Multistage Nonlinear Estimation with Applications to Image Based Parameter Estimation

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2b-29 2

I :lrh Triennial World Congre~s. San Franci sco. USA

MULT ISTAG E NONL INEAR ESTIM ATION WITH APPLI CATIO NS TO IMAG E BASE D PARA METE R ESTIM ATION Bijoy K. Ghosh '.1 , E. P. Loucks " and M. Jankov ic'" • Department of System .. Science fj Math., Washington Univ., St. Louis, MO 63130 •• Chiron Corporation, J,777 Le Bourget fmve, St. Louis, MO 63134 ••• Ford Motor Co., Scientific Research Lob., Dearbom, MI48121

Abstra ct. In this paper we apply recently developed results in single stage nonlinear observer design to estimat e motion and shape parame ters of a rigid object undergoing rotation and translat ional motion. We also extend results in single stage observer design problems to a cascade of single stages and introduce a new theory of multistage nonlinear observer design. The multistage observer design problem has been ~pplied to the problem of estimat ing motion and range parameters for a rigid object falling freely under gravity. All the proposed applications of the observer have been simulated and the results are described in this paper. Keywo rds. Nonlinear Estima tion, Machine Vision, Observer Design.

l. INTRODU<.,'TION

In this paper we propose to study the),rob lem of dynamic observer design for a suitable c1a.ss of nonlinea r systems that arises from various problem s in machine vision. Such a class of systems has antecede nts in the literatur e on Paramet er Identific ation (Mor· gan and Narendr a (1911)). (Sa."ry and Bodson (1989)). ('!'eel, Kadiyala , Kokotov ic and Sastry ( 1991», Machine Vision (Gb.06h, Jankovic and Loucks (1994), (Jankovi c and Ghos" (1995», Robotics (Jankovi c (1995), Biological Reactors (Gauthle r, Hammou ri, Othman (1992»), and Distillat ion Columns (Deza, Bossane, Busve1le , Gauthie r and Rakatop ara. (1993)), although we choose to focus in this paper on machine vision. The class of dynamic al system tha' we propose to consider has the following general structure , %1 =Wr(Xl . t)%2+ 4'I(~1,t) X2 =w!(Xt ,X2,t)X 3 +tP2(Xl , X'l,t) (1.1) in =w;:'(X/ .X2, ... ,X Tl .l)x,,+1 + r;of1(Xl , ·· ·. X" , t) %,,+1 = tPft+I(Xl , .. . ,X.. +l,t)

where we asaume that Wj(·),j::;::: l, ... ,n and
The main focus of this paper is to derive computa.ble algorithm s for recursive estimatio n of unknown paramete rs for a class of problems in machine vision (see Ghosh and Loucks (1995), Gbosh, Jankovic and Wu (1994)). These problems have been posed. a.nd studied in the literatur e pertainin g to compute r vision for at least fifteen years with emphasia in deriving computa ble algorithm s to estimate the paramete rs (see Tsai, Huang (1981), Tsa.i, Huang and Zhu (l982), Tsai and Huang {1984}, Wa.xman and UlIman (1985), for details).

I Research supporte d in part by DOE under grant DE-FG0 2· 90ER14140.

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2_ DESIGN ING OBSERV ERS FOR A SINGLE STAGE NONLIN EAR SYSTEM Conside r the problem of observer design for a nonJinear system given by i-

= J(~)

y

=

h{z)

Definiti on 2.1 The regressor m4trizw T(x) i3 3aid lo be penia. tently ezciting rp.E.) if there e.Z'ist positive constant s 01,0'2 and q .such that for aU t, Xo we htltle.

all

-s

(2_1)

where the state vector :z: E D C Rn a.lId the output vector y E am, We shall also assume that / : RO .... RR and h ; Rn ~ RD'I are Coo function s of their iU'gume.nts and that the solution ::t:(t) of (2.1) exists for &11 time t; i.e. the vector field 10 is complete . FUrthen nore we shall assume that only the output vector y is available for meaBure ment.

~ a21

(2_2)

this clasa of Donlinear systems, th.ough importan t. has never been consider ed in the literatur e in systems identi6ca tion and adaptive control. Specifically, we consider a plant described as

x = wT(x ;tJ

(2.8)

where as before, x ERn, e € RP, w(·) ia a p x n matrix of COO functioll8 in x. Note in particula r that (2.8) is already a single stage dynamic al syslem of the form (1.1). An observer with a high gain paramet er 9 has been introduc ed for the class of system (2_8) by Janlrovic (I992) (see also Jankov;c and Ghosh (1995)) and is given as follows.

=

i = gA(;; -

L If z{O) .(0) then x(t) *) fM all , ": 0_ 2_ ~(O) - z(O) E V implies that x{t) - z(t) E V and IIz {t)-

flx{t) - z(t)1I ::; ae-"lIx{ O) - ,(O)II, then we shall say that (2.2) is an exponen tial identity observer . In this paper we would like to consider an importan t class of observers tha.t are designed based on methods used in identifica tion of systems with constant paramet ers Sas'ry. Bod80n (1989) with the high~ga.in nonlinea r observer s Bornard , Couene. and Celle (1988), Jankovic (1992). Conside r a nonlinea r system of the form

x)

+ wT(x)1i x ) + ~(z, 9).

8 = -g'w(.)P {i: -

*)11 --; 0 as , --; 00_

If. in addition to conditio ns 1. and 2 .. we have thai, for x(O) %(0) E V and for some a > 0, b> O. the follOwing holds

(2_7)

An importan , extenslon for the nonlinea r systems of the form (2.3)

11 = .p{x, 91

where z E Rn and 9 : Rn X am is a C= function of its argumen ts. We shall call the llonlinea r system (2,2) an identity o~erver for the system (2.1) if there exU;:t an open set V C Rn suc.h that the following two conditio ns are satisfied.

=

w(x(r»)t oT(x(r») dT

i8 the case where the paramet er f) is not constant . It turns out that

Let us now consider another dynamic al system given by .i = g(z , Y)

f

,~

(2_9)

In (2.9), A and P are chosen as in (2.4). The paramet er 9 is a suitably chosen scalar bigh gain. Finally i is defined as 6 :; sateS) where sateS) is described as follows. Let U and V be two compact and convex sets such that U C V and assume that tJ(t} E U '" t ~ O. We define the function sat(tJ) to be a. radially monoton e smooth function such ~hat sat(B)

= 9 for 8 € U = constant for 6 outside

v:

(2_10)

It has been shown in Jankovic and Ghash ( 1995). that the observer (2.4) is exponen tially converge nt if

(2_3)

(2. 11) where x E Rn . 0 E RP . w(·) is a p x n"fnatrix of C(>Q functions in x. The matrix w T (%) is called the regresso r ma.trix. Tbe paramet er vector (} is unknown , but asaumed to be constant . An observer that identifies the unknown (J is given by

i: = A(i - x) + w-!"{x)<1 <1 = -w{x)P (± - x)

dim x (2.4)

where A is an n x n Hurwitz matrix (Le. has all its eigenvalu es with negative real parts) and P is a positive definite solution of

th.e Liapuno v matrix equation

_~Tp+

P_4 = _Q_

with Q a positive definitt: symmetr ic matrix. If we define e = 8 - (J , tbe error equation takes the form

and r/J :::;

e = Ae+wT (x)
for all x in the domain. Note in particula r that the rank conditio n (2.11) is Stronger than the P.E. condition expresse d by (2.7). In particula r, the rank (."(InditiOD (2.LL) implies tha.t

(7.S)

x-z

~

dim

1].

(2_12)

Thus a. necessary condition for the observer (2.9) to converge is that the number of states is no less than the number of unknown paramete rs. If the dimensio nality conditio n (2.12) is not satisfied, the observer (2.9) may not converge. However , if this t.he case, one can modify t.he high gain observer and the techniqu e ia illustrate d as follows. Consider a. dynamic al system of t he form (2.13)

(2_6)

It has been shown Morgan and Narendr a (1917), that (2.6) is exponen tially stable iff the regressor matrix w T (xl is Pefflistently Exc:iting (P.E.), a concept which we shall now introduc e.

when! we assume for simplicit y that x is a scalar. Let us define the vector O= (9,, ___ ,B,)T and assume as befor-e that (2_14)

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The pair (2.13). ('2.14) is not a dynamic al system in any of the standard forms consider ed so far . If we now define

er

==f.(x.8 \, . . . ,@,,)

"

~

<1

We shall assume t.hat the point object is being observed. with the aid of a CCD camera which produ.ces a scalar observat ion yt{t) described as Z Yl::;' - .

(2.15)

and a.ssume t hat the Ronlinea r equation s (2.15) can be solved in terms of the paramet ers llt> .•. , @,, > then for a suitably defined function we can write

w.

~. ~ ~( ••<1 ... ·.{.)

(2.16)

Let us define

(3.2)

Y

Such an observat ion is a. result of perspect ive projectio n of the two dimensio nal vector (:z:,y} T . These projectio ns have heen considered in the literatur e on Machine Vision and we would like to refer to Kanatan i (1990), and Gb08h and Loucks (1995), for a detailed reference. Our objec&ive in this section is to identify the motion paramet ers (to the extent poesible) from the observat ion Yl( t) . It can be easily seen t hat the observat ion function Yl (t) satisfies the ordinary differential equation (3.3)

(2.11)

C ~ (l

o.. 0)

B ~ (0 .. . 0

(2.18)

l)T .

(2.19)

where Cl = bl / Y and C2 = b2/1J can be viewed as a new pair of time varying motion pa.rameters together with tbe angulax velocity w. Note that although (3.3) is to the fonn (2.3) , the identifie r (2.4) does not a pply because C l and Cl are not constant paramete rs. The observer (2.9) doesn 't a.pply either, because the dirnensionaLity conditio n (2. 12) is not met. Therefor e, we try to reduce (3.3) to a standard form of a multiple integrato r (2.16). We define

For the dynamic al system (2. 13) an observer can be construc ted as follows, Let L be such that all the eigenvalues of A - Le are in the left half of the complex plane and r be such t hat

(3.4) and compute

(2.20) where 9 is a high gain paramet er, The observer is given by

i

= (A -

rLC)i

+ rLy + B",(X.~'. " . •~.)

where i denotes the vector define via E ==

(j ==sat({ j)

lx ( 1

(2.21)

In computin g (3.5) we bavt! utilized (3.4) in order to eliminat e the varia.ble Cl. Next we define

. . . (pJT and

j == 1.2 • ... , p

Cor a suitably chosen saturatio n functio n of the fo rm (2 .10). The high-gain observer (2,21) for the syatem (2.16) achieves exponen tial converge nce of the sta.te estimate s provided that t he constant 9 is chosen sufficien tly high and the States of the system are bounded in a. known compact set which is used in saturatin g the estima&e S of';j Jankovic (1 992).

To summari ze, we have introduc ed observer s (or nonUnear systems of the form (2.3), (2.8) and (2.16). We shall call such forms of the dynamic al ~tema as 'standar d forms ' . For the second and third category of nonlinea r systems , the associated. observer has a high gain paramet er 9. In the next section, we shall apply the two observer s to identify motion and shape paramet ers of moving rigid objects that Ilre observed by a. C CD camera..

3. OBSERVING MOTIO N PARAM ETERS IN A PLANE

and compute lis -=3w(y1 +!l1Y3) -W 2Y2(1+ 3y;)+

2c, (WYlll' -

g,l + (WC,Yl

- cj)[w(1

+ yiJ -

37 2y, j ( . )

Let us now define the right hand side of (3.7) as y" . From (3.4), (3.6) and (3.1) we obtain Y4 -=

Y3 = 3w(yj + Y11/3) - w'y, (l + 3y')+ 3("'lI1" - " )('3 - 3wy'!/2 + W\Y1 (1 + y1J) (3.8) w(l + Y~) - 2Y2

Finally one compute s 94 ::::; w(l + +W 2y2 ......

Y4

as follows.

1

yD _ 2?/2 {(WYIY2 -

,

Y3)(Y4 - 3wY2 - 3WYIYS + 3(113 - 3wYIY2 + W2Yl (1 + yD) (3.9) (wyi + WY IY3 - Y4)} + {9wY2Y3 + 3wYIY4 "ly,(1 + Jy~) - 6w 1 Ylyn

We consider a point object wh.ich is moving in a plane. The dynamics of the object is assumed to be given by

2 3w yiY2)

Note that (3.S) defines a cu bic equation in UJ with paramet ers tba.t are functions ofYl,Y2, Y3 and Y-4 . We tberefore have

lil = Y2 Y2 ::::; Y3

where (x , y )T is the co-ordin ate vector in the plane. The motion paramet ers are given by w. b1 , b2 which we assume time invariant .

il3 =- Y4

Y4 == 4l(Y1.lI2 , Y3,Y4)

2845

(3. 10)

where the last equation in (3.1 0) is the "'
Yl ;:: yz !i2 :;;

Y3

(3.11)

Y3 ;:: Y4 y" ==41j(Yl,Y2,Y3,Y4)

J

== 1, 2.3,

only one or which would match the solut ion !lI of (3.3) , provided of course the paramet ers w, Cl, Cl are identifiab le from Yt. We know rrom Ghosh, Jankovic and Wu (1994) . that tbe paramet ers are identifiab le if w :# 0 and (b}, b'l )T ::f: O. In fact it has been shown t hat the paramet. ers w, Cl, Cl are identifiable from 9t if Tank

(b1

wb 2 ) b2 -Wbl

;

2

(3.12)

where XI E Xl C Rn.l,X2 E X2 C R 1I 2, and X3 E X3 eRns . In order to prove the converv nce of the observer which we shall propose later, we propoElC to impose the dimensionaLity conditio n that nl ~ n2 ~ D3. We also assume that; 'Wi and q,j are smooth functions of their argumen ts and that they, together with are bounded as functions of t. Addition ally, we impose the following two assumpti ons.

W-,

Assump tion 4.1: The state or the system (4.1) is bounded in a known, compa.ct and convex set n. Le. ~ [:rf xrJT E n c Rn

(n ~ nl

x

+ rl2 + D3)

for aB t >

A.sump tian 4.2: The matrices 10t and 102 have full column ranks unitbnnl y jn their respectiv e argumen ts, Le. there exists a positive constant e: such that )'mill(,W I(:tl,

Let u.s now construc t the triplet of high gain observer s (2.21) for the system (3.11) given by

y" = g 4l 4(Yl

-

b\)' $ -L -.')' + (y(O) + _ b' . w W

",,2

(3. 14)

It follows that the observer (3.13) cc:Tnverges for one of the three va.iuea of j provided that the value of the paramet er Ig' is sufficiently large and the saturatio n function s are suitably chosen. Extensiv e simuJations have been performe d to evaluate the performance of the high gain o~rvet" (3.13). These have been reported in (GhO$h, Jankovic and Loucks (1996)).

4. MULTISTAGE HIGH-GAIN NONLINEAR OBSERVER In this sedion we iotroduc e the main technical remIt of this paper. We show the existenc e of observer s for the general class of nooHnear systems (1.1) and obtain a suitable set of sufficiency conditio ns under whicb these observer s are guarante ed to converge. The main result of this section has been stated for dynamic al systems of the fonn ( L.t) with n = 2. but extensio n to the gener

2 is obvious. Thus tbe dass of systems we consider here is given by

Xl =1O~(:rI,t)X2+.pd Xl ,t) == (:Q,X2,t )X3 + tP'l(l:t,X2 , t)

10,

%,::


Y

==

Xl

L

>e

(4.2)

> O. .\min{·) denotes the smallest eigenvalue

In order to design an observer we choose two constant Hurwitz

j=I,2,3

Theore m 3.1: Conside r the dynamic al system (3.1) with un· known paramet ers w. 6 1 ,62, Let the paramet ers be such that w :r: 0 and (bl'~) i:- 0 and assume that the initial conditio n on (3.1) is such that

X2

>e

(3.13)

rh) + rPj(Yl ,jh,Y3,JJ4),

with ih-. y". Y. obtained from suitably chosen saturatio n functions . Note that there are three observer s for each of the three models (3.11) . The problem is to deteet. the COIT~Ct. value of j, Le. to decid.e which of t.he three sets of estimate s Ut, Y2, rh. fj" is the correct one. The following theorem summari zes t he main result of this section.

(x(O ) -

t)wi{Xl. t.l)

-Xmill(W 'l(Xt,%2 ,l)wf(xt, X2,t»

for all x E n and all of a matrix.

'fh ;:: in + gl t(Yt - ill) ¥2 =!i3 + g2t:. (Yl - yd ila == Y4 + g'l, (Y1 -!id

x5

o.

(4.1)

matrices : Ht of dimensio n rll x ni, and H2 of dimensio n n 2 X n2. We denote by P1 and P! the p06itive definite solutions of the

follOWing Liapunov matrix equation s: P,H, P2H2

+ H~1't ;-1 + H'J P2 == -J

Denote by nO' the projectio n ofn on X2 and by Q" the projectjo n of non X3. Then we know that as long as:r E n the veetor %1, respectively l:3 . is in n.,., respectively n.,. Now the two stage nonlinea r observer is given by the following equatioru!; 0-1 =fH(((TI-Xt}+Wr(XI,t)0"2+~t(Zl.t) 0-2 == - /lWl(Xl , l)Pt(O'l - Xl) + WT (X l ,"2. t)1J2 +4>2(%\,b"t) ii, ; ""n.(u, )

(4.3)

Tit :: 9H2(1/1 - U'l) + wi (Xl, 0'2, t)'1'2 + 1>2(%1, "'::I, t) in ::: _g2 W2 (.J:l,U'l, t)P-:!(fJt - a:z) + 4'3(:Ja,C12, rh) !h = sato'l (f12) where f and 9 are two positive constant3 and sato". (.) is a saturation function ; Le. 3&to". (0"1) == U'l if (72 E !1a and saturate s outside 0 .... The same holds for sato"l (.). For technical reasons we have to assume that these saturatio n fUnctions a.re at !eaat once continuously diffenmtiable. Since (la and 0'1 are compact and convex sets such saturatio n functions are not difficult to construc t. Note that t he observer (4.3) has more states than the original system. The reasoo is that both eT,! and 1]1 estimate the same state X2. Therefor e it is not one of the identity observer s describe d in (2.2). Wt!. define twO vectors of estimatio n eITOr.I: and

e1J

~

(4.4)

and define the vector e via e ~ e;J T . Now we can state the following theorem which gives the conditio ns for the convergence of e to O.

re;

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Main Theore m 4.4: If Assumpt ion 4. 1 holds for some known compact and convex set n and Assumpt iOn 4.2 holds for some £. o then there exist a positive constant 90 and a. strictly increaain>g funct ion :F ; R+ -;. R+ 3uch that the estimMi on error e converge s to 0 exponen tially provided. that 9 > 90 and f > ,]"(9)·

to be monitore d on line, However, even when it is not satisfied , we can get approxim ate estimate s of Y3 a.nd Y4 wbich would allow us to caiculate the velocity vector up to a 2-Plane in R3 . An importan t special case is wben !.be gra.vity is zero, i.e. 91 :;;;

!/2 ::;:: !l3 :: O. The equation (5.2) reduces to

(~: ) (~: ) . (~: )

S. MOTIO N AND RANGE ESTIMA TION FOR A RIGID OBJECT S FALLING FREELY UNDER GRAVIT Y We now consider an estimati on problem for which One needs to consider the multista ge nonlinea r observer s describe d in Section 4. We consider a rigid body which is {ailing freely under gravity. A point on the body i8 observed with the aid of a CCD camera. The problem is to estimate the motion paramet ers and range infonna· tion from this data.. Let us assume that a point m8.S!l P moves in a gravity field with an unknown velocity and is observed by a stationa ry camera with known orientati on with respect to the gravity field. The equation s of motion are gi ven by:

x == V:r; if ::;:: VII Z = v:

v.

v,

&,

==

91

::;:: g2

(~.l)

:::; 93

where :1:, y, z denote the coordina tes of the point in the camera reference frame, V:z:, VlI' tl l are compone nts of unknown velocity. and 9i'S are compone nts of the known vector of gravitati onal ac· celeratio n. We now assume that the point

?l6

g

y,

= -.; (5.3)

two stages with one of the stages hayjng a double integrato r and. therefore , is in the form for which 'multisI;a ge nonlinea r observer ' is applicab le. It is easy to check that the ,)bservability rank. conditio n of Assumpt ion 4 .2 reduces to vi + y~ > ~ > O. This condition has an easy interpret ation because Y5 + Y.i 0 means that the point is moving right towards the focus and. therefore , its image on the screen is stationar y. The velocity is th'.~refore unobserv able in this case,

-=

SimuJati on of the MNO has; been carried out for the dynamic s (5.2). In order to design an MNO of the form (4.3) , we choose PI == Pz :; 1 and Hurwitz matrices HI == Hz :: -1 / 2 in (4.3). The two stage ;\1NO given by (4.3) is ,)btained as follows:

it.

= - .5/(!i. -

it' =- .5f(y, -

yt)

+ Y3

+ !i. - /'"('Yt - yI) + (-2Y3) i 3 + (91 - f2(fn - Y2) + (-2Y4}Z3 + (92

-=

l/2 )

¥3 - 93yd z 4 ¥4 :: - 93Y2)Z4 il :: - .59(i. - Y3 ) + (-2Y3)£ ' + (91 -gJ;Yl)i 4 %2 :: - .•iy{h - Y4) + (-2Y.c)i3 + (g:z - 93YZ )Z4 Z, -9' [(-2., )« , - y,) +( -2Y4) (i, - •• )]

camera and that the observed variabt~ are Yl ::: x/ z and ?/2 = y/ :.. It ia not difficult to calculate the dynamic s for the observed variables !ll and Jj'l . They are given by YI = Pl - YtY6 "!i2 P'l - ?!2YS where Pi 'S are compone nts veloci ty scaled by the

-=

=

~ E;, P2 Y!i ~ 7 ' Define t and consider the dynamic s in the coordina tes given by YI,

depth of the point. Le. Pt

y"

Note that the paramet er Y6 is not identifiab le and therefore drops out from the dynamic s. In this case the vector (x, y, Z. ti:z: , tiy, VI:) is identifia ble up to depth ambiguit y. Note once again tha.t (5.3) has

P is observed with Lhe &id of a CCD

or E: 7-, and

= - 2 (~: )

+93%4 -

Z4

Y2, Y3 ~ Pt - YIYS, y4. ~ P2 - Y2YS, ys, a.nd yr.. The dynamic s is gi ven by

=

(S.4)

i5

2 -9 [(91 - Y3yd(.h - !i3 l +(9Z - 9:'1/2)(Z2 -

Y4)J - Z3£4

In (5.4), t he vector (9101/2, j)a, Y4) is an estimate of (1/1, Y2, Y3,Y4.) and the vector (ZI,Z2 , Z3.%4) is an estimate of (V3d/4,Y5,YS).

6. CONGLUSION From the form of the a.bove equation s it is obvious that a two stage nonlins u observer describe d in section 4 can be applied to estimate Y3t Y"" y~, and Y6. If these estimate s are available. one can easily solve for the estimate s of x , y, Z , V%, tilt" V:. In order for the estimate s to converge we have to satisfy Assumptions 4.1 and 4.2 in section 4. Assumpt ion 4.1 is satisfied if the point mass doesn't come too dose to the focal point (origin). Obviousl y, the rank conditio n of Assump tion 4.2 for the first stage is satisfied because Wt = hx'l' If the staJ,es Vi are bounded then the rank conditio n for the second stag~ is ~ui va.lent to the follow· ing: det (-2 Y3 91 - Ylg3 ) - 2Y4 92 - YZ93

>

<

{or some positive € . This o bservabi lity conditio n will not be sati6~ lied for every combina tion of velocity and initial p08itions and has

In this paper we propose a new class of '!'dultista ge Nonlinea r Ob-server' for a cascade of nonHnear systems. The proposed observer generaliz es on earlier results in the theory of "single sta.ge nonlinear observer s" and is of independ ent interest. As an applicati on of the proposed single and multistag e observer s, we bave consider ed, in sufficient details, paramet er estimatio n problems from machine vision. It is well known that these estima.tion problems are ex· amples of i1J.posed. inverse problem s and one needs both spatial and time averagin g to obtain a robust 6lter. We claim in this paper that the proposed multista.ge non· linear observer theory provides an appropri ate foundati on for time averagin g with guar· anteed convergence propertie s. We show, via simulatio n, that tbe converge nce propertie s of the observer s are essential ly immune to the choice of the initial condition s and initial estimate s of the pa.rameters . IuJ a shortcom ing, th.e proposed obserwr theory presupposes that the states of the dyna.mical system are a.·priori bounded. within a known closed and bounded set. T his assumpti on may not always be satisfied in pr
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