Table Of ContenterutceL Notes ni
Control and
noitamrofnI Sciences
Edited by .V.A and M.Thoma Balakdshnan
65
II III
vokaaY nivaY
Studies Numerical
ni Nonlinear gniretliF
II I
galreV-regnirpS
nilreB grebledieH weN kroY oykoT
Series Editors
A.V. Balakrishnan • M. Thoma
Advisory Board
L. D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L. Massey • Ya Z. Tsypkin - A. .J Viterbi
Author
Yaakov Yavin
c/o NRIMS
CSIR
P.O. Box 395
Pretoria 0001 - South Africa
ISBN 3-540-13958-3 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-13958-3 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data
Yavin, Yaakov
Numerical studies in nonlinear filtering.
(Lecture notes in control and information sciences; 65)
Includes bibliographies.
1. System analysis.
2. Filters (Mathematics).
3. Estimation theory.
.I Title.
II. Series.
QA402.Y3788 1985 003 84-23567
This work is subject to copyright. All rights are reserved, whethetrh e whole or part of the matedal
is concerned, specifically thoseo f translationr,e printing, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin, Heidelberg 1985
Printed in Germany
Offsetprinting: Color-Druck, Baucke, G. Berlin
Binding: L(Jderitz und Bauer, Berlin
2061/3020-543210
ECAFERP
State estimation techniques were developed for situations in engineering
in which, based no nonlinear dna noise-corrupted stnemerusaem of a process,
dna no a good model of the process, the process is estimated either on-
line or off-line using the available .stnemerusaem
esehT techniques emaceb nwonk in the early sixties under the celebrated
eman of namlaK filtering, dna were applied mainly to linear problems.
Later developments yb Kushner, mahnoW dna others led to solutions to
nonlinear state estimation problems, in which, in general, infinite-
dimensional filters are required. Practical algorithms, such sa the
linearized dna extended namlaK filters, which involve only finite-dimen:
sional filters, have been most frequently used sa approximate solutions
to these nonlinear state estimation problems.
ehT present work offers emos wen approaches to the construction of finite-
dimensional filters sa approximate solutions to nonlinear state estima=
tion problems. Numerical procedures for the implementation of these
filters are given, dna the efficiency dna applicability of these proce=
dures is demonstrated by snaem of numerical experimentation.
It is ym pleasant duty to record here ym sincere thanks to the National
hcraeseR Institute for Mathematical Sciences of the CSlR for encouraging
this research. I gratefully egdelwonkca the contribution edam yb Mrs.
H C Marais, Mrs. R ed Villiers dna Miss T~nsing, M H ohw wrote the :moc
puter programs for the examples presented here.
Finally, I should like to thank srM M wuossuR for her excellent typing
of the manuscript.
vokaaY Yavin
Pretoria, June 4891
STNETNOC
RETPAHC 1 : SEIRANIMILERP
1.1 NOITCUDORTNI I
2,1 ATINUK -ERUHPTNAILLAK-IKASIJUF GNIRETLIF ALUMROF 8
3.1 EHT 'DRADNATS' RAENILNON GNIRETLIF MELBORP ii
4.1 SECNEREFER 42
RETPAHC 2 : NOITAMITSE FO SRETEMARAP VIA ETATS NOITAVRESBO
1.2 NOITCUDORTNI 82
2.2 NOITAVIRED FO EHT RETLIF 92
3.2 MHTIROGL ANA R OGFNITUPMOC x n 13
4.2 ELPMAXE 1.2 EVA: W-ENIS ROTALLICSO 33
5.2 ELPMAXE SE 2.2VAW-RALUGN: AIRT ROTARENEG 53
6.2 NOITAMITSE FO VOA KRAM NIAHC 63
7.2 EHT SNOITAUQE FO LAMITPO GNIRETLIF 83
8.2 ELPMAXE 3.2 NO: SSIOP SSECORP 14
9.2 ELPMAXE 4.2 MO: DNAR HPARGELET LANGIS 05
01,2 SKRAMER 85
11.2 SECNEREFER 95
CHAPTER 3 ' VIA FILTERING VOKRAM SNIAHC NOITAMIXORPPA
.3 NI OITCUDORTNI 06
2.3 NOITCURTSNOC FO EHT VOKRAM NIAHC 26
3,3 EHT SNOITAUQE EH TFO LAMITPO RETLIF 46
"h,y
4.3 NMAHTIROGLA RO FGNITUPMOC xC 76
5,3 SELPMAXE : EHT ESAC l=m 07
6,3 SELPMAXE : EHT ESAC 2=m 08
7.3 YLLAITRAP ELBAVRESBO SMETSYS 09
3,8 SKRAMER 89
9.3 SECNEREFER 001
RETPAHC 4 : A NAMLAK FILTER ROF A SSALC FO RAENILNON CITSAHCOTS
SYSTEMS
1,4 NOITCUDORTNI 301
2.4 EHT EMIT-ETERCSID LEDOM 401
3.4 EHT EMIT-ETERCSID RETLIF 701
4.4 ELPMAXE 1.4 YCNE: UQERF DEBRUTREP EVAW-ENIS ROTALLICSO 801
5.4 ELPMAXE 2.4 : A EERHT ESAHP SEVAW-ENIS ROTARENEG 311
6.4 NOITAMITSE HTIW NIATRECNU SNOITAVRESBO 711
7.4 SKRAMER 621
8.4 SECNEREFER 821
RETPAHC 5 : GNITAMIXORPPA FILTERS ROF EMIT-SUOUNITNOC SMETSYS
WITH INTERRUPTED OBSERVATIONS
1.5 NOITCUDORTNI 031
2.5 NOITCURTSNOC FO EHT VOKRAM NIAHC 231
3.5 EHT SNOITAUQE FO EHT LAMITPO RETLIF 531
4.5 N AMHTIROGLA ROF GNITUPMOC (c~'Y, G h'y) 831
5.5 SELPMAXE : EHT ESAC l=m 041
6.5 SELPMAXE : EHT ESAC 2=m 841
7.5 SKRAMER 561
8.5 SECNEREFER 661
RETPAHC 6 : NOITAtVIITSE NI A TEGRATITLUM TNEMNORIVNE
1.6 NOITCUDORTNI 861
2.6 EHT SNOITAUQE FO EHT LAMITPO RETLIF 961
3.6 N AMHTIROGLA ROF GNITUPMOC (e,~) 471
6.4 SELPMAXE 571
5.6 SECNEREFER 181
vI
RETPAHC 7 : ETATS DNA RETEMARAP NOITAMITSE
1.7 NOITCUDORTNI 281
2.7 NOITCURTSNOC FO EHT VOKRAM NIAHC 481
3.7 EHT SNOITAUQE FO EHT LAMITPO RETLIF 681
.sh,y ~hl,y)
4,7 N AMHTIROGLA ROF GNITUPMOC ~x ~' 091
5.7 SELPMAXE : EHT ESAC l=m 191
6.7 SELPMAXE : EHT ESAC 2=m 602
7.7 SKRAMER 412
8.7 SECNEREFER 412
RETPAHC 8 : ETATS NOITAMITSE ROF SMETSYS NEVIRD YB RENEIW
DNA NOSSIOP SESSECORP
1.8 NOITCUDORTNI 512
2.8 NOITCURTSNOC FO EHT VOKRAM NIAHC 612
8.3 EHT SNOITAUQE FO EHT LAMITPO RETLIF 912
8.4 SELPMAXE : EHT ESAC FO l=m 222
5.8 SELPMAXE : EHT ESAC 2=m 822
6.8 NA NOISNETXE FO NOITAUQE (8.1) 632
7.8 SECNEREFER 832
RETPAHC 9 : NOITCIDERP VIA VOKRAM SNIAHC NOITAMIXORPPA
1.9 NOITCUDORTNI 042
2.9 EHT SNOITAUQE FO LAMITPO NOITCIDERP 142
3.9 N AMHTIROGLA ROF GNITUPMOC E~'Y(t,s) 442
4.9 SELPMAXE 642
RETPAHC i0: EMOS SNOISNETXE FO LINEAR GNIRETLIF
10.1 RAENIL GNIRETLIF HTIW NAISSUAG-NON INITIAL SNOITIDNOC 452
10.2 NOITAMITSE GN IFROEVUENAM STEGRAT 262
3.01 NOA ITCETED MELBORP 762
VII
10.4 ETATS DNA ECNAIRAVOC NOITAMITSE 072
10.5 SNOISULCNOC 272
10.6 SECNEREFER 372
RETPAHC 1
SEIRANIMILERP
1.1 NOITCUDORTNI
ehT problem of nonlinear filtering or state estimation nac eb described
as follows. ~x = {~x (t)' t m 0}, called the signal or the state of the
system, is anRm-valued stochastic process, direct observation of which
is not possible. ehT data related to x~ are provided by anRP-valued
measurement process Y = {y(t), t ~ 0}. ehT minimum variance estimate
of ~x(t), based no the measurements yt : {y(s), 0 ~ s ~ t}, is given by
the conditional expectation E[~x(t) Iyt]. This work deals with the pro=
blem of finding implementable approximations to E[~x(t) Iyt]. ehT efforts
here have been directed exclusively towards the derivation of finite-
dimensional filters for computing approximations to {E[~x(t) Iyt],
0 < t T < = T min(T,T)}, where • is the first exit time of ~x(t) from a
given open dna bounded domain D cR ,m dna T is a given positive number.
ehT following nonlinear filtering problems have been considered here:
(a) Estimation of parameters via state observation.
ehT process x~ satisfies the following equation
t
~x(t) = x + f [n(s)f(~x(S)) + g(~x(S))]ds + )t(WB , t ~ ,O x ER m
o (i .1)
where f : m~ ~m dna g m : R +~m are given functions; {W(t), t ~ }O is
an~m-valued standard Wiener process; dna {n(t), t m }O is a continuous
time Markov chain with a state space S which is at most countable. (The
special case where n(t) = n, t ~ ;O i.e. n is a random element, is con=
sidered first). ehT tnemerusaem process Y is given yb
y(t) = ~x(t) , t ~ 0 . (1.2)
ehT problem is to find {E[n(t) I~x(S ), 0 ~ s ~ t] , t • (O,T]}.
(b) ehT 'standard' nonlinear filtering problem.
ehT process x~ satisfies the following equation
t
~x(t) = x + [ f(~x(S))ds + )t(WB , t -> 0 , x E~m (1.3)
o
dna the tnemerusaem process Y is given yb
t
[
y(t) = g(~x(S))ds + FV(t) , t ~ 0 (1.4)
0
erehw f +~m m : R dna g : m~ ÷~p are given functions; B dna P are given
mxm dna pxp matrices respectively; {W(t), t ~ }O dna {v(t), t m }O are
~m-valued dna ~P-valued standard Wiener processes respectively. Let
D c~ m eb na open dna dednuob domain. ehT problem is to find =amixorppa
tions to {ix(t ) ~ E[~x(tATT-)Iy(s), 0 ~ S ~ tATT-], t e [O,T]} (aab
min(a,b), T = T TAT erehw T is the first exit time of Qx(t) from D).
(c) A modified namlaK filter.
ehT process xC satisfies the following equation
t
[
~x(t) = x + A(¢x(S))¢x(S)ds + )t(WB , t >- 0 , x R~e m (1.5)
o
dna the tnemerusaem process Y is given yb
Y(tk) : Y(tk)H(tk)~x(tk) + V(tk) , k t : Ak , k:O,1,... (1.6)
erehw A(x), xE~m dna B are given m×m matrices; {H(tk), k=0,1 .... } are
given mxp matrices; {W(t), t ~ }O is ahem-valued standard Wiener process
dna {V(tk), k=0,1 .... } is a sequence of independent RP-valued modnar
naissuaG elements. owT cases are considered:
(c-1) Y(tk) = I , k=0,1 .... (1.7)
(c-2) Y(tk) e {0,1} according to
p(k) = P(Y(tk) = i) , k=0,1,2 .... (1.8)
q(k) = i - p(k) = P(Y(tk) = )O , k=0,I,2 .... (1.9)
where the sequence {p(k)} is given.
Let ky = {Y(to),Y(tl) ..... Y(tk)}. ehT problem is to find approximations
to {E[~x(tk) IYk], k=1,2 .... } for cases (c-1) dna (c-2) respectively.
(d) State estimation for systems with interrupted observations.
ehT process x~ satisfies the following equation
t
~x(t) = x + f f(~x(S))ds + )t(WB , t ~ 0 , x e~m (1.3)
o
dna the tnemerusaem process Y is given yb
t
y(t) = f e(s)g(~x(S))ds + rv(t) , t ~ 0 (1.10)
o
where f,g,B,r, {W(t), t ~ }O dna {v(t), t ~ }O are the emas sa described
in Problem (b). {e(t), t ~ }O is a suoenegomoh jump Markov process with
state space {0,1}. Let D c~m eb na open dna dednuob domain. ehT pro=
melb is to find approximations to
{E[(~x(tATT-),8(tATT-))Iy(s) , 0 ~ S ~ tATT-] , t E [O,T]}.
(e) Estimation in a multitarget environment.
neviG L disjoint cells in the (Xl,X2)-plane. In each of the cells
there yam eb at most eno target. Let e = 18( ..... )L0 eb a modnar element
such that, for j:l ..... L : 0j=1 if there is a target at the j-th cell,
dna 8j=0 otherwise.
ehT tnemerusaem process Y is given yb
t
Yi(t) = ~ ) Cin(u Ud)u(nB + Yivi(t) , t e [O,T], i=1,2 (1.11)
o
