocity of deflection (ms)0 02 04 06

Fault 2

Free of faults

Fault 1

Nonlinear viscous damping

Saturation

minus06 minus04 minus02

Forc

e (N

)

0

200

400

600

800

minus200

minus400

(b) FV characteristic curve

Figure 1 Experimental characteristic curves of a damper subject to leakages Fault 1 represents 5 of oil leakage and fault 2 10The damperdoes not have control input value

ms zs

FMR

mus

ks

ktzr

zus

Figure 2 QoV model of an automotive semiactive suspensionsystem

remarks are presented in Section 6The variables of this studyare defined in the Abbreviations

2 Problem Statement

The QoV model is the most basic system to represent anautomotive suspension Figure 2 Its use assumes an equiv-alent load distribution among the four corners and a lineardependency with respect to the translational and rotationalchassis motionsThe lateral and longitudinal wheel dynamicsis not considered while the wheel-road contact is ensuredThe vertical dynamics of a QoV model is governed by

119898119904119904(119905) = minus 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119865MR (119905)

119898119906119904119906119904(119905) = 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119896

119905[119911119906119904(119905) minus 119911

119903(119905)]

+ 119865MR (119905)

(1)

The system considers a sprung mass (119898119904) and an

unsprung mass (119898119906119904) A spring with stiffness coefficient

119896119904and an MR shock absorber represent the suspension

between both masses The spring is considered as linearbecause around 95 of its operating zone in an automotiveapplication is linear however the MR damping force (119865MR)depends on a control input variable (electric current) and itis highly nonlinear with respect to the suspension motionThe stiffness coefficient 119896

119905models the wheel tire The vertical

position of the mass 119898119904(119898119906119904) is defined by 119911

119904(119911119906119904) while

119911119903corresponds to the unknown road disturbance Because

the MR damper represents the key element in the semiac-tive suspension control system it is essential to model itsnonlinearities and actuation properties with high accuracy todesign model-based controllers even more if the damper isfaulty

Consider an oil leakage on the MR damper in a QoVmodel which induces a lack of forcemodeled as amultiplica-tive fault [33] the faulty force expressed as a reduction of thenominal semiactive force 119865MR is given by

119865MR (119905) = 120572 (119905) 119865MR (119905) (2)

where 119865MR is the reduced damping force and 120572 isin (0 1] isthe fault parameter associated with the oil leakage degree forexample 120572 = 07 means that the damping force will be of70 of 119865MR due to a lost force of 30 Thus if 120572 = 1 thesemiactive damper is free of oil leakages According to thenature of the oil leakage the fault can be considered as abrupt(sudden leakage caused eg by a seal worn out) or gradual(slow leakage caused eg by wear)

To discriminate an operating change in the damping force(eg a change in the damper velocity applied electric currentetc) from a fault a redundancy is needed (analytical orphysical) In this paper the analytical redundancy consists inusing a damper model and a nominal damper model whereboth depend on the same variables except the fault Anotheralternative is using redundant sensors (force velocity accel-eration pressure etc) to make the fault discrimination


Inspired on the parametric damper model of Guo etal [13] which has good balance between performance andcomplexity the damper fault in (2) can be represented by

119865MR (119905) = 120572 (119905) [119868 (119905) 1198911198881205881 (119905) + 1198871def (119905) + 1198872119911def (119905)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119865MR(119905)

= 119865MR (119905) minus 119865120575(119905)

(3)

where 119868 is the electric current to control the MR force and1205881(119905) = tanh[1198861def (119905) + 1198862119911def (119905)] isin [0 1] represents thenonlinearities of the shock absorber For 119868 = 0 119865MR reducesto the passive damping force of the suspension system 119865

120575

is then interpreted as the lack of damping force from thenominal one caused by the oil leakage

By using the degree of leakage 120572 in an MR damper asa varying parameter in the QoV model dynamics whichis estimable by physical features of the shock absorber(pressure sensors accelerometers etc) an LPV observer canbe used to estimate the faulty force 119865

120575instead of using a

parameter estimation algorithm for (3) that is sensitive tonoise measurements

By substituting (3) into the verticalmodel dynamics of (1)a state-space representation of the QoVmodel with a damperleakage can be obtained as

[[[[[[[[

[

119904

119904

119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

=

[[[[[[[[[[[

[

0 1 0 0

minusℎ (120572)

119898119904

minus1205721198871119898119904

ℎ (120572)

119898119904

1205721198871119898119904

0 0 0 1

ℎ (120572)

119898119906119904

1205721198871119898119906119904

minusℎ (120572) + 119896

119905

119898119906119904

minus1205721198871119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

[[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

+

[[[[[[[[[[[

[

0 0

0minus120572 sdot 1205881119891119888

119898119904

0 0

119896119905

119898119906119904

120572 sdot 1205881119891119888119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119861

[

[

119911119903

119868

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906

[

[

119911def

def

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910

= [

[

1 0 minus1 0

0 1 0 minus1]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862

[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]

]

(4)

with ℎ(120572) = 119896119904+ 1205721198872

By defining a generalized LPV plant Σ(120579) of the form of(5) using the faulty system matrices in (4) with 120579 120572 1205881and augmented by the weighting function states as

Σ (120579) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120579) B1 (120579) B2 (120579)C1 (120579) D11 (120579) D12 (120579)

C2 (120579) D21 (120579) D22 (120579)

]]

]

[[

[

120585

119908

119906119888

]]

]

(5)

it is possible to design an LPV119867infin

observer of 119865120575asymptot-

ically stable in all variations of 120579 whose robust performance(estimation error) is defined as a control approach

On considering that only one MR damper is faulty theother three healthy MR dampers can be used for the faultcompensation Using the fault estimation (eg 119865

1205751if the

front-leftMR damper is faulty) it is possible to determine thecompensation forces 119865

119888119895 119895 = 2 3 4 from the vertical vehicle

dynamics in the center of gravity of a 7-DOFmodel such that

119898119904119888119904119888= minus (119865

1198961+119865MR1

minus1198651205751+119865

1198962+119865MR2

+1198651198882+119865

1198963

+119865MR3+119865

1198883+119865

1198964+119865MR4

+1198651198884)

119868119909119909

120579 = (1198651198961+119865MR1

minus1198651205751minus119865

1198962minus119865MR2

minus1198651198882) 119905119891+ (119865

1198963

+119865MR3+119865

1198883minus119865

1198964minus119865MR4

minus1198651198884) 119905119903

119868119910119910

120601 = (1198651198964+119865MR4

+1198651198884+119865

1198963+119865MR3

+1198651198883) 119897119903minus (119865

1198962

+119865MR2+119865

1198882+119865

1198961+119865MR1

minus1198651205751) 119897119891

(6)

where 119865119896119894= 119896

119904119894(119911119904119894minus119911119906119904119894) are the stiffness forces of the springs

and 119865MR119894 with 119894 =isin [1 4] are the nominal MR dampingforces used in a free-fault case and the compensation forcesare specifically used to keep the balance in the vertical vehicledynamics once a damper fails such that

1198651205751minus119865

1198882minus119865

1198883minus119865

1198884= 0 (7a)

minus 1199051198911198651205751minus 1199051198911198651198882+ 1199051199031198651198883minus 1199051199031198651198884= 0 (7b)

1198971199031198651198884+ 1198971199031198651198883minus 1198971198911198651198882+ 1198971198911198651205751= 0 (7c)

that is the equilibrium only depends on the vehicle dimen-sion parameters to make the load distribution

The controlled force with fault adaptive features in aQoV is then composed by a nominal actuation (119868

119899(119865MR)

free-fault case) and by a compensation actuation (119868119888(119865120575) at

presence of faults) Using an LPV controller it is possible todesign anAVCS that fulfills the semiactiveness and saturationconstraints of the damper when the compensation force 119865

119888119895is

added such that the closed-loop system is

= ACL (120588) sdot 119909 +B1CL sdot (119868119899 + 119868119888)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870(120588)sdot119909

+B2CL sdot 119911119903

119910 = CCL (120588) sdot 119909 +D1CL sdot (119868119899 + 119868119888) +D2CL sdot 119911119903

(8)


with 119870(120588) = sum119873

119894=1 120585119894(120588)119870119894 by appropriately choosing thegains 119870

119894 119894 = 1 119873 such that the closed-loop system

(ACLB1CLB2CLCCLD1CLD2CL) is asymptotically stablefor all parameter variations Two varying parameters are usedin the controller design 120588lowast1 represents the semiactiveness ofthe MR damper and 120588

lowast

2 its saturation that depends on themaximum damping force available for the compensation

3 LPV119867infin

Fault Observer

On using the bounded parameter 120572 which represents amultiplicative damper fault in (2) it is possible to design anLPV119867

infinobserver to determine the lack of force caused by

the damper leakage

31 LPVModeling From (4) an LPVmodel structure can beexpressed as

= 119860 (120572) sdot 119909 +119861 (120572 1205881) sdot 119906

119910 = 119862 sdot 119909

(9)

However the model of (9) does not respect the polytopicsolution because the system is parameter dependent on theinput In order to make 119861 parameter independent and get aproper structure for the LPV based controller synthesis [22]themodel in (9) is a state-augmented system by adding a low-pass filter119882filter = 120596

119891(119904+120596

119891)with state119909

119891andmatrices (119860

119891

119861119891 119862119891) such that the new LPV model is given by

[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

= [

119860 (120572) 1198611 (120572 1205881) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119901(1205721205881)

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199011

119911119903

+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199012

119868

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119901

[

119909

119909119891

]

(10)

where 1198611 is the column matrix of 119861 associated with 119868 and 1198612is the column matrix of 119861 associated with the road profile 119911

119903

in (4)

Scheduling Parameters The LPV system in (10) includes 2time-varying parameters assuming that both can be esti-mated directly from measurements or by additional sensorsThe formed polytope represents a quadrilateral polygonwhose vertices are given by the bounded values of 1205881 isin [minus1 1]and 120572 isin (0 1]

By defining the set of four Linear Time-Invariant (LTI)systems the main idea is to design an LPV observer ofthe missing force caused by the damper leakage whoseconvex combination is stable for all trajectories of the varyingparameters by solving an LMI problem

32 LPV119867infin

Observer Design Based on the LPV model of(10) an LPV119867

infinobserver is designed to estimate the damp-

ing force lost by the oil leakage in the shock absorber addingrobustness to unknown road disturbances The frequencyspecification performance is to reduce the estimation errorof the faulty force (119890 = 119865

120575modelminus 119865

120575(119910)) 119882

119890represents the

weighting function used to minimize the estimation error inthe frequency range of interest for the suspension motionwhile119882

119911119903shapes the road irregularities in the observer design

in a frequency band of interest even narrow-banded

119882119903=119870119903120596119903119904

119904 + 120596119903

to add robustness to road disturbances(11)

119882119890=

119870119890(119904

2+ 2120577

11989011205961198901119904 + 120596

21198901)

1199042 + 212057711989021205961198902119904 + 1205962

1198902

to add robustness to noisy measurements

(12)

By considering the filtering specifications the generalizedplant Σ(120572 1205881) used for the synthesis of the 119867

infinobserver is

given by (13) Figure 3 shows the structure of its design

Σ (120572 1205881) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120572 1205881) B1 B2 0Cinfin(120572 1205881) 0 0 D

infin

C 0 C119868

0

]]

]

[[[[

[

120585

119908

119868

119906119888

]]]]

]

(13)

with

A (120572 1205881) = [

119860119901(120572 1205881) 00 119860

119908

]

B1 = [

1198611199011119882119903

0]

B2 = [

1198611199012

0]

C = [

119862119901

00 0

]

C119868= [

01]

(14)

where 120585 = [119909119901

119909119908]119879 such that 119909

119901are the states in the

vertical dynamics of the augmented QoV model of (10) and119909119908

are the vertical weighting functions states Cinfin(120572 1205881)

and Dinfin

represent the controlled output 119911infin

with frequencyspecifications in 119882

119890to minimize the estimation error 119890(119905)

in the frequency band of interest 119910 = [119911def def 119868]119879 is


observer

ldquoControlledrdquooutput

Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr

PLPV(120579) F120575model

y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z

Figure 3 LPV119867infin

observer design in a QoV model to estimatethe faulty force caused by a damper leakage (inspired as a controlapproach)

the measured output 119908 is the unknown disturbance 119868 is theelectric current (manual actuation in open-loop) and 119906

119888=

119906119867infin is the observer output in this case the estimated faulty

force119865120575which is considered as control input into the observer

design Figure 3The LPV119867

infinobserver is given by

[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)

and is quadratically stable by solving an optimization prob-lemwith LMI techniques [42]Theobserver reduces the effectof the road disturbances and avoids drifting in the estimationof the missed force by decreasing asymptomatically the errordynamics

Remark 1 LetΣ(1205791 120579119903) be the generalized plant of an LPVfaulty model with matricesA(1205791 120579119903)B119894

C119895 andD

119894119895as

in (13) where the input matrices are parameter independentas well as the output matrices then it is possible to build aset of LMIs by applying the Bounded Real Lemma To reducethe optimization problem into a finite number of LMIs thesolution is reached at each vertex of the polytope given by(16) by ensuring quadratic and robust stability in the closed-loop system [42 43]

Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)

where 120596119894are the 2119903 vertices of the polytope given by all

combinations of the limit values of the varying parameters

and 120575119894(120579) is the weighting parameter among the LTI systems

defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)

whereC(120596119894)119895is the 119895th component of the vectorC(120596

119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)

By defining the performance criterion as the minimiza-tion of the estimation error of the faulty force (119890 = 119865

120575modelminus

119865120575(119910)) in terms of an 119867

infinproblem the global LPV119867

infin

observer will be the convex combination of 2119903 local observersof the form of (19) by solving the set of LMI problemswhose solution is the extended result from the LTI systemsproposed by [42]

119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)

4 Adaptive Vibration Control System

Once a damper leakage affects the vehicle suspension thefirst step is to determine as soon as possible which cornerhas suffered the lack of force Afterwards this fault must beestimated to compute the required compensation force in thehealthy dampers

By using an LPV119867infin

observer at each vehicle corner itis possible to monitor online the behavior of the four MRdampers If there are no faults on the dampers 119865

120575119894sim 0 and

consequently all compensations forces 119865119888119895

sim 0 for 119894 = 119895However when a fault occurs in one MR damper the faultisolation detection and estimation are naturally possiblebecause the four LPV observers are decoupled thanks to thevehicle dynamics

Since 119865120575119894is not measured but estimated from an LPV

observer using (19) this allows the straightforward computa-tion of the compensation forces 119865

119888119895119895 =119894according to (7a) (7b)

and (7c) such that the full matrix of compensation obtainedby the equilibrium of the load distribution when one MRdamper fails is given by

[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794

Figure 4 Operating region of the scheduling parameters 120588lowast1 and 120588lowast2

For multiple faults that is faults in two or three dampersof the vehicle two aspects must be considered

(1) The equilibrium of the load transfer in the chassisincluding the simultaneous faultsmust be guaranteedusing the compensation force matrix of (20)

(2) Mechanically the MR damper must be designed withenough damping force (plus the nominal force usedin the fault-free case) in order to have faculty toaccommodate the lack of damping force of two ormore faulty actuators

41 Fault Tolerant Control Based on LPV The control ofthe vertical dynamics adaptive to the damper fault can beensured by the LPV control theory to achieve frequencyspecification performances [44] Based on [45] that embedsinto the control design the constraints of dissipativity andsaturation of a semiactive damper with two schedulingparameters here an LPV119867

infincontroller is designed at each

corner for (a) compensation of lack of damping force of afaulty damper (fault accommodation) and (b) performancesof comfort and road holding against road disturbances anduncertainties when a damper leakage occurs

Scheduling Parameters Two varying parameters are used inthe LPV controller synthesis for each corner 120588lowast1 includes thenonlinearities of the damper and is given by

120588lowast

1 = tanh (1198861def + 1198862119911def) sdottanh (119868

119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)

where 1198681198990 and 1198681198880 are the average of electric currents dedicated

for the nominal suspension control and compensation 119868119891=

119868119899+ 119868119888is the electric current (controller output) bounded by

the saturation constraint

0 le 119868min lt 119868119891le 119868max (22)

120588lowast

2 is used directly to saturate the controller output given by

120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)

Figure 4 shows the operating region for both schedulingparameters when the QoV model of one corner is subjectto a chirp road signal with enough frequency content Theoperating region of the polytope is similar to a half ellipsoidnote that there are some areas where the varying parametersare not contained (close to the polytopesP1 andP2) whichcould be used to reduce the conservatism

Because the polytopic LPV controller is defined by thescheduling parameters whose limit values are associatedwiththe minimum and maximum damping force the transientresponse of the damper in an online operation is inside of thepolytope region such that the response speed of the MR fluiddoes not affect the frequency specification performances inthe controller

Note that there is a compromise between the controlperformances and the compensation because 119868

1198990 + 1198681198880 =

119868max2 the capability to compensate for a faulty damper islimited by

119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865

119888sdot coth(1198861def + 1198862119911def )]119891119888 is the inverse

dynamics of the MR damper model (3) with 120572 = 1 whichdepends on the compensation force (20) and 119868

119888max= 2 sdot 119868

1198880is the maximum electric current available for compensationIf the faulty damper requires a greater force than the onethat could be generated by 119868

119888max the fault will not be well

accommodated In the fault-free case 119868119888= 0 and the nominal

suspension controller works inside the saturation constraint[119868119899min

119868119899max

] Figure 5 shows a scheme to represent the electriccurrent used for the compensation and its interaction withthe nominal control actuation

LPV Modeling Formulation Following [45] an LPV con-troller for each healthy MR damper is designed to accommo-date the fault of other dampers The LPV controller is able tohandle the nonlinearity of the semiactive damper model andthe saturation constraint (represented by the maximum limitof the input electric current 119868

119891) The considered LPV model

in this study also is a state-augmented system by adding alow-pass filter to get a proper structure for the LPV controllersynthesis [22] The new LPV QoV model is given by

[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In

Electric current (A)

Forc

e (N

)

Figure 5 Force-electric current diagram in a healthy MR damperused for the adaptive control

with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903

in (4) Note that the matrix 119860 does not depend on 120572 becausethe compensation purpose is based on healthy dampers

LPV119867infinControl DesignThe extension to themethod in [45]

lies into the parameter120588lowast2 that incorporates the damping forceused to compensate the faulty damper The control designsatisfies the actuator constraints Indeed depending on thevalue of the fault the semiactive suspension is adapted tomeet the required robust performances in comfort and roadholding according to the following frequency specifications

119882119903=119870119903120596119903119904

119904 + 120596119903

to add robustness to road disturbances

119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904

to reduce the vertical acceleration of 119898119904in its natural frequency

119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904

to reduce the vertical displacement of 119898119906119904

at high frequencies

(27)

where119882119904is a high-pass filter with cut-off frequency greater

than the natural frequency of the sprung mass 119882119911119906119904

is alow-pass filter with cut-off frequency lower than the naturalfrequency of the unsprung mass and119882

119903can be a first-order

filter to shape the road disturbances or a band pass filter toincrease the sensitivity in a frequency band of interest

Taking into account the control specifications the gener-alized system Σ(120588

lowast

1 120588lowast

2 ) used for the LPV control synthesis isgiven by

Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909

119904are the states in the vertical

dynamics of the augmented QoV model of (25) and 119909119908are

the vertical weighting functions states with dynamics 119860119908

119911infin

= [1199111 1199112]119879 is the controlled output with dynamics C

infin

119910 = [119911def def ]119879 is themeasured output with dynamics119862 that

represents the state feedback and 119906119888= 119906

119867infin is the controlleroutput

Problem Solution The LPV119867infin

controller is synthesized inthe LMI framework for polytopic systems according to [42]all varying parameters are bounded in 120588

lowast

1 isin [minus1 1] and120588lowast

2 isin [0 119868max2] with 119868max = 6A The resulting globalLPV119867

infincontroller is a convex combination of the local

controllers obtained by solving the set of LMIs only on eachvertex of the polytope formed by the limit values of thevarying parameters Since 2 varying parameters are usedeach QoV considers a polytope of 4 vertices that is 4 localLTI controllers are obtained such that

119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)

The controller stability is guaranteed for all trajectories ofthe varying parameters even if an extra compensation forceis demanded by solving the LMI problem at each vertexexpressed by the Bounded Real Lemma [42 46] as

[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)


5 Simulation Results

The evaluation of the proposed AVCS for MR damperfaults has been separated into two sections (a) firstly theperformance of the fault observer is analyzed using differentsimulation tests and then it is compared with respect toa parameter estimation method considering noise on themeasurements (b) secondly the fault tolerant controller isevaluated when one damper fails and the other three MRdampers are used for the fault compensation

51 Estimation of MR Damper Faults To evaluate the per-formance of the LPV119867

infinobserver the front-left corner of

a medium payload pickup truck has been used as test-bedFigure 2 whose experimental parameters obtained by a test ofkinematics and compliance are 119898

119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm

Experimental data obtained from a commercial MRdamper are used to model the nonlinearities of this actuatorby considering 120572 = 1 in (3) according to the methodology ofcharacterization of MR dampers in [34] De-J Lozoya-Santoset al [34] present a discussion about the required experimen-tation to characterize MR dampers exclusively to describethe physical phenomena or according to a target application(control design purposes study of the vehicle dynamicsmechanical configuration etc) In this study the usedexperiments were chosen to describe the different physicalMR phenomena of the shock absorber (stiffness nonlinearviscous damping friction and saturation) in the full range offrequencies of interest with common rod displacements in acarTheDesign of Experiments (DoE) used to characterize thehysteresis loops and nonlinear transient response of the MRdamping force includes different sequences of displacement(suspension deflection) and electric current (manipulationvariable) in a composite design

The commercial MR damper used to perform differentexperimental tests of identification was manufactured byDelphi MagneRide for SUVs The range of force is plusmn8000N(peak to peak) the stroke is around 40mm with a timeconstant of 14ms and settling time of 84ms This shockabsorber has continuous actuation (from 0 to 6A) and con-siderable hysteresis at high frequencies with high deflectionsFigure 6(a) shows the FV curve of the MR damper using anexperiment with excitation of displacement up to 15Hz atdifferent 119868 values note that the yield stress of the MR fluidreaches the saturation of 8000N when 119868 = 6A and thefrequency of motion is 15Hz

Figure 6(b) shows the performance of the MR dampermodel used in the simulations (3) whose parameters wereobtained from a standard ISO road profile type D119891

119888= 6009

1198861 = 378 1198862 = 221 1198871 = 28308 and 1198872 = minus78972In the simulation tests to evaluate the FDI module based

on the LPV observer the road profile is composed of a chirpsignal from 0 to 8Hz Gaussian noise is assumed in the119911def and def measurements Two fault scenarios have beenstudied

(i) An abrupt damper leakage from 119905 = 5 to 10 s with20 of lack of force (120572 = 08) and from 119905 = 13 to 18 s

0 1

0

4000

8000

Forc

e (N

)

Velocity of deflection (ms)minus1

minus8000

minus4000

(a) Experimental damping force

0 005 01

0200400600800

1000

Velocity of deflection (ms)

Forc

e (N

)

Experimental data(ISO 8608 road profile)

Modelminus200minus400minus600minus800minus1000

minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A

(modeling error = 17)

(b) MR damper model performance

Figure 6 (a) Experimental data of the commercial MR damperwhen the damper rod is excited by a chirp signal from 05 to 15Hzat different stepped values of electric current from 0 to 6A (b) MRdamper modeling for different 119868 values when 120572 = 1

with 50 of lack of force (120572 = 05) during whole testthe electric current on the MR damper is constant119868 = 1A This test studies the property of the LPVobserver to schedule the fault estimation according tothe 120572 value

(ii) A gradual damper leakage from 0 of lack of force(120572 = 1) at 119905 = 5 s until 50 of lack of force (120572 = 05)at 119905 = 20 s in this test the electric current is a randomsignal from 0 to 25 A that mimics a controller outputon the MR damper This test studies the transientresponse of the fault estimation and its insensitivityto the manipulation variable

The robust fault estimation of the LPV observer wascompared with respect to a static fault estimation using (3)which depends on the reliability of themodel parameters andit can be affected by the noise in themeasurements of 119911def anddef

Figure 7 shows the performance of the LPV119867infinobserver

in both tests Note that the transient response of the faultestimation with the LPV observer is robust to the noiseof measurements however the static estimation is noisyat all frequencies even when the damper is free of faultsIndeed the fault estimation with the static equation has moreamplitude than the correct one as the frequency increases

Because the transient response of the LPV observerrepresents correctly the implemented fault in both tests theproposed observer can estimate abrupt or gradual damperleakages of any magnitude for all trajectories bounded by120572 that belong to a convex LMI solution Also Figure 7illustrates that the LPV119867

infinobserver is robust to road


0 2 4 6 8 10 12 14 16 18 20

0123

Force free of faults

Static estimation (equation)

Robust estimation (LPV observer)

Nonfault 120572 = 10 Faulty force 120572 = 08 120572 = 10 120572 = 10Faulty force 120572 = 05Forc

e (Ntimes10

3)

minus1minus2minus3

Time (s)

(a) Test with abrupt damper leakages

Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force

Force free of faults Static estimation (equation)Robust estimation (LPV observer)

Nonfault 120572 = 10120572 = 10

minus1minus2minus3minus4

Forc

e (Ntimes10

3)

05

(b) Test with a gradual damper leakage

Figure 7 Fault estimation performance with the LPV119867infinobserver

disturbances (road profile at different sinusoidal frequenciesand amplitudes) and the fault estimation is insensitive to thevalue of electric current on the MR damper that is a closed-loop implementation does not affect the FDI performanceIndeed 119868 is only a gain in the semiactive force such that

119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)

where both terms in right side are dependent on 119868 and itsdifference has only dependence on 120572

A correct estimation of 119865120575in the FDI module avoids

unrealistic control reconfigurations or fault accommodationsfor fault tolerant control purposes For instance the staticestimation of the fault in Figure 7 overshoots around 800Nfrom the real fault in some frequencies this incorrect faultinformation will demand more damping force of compen-sation than the required one by the actuator or by externalactuators (other healthy dampers in the vehicle) to reduce thefault effect

52 Control Adaptation for MR Damper Faults When an oilleakage occurs in one of the dampers and the car is movingthe lack of damping force increases the vertical vehicle bodymotion To reduce the fault propagation into the verticaldynamics the obtained FDI information is used by the AVCSto compensate for the faulty damper and thus maintain thecomfort in the vehicle

Figure 8 schematizes the proposed AVCS in particularthe interaction between the FDI module and the LPVcontroller Once a damper leakage is detected and estimatedthe load distribution analysis determines the compensationforce that the healthy dampers must deliver Finally at eachcorner the LPV based controller is adapted to accommodatethe fault through 119865

119888119894 fulfilling the MR damper constraints

(semiactiveness and saturation)A vehicle CarSim model was used to evaluate the pro-

posed AVCS the suspension model was characterized from

real data curves (camber angles caster angles dampingforce jouncerebound stops spring tire stiffness etc) struc-tural parameters (mechanical ratios compliance coefficientsKingpin geometry etc) and physical dimensions (weightlength width height wheel base front and rear track andso on) obtained by a KampC test on a commercial pickup truckof medium payload Figure 9 illustrates some of the physicalparameters used in the vehicle model characterization inCarSim more parameters and data curves used in the vehiclecustomization are in [33]

In CarSim the model was customized as a generic full-size load vehiclemodel which is used as Software-in-the-Loop(SiL) in a MatlabSimulink environment The customizationof the suspension system considers independent wheel sta-tions at the front side and a rear solid axle at the back SinceCarSim is well accepted in the automotive control community(research and manufacturing) the customized model is avery useful tool to analyze the vehicle dynamics and evaluatethe performance of the proposed control prototype

Two simulation tests have been used to evaluate theperformance of the AVCS In both tests an oil leakage with120572 = 05 at 119905 = 0 has been implemented in the MR damper ofthe front-left corner of the vehicle

Test 1 Two Bumps in SeriesThe vehicle is driven straightfor-wardly at constant speed of V

119909= 30KmhThe road profile is

composed of two bumps in series of 5 cmheight Figure 10(a)This test allows studying the AVCS performance when thevehicle is driven over abrupt (sudden) road disturbances

Figure 11 shows the performance of the proposed AVCSversus an uncontrolled suspension (US) considered as apassive system and versus a controlled suspension that doesnot include the adaptation strategy to the fault (LPV nominalcontrol LPVNC) SFF (Suspension Free of Faults) refers to thesimulation of the LPV119867

infinsuspension control when no fault

is considered By using the Root Mean Square (RMS) value ofthe pitch rate ( 120601) sprung mass acceleration (

119904119888) and heave

(119911119904119888) motion of the vehicle as comfort performance indexes

normalized with respect to the uncontrolled suspension theproposed AVCS improves 120601 28

11990411988817 and 119911

1199041198889 Note

that the LPV nominal control has better comfort perfor-mance than US however the AVCS improves the comfort upto 10when a damper fault occurs in comparison to the LPVNC

Test 2 Bounce Sine Sweep (BSS) Test Chirp road sequencewith decreasing road elevation (10 to 1 cm) and span offrequency (05 to 10Hz) the vehicle velocity is 30KmhFigure 10(b) This test allows the excitation of the automobilesuspension system in the frequency band of interest forride comfort The test allows the monitoring of the vehiclebehavior at each particular frequency in contrast to a randomstandard road profile (ISO 8608)

Figure 12 illustrates the pitch angle and the chassis verticalacceleration as performance indexes of comfort In bothsignals the AVCS has better comfort performance thanthe LPV NC improvement of 24 in the pitch angle and10 in the chassis vertical acceleration according to theRMS index With respect to the US the AVCS increases


zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )

120588lowast1 120588lowast2

I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim

Load distributionanalysis

120572

zr

Adaptive control in front-le corner

Adaptive control in front-right corner

Adaptive control in rear-le corner


Figure 8 Adaptive suspension control system for MR damper faults

the passenger comfort up to 64 Indeed in both signalsthe AVCS presents similar comfort performance than the SFFcase see bar graphs in Figure 12 Qualitatively the US has theworst comfort performance below the 15 s (from 05 to 5Hz)that is around the first frequency of resonance of the chassisFigure 12(b) shows that the main differences between AVCSand the nominal control are in the resonance frequenciesthat is the AVCS seeks to keep as much as possible thesame vibrations like the SFF system mainly in the naturalfrequencies of the suspension system

Figure 13(a) shows how the MR damping force increasesin a healthy damper for example at the front-right actuatorto compensate for the missing force of the faulty damperThe semiactiveness constraint of the MR damper is thenensured with the proposed AVCS Figure 13(b) shows thatthe nominal LPV controller whose performance is goodwhen the suspension is free of faults operates between 0 and2A without the compensation when the compensation isused the LPV controller utilizes the full range of actuationto generate more damping force and satisfies the saturationconstraint of the MR damper

Remark 2 The most important restriction to implement theproposedAVCS approach is the extra damping force availableto compensate the lack of force caused by oil leakagesMechanically the semiactive damper must be designed withenough damping force (plus the nominal force used in thefault-free case) in order to have faculty to accommodate thelack of force of other faulty actuators

In an implementation the fault estimation and itscompensation depend on the sample frequency Usuallythe most important bandwidth of control in automotivesuspensions is from 0 to 25Hz such that the micro-controller must use at least 100Hz as sample frequencyAlthough at high car velocities the frequency of motion isgreater than 20Hz at these frequencies the vehicle bodyfilters the road disturbances such that there is no differ-ence in the vertical vehicle dynamics when a damper isfaulty Indeed the fault estimation is frequency dependentsuch that the best frequency to estimate this malfunctionis around the resonance frequency of the sprung mass[47]


Height foranimator

animator

1722

2000

Width for

Left Right260 260

Sprung masscoordinate system

1560

760

0 Lateral coordinate sprung mass center

3670

4910

All dimensions and coordinatesare in millimeters

Lateral coordinateof hitch

208

Rear solid axle suspension

1700

Sprung mass origin

Front independent suspensionUnsprung mass (both sides)

Fraction steered (0-1)163

08(kg)

Spin inertia for each sideLeft Right11 11

1700

Wheel centersDimensions are in

Sprung mass origin

CG

260

260 260Left Right

Mass center of sprung mass 0

x

millimeters

mdash

(kg-m2)

z

Figure 9 Parameters for the pickup vehicle model in CarSim

(a) Two bumps in series (b) BSS test

Figure 10 Simulation tests in CarSim used in the evaluation of the AVCS

6 Conclusions

Based on a full vehicle suspension model a new Adap-tive Vibration Control System (AVCS) based on the LinearParameter-Varying (LPV) control theory has been presentedfor a magnetorheological (MR) damper leakage The mainidea is to use the damping force of the remaining threehealthy shock absorbers to avoid the fault propagation intothe vertical dynamics of the vehicle the control algorithmis two levels Firstly the faulty Quarter of Vehicle (QoV) isisolated and the lack of force is estimated Secondly the loaddistribution analysis determines the compensation force thateach healthy MR damper must add to reduce the effect ofthe lack of force of the faulty damper This AVCS is basedon the LPV framework to include the semiactiveness andsaturation constraints of an MR damper into the controllerdesign

Inspired on the 119867infin

control problem an LPV robustobserver is proposed to estimate the missed force caused byan MR damper leakage the polytopic LPV approach is usedto solve a finite number of linearmatrix inequalitieswhere theleakage degree is a varying parameter together with anothervarying parameter used to represent the MR nonlinearitiesSimulation results show that this fault observer has goodperformance with robustness to noisy measurements androad disturbances in contrast with a static equation used tomodel the damping force obtained from the least squaresestimation algorithm The estimation error of the fault wasdecreased up to 20 in some frequencies using the faultobserver Indeed the LPV observer can diagnose and esti-mate the faulty force caused by an abrupt or gradual damperleakage without considering the degree of the oil leakage

Because the proposed Fault Detection and Isolation (FDI)system is based on a QoV model four LPV robust observers


5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z

(b) Sprung mass acceleration

0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC

(c) Sprung mass position

Figure 11 Performance of the AVCS and comparison with other suspension systems in test 1 by using a normalized RMS with respect tothe uncontrolled suspension

can be used to develop a global FDI strategywith independentcornersWhen a fault occurs in one vehicle corner the associ-ated observer estimates the lack of force caused by the damperleakage which is used to compute the compensation forcesthat the healthy dampers must add Simulation results in

CarSim used as Software-in-the-Loop show that when an oilleakage in the front-left MR damper is present the proposedAVCS improves the comfort up to 24 and 64 with respectto a nominal controlled (without fault compensation) anduncontrolled suspension system respectively


0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)

LPV control withcompensation

LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)

LPV control with compensation

LPV nominal control

Time (s)

(b) Controller output

Figure 13 (a) Semiactive damping force in the front-right corner with and without fault compensation and (b) the corresponding controlleroutput


Abbreviations

120572 Ratio of the faulty force and thenominal force

120577119894 Damping factor of a second order filter

(weighting function)120579 120579 and 120579 Roll angle roll rate and roll

acceleration (rad rads and rads2)1205881 Nonlinear part of the damper model120601 120601 and 120601 Pitch angle pitch rate and pitch

acceleration (rad rads and rads2)Σ(120579) Generalized LPV plant in an119867

infin

control problem120596119894 Cut-frequency of the filter (weighting

function) (rads)1198861 1198862 Preyield viscous damping coefficients of

Guomodel (sm 1m)1198871 Viscous damping coefficient of Guo

model (Nsm)1198872 Stiffness coefficient of Guomodel

(Nm)119890 Estimation error of the fault (N)119891119888 Dynamic yield force of Guomodel (N)

119865120575 Loss of force (N)

119865MR Force of an MR (healthy) damper (N)119865MR Force of a faulty MR damper (N)119868 Control effort of an MR damper

(electric current) (A)119868119909119909 119868119910119910 Roll and pitch inertia (Kgsdotm2)

119870(120588) LPV controller119896119904 Spring stiffness in a QoV (Nm)

119896119905 Tire stiffness in a QoV (Nm)

119871(120579) LPV fault observer119897119891 Distance from center of gravity (CoG) to

front track (m)119897119903 Distance from CoG to rear track (m)

119898119904 Sprung mass in a QoV (Kg)

119898119904119888 Total sprung mass (chassis mass) (Kg)

119898119906119904 Unsprung mass in a QoV (Kg)

119905119891 Distance from CoG to front-left tire (m)

119905119903 Distance from CoG to rear-left tire (m)

V119909 Longitudinal vehicle velocity (ms

Kmh)119882119894 Weighting function

119911def def Displacement and velocity of thedamper piston in a QoV (m ms)

119911infin Controlled variables in an119867

infincontrol

problem119911119903 Road profile (m)

119911119904 119904 and

119904 Sprung mass displacement velocity and

acceleration in a QoV (m ms andms2)

119911119904119888 119904119888 and

119904119888 Chassis displacement velocity and

acceleration in the CoG (m ms andms2)

119911119906119904 119906119904 and

119906119904 Unsprung mass displacement velocityand acceleration in a QoV (m ms andms2)

Conflict of Interests

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

Acknowledgments

The authors thank CONACyT (PCP Project 032010) for itspartial support Also the authors thank Dr Jorge de JesusLozoya-Santos for sharing his experience and knowledge toimprove this paper along with some discussions they thankhim for his time

References

[1] J Dixon The Shock Absorber Handbook Wiley 2nd edition2007

[2] B T Fijalkowski Automotive Mechatronics Operational andPractical Issues vol 2 Springer 2011

[3] J Lozoya-Santos Control of automotive semiactive suspensions[PhD thesis] Tecnologico de Monterrey Campus Monterrey2013 chapter 2ndash4

[4] T Gillespie Fundamentals of Vehicle Dynamics chapter 5Society of Automotive Engineers 1992

[5] R Stanway J L Sproston and N G Stevens ldquoNon-linearmodelling of an electro-rheological vibration damperrdquo Journalof Electrostatics vol 20 no 2 pp 167ndash184 1987

[6] D R Gamota and F E Filisko ldquoDynamic mechanical studies ofelectrorheological materials moderate frequenciesrdquo Journal ofRheology vol 35 no 3 pp 399ndash425 1991

[7] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997

[8] N M Wereley L Pang and G M Kamath ldquoIdealized hys-teresis modeling of electrorheological and magnetorheologicaldampersrdquo Journal of Intelligent Material Systems and Structuresvol 9 no 8 pp 642ndash649 1998

[9] G Yang B F Spencer Jr J D Carlson and M K Sain ldquoLarge-scale MR fluid dampers modeling and dynamic performanceconsiderationsrdquo Engineering Structures vol 24 no 3 pp 309ndash323 2002

[10] S-B Choi and S-S Han ldquo119867infin

control of electrorheologi-cal suspension system subjected to parameter uncertaintiesrdquoMechatronics vol 13 no 7 pp 639ndash657 2003

[11] D Guo and H Hu ldquoNonlinear stiffness of a magneto-rheological damperrdquoNonlinearDynamics vol 40 no 3 pp 241ndash249 2005

[12] L X Wang and H Kamath ldquoModelling hysteretic behaviour inmagnetorheological fluids and dampers using phase-transitiontheoryrdquo Smart Materials and Structures vol 15 no 6 pp 1725ndash1733 2006

[13] S Guo S Yang and C Pan ldquoDynamical modeling of magneto-rheological damper behaviorsrdquo Journal of Intelligent MaterialSystems and Structures vol 17 pp 3ndash14 2006

[14] NMKwokQ PHa THNguyen J Li and B Samali ldquoA novelhysteretic model for magnetorheological fluid dampers andparameter identification using particle swarm optimizationrdquoSensors and Actuators A Physical vol 132 no 2 pp 441ndash4512006


[15] S Cesmeci and T Engin ldquoModeling and testing of a field-controllable magnetorheological fluid damperrdquo InternationalJournal of Mechanical Sciences vol 52 no 8 pp 1036ndash10462010

[16] V S Atray and P N Roschke ldquoDesign fabrication testingand fuzzy modeling of a large magnetorheological damper forvibration control in a railcarrdquo in Proceedings of the IEEEASMEJoint Rail Conference pp 223ndash229 Chicago Ill USA April2003

[17] K K AhnM A Islam andDQ Truong ldquoHysteresismodelingof magneto-rheological (MR) fluid damper by self tuningfuzzy controlrdquo in Proceedings of the International Conference onControl Automation and Systems (ICCAS rsquo08) pp 2628ndash2633October 2008

[18] A C Shivaram and K V Gangadharan ldquoStatistical modelingof a magneto-rheological fluid damper using the design ofexperiments approachrdquo Smart Materials and Structures vol 16no 4 pp 1310ndash1314 2007

[19] S-B Choi S-K Lee and Y-P Park ldquoA hysteresis model forthe field-dependent damping force of a magnetorheologicaldamperrdquo Journal of Sound andVibration vol 245 no 2 pp 375ndash383 2001

[20] K-S Hong H-C Sohn and J K Hedrick ldquoModified skyhookcontrol of semi-active suspensions a new model gain schedul-ing and hardware-in-the-loop tuningrdquo Journal of DynamicSystems Measurement and Control vol 124 no 1 pp 158ndash1672002

[21] H Du K Y Sze and J Lam ldquoSemi-active 119867infin

control ofvehicle suspensionwithmagneto-rheological dampersrdquo Journalof Sound and Vibration vol 283 no 3ndash5 pp 981ndash996 2005

[22] C Poussot-Vassal O Sename L Dugard P Gaspar Z Szaboand J Bokor ldquoA new semi-active suspension control strategythrough LPV techniquerdquo Control Engineering Practice vol 16no 12 pp 1519ndash1534 2008

[23] C-C Chang and L Zhou ldquoNeural network emulation ofinverse dynamics for a magnetorheological damperrdquo Journal ofStructural Engineering vol 128 no 2 pp 231ndash239 2002

[24] S M Savaresi S Bittanti and M Montiglio ldquoIdentificationof semi-physical and black-box non-linear models the case ofMR-dampers for vehicles controlrdquoAutomatica vol 41 no 1 pp113ndash127 2005

[25] H Du J Lam and N Zhang ldquoModelling of a magneto-rheological damper by evolving radial basis function networksrdquoEngineering Applications of Artificial Intelligence vol 19 no 8pp 869ndash881 2006

[26] M Zapateiro N Luo H R Karimi and J Vehı ldquoVibrationcontrol of a class of semiactive suspension system using neuralnetwork and backstepping techniquesrdquoMechanical Systems andSignal Processing vol 23 no 6 pp 1946ndash1953 2009

[27] H Metered P Bonello and S O Oyadiji ldquoThe experimentalidentification ofmagnetorheological dampers and evaluation oftheir controllersrdquoMechanical Systems and Signal Processing vol24 no 4 pp 976ndash994 2010

[28] M J L Boada J A Calvo B L Boada and V Dıaz ldquoModelingof a magnetorheological damper by recursive lazy learningrdquoInternational Journal of Non-LinearMechanics vol 46 no 3 pp479ndash485 2011

[29] J C Tudon-Martınez J J Lozoya-Santos R Morales-Menendez and R A Ramirez-Mendoza ldquoAn experimentalartificial-neural-network-based modeling of magneto-rheological fluid dampersrdquo Smart Materials and Structures vol21 no 8 Article ID 085007 pp 1ndash16 2012

[30] Sachs ldquoShock absorbers leak potential causesrdquo Tech RepSachs 2008

[31] S Fergani O Sename and L Dugard ldquoA LPVHinfinfault tolerant

control of vehicle roll dynamics under semi-active dampermalfunctionrdquo inProceedings of the AmericanControl Conference(ACC rsquo14) pp 4482ndash4487 IEEE Portland Ore USA June 2014

[32] J C Tudon-Martınez S Varrier O Sename R Morales-Menendez J-J Martinez and L Dugard ldquoFault tolerant strat-egy for semi-active suspensions with LPV accommodationrdquo inProceedings of the 2nd International Conference on Control andFault-Tolerant Systems (SysTolrsquo 13) pp 631ndash636 Nice FranceOctober 2013

[33] J Tudon-Martınez Fault tolerant control in automotive semi-active suspensions [PhD thesis] Tecnologico de MonterreyCampus Monterrey Monterrey Mexico 2014

[34] J De-J Lozoya-Santos R Morales-Menendez R Ramirez-Mendoza J C Tudon-Martinez O Sename and L DugardldquoMagnetorheological dampermdashan experimental studyrdquo Journalof Intelligent Material Systems and Structures vol 23 no 11 pp1213ndash1232 2012

[35] A Yetendje M Seron and J De Dona ldquoDiagnosis andactuator fault tolerant control in vehicle active suspensionrdquo inProceedings of the 3rd International Conference on Informationand Automation for Sustainability (ICIAFS rsquo07) pp 153ndash158Melbourne Australia December 2007

[36] A Chamseddine and H Noura ldquoControl and sensor faulttolerance of vehicle active suspensionrdquo IEEE Transactions onControl Systems Technology vol 16 no 3 pp 416ndash433 2008

[37] P Gaspar Z Szabo and J Bokor ldquoLPV design of fault-tolerantcontrol for road vehiclesrdquo in Proceedings of the 1st Conferenceon Control and Fault-Tolerant Systems (SysTol rsquo10) pp 807ndash812Nice France October 2010

[38] D Fischer and R Isermann ldquoMechatronic semi-active andactive vehicle suspensionsrdquoControl Engineering Practice vol 12no 11 pp 1353ndash1367 2004

[39] J Kim and H Lee ldquoSensor fault detection and isolationalgorithm for a continuous damping control systemrdquo Journal ofAutomobile Engineering vol 225 no 10 pp 1347ndash1364 2011

[40] HWang andG Song ldquoFault detection and fault tolerant controlof a smart base isolation system with magneto-rheologicaldamperrdquo Smart Materials and Structures vol 20 no 8 ArticleID 085015 9 pages 2011

[41] Y Vidal L Acho F Pozo and J Rodellar ldquoFault detectionin base-isolation systems via a restoring force observerrdquo inProceedings of the Conference on Control and Fault-TolerantSystems (SysTol rsquo10) pp 777ndash782 IEEE Nice France October2010

[42] C Scherer P Gahinet and M Chilali ldquoMultiobjective output-feedback control via LMI optimizationrdquo IEEE Transactions onAutomatic Control vol 42 no 7 pp 896ndash911 1997

[43] P Gahinet P Apkarian and M Chilali ldquoAffine parameter-dependent Lyapunov functions and real parametric uncer-taintyrdquo IEEE Transactions on Automatic Control vol 41 no 3pp 436ndash442 1996

[44] S Savaresi C Poussot-Vassal C Spelta O Sename and LDugard Semi-Active Suspension Control for Vehicles ElsevierButterworth-Heinemann 2010

[45] A-L Do O Sename and L Dugard ldquoLPV modeling andcontrol of semi-active dampers in automotive systemsrdquo inControl of Linear ParameterVarying SystemswithApplications JMohammadpour and CW Scherer Eds pp 381ndash411 SpringerNew York NY USA 2012


[46] P Apkarian and P Gahinet ldquoA convex characterization of gain-scheduled 119867

infincontrollersrdquo IEEE Transactions on Automatic

Control vol 40 no 5 pp 853ndash864 1995[47] J Lozoya-Santos J Tudon-Martınez R Morales-Menendez R

Ramırez-Mendoza and L Garza-Castanon ldquoA fault detectionmethod for an automotive magneto-rheological damperrdquo inProceedings of the 8th IFAC International Symposium on FaultDetection Supervision and Safety for Technical Processes pp1209ndash1214 Mexico City Mexico 2012

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



presented in [3] Indeed without considering the natureof the control strategy the interesting challenges of thesemiactive suspension control problem are as follows [4]

(1) Realistic design according to the nonlinearities of thesemiactive damper

(2) Constrained control according to the dissipativityand saturation of the damper to avoid impracticalsolutions even if this actuator is partially faulty

(3) Multiobjective control design that is capable ofimproving the vehicle comfort with no compromise(and perhaps enhancement) for the road holding

(4) Feasibility of implementation with adaptation to dif-ferent damper technologies small number of controlinputs and low computing cost to be programmed inmicrocontrollers

An extensive work on semiactive suspension systemsemphasizes that MR dampers are one of the best solutionsto manage the compromise between the objectives of anautomotive suspension control comfort and road holdingMR dampers are as fast as the EH actuators with less powerrequirement than ER dampers and higher robustness totemperature variations they are easier to manufacture andimplement than EH dampers and to control than PA Indeedmost of the existing work on semiactive damper modelingand control is based on the MR damping technology [1]

Several mathematical models are available to representthe nonlinear behavior of MR dampers In general theycan be grouped as parametric and nonparametric modelsParametric models explain the physical phenomena in math-ematical expressions for instance the classical viscoelastic-plastic models [5 6] the phenomenological evolutive Bouc-Wenmodels [7ndash12] and the algebraic models [13ndash15] On theother hand coefficients of nonparametric structures do nothave a physical meaning for instance fuzzy models [16 17]statistical models [18] polynomial models [19ndash22] or modelsbased on artificial neural networks (ANN) [23ndash29] Usuallyall aforementioned structures have presented good modelingresults andor good properties to design control modules butwhen the damper is faulty nothing can be said

The most common fault in MR dampers is oil leakage[1] There are different aspects that cause a damper leakage[30] which are related to bad installation excessive usedamage by external devices or bad design of control systemthat saturates the manipulation The direct consequence ofa faulty shock absorber is the loss of damping force thatcauses a performance reduction in the suspension capabilities(comfort and safety for passengers) at any driving situationRecently the Linear Parameter-Varying (LPV) control theoryhas been extended to fault tolerant control (FTC) systems tomaintain the desired control goals as much as possible whena damper fault occurs A scheduling parameter based on theload transfer is used to keep the equilibrium (comfort) whena semiactive damper fails in [31] in [32] a supervisory systemcomputes the compensation force that each healthy dampermust add to reduce the fault effect considering the semiactiveconstraints

Recently Tudon-Martınez [33] studied the effect of the oilleakage degree in the damping force using a commercial auto-motive shock absorber Several degrees of oil leakage wereimplemented to analyze how the dynamic behavior of theMRdamper is modified through the characteristic curves force-velocity (FV) and force-displacement (FD) Figure 1 showsthat the friction effect is reduced proportionally to the oilleakage degree the linear stiffness also is reduced when theoil leakage increases the nonlinearity in the viscous dampingis modified and evidently the saturation force decreasesinversely proportional to the damper leakage

Experimental data of a Design of Experiments (DoE)with central composite design [34] illustrate that the damperleakage can be represented as a fault model of multiplicativenature [33] By using an extension of a parametric modelstructure for instance the algebraic model of Guo et al[13] a faulty MR damper can be modeled by a parameteridentification algorithm as a static equation In this modelrepresentation the nominal parameters do not change onlythe fault parameter modifies the force characteristics

Because the parameter estimation approaches are sensi-tive to noise measurements and their online adaptation canrequiremany computational resources a robust estimation ofthe faulty force is essential Indeed the accuracy of the faultestimation not only determines the damper condition thisinformation can be used also to design Adaptive VibrationControl Systems to accommodate the lack of force caused bythe faulty MR damper [32 35ndash37]

Most of the work in Fault Detection and Isolation (FDI)modules for automobile suspensions is based on analyticalredundancy such as the parity space theory [38 39] robustfiltering [40] and nonlinear-robust observers [35 36 41]The above approaches perform well for sensor faults or evenfor actuator faults considering active dampers that do nothave any constraint on the force characteristics in contrastto an MR damper whose semiactiveness friction and forcesaturation constraint the actuator properties

An Adaptive Vibration Control System (AVCS) based on arobust LPV observer used to estimate faults in anMRdamperis proposed in this paper By using an 119867

infindesign the LPV-

AVCS system is robust to the road disturbances and noisymeasurements The LPV approach enables embedding thenonlinearities of the vehicle dynamics into varying param-eters such that the observer becomes linearly parameterdependent with internal stability and robust performanceThe inclusion of the faulty damper constraints as schedulingparameters into the LPV119867

infinobserver makes the fault

estimation fulfill the semiactiveness of the shock absorber ina practical implementationTheAVCS based also on the LPVframework uses the fault estimation to compute the neededcompensation force in the healthyMR shock absorbers of thevehicle to reduce the effect of lack of force caused by the faultydamper (comfort deterioration)

The outline of this paper is as follows in the next sectionthe problem statement is formulated Section 3 details thedesign of the FDI system used to diagnose damper faults andSection 4 describes the AVCS Section 5 illustrates the casestudy to validate the proposed FDI strategy under differenttests and discusses the results in the AVCS Concluding


Displacement (m)

Forc

e (N

)

0 0005 001 0015

0

200

400

600

800

Friction

StiffnessStiffness

minus200




Fault 2

Free of faults

Fault 1


Saturation


Forc

e (N

)

0

200

400

600

800

minus200

minus400



ms zs

FMR

mus

ks

ktzr

zus



2 Problem Statement


119898119904119904(119905) = minus 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119865MR (119905)

119898119906119904119906119904(119905) = 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119896

119905[119911119906119904(119905) minus 119911

119903(119905)]

+ 119865MR (119905)

(1)







119904(119911119906119904) while




119865MR (119905) = 120572 (119905) 119865MR (119905) (2)





119865MR (119905) = 120572 (119905) [119868 (119905) 1198911198881205881 (119905) + 1198871def (119905) + 1198872119911def (119905)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119865MR(119905)

= 119865MR (119905) minus 119865120575(119905)

(3)


120575






[[[[[[[[

[

119904

119904

119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

=

[[[[[[[[[[[

[

0 1 0 0

minusℎ (120572)

119898119904

minus1205721198871119898119904

ℎ (120572)

119898119904

1205721198871119898119904

0 0 0 1

ℎ (120572)

119898119906119904

1205721198871119898119906119904

minusℎ (120572) + 119896

119905

119898119906119904

minus1205721198871119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

[[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

+

[[[[[[[[[[[

[

0 0

0minus120572 sdot 1205881119891119888

119898119904

0 0

119896119905

119898119906119904

120572 sdot 1205881119891119888119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119861

[

[

119911119903

119868

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906

[

[

119911def

def

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910

= [

[

1 0 minus1 0

0 1 0 minus1]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862

[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]

]

(4)

with ℎ(120572) = 119896119904+ 1205721198872


Σ (120579) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120579) B1 (120579) B2 (120579)C1 (120579) D11 (120579) D12 (120579)

C2 (120579) D21 (120579) D22 (120579)

]]

]

[[

[

120585

119908

119906119888

]]

]

(5)





1205751if the




119898119904119888119904119888= minus (119865

1198961+119865MR1

minus1198651205751+119865

1198962+119865MR2

+1198651198882+119865

1198963

+119865MR3+119865

1198883+119865

1198964+119865MR4

+1198651198884)

119868119909119909

120579 = (1198651198961+119865MR1

minus1198651205751minus119865

1198962minus119865MR2

minus1198651198882) 119905119891+ (119865

1198963

+119865MR3+119865

1198883minus119865

1198964minus119865MR4

minus1198651198884) 119905119903

119868119910119910

120601 = (1198651198964+119865MR4

+1198651198884+119865

1198963+119865MR3

+1198651198883) 119897119903minus (119865

1198962

+119865MR2+119865

1198882+119865

1198961+119865MR1

minus1198651205751) 119897119891

(6)

where 119865119896119894= 119896



1198651205751minus119865

1198882minus119865

1198883minus119865

1198884= 0 (7a)


1198971199031198651198884+ 1198971199031198651198883minus 1198971198911198651198882+ 1198971198911198651205751= 0 (7c)



119899(119865MR)



119888119895is


= ACL (120588) sdot 119909 +B1CL sdot (119868119899 + 119868119888)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870(120588)sdot119909

+B2CL sdot 119911119903


(8)


with 119870(120588) = sum119873




lowast


3 LPV119867infin

Fault Observer



the damper leakage


= 119860 (120572) sdot 119909 +119861 (120572 1205881) sdot 119906

119910 = 119862 sdot 119909

(9)


119891(119904+120596

119891)with state119909


119891


[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

= [

119860 (120572) 1198611 (120572 1205881) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119901(1205721205881)

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199011

119911119903

+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199012

119868

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119901

[

119909

119909119891

]

(10)


119903

in (4)



32 LPV119867infin





120575(119910)) 119882





119882119903=119870119903120596119903119904

119904 + 120596119903


119882119890=

119870119890(119904

2+ 2120577

11989011205961198901119904 + 120596

21198901)

1199042 + 212057711989021205961198902119904 + 1205962

1198902


(12)


infinobserver is


Σ (120572 1205881) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120572 1205881) B1 B2 0Cinfin(120572 1205881) 0 0 D

infin

C 0 C119868

0

]]

]

[[[[

[

120585

119908

119868

119906119888

]]]]

]

(13)

with

A (120572 1205881) = [

119860119901(120572 1205881) 00 119860

119908

]

B1 = [

1198611199011119882119903

0]

B2 = [

1198611199012

0]

C = [

119862119901

00 0

]

C119868= [

01]

(14)

where 120585 = [119909119901

119909119908]119879 such that 119909




and Dinfin






observer


Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr


y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z




119888=





[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)



C119895 andD

119894119895as


Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)




defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)


119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)


120575modelminus

119865120575(119910)) in terms of an 119867


infin


119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)





120575119894sim 0 and





119888119895119895 =119894according to (7a) (7b)


[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Displacement (m)

Forc

e (N

)

0 0005 001 0015

0

200

400

600

800

Friction

StiffnessStiffness

minus200




Fault 2

Free of faults

Fault 1


Saturation


Forc

e (N

)

0

200

400

600

800

minus200

minus400



ms zs

FMR

mus

ks

ktzr

zus



2 Problem Statement


119898119904119904(119905) = minus 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119865MR (119905)

119898119906119904119906119904(119905) = 119896

119904[119911119904(119905) minus 119911

119906119904(119905)] minus 119896

119905[119911119906119904(119905) minus 119911

119903(119905)]

+ 119865MR (119905)

(1)







119904(119911119906119904) while




119865MR (119905) = 120572 (119905) 119865MR (119905) (2)





119865MR (119905) = 120572 (119905) [119868 (119905) 1198911198881205881 (119905) + 1198871def (119905) + 1198872119911def (119905)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119865MR(119905)

= 119865MR (119905) minus 119865120575(119905)

(3)


120575






[[[[[[[[

[

119904

119904

119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

=

[[[[[[[[[[[

[

0 1 0 0

minusℎ (120572)

119898119904

minus1205721198871119898119904

ℎ (120572)

119898119904

1205721198871119898119904

0 0 0 1

ℎ (120572)

119898119906119904

1205721198871119898119906119904

minusℎ (120572) + 119896

119905

119898119906119904

minus1205721198871119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

[[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

+

[[[[[[[[[[[

[

0 0

0minus120572 sdot 1205881119891119888

119898119904

0 0

119896119905

119898119906119904

120572 sdot 1205881119891119888119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119861

[

[

119911119903

119868

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906

[

[

119911def

def

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910

= [

[

1 0 minus1 0

0 1 0 minus1]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862

[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]

]

(4)

with ℎ(120572) = 119896119904+ 1205721198872


Σ (120579) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120579) B1 (120579) B2 (120579)C1 (120579) D11 (120579) D12 (120579)

C2 (120579) D21 (120579) D22 (120579)

]]

]

[[

[

120585

119908

119906119888

]]

]

(5)





1205751if the




119898119904119888119904119888= minus (119865

1198961+119865MR1

minus1198651205751+119865

1198962+119865MR2

+1198651198882+119865

1198963

+119865MR3+119865

1198883+119865

1198964+119865MR4

+1198651198884)

119868119909119909

120579 = (1198651198961+119865MR1

minus1198651205751minus119865

1198962minus119865MR2

minus1198651198882) 119905119891+ (119865

1198963

+119865MR3+119865

1198883minus119865

1198964minus119865MR4

minus1198651198884) 119905119903

119868119910119910

120601 = (1198651198964+119865MR4

+1198651198884+119865

1198963+119865MR3

+1198651198883) 119897119903minus (119865

1198962

+119865MR2+119865

1198882+119865

1198961+119865MR1

minus1198651205751) 119897119891

(6)

where 119865119896119894= 119896



1198651205751minus119865

1198882minus119865

1198883minus119865

1198884= 0 (7a)


1198971199031198651198884+ 1198971199031198651198883minus 1198971198911198651198882+ 1198971198911198651205751= 0 (7c)



119899(119865MR)



119888119895is


= ACL (120588) sdot 119909 +B1CL sdot (119868119899 + 119868119888)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870(120588)sdot119909

+B2CL sdot 119911119903


(8)


with 119870(120588) = sum119873




lowast


3 LPV119867infin

Fault Observer



the damper leakage


= 119860 (120572) sdot 119909 +119861 (120572 1205881) sdot 119906

119910 = 119862 sdot 119909

(9)


119891(119904+120596

119891)with state119909


119891


[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

= [

119860 (120572) 1198611 (120572 1205881) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119901(1205721205881)

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199011

119911119903

+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199012

119868

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119901

[

119909

119909119891

]

(10)


119903

in (4)



32 LPV119867infin





120575(119910)) 119882





119882119903=119870119903120596119903119904

119904 + 120596119903


119882119890=

119870119890(119904

2+ 2120577

11989011205961198901119904 + 120596

21198901)

1199042 + 212057711989021205961198902119904 + 1205962

1198902


(12)


infinobserver is


Σ (120572 1205881) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120572 1205881) B1 B2 0Cinfin(120572 1205881) 0 0 D

infin

C 0 C119868

0

]]

]

[[[[

[

120585

119908

119868

119906119888

]]]]

]

(13)

with

A (120572 1205881) = [

119860119901(120572 1205881) 00 119860

119908

]

B1 = [

1198611199011119882119903

0]

B2 = [

1198611199012

0]

C = [

119862119901

00 0

]

C119868= [

01]

(14)

where 120585 = [119909119901

119909119908]119879 such that 119909




and Dinfin






observer


Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr


y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z




119888=





[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)



C119895 andD

119894119895as


Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)




defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)


119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)


120575modelminus

119865120575(119910)) in terms of an 119867


infin


119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)





120575119894sim 0 and





119888119895119895 =119894according to (7a) (7b)


[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119865MR (119905) = 120572 (119905) [119868 (119905) 1198911198881205881 (119905) + 1198871def (119905) + 1198872119911def (119905)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119865MR(119905)

= 119865MR (119905) minus 119865120575(119905)

(3)


120575






[[[[[[[[

[

119904

119904

119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

=

[[[[[[[[[[[

[

0 1 0 0

minusℎ (120572)

119898119904

minus1205721198871119898119904

ℎ (120572)

119898119904

1205721198871119898119904

0 0 0 1

ℎ (120572)

119898119906119904

1205721198871119898119906119904

minusℎ (120572) + 119896

119905

119898119906119904

minus1205721198871119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

[[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909

+

[[[[[[[[[[[

[

0 0

0minus120572 sdot 1205881119891119888

119898119904

0 0

119896119905

119898119906119904

120572 sdot 1205881119891119888119898119906119904

]]]]]]]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119861

[

[

119911119903

119868

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906

[

[

119911def

def

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910

= [

[

1 0 minus1 0

0 1 0 minus1]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862

[[[[[[[

[

119911119904

119904

119911119906119904

119906119904

]]]]]]]

]

(4)

with ℎ(120572) = 119896119904+ 1205721198872


Σ (120579) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120579) B1 (120579) B2 (120579)C1 (120579) D11 (120579) D12 (120579)

C2 (120579) D21 (120579) D22 (120579)

]]

]

[[

[

120585

119908

119906119888

]]

]

(5)





1205751if the




119898119904119888119904119888= minus (119865

1198961+119865MR1

minus1198651205751+119865

1198962+119865MR2

+1198651198882+119865

1198963

+119865MR3+119865

1198883+119865

1198964+119865MR4

+1198651198884)

119868119909119909

120579 = (1198651198961+119865MR1

minus1198651205751minus119865

1198962minus119865MR2

minus1198651198882) 119905119891+ (119865

1198963

+119865MR3+119865

1198883minus119865

1198964minus119865MR4

minus1198651198884) 119905119903

119868119910119910

120601 = (1198651198964+119865MR4

+1198651198884+119865

1198963+119865MR3

+1198651198883) 119897119903minus (119865

1198962

+119865MR2+119865

1198882+119865

1198961+119865MR1

minus1198651205751) 119897119891

(6)

where 119865119896119894= 119896



1198651205751minus119865

1198882minus119865

1198883minus119865

1198884= 0 (7a)


1198971199031198651198884+ 1198971199031198651198883minus 1198971198911198651198882+ 1198971198911198651205751= 0 (7c)



119899(119865MR)



119888119895is


= ACL (120588) sdot 119909 +B1CL sdot (119868119899 + 119868119888)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870(120588)sdot119909

+B2CL sdot 119911119903


(8)


with 119870(120588) = sum119873




lowast


3 LPV119867infin

Fault Observer



the damper leakage


= 119860 (120572) sdot 119909 +119861 (120572 1205881) sdot 119906

119910 = 119862 sdot 119909

(9)


119891(119904+120596

119891)with state119909


119891


[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

= [

119860 (120572) 1198611 (120572 1205881) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119901(1205721205881)

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199011

119911119903

+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199012

119868

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119901

[

119909

119909119891

]

(10)


119903

in (4)



32 LPV119867infin





120575(119910)) 119882





119882119903=119870119903120596119903119904

119904 + 120596119903


119882119890=

119870119890(119904

2+ 2120577

11989011205961198901119904 + 120596

21198901)

1199042 + 212057711989021205961198902119904 + 1205962

1198902


(12)


infinobserver is


Σ (120572 1205881) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120572 1205881) B1 B2 0Cinfin(120572 1205881) 0 0 D

infin

C 0 C119868

0

]]

]

[[[[

[

120585

119908

119868

119906119888

]]]]

]

(13)

with

A (120572 1205881) = [

119860119901(120572 1205881) 00 119860

119908

]

B1 = [

1198611199011119882119903

0]

B2 = [

1198611199012

0]

C = [

119862119901

00 0

]

C119868= [

01]

(14)

where 120585 = [119909119901

119909119908]119879 such that 119909




and Dinfin






observer


Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr


y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z




119888=





[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)



C119895 andD

119894119895as


Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)




defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)


119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)


120575modelminus

119865120575(119910)) in terms of an 119867


infin


119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)





120575119894sim 0 and





119888119895119895 =119894according to (7a) (7b)


[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










with 119870(120588) = sum119873




lowast


3 LPV119867infin

Fault Observer



the damper leakage


= 119860 (120572) sdot 119909 +119861 (120572 1205881) sdot 119906

119910 = 119862 sdot 119909

(9)


119891(119904+120596

119891)with state119909


119891


[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

= [

119860 (120572) 1198611 (120572 1205881) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119901(1205721205881)

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119901

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199011

119911119903

+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199012

119868

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119901

[

119909

119909119891

]

(10)


119903

in (4)



32 LPV119867infin





120575(119910)) 119882





119882119903=119870119903120596119903119904

119904 + 120596119903


119882119890=

119870119890(119904

2+ 2120577

11989011205961198901119904 + 120596

21198901)

1199042 + 212057711989021205961198902119904 + 1205962

1198902


(12)


infinobserver is


Σ (120572 1205881) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120572 1205881) B1 B2 0Cinfin(120572 1205881) 0 0 D

infin

C 0 C119868

0

]]

]

[[[[

[

120585

119908

119868

119906119888

]]]]

]

(13)

with

A (120572 1205881) = [

119860119901(120572 1205881) 00 119860

119908

]

B1 = [

1198611199011119882119903

0]

B2 = [

1198611199012

0]

C = [

119862119901

00 0

]

C119868= [

01]

(14)

where 120585 = [119909119901

119909119908]119879 such that 119909




and Dinfin






observer


Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr


y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z




119888=





[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)



C119895 andD

119894119895as


Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)




defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)


119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)


120575modelminus

119865120575(119910)) in terms of an 119867


infin


119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)





120575119894sim 0 and





119888119895119895 =119894according to (7a) (7b)


[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










observer


Measuredoutput

Exogenous inputs

input

w

I

F120575

Wr

zr


y

Σ(120579)

+minus

We

zdef def I

zinfin

1205881

120572

ldquoControlrdquo

e

LPVHinfin

z




119888=





[

119900(119905)

119865120575(119905)

] = [

119860119900(120579 (119905)) 119861

119900(120579 (119905))

119862119900(120579 (119905)) 119863

119900(120579 (119905))

] [

119909119900(119905)

119910 (119905)] (15)



C119895 andD

119894119895as


Σ (120579) [119860 (120579) 119861 (120579)

119862 (120579) 119863 (120579)]

=

119873

sum

119894=1120575119894(120579) [

119860 (120596119894) 119861 (120596

119894)

119862 (120596119894) 119863 (120596

119894)]

isin 119862119900[1198601 11986111198621 1198631

] [119860119873

119861119873

119862119873

119863119873

]

(16)




defined by

120575119894(120579)

prod119903

119895=1100381610038161003816100381610038161003816120579119895minusC (120596

119894)119895

100381610038161003816100381610038161003816

prod119903

119895=1 (120579119895 minus 120579119895)

119894 = 1 119873 where 120575119894(120579) gt 0

119873

sum

119894=1120575119894(120579) = 1

(17)


119894) given

by

C (120596119894)119895

120579119895= 120579

119895if (120596

119894)119895= 120579

119895

120579119895= 120579

119895otherwise

(18)


120575modelminus

119865120575(119910)) in terms of an 119867


infin


119871 (120579) [

119900

119865120575

] = [

119860119900(120579) 119861

119900(120579)

119862119900(120579) 119863

119900(120579)

] [

119909119900

119910] (19)





120575119894sim 0 and





119888119895119895 =119894according to (7a) (7b)


[[[[[

[

1198651198881

1198651198882

1198651198883

1198651198884

]]]]]

]

=

[[[[[[[[[[[[[

[

0 1119905119903

119905119891

minus119905119903

119905119891

1 0 minus119905119903

119905119891

119905119903

119905119891

119905119891

119905119903

minus

119905119891

119905119903

0 1

minus

119905119891

119905119903

119905119891

119905119903

1 0

]]]]]]]]]]]]]

]

[[[[[[

[

1198651205751

1198651205752

1198651205753

1198651205754

]]]]]]

]

(20)


0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 10

010203040506070809

1

minus1 minus05

120588lowast 2

120588lowast11199793

1199792 1199791

1199794









120588lowast


119891 (119868

1198990 + 1198681198880))

119868119891 (119868

1198990 + 1198681198880)

isin [minus1 1]

(21)





0 le 119868min lt 119868119891le 119868max (22)

120588lowast


120588lowast

2 =tanh (1198861def + 1198862119911def)

1198861def + 1198862119911defsdot (119868119899119900+ 119868119888119900) isin [0

119868max2

] (23)




1198990 + 1198681198880 =


119868119888=

119868119888= 2 sdot 119868

1198880 if 119868 (119865119888) ge 119868

119888max

0 lt 119868119888lt 2 sdot 119868

1198880 if 119868119888min

lt 119868 (119865119888) lt 119868

119888max

119868119888= 0 if 119868 (119865

119888) lt 119868

119888min

(24)

where 119868(119865119888) = [119865



119888max= 2 sdot 119868





119868119899max





[

119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

= [

119860 + 120588lowast

2 11986111986801198621198680 1198611 (120588lowast

1 ) 119862119891

01times4 119860119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860119904(120588lowast1 120588lowast2 )

[

119909

119909119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119904

+[

1198612

0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199041

119911119903+[

04times1119861119891

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198611199042

119906119888

119910 = [119862 02times1]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119862119904

[

119909

119909119891

]

(25)


Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Fsamax

Fp

minusFp

minusFsamax

Icmin

Imin

Inmin

In0 + Ic0Inmax

Imax

Icmax

If

Ic

In


Forc

e (N

)


with

1198611198680 = [minus

1198681198990 + 119868

1198880119898119904

01198681198990 + 119868

1198880119898119904

0]119879

1198621198680 = [1198861 1198862 minus1198861 minus1198862]

(26)


119903




119882119903=119870119903120596119903119904

119904 + 120596119903


119882119904=

119870119904(119904

2+ 212057711199041205961119904119904 + 120596

21119904)

1199042 + 212057721199041205962119904119904 + 12059622119904


119882119911119906119904

=

119870119906119904(119904

2+ 212057711199061199041205961119906119904119904 + 120596

21119906119904)

1199042 + 212057721199061199041205962119906119904119904 + 12059622119906119904


at high frequencies

(27)







lowast

1 120588lowast


Σ (120588lowast

1 120588lowast

2 ) =[[

[

120585

119911infin

119910

]]

]

=[[

[

A (120588lowast

1 120588lowast

2 ) B1 B2Cinfin

0 0C 0 0

]]

]

[[

[

120585

119908

119906119888

]]

]

(28)

with

A (120588lowast

1 120588lowast

2 ) = [

119860119904(120588lowast

1 120588lowast

2 ) 00 119860

119908

]

B1 = [

1198611199041119882119911119903

0]

B2 = [

1198611199042

0]

C = [119862 0]

(29)

where 120585 = [119909119904119909119908]119879 such that 119909




119911infin


infin






lowast





119870(120588) [

119860119888(120588) 119861

119888(120588)

119862119888(120588) 119863

119888(120588)

]

=

22

sum

119894=1120585119894(120588)119870

119894=

22

sum

119894=1120585119894(120588) [

119860119888119894

119861119888119894

119862119888119894

119863119888119894

]

(30)


[[[

[

A (120588)119879

119870(120588) + 119870 (120588)A (120588) 119870 (120588)B C119879infin

B119879

119870(120588) minus1205742infin119868 0

Cinfin

0 minus119868

]]]

]

lt 0 (31)







119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119904= 630Kg 119898

119906119904= 815Kg

119896119904= 43500Nm and 119896

119905= 230000Nm




119888= 6009




0 1

0

4000

8000

Forc

e (N

)


minus8000

minus4000


0 005 01

0200400600800

1000


Forc

e (N

)



minus01 minus005

I = 20AI = 15AI = 10AI = 05AI = 0A












0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20

0123





e (Ntimes10

3)

minus1minus2minus3

Time (s)


Time (s)0 2 4 6 8 10 12 14 16 18 20

01234

Faulty force


Nonfault 120572 = 10120572 = 10


Forc

e (Ntimes10

3)

05




119865120575= 119865MR (119868) minus 120572119865MR (119868) (32)


















119904119888) and heave



11990411988817 and 119911

1199041198889 Note





zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










zsms

Fault

muszus

zdef1 zdef1

K1(120588lowast1 120588

lowast2 )


I1

F1205751

F1205752

F1205753

F1205754

L1(120579)

zdef119894 zdef119894

Ii

observer

Signals from CarSim

Fc1

FMR1

from CarSim


120572

zr











Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Height foranimator

animator

1722

2000

Width for

Left Right260 260


1560

760


3670

4910



208


1700

Sprung mass origin



08(kg)


1700


Sprung mass origin

CG

260

260 260Left Right


x

millimeters

mdash

(kg-m2)

z




6 Conclusions






5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 6 7 8 9 10 11 12 13 14 15

0

5

10

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

72

minus5

minus10

Nor

mal

ized

RM

S

US

LPV NC

AVCS

SFF

120601(d

egs

)

(a) Pitch rate

5 6 7 8 9 10 11 12 13 14 15Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

830

49

98

147

minus49

minus98

s 119888(m

s2) US

AVCS

SFF

LPV NC

Nor

mal

ized

RM

S

z


0

001

002

003

004

0

02

04

06

08

1U

SLP

V N

CAV

CS SFF

091

minus002

minus001

z s119888

(m)

5 6 7 8 9 10 11 12 13 14 15Time (s)

Nor

mal

ized

RM

S

US

AVCS

LPV NC






0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30

005

115

225

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVCS SF

F0

76

minus05

minus1

minus15

minus2

minus25

120601(d

eg)

USAVCS

LPV NC

SFF

Nor

mal

ized

RM

S

(a) Pitch angle

0 5 10 15 20 25 30

0

05

1

15

Time (s)

0

02

04

06

08

1

US

LPV

NC

AVC

SSF

F0

36

minus05

minus1

minus15

US

AVCS

LPV NC

SFF

Nor

mal

ized

RM

S

s 119888(m

s2)

z



0 05 15 25

0

2000

4000

6000

8000

minus8000

minus6000

minus4000

minus2000


Forc

e (N

)


LPV nominalcontrol


(a) FV diagram

0 2 4 6 8 100

1

2

3

4

5

6

Elec

tric

curr

ent (

A)


LPV nominal control

Time (s)




Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


References

























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Abbreviations






infin






















infincontrol


119911119904 119904 and



119911119904119888 119904119888 and



119911119906119904 119906119904 and




Acknowledgments


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
















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









Documents

Research Article Adaptive Vibration Control System for MR …downloads.hindawi.com/journals/sv/2015/163694.pdf · 2019. 7. 31. · frequency switched ,andmeasurement-based ), type