Disturbances in the control of a dynamic system track-railway vehicle

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ZE SZ Y T Y N A U K O W E II IN T E R N A T IO N A L C O N FE R E N C E ________________ P O LITEC H N IK I ŚLĄ SK IEJ 2002 T R A N SP O R T SY ST E M S T E L E M A T IC S T S T '02 T R A N SP O R T z.45, nr kol. 1570

disturbances, system, control, railway vehicle, status model T ad eu sz C IS O W S K I1

D IST U R B A N C E S IN T H E C O N TR O L O F A D Y N A M IC SYSTE M T R A C K -R A IL W A Y V E H IC L E

In the paper is presented an application of theory of wave disturbances to the problems of dynamic interactions between the tracks and rail vehicles. In the given wave description o f unknown disturbances.

The present applications theory wave disturbances for projection of optimum system “e n g in e -d riv e r “ .

Z A K Ł Ó C E N IA W S T E R O W A N IU U K ŁA D U D Y N A M IC Z N E G O T O R -PO JA Z D SZY N O W Y

W referacie przedstawiono alternatywne podejście do opisu zakłóceń w sterowaniu systemów transportowych. Zdefiniowano pojęcie zakłóceń falowych i dano ich matematyczną interpretację.

Pokazano przykłady modeli realnych zakłóceń falowych, występujących w realnych układach dynamicznych tor-pojazd szynowy.

1. IN TR O D U C TIO N

T h e disturbances are an indispensable and often underestim ated elem ent o f transport system control. Instances o f typical disturbances occurring in real track-railw ay vehicle interactions are: w ind blow s and other aerodynam ic forces influencing the vehicle, frictions and clearances in the suspension system , uneven railw ay tracks, m oved w eight center, eccentric traction force in the m ovem ent along a curve and o ther undefined effects o f d islocations w ithin the m echanical vehicle system .

T he classic control m ethods are based on either determ inistic o r random interpretation o f distu rb in g signals. I f the first approach represents an all-too-sim plified understanding o f the n atu re o f disturbances, the second one renders it overcom plicated.

T his paper proposes an alternative approach to the description o f disturbance signals, based on a theory o f w ave disturbances and enabling description o f a large range o f d isturbances occurring in the real track-railw ay vehicle system s.

1 Faculty o f Transport, Technical University of Radom, 26-600 Radom, Poland, Lukovnm@wp.pl

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360 T ad eu sz C ISO W SK I

2. W A V E IN T E R P R E T A T IO N O F D ISTU R B A N C ES IN T H E T R A C K -R A IL W A Y V E H IC L E SY STEM

T he disturbances featuring a w ave structure m ay be described in a m athem atical w ay using sem i-determ inistic analytic equations o f the follow ing type [1]:

MO = w[f/t),f2(t),...lf M(t);.cv.,cL\, (1)

w here f ( t ) , i= l,2 , a finite value) - know n tim e functions, c*, k= ],...,L - unknow n param eters th at m ay discreetly change their m eaning in a partially fixed way. M athem atical m odels o f type (1) w ill be further on referred as the „w ave” interpretation o f a disturbance

MO-

T h e lin ear description (1) m ay be considered as an interpretation o f

MO

w ithin a functional space, w here the set o f functions

{ffO.-fMO}

constitutes a basis and c, - are partially fixed w eight factors. In other w ords, disturbances w(t) represent a linear w eight com bination o f know n basic function fi(t) and unknow n w eight factor c,- that random ly and in a partially fixed w ay change their m eaning [3].

A n instance illustration o f the equation (1) is show n on F ig .l.

Fig .l. Disturbances with wave structure

T he proposed interpretation o f disturbances vvffj considerably differs from its traditional, random interpretation. S pecifically the scope o f inform ation contained in the equation ( I ) is o f another quality than the inform ation contained in traditional statistical term s, such as m ean value, variance, spectral density and other. M eaning o f c, factors in the equation ( I ) are perfectly unknow n (except o f the fact that they are changing in a partially fixed way). W ave interpretation does n o t use the traditional sta tistica l p ro p e rtie s o f disturbances a n d d o es n o t define them.

In case o f disturbances, an efficient control requires inform ation about their effect on a current basis. S tatistical inform ation based on a long-term observation does n o t fulfill the criteria fo r cu rren t inform ation and becom es q uite superfluous in th e operation control.

T hus, interpretation (1) fills in the „inform ation void” in the description o f disturbances actually occurring in the dynam ic track-railw ay vehicle system .

In particular, equation (1) enables description o f a w ide range o f likely w ave form s covering any unknow n realization o f a disturbance w(t) at the m om ent t. W h at is m ore, each separate realization w (t) in the w ave interpretation may have its "ow n” set o f statistical properties, and

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D isturbances in the control o f a dynam ic system track-railw ay vehicle 361 thus m ay be used for description o f non-ergodic disturbance functions w(t), especially in the case w hen each realization w(t) is a fixed random value.

3. M O D ELS O F W A V E D IST U R B A N C E STA TES

D eterm ination o f a system o f basic functions ff( t) ) , is the first step in use o f wave interpretation o f disturbances as an instrum ent o f regulation system . It m ay be done using visual and m athem atical analysis o f experim ental records o f w(t) o r through an analysis o f dynam ic characteristics o f a physical process generating the w(t).

T h e second step includes determ ination o f „state m odel” for the equation (1). This m odel is a differential equation fulfilled by function (1). In other w ords, equation (1) shall be considered as a know n „general solution” o f the differential equation needed. L e t’s assum e th at each chosen function f ( t ) features L aplace transform /(T ), in the follow ing form:

/ , ^ (2)

Q„,(s)

w here Pm (s), Q n (s) - polynom ials o f m-th and n-th degree, wi, <n,. If w e assum e m om entarily c-, as fixed values, the Laplace transform o f equation (2) takes the follow ing form :

m p (s )

w( s) = c , f , ( s ) + c 2f 2(s) + ... + c „ f j s ) = £ c, , (3)

finally

w (s) = f ^ - , (4)

Q(s)

w here the n u m erato r’s polynom ial P (s) contain c; factors, w hile the denom inator’s polynom ial Q (s) is the low est general denom inator in the set o f d en o m in ato r’s polynom ials {Q„1(s),Q „2( s )„ ..,Q nM(s)} o f equation (3). This interpretation guarantees a m inim um size o f th e final state m odel o f w(t), w hich is very im portant from the point o f view o f costs and com plexity o f equipm ent. T hus, w e m ay assum e that the d en o m in ato r’s polynom ial Q(s) in the equation (4) has the follow ing form:

Q(s) = s p + qps p-' + qp_ ,sp~2 + ... + q 2s + q , (5)

M

w here p < ^ T n * . E quation (4) show s that the disturbances w(t) m ay be treated as „output i

v ariable” o f a fictional linear dynam ic system w ith the operational transm ittance:

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362 T ad eu sz C IS O W SK I

G( s ) = — — , (6)

Q( s )

at the initial conditions equal to

(w(o), w(o),

w (0),...}. T hen disturbances (1) w ith consideration to (3)-(5), fulfill the follow ing uniform linear differential equation w ith fixed param eters:

d p w d p~'w d p~2w dw

—_{d , P} + 'I? d t P-> --- r + <?„ ; * P - > d t P -2r + —+ <?2 — + q ,w = 0 , * * d t * / (7)

w here factors i- l,2 ,...,p , are know n, as they are independent o f c, and determ ined by the set o f basic functions ff( t) ) , assum ed as known.

In order to take into account discreet changes o f c, factors in the equation (7) w e will add to it an external enforcing function cu(t), being a sequence o f unknow n and appearing and random p ulse functions w ith random intensity (single, double, triple etc D irac function type).

T hus the state m odel w(t) w ill finally take the follow ing form :

d pw d p~'w d p~2w dw

W e have to em phasize that the pulse enforcing function a(t), is unknow n and introduced to th e state m odel (8) in a sym bolic w ay only to describe the c, leaps in the equation (1) m athem atically. M oreover, m om ents o f appearance o f n eig h b o r p u lse functions are separates from one another w ith a certain m inim um positive interval p > 0.

T hus, if the basic functions f ( t ) in the equation (1) feature the L aplace transform o f type (2), then in o rd er to estab lish a state m odel for equation (1) w e have b u t to determ ine the factors (qi, q2,—,qp} from equations (1) and (5), and subsequently use the general state m odel (8).

A differential equatio n o f p -th degree (8) m ay be represented as a system o f differential linear equations. F or instance, equation (8) m ay be w ritten in an eq uivalent w ay as a know n and „fully traceab le” ca nonical form . .

w = z,,

ZI = z 2 + d , ( t ) ,

Z2 = Z j + a 2( t ) ,

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Z P- l = Z p + C J p_; ( / j ,

Zp = - q , z , - q 2 Z2 - . . . - q llZ p + ^ f ( t )

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D isturbances in the control o f a dynam ic system track-railw ay vehicle 363

w here the sym bolic influence o f w(t) in the equation (8) is in the equation (9) replaced by the functions a,(t), i= l,2 ,...,p , constituting sequences o f unknow n, random D irac functions, w hile the d o t m eans d/dt operator.

In the general case w e m ay expect that the differential equation (8) o r a system o f d ifferential equations (9) w ill contain variable factors q, and/or n on-linear term s in relation to vv, d w /d t etc. T hus, the searched „state m odel” for disturbances w(t), having a w ave structure m ay be represented as a single differential high degree equation:

d pw d w d p~'w . . . / i m

+ f ( vv,— ,...,--- - , r ) = c o ( ij, (10) d t p J d t d f

or as a system o f differential linear equations:

co = W {z,t),

Z = (Zl. Z2, ... Zp). ( I I )

L e t’s em phasize that the m odel ( l l ) has certain advantages o v er that o f (10), as it uses state v ariable m ethods. I f w(t) is a m ulti-dim ensional disturbance containing p com ponents vv = (wj,W2,...,wp), then the state m odel shall be determ ined for each independent com ponent

W i(t).

4. IN STA N C ES O F STA T E M O D ELS FO R A C T U A L D ISTU R B A N C ES IN T H E T R A C K -R A ILW A Y V E H IC LE SY STEM

T h e state m odels expressed by equations: (10) and ( l l) for the actual disturbances appearing in the track-railw ay vehicle system may be determ ined based upon experim ental oscillogram s show n on F ig.2

Insta n ce 1. In the case show n on F ig.2a the disturbances fulfill the follow ing differential equation

£

_dt ^{= 0 ,} ⁽¹²⁾

w here c in this case is a fixed value. In order to take into account random variables, discreet changes o f c, w e have to add a (t) term , consisting o f an unknow n sequence o f Dirac functions, to the equation (12).

S tate m odel vv (t) o f disturbances show n on Fig.2a. F inally assum es the follow ing form:

$- ₄ ) ^{( .} ³ ^.

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364 T ad eu sz C IS O W SK I

w(t)=C dw dt O(t)

w(t)= ¡ą +cye

u dw

3 dt + k4w = co(t)

Fig.2. Disturbances with wave structure in the real track-railway vehicle systems

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In sta n ce 2. T h e disturbances show n on F ig.2b. are described by the follow ing equation w(t) = ci + C2t, fulfilling a quadratic differential equation:

d 2w / \

= (14)

w here co(t) m eans an unknow n sequence o f random single and double pulses w ith random intensity. A n equivalent m odel in the form o f system (9) looks as follow s:

0 5 )

z, = z 2 +cs,(t), z 2 = 0 + <j2{t),

Insta n ce 3. D isturbances w(t), show n on Fig 2c. w ith the follow ing L aplace transform :

( .7 ,

m ay b e presented as an equation (4) in the follow ing form :

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s{s + a )

In this case Q (s) = s2 + as, and the equation (5) and (8) show that \v(t) fulfills the follow ing quadratic differential equation:

d 2w dw

T h e equivalent state m odel has the follow ing form:

W - ] ⁽²⁰⁾

z, = z 2 + cr,(t), Z2 = a z 2 + c r2(f)> (21)

In sta n ce 4. Short w ave-type disturbances w(t) as show n on F ig.2d. have th e follow ing L aplace transform :

» < * )= ~ n — P2 \ n • <2 2 )

s(s 2 + y ){s2 + /3 )

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366 T adeusz C IS O W SK I

T hus, in accordance w ith the equations: (5)-(8), w(t) fulfills the follow ing quintic differential equation:

d 5w

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An eq uivalent presentation o f equation (23) has the follow ing form:

vv{r)=(;,0,0,0,0)

V

Zj

\ z i J

¿1 = z 2 + cr.it), z2 = z 3 +cr2{t), h = z4 + a 3{t), i 4 = z s + a 4{t),

żs = -irP )2 z 2 - (r 2 + P *2 )< + ^(O-

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Instance 5. P ulse-type disturbances show n on Fig.2e. w ith the follow ing L aplace transform :

J s) = _CSLŁ_

P{s)

(s'* + k . s 3 + k 2s 2 + k 3s + k 4) (26)

w here k> - are functions o f tw o know n param eters a and /?, and fulfill the follow ing differential equation:

d 4w , d 3w , d 2w , d w ,

— t + k , — r + k 2 — r + k , — + k 4w = co(t)

d t 4 ' d t 3 2 d t 2 3 dt 4 W

or

w(r) = {1,0,0,0) ' z , '

Z-2 Z-3

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Ż, = Z 2 +cr,(t), Z2 = z 3 + a 2(t),

i 3 = Z4 +cr3{t),

¿4 = - ^ 4 Z , - k 3z 2 - k 2z 3 - k , z 4 +<J4{t)

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w here k, are know n coefficients depending o f param eters aand ¡5.

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B IB L IO G R A PH Y

[ 1 ] EPAHCOH A . , Xo- I O - U Ih, r ip iiK jia ;» ia « T e o p iw o rro tM a jib H o ro y iip a n jie m is i.-M .: M n p , 1972.

[2] KBAKEPHAAK X., CHBAH 3., Jlmieimbie oirrHManbHbie citcreMbt ynpaBnemw.-M . Mnp, 1977, -650 c.

[3] JOHNSON C.D., Theory of Disturbance-Accomodating Controllers. Chapter in the book, Control and D ynamic Systems; Advances in Theory and Applications, Vol. 12, etited by C. T. Leondes, Academic Press., Inc., New York, 1976, p. 627.

[4] JOHNSON C.D., Accomodation of External Disturbances in Linear Regulator and Servomechanism Problems, IEEE Trans. Automat. Control, AC-16,pp. 635-643, 1971

R eview er: Prof. A leksander Sładkow ski