Model-Based Identification of a Vehicle Suspension Using Parameter Estimation and Neural Networks

Copyright © 1996 IFAC 13th Triennial World COll gres~. S,m Fr,mcisco. USA

3a-18 3

MODEL-BASED IDENTIFICATION OF A VEIDCLE SUSPENSION USING PARAMETER ESTIMATION AND NEURAL NETWORKS Thomas Weispfenning, Steeren Leonhardt IIISlilllle ojAutmnali(' COlltrol. Laboratory of C(mlrul EIl}!,il1eerillg and Process Automation Techn ical U"il'erxily of IJarmstadt, Lmu/p,rq(Ceorg-Str. 4 f)·('.f283 n"nn,I'/(Idl, Germany; Phone: +49(6151) 16· 7-412. FAX:+49(0/5/ J2936U4 E -MAIL,

fJEIS@IRT2 .RT,E- TECHNIK.TH-DARMSTADT.DE

Ahslrat.;t : A twn-step ...c he me fo r idenlificati o n of ,0} vehicle suspensio n is pft!sc ntcd which (omh ines pa rameter estimation a nd neural nctworks fo r appro ximati o n. At tirst. the paral11eters o f the di'>Crete time transfer fun ctio n are est imated using Cl RLS-nlgo nthm . These pnmtnC1Crs are noniinear funct io ns of the physical coefficienls. but a direc t calculation of these is oftl.!n not possibl e or leads 10 Jarge errors due to the nOlllinear amplificatil,n of noise. Therefore. to approximat e the coefficients, a non linear mapping using 11 RllF network is perfo rmed. For training of the network anti to test generali zation abilities, the cOl! rficients of .t vchic.:le suspemion were varied. '[ne slUdy shows th at an approximati o n of the physical It.:oeffic.:iel1[s hy application of the presented scheme is possible. The me thod was tested by s imui

or

Keyworus : iJcntificati un. ne ural network s. paramete r esti mat io n. vchicle fault diagnosis

s u s ren ~ i on,

~upcrvi s ion.

I , tNTRODUCTION

Tht! suspens ion of a vd tide is mainl y res ponsible for driving safety and co mfort. Besides the spring and other components the shock absorbe r and the tire pressure are most important. Dur ing its life cycle 1111.: performance nf tht: damper declines d ue to we.ar and luss of oil. This is <.I vcry slow process and therefore is us uall y nOI noticed in linK' by the driver who gets accuslOmcd to the cha ng ing hcha viour of the (.;::tr. Nute that weak. d:'lmpc rs mny cau se 20 pen.:e nt longer breaking t.li S lClIlCt:S and worsen Ihe dri vin g hand lin g as well. FUrlhermore . Iow tire pressure shortl!.ns the life cyde of the lire sig nitkanll y and m.lY \cad to flat tires causing dangerous situations. Most urivl.::rs do not superv ise it regularly. Therefore a meth od In de.termine the physical coeffi cients of [he vehicle suspe nsion is reqlllred. Recent ly, a model-based merlIod In estimate the coeffic Ient s of a suspensIon has been

developed by Butlhardt ( 1995), It is based o n continuous time paramete r estimation using sla te variable filters (SVF) to ob tain the deri vatives. H owev~r. the corner frequency of Ihese filte rs strongly affects the result of the parameter estimation. Discrete time para me ter estimation circumvents this probkm but the prke b a lTlore complex re lations hip betw ee n estimated parame te rs ~lnd physical coefficients. A direct m
In this Mudy. a scheme for mode l-based identifi cation using a combination of discrete parameter estimation and neural networks is presented. By app li cation of the proposed scheme, on -b oard supervision of a drivi ng car as well as technical inlipection are poss ibl e .

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2. THEORETICAL AND EXPERIM ENTAL MODELS

For investigations of th e vertical car dynami cs, in most cases

shown in Fig. 2. was cu nstructed at the Department uf C o ntrol Engineering at the Technical University of Durmstadt.

..

a simplified model. u quarter-car. is sufficient, see Fig. 1.

z. __

~'

t, - u,

.. Zw . r

,r

-~ i

~

0

II

Pw,Sw

.. . 0

IlHl(.h ..d

er

T he fIlove ulI;!llls arc I j ll ~<.lr i zcd around Ih~ operatio n pOlm and sprin g. damper and ti re is Ihe no nl incar ~h;Jviour approx imated \.... il li lin c.lr elements. Fmm tirst pr inci ples, the fo ll o wing equation s ~ I rt: obtained:

or

( I)

The body and wilt',el ImlSS are rcprcsl' ntcd by m/l and //l w, respectively, till count s for the damping ..:oefficicllt, ell for tne spring stiffness. ( ' 11' lknotes the tire stiffness which is primarily influe m;ed oy the tm: prt::s ... ure. Using Laplace rra nsform, th e foll owi ng transfer /"unl:ti' ln s It)f the subsystem

I

- -- - - : - -- -:---- - --,0.08 E

5

o

"

=

CR

dB

=

1 + . -s c,

K_ +

T,'

(3)

0

I~

r-...J\'oo'lI'....-,\O....~1""''''''IJ"..~''''rff''rMl'''"'~''''l 0.06

o

body acceleralion

0 ·, 0

c

.

0

o

Fig. 3: (4 )

1 + 2DTws +

1E

oru ~

.

~ -1 5

E ·20

~

_1

~sP9nsion deflaclion z,. - l..,

Z.

m

and the subsy stem wil ee l

Kw

I

The input (road displacement) is generated by a hydraulic ser vo cylinder. Adjustable e lements are the continuously ~dj u ~t <.lble Jamper and an air spring which is positi oned pnra\leJ 10 tht: steel spring. A typ ical mea.... urement is shown in Fig .,. Here. tbe input signa l wm; a pseudo rand om binary signal ( PRI3S), with a Ii'cq uc m:y spectrum similar ro rhe road spectnll1l.

I:""" zW - ZR

fl

.,...

Fig. 2: Quarter car test-slanu

body

G ,.(s)

.Fs

0

'Vi

G_(s)

iCD Ps

,z. -Zw .-,

c,

Fig. I : Quarler-GIr

r- r

-"--

Fo

~

.~

~

~-:-~-:-~--:----' - 0.02 ",

2

3

,

5

~

time [sec] Me;l'.;ured o utput s igna b body uccclcration and suspe nsion deflecti o n

r;.s 2 J. A TWO-STEP SCHEME FOR APPROXIMATION Or PHYSICAL COEFFICIE NTS AND FAULT DIAGNOSIS

can be deri ved. Appl ying the I.- transform wi th a zero-order ho ld. the di screte lime Intnsft''.f '-unelion:-. ..Ire

Tll s imulate the vc hid e vertical dynamics and 10 lest semi~\ c ti v e sUSpe n' lilll ,.: o nt: e pls ( BuBhan.h . 1995). a test stand,

As shown in Fig . 4 Ihe genc r.JI st,.:hc me includes two sleps. In [he first step . inforllJalion on the c urre nt process status is ex lrrlcted . ror thi s, m otle l - ba ~cd methods li ke parameter est imation (l sermann , 1992). st ate estimation or signal-based mcthods like signal spectra estimation (Neumann. 199 1) are app licab le. (n the followin g step. coefficients Or t~\Ults can be dc:rived from the sympto ms via so me nonlinear mappi ng. This mapping Gln be approximated by neural network s ( Harsc ho tf.

4511

1 ~90),

ex.pert systems or fu zzy log ic (i.<-ermann . 1995 ).

Input u[k]

described as an e ncapsulated mapping

output y[k]

O'w =

G,(F,(O,.,.»)

=

H,(O,.,.)

0", ~

G,(F,(O,.,.»)

=

H,(O".,.)

(10)

1. parameter estimation 2. state

For disc relt: timc parameter-estimation. a recursive LSa lgori thm using tht: discrete square root fillering approach in info rmation form ( DSFI) was used. For rm implementation. see Leonhaldt (1996).

estimation 3. signal analysis

ns features, symptoms 3.2 Nonlillear mapJ}ing wiTh neural networks

r-....;..l....;..l....,l I I I I

n, parameters, coefficients

n, faults

Fig. 4 : Ge neri.ll scheme for rault-ddccti on and diagnosis Note that for the appro ximation of the system coefficients. the symptoms are map ped on 11, coeffici e nt s with a non linear I.;ontinuolls lIJapping. Fault diagnosis. on the other hano, implies "I di stin\:ti o n bt.::tween a fault free case and faulty cases , which in f,lc t is a da..si fi cation pmhl e m with a discrete mappi ng of 11 , sy mptoms on n, faults.

T he second step of the proposed schcme consists of a no nlinear mapping. Several studies hav~ shuwn that neural networks are a valuable too l to approximate multidimensional fun c tions (e.g. Lippmann, 1987). Many different artificial ne ural nelwork types are known . each of them wilh different ad vuntagcs and di sadvantages. The types can be dislingui shed by the following three specific features

11,

apprQximatiQ~

ill:/I· ·, R"· • diagnosis : R'" - IO,U"f

(6)

Fi rs t results o n comronent fault dassi tkation of the: suspension system wen! presented by Leonhardt ct al. (1993). This study deal s with the: approximation of the coefficients.

central process ing element (netwurk node) network topology Ie.urning rule

• • •

Wilhin .he l:e nlral process ing ele ment , the sum o f input sig nals Xi lIlulliplitx.l wi th the wci hg ts Wi .i is calcul a ted and the threshold fisubstracled. The o utput is calculated apply ing rhe usually no nlinear activati on fu nction/m' TIle resulting ne uron i . . known as the McCulloch-Piult neuron (1943), see Pig. 5,

3. 1 Felltlfrt' eXll"(1("/;OIl (lsillX porometer C:.Himcrtiml

For femure ex rr:.lctlo n. :1 mode l-hased me thod is used and parameter estimatiOll
relationship. Let

a::s,p. m ==

rm8 .m"",cB ,cW ,d

lJ

I (: RS re.~. (7)

outputy

input X

Fig. 5: Artificial neuron The network lupo logy can be desc ribed graphi<.:ally or using m,ltri x notation . Consider a ne two rk with nE layers and Ihe nc ti vity func [ion J.~,.,. The d imenSio n o f each layer (the number of neumns) is 11, with the dimenSion of the output layer 11", and the input layer, whi ch cont ain s no neurons, nil' A weight mat rix containing the weights between the (k-I )th and the kth layer ca n be defined

and w

Wllll ..l"i

wc ll as

O;B

==lc 1 , d 1 J c R2 resp.

e!'w :7[a 1,al'bl'b1J

E" R 4

(l)

In real applicati o ns. the physicalcoeffl<.:ients rnit' ant! m ll arc often known a-priori <.~u n e asily he mcasureci. The remaining L'oefficients, '-'I!' C W • d",." arc combined int o a three-dimensional vt;:dor tl r>h\" as shtlwn ill eq. (7). The whole ITI
f ~

;,

k,

w l .... _ I

TJ k w

"~"'"

w» ]

E

U •...• m ~

(11 )

1

in which (he weight co nner ts the input of the jth neuron of the kth layer with the o utput of the i th neuron of the: (kI )th layer. The vector a E JR'" d l..!s<.:ribes the acti vati o n status of eac h layer. The activation ve~ ror is given by F ",.,."

4512

( 12)

The prohably best kllown type of neural netwnrks is the multi layer perceptron. a statit.: feeJforward network well suiled for c1assifi';'ltion purposes (Lippmann 1987), see Fig. 6 for the m=3 ve r~ion ,

The weighting factors I~J/ I/ art:' usually spec ified in advance and shape rhe gaussian curves. The activation status of the ith neuron IS

aI = i.(X-X )'(Wl'~'(x-X'(/J) = i.2X '·(W 1'1)'X - e,. ( 15) 2 <7f.t Therefore. the topology ca n be desl:ribed by Y = Fhll",JWf2IFm/,1I1(a»)

(16)

with dim(X)

,

x JOI

dim(W I21)

""ut

1. hidden

2. hidden

layer

layer

= (.,Hn,).

dimlY)

( J7)

= n,

To adjust the weighting matrices all neural networks have to be trained with training data. Different algorithms have been

1 X

dim(Wl'l) = (.oHn).

n,.

oulputy

deve loped. see RUlJlmelhardl "nd McCleland ( 19 ~6) for MLP ne twork s. for RBfs a standard LS-algorithm can be used.

rig. 6: Multi layer perceptron Us ing vet:tor notati on. its topology is described by

4. RES ULTS ( 13)

4. 1. Simularion with dim(X)

djm(WI.i:J)

=

no'

dim(SIKJ)

= nJ;"Ir _1'

dim(Y)

..;. ".

=

nu

( 14)

In a first step the applicability of the proposed method was investigated using simulated data. mH and mw were set to fixed values. whereas d IJ , rH' <.Jnd ("w were varied systematk:all y. Table I gives the coeffic ients.

Iablt: I · Coefficienl" for simulation Rct.:cntly. radilll -bllsis-JlIlll"liOlt nl!lworks (RBF) have been investi gated for use in genenll approx:im(llion tools (Poggio and Girosi. 1990). system identification find control (Sanner and Slotinc, 1991 ,). A RBF network is a feed forward network !":onsisting of two laye rs with radial-basis-fum.:tions in the first layer and linear functions in the second lilyer as activation fU llcti o ns. sec· Fig 7.

coeffi cien ts

value

variation

11//1

253 kg

const. con st. ± 650 Ns/rn ± 5000 N/m ± 50.000 N/m

IIlw

d/l

32 kg II~O N,/m

('n

35.000 N/m

(' IV

2111.(){X) N/m

The tire stillness C II wmi varied in 9 steps . the spring stiffnes (·If in ;) sleps and the damping coeftici ent d H ;n 6 steps. In ~lddiTjon , some points in between were considered leading to a total of 12 1 operating poinb. [n discrete rime parameter dt)[nuin. the variation of the physical coefficients results in speci tic trajectories. For examp le. Fig. 8 shows the pa ralneters (If the Sllbsyst Clll hlh/)'. InpUlX

Pig.7:

/bdden Layer

Topology of a RBF nctwl)rk

Output Y

Note Ihat the tmjcelories are 110 straight lines, hut part of nonlinear c ur ves. Fig. K re vea ls Ihat a variation of the physical eocfficien l5; intluen ces Ihe parameters of the z ~ domain transfer function in an unambiguous way. According to cq . (3). a changing tire stiffness does not influence the parameters of the subsy stem hody.

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,----..--'

022

- ..

- ~- .-J-

02

('UB

behaviour of tile air spring . According to the theoretical consi dermions, the tire pressure had no intluence on the trajectories in the subsystem hudy. see Fig. 10.

,

0.15

d,

0.14

O.9!io r

(H~ ~

0 .12

O~I

0'

d,

OM

O.

Q.75·

0.00\

--~

·O~

·(L8

~.R!'i

-0.75

...- -.--

07

~.7

/ '"

C, Fig. R:

Subsystem hoc/y, Cl/ ( 00 ... 4() kN/m ). d h ) ROO Ns/m , ( '11' = 2 10 kN!m I

.<~)

Oll!>

C

(500 ...

0 .• OM

.....,.

.(1.97

For (he appro lt ima lion. the 121 data records were d ivided into 150 training and 17 1 te.'\t records. The dimension of the no nlincar mappin gs HI 1( 11 4 - . R ' ) and //e· 1 (R 2 - R.l ) (see eg. ( 10») specifies the number of input el e me nt s anu the number

. ;0:'

~) ""~ :

,.

-"

., . . -O$Wii

C, Fig. 9:

of output neuron s. D urin g the trainin& a RBF neuron wa., placed 011 every sei.:n nd rrainill!l- point Accordingly. the

Subsystem bocr)'. var iarion of In and P r;, Pw= 2.2 bar

,I 0.95 !

number of neuron . . in rhe hidden la yer of the RBF was 75.

no influence

0.9 '

ror the subsyste m \\'h l'C'/, (he mean relat ive error wilh simulated d ata for rfJinin g as well as lor lest da ln is below 0.25 0/,'., [
4.2 Measuremellts

(f(

OS5 !

d,

; ) Pw

0.8: 0 .7!>

0 .7 0.85

"n' tesT rig

0.6

The measurement s w~ rt' take n at (he (e st stand described in section 2. Due In tec hni(:ul reasons. the ph ysica l coeffi c ie nts ill the [esl stand c ..lIlnol be varied dircctl ),. Instead. the control inpu ts of the 'Ktumor:-. il nd th~ tire pressure were changed. For the damper. a hi g li er con trol (UTn;' llt In causes a lower damping c()e1li~icnl. f(lr the
'/I

OMI

. ~- - ---- - -

-()..97

.(),98

-().95

.0.94

C, Fig. 10:

Subsystem body , variali o n of Ps and Pw. 1,1:: a,R A

In the nex t step, tht, appro ximation with measured data was perf( )rmed. Again the 321 daw sets were divided into 150 training a nd 171 test :sets a nd tl l ~ number of neu rons in the

hidden layer of the Rill'

W;1S

75. The quality of the

approximatio n W; IS no t itS good as with s imulated data. Some bad trai ning daca had 10 be removed manuall y. For the subsystem hod), with test data. fo r III e,l" was 12 %, for Pt 20 qv,. (;ompared to K % and 12 (Yr; with trainin g data. Fi g. 11 shows the results of the appro ximation. The leng th of the lines corresponds to the geome Lric errur at each data point. Si nce Pw cannot he appro ximated for thi s subsystem the v,\l ues were set to the re ference va lues. For Ihe suhsyste lll ."'Ieel. bette r result" were obta ined. For 1'1 and Pw the mean rel ati ve error was K % and 6 %, respec ti vel y, with test data, comparc:d to 4 (]f. w ith training data. The res ults for 1'1- were not s ati sfy ing because the intluence of /1) covers the intluence of PF' The re fore. in Fig. 12 the va lues of PF were set to reference va lues.

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diagnosis in a driving car as well as technical inspection are interesting areas for appli catio n of the presented technique. o training data • reference values approximated values

3 2

6 ".

Acknowledgement: T his work is supported by Deutsche Forschun gsgemeinschaft (DFG) throug h Sonderforschungs~ bereic h 24 1, 'Innovative Mechatronic Systems' (IMES)

REFERENCES

1,5 . '

.:,,". 2

p, [barJ

... 0,5

o" 0

I, [AJ

Fig. 11 : Approx imati on of H ~ ' (subsystem hody) with RBF network, measured uma. Pl~" set 10 reference values

o !raining OO1a • reference values + approximated values 3 2 B 1,5

6 4

Ps [bolJ

00

Fig. 12 :

Approximali l)n of HI (sll bsyslern H!heei ) with R [l r: netwmk . Illcusurcd d;'lt
h can be ~u lTlrnarizcd th LIt with RBF nt::twor ks a rec onstru<.> tion of Ihe ph ysical I: t)e tlicicnts ,." /{, d ll and Cif', or Ihe correspo nding (.;:onl ro l v,llues I". P.'· and PIV respecti vely. with a mean error of about 10 ok (for PF about 20 %) is possi ble. Therefore a supervisi on of the damper and the tire press ure can be perfnrmed. 1'or h'llllt tletcl:tioll. a classification of the obtained fea ture., is necl!ssary, see Leo nlmrdt et al. ( 1(93).

5. CONC I.US IONS

Bursc hdorff. D. (1990) . Case Studies in Adaptive fault Diagnosis using Neur
Third European Congress

0 11

Fuzzy and

/ll fe lJi&enl

Ta/lIl%gies, t;UF/T {Yf)5. Aachell, Germany, Aug. 28·3 11995 Leunhardt. S. (199'» . RuBhard. , l ., Rajllmani , R. , Hed ric k, K .. [scrmann . R . Parameter Estimation of Shock Abs(lrOCrS with Artificial Neura l Networks. American Conf rol COl~rereltc:e (ACe) 1993, San Francis(:o. CA, .11111 (' 2. -4. /993 Leon hmdt, S. (11.)96). Modellgcstiit7.te Fe hlererke nnung mit Uberwachung vun Neuro nalen Nctzcn Radaulhangungen und Diese l-E in spritzanlangen. VDI Fo/"{sc/t ritf-fieric:lire. Rdlte J2, VDI - Verlag. O{issddOlfr to be published). Lippmann. R.P, (1987). An Introduct ion to Computing with Nellral Nets. 1£££ ASS? Ma,~a,i" e , No. 4, pp. 4·22. Neum ann . D. (1991). Fauh Di agnosis of Machim:-Tools by Est imation o f S ig nal Spectra . IFA CI/MA CS-Symposiu", SAFEPROCESS '91, Baden Raden, Germany, Sep. 1991. Pogg io. T. and fiiros i, F (1990). Networks for approximation and ie::lfning. Pr(J(·eedilJ.g.'i of the /EEE, Vol . 7R, N o. 'I, pp. 14NI·1497. Rosenbhttt . F. (I (}() I ). I'r;nciple\· of NeurodYllamics. Spartan Books, Washington , DC. Rumelhart , D.E. and McClelanLl, 1.L. (19H6). Parallel ni,l"{rifJUted }Jroce.c;silll!" ~UT Press, Cambridge, Mass. Sanner. K.M. and S lo tinc, J.·1 . ( 1992). Slable Adaptive Con tflll and Recursivc Ide ntificatio n using Radial (lall ss ian Networks. 30rh COl~f. rm nee. and Contr., Uri.r.:h1on, En ~ l and. Dee. 1991

In this pape r. an appro<.lc h for vehic le suspension id e ntifi ~ cation W~\S presentetl . I, (;omhines di screte time paramete r estimatio n and ne ural n l~ lwo rks . Th ... method ha~ .,11Own 10 work SlIc(;cssfull y with slIllul
4515