ISSC 2004, Bel as , June 30 - July 2
A Ja a Tool o Explo ing S a e Es ima ion using he Kalman
Fil e
Declan Delaney
1
and Tomas Wa d
2
1
Depa men o Compu e Science,
Na ional Uni e si y o I eland, Maynoo h
IRELAND
E-mail: [email protected]
2
Depa men o Elec onic Enginee ing,
Na ional Uni e si y o I eland, Maynoo h
IRELAND
E-mail: omas.wa [email protected]
__________________________________________________________________________________________
Abs ac – This pape desc ibes a no el so wa e applica ion
ha assis s in unde s anding he p ocess o es ima ing he s a e o a
linea dynamic sys em based on noisy ou pu signal measu emen s.
An in e ac i e Ja a ool, based on he Kalman Fil e , is desc ibed.
This consis s o wo main pa s, a simple one-dimensional il e and
a mul i-dimensional il e ool. The use can se all inpu
pa ame e s h ough a single in e ace o by ollowing a se ies o
guided s eps. Raw measu emen da a can be gene a ed
au oma ically o inse ed manually by he use and he esul s o
es ima ing he ue sys em s a e a e hen p esen ed as a se ies o
g aphs in eal- ime. The applica ion is desc ibed h ough he use o
wo simple examples. Such an applica ion could be used o each
signal and sys ems enginee ing.
Keywo ds – Kalman Fil e , Ja a applica ion, Signal and sys ems
enginee ing, Compu e assis ed lea ning
________________________________________________________________________________________
I INTRODUCTION
Fo mal enginee ing educa ion appea ed o he i s
ime a ound he middle o he 18
h
cen u y [1]. As
he discipline e ol ed, a ious in en ions se he
s anda ds o ‘mode n’ enginee ing. These
in en ions called in o exis ence new indus ies and
new s anda ds o educa ion and aining [2]. P io
o Wo ld Wa II he ypical elec ical/elec onic
enginee s udied e y li le undamen al science and
ma hema ics, wi h he emphasis es ing on he
acquisi ion o expe ience [3, 4]. By he end o
Wo ld Wa II all his had changed as i was ealised
ha undamen al knowledge o science led di ec ly
o new applica ions in he mili a y domain, wi h
consequen ial e ec s on enginee ing cu icula [5]
[1]. Du ing he pas decade he Wo ld Wide Web
and compu e -assis ed lea ning has again posed
in e es ing challenges and possibili ies o
app oaching he educa ion o elec ical and
elec onic enginee s [6 -9].
Fundamen al unde s anding o key concep s in
signals and sys ems is o en di icul o each
success ully. In dynamic linea sys ems, he concep
o es ima ing he s a e a iables om a se ies o
noisy measu emen s is one such example. The mos
widely accep ed algo i hm o pe o ming an
es ima ion o his kind is he Kalman Fil e . This
algo i hm is decep i ely simple and i s ho ough
unde s anding encapsula es a numbe o enginee ing
opics such as s a is ical modelling, enginee ing
compu a ion, linea sys em dynamics, measu emen
science and digi al signal p ocessing [10]. S udying
he ope a ion o he Kalman il e he e o e leads o
an app ecia ion o he in e -disciplina y na u e o
elec onic enginee ing. Howe e , a deepe
unde s anding o he heo y, and awa eness o
p ac ical implemen a ion issues, can only be
expe ienced by employing he il e in p ac ical
si ua ions. A compu e -based ool ha allows
expe imen s based on ‘wha i ’ scena ios would
challenge use s, mo i a e hem o ask ques ions and
acili a e he consolida ion o hei enginee ing
knowledge.
This pape desc ibes a no el Ja a implemen a ion o
he Kalman Fil e , which aims o assis use s in
unde s anding he e ec s o noise on he ou pu
signals gene a ed by a linea dynamic sys em and on
es ima ing he s a e o such a sys em. Two dis inc
implemen a ions a e desc ibed - a wo-dimensional
and a mul i-dimensional Kalman Fil e – and wo
illus a i e examples a e p o ided.
In sec ion II we b ie ly examine he o igins o he
Kalman Fil e and desc ibe he heo y unde lying i s
ope a ion. Sec ions III and IV desc ibe he wo-
dimensional and mul i-dimensional Ja a
implemen a ions espec i ely oge he wi h examples
ha illus a e hei use. Inspi a ion o he
ISSC 2004, Bel as , June 30 - July 2
applica ion de i ed om one o he au ho s’
indus ial expe ience in seismic explo a ion and om
expe ience in eaching elec onic enginee ing
s uden s.
II THE KALMAN FILTER
The i s me hod o o ming an op imal es ima e
om noisy da a is he me hod o leas squa es, a
disco e y a ibu ed o Ca l F ied ich Gauss in 1795
[10]. Du ing Wo ld Wa II No be Wiene wo ked
on he p oblem o au oma ically con olling he
di ec ion o an i-ai c a i e using noisy ada
in o ma ion and de i ed he solu ion o he leas -
mean-squa ed p edic ion e o in e ms o he
au oco ela ion unc ions o he signal and noise. By
combining p obabili y heo y and he idea o s a e
a iables Kalman subsequen ly de i ed he Weine
Fil e , which became known as he Kalman Fil e
a) Wha is he Kalman Fil e ?
The Kalman Fil e is named a e Rudol Kalman
who i s in oduced he algo i hm in 1960. I has
been employed in my iad applica ions including
p ocess con ol sys ems, ehicle acking, ma ine
na iga ion, geology, demog aphic es ima ion and
s ock p ice p edic ion. I es ima es he ins an aneous
s a e o a linea dynamic sys em pe u bed by
Gaussian whi e noise by using measu emen s ha a e
linea ly ela ed o he sys em s a e bu ha a e
co up ed by Gaussian whi e noise. The il e
ecu si ely minimizes he mean squa e es ima ion
e o wi hou di ec ly obse ing he sys em s a e o
knowing he na u e o he modelled sys em. We
only obse e some measu emen s using an a ay o
noisy senso s – see Figu e 1.
The sys em s a e i sel e ol es wi h ime unde he
e ec o andom pe u ba ions o con ol inpu s. The
Kalman Fil e hen p o ides an op imal es ima e o
he unobse ed sys em s a es and hei unce ain ies
based on noisy measu emen s o he p ocess. The
Kalman il e ope a es online so ha he bes es ima e
o he sys em s a e and i s unce ain y can be
compu ed by upda ing he p e ious es ima es wi h
new measu emen s.
b) Kalman Fil e Theo y
An ou line desc ip ion o he Kalman Fil e algo i hm
is p esen ed he e in equa ions (1) o (7). Fo a
de ailed heo e ical explana ion o he Kalman Fil e
e e o [10] and [11].
The Kalman Fil e a emp s o es ima e he n-
dimensional s a e ec o x o a i s o de , disc e e-
ime con olled p ocess wi h inpu s go e ned by a
linea s ochas ic di e ence equa ion, and an m-
dimensional measu emen ec o :
κκκκκ
wBuxx
+
+
Φ
=
−−− 111
(1)
κκκκ
xHz
+
=
(2)
whe e k is he ime index, w
k
is he p ocess noise,
k
is he measu emen noise, Φ is he s a e ansi ion
ma ix, B is he con ol ma ix, u is he con ol inpu
ec o , H is he sensi i i y ma ix and z is he
measu emen ec o .
Equa ion (1) de ines he sys em dynamic model and
equa ion (2) de ines he measu emen model.
The Kalman Fil e i e a i ely applies wo s ages o
compu a ions using eedback con ol:
1. ime upda e compu a ions ( he p edic ion) –
equa ions (3) and (4);
2. measu emen upda e compu a ions ( he adjus ed
p edic ion) – equa ions (5) o (7)
111 ˆˆ −−−
−+Φ=
κκκκ
Buxx
(3)
1111 −−−−
−+ΦΦ=
κκκκκ
QPP T
(4)
(
)
1−
−− +=
κκκκκκκ
RHPHHPK TT (5)
(
)
−−
−+=
kkkkkk
xHzKxx ˆˆˆ
(6)
(
)
−
−=
κκκκ
PHKIP
(7)
Figu e 1: The Kalman il e es ima es he
sys em s a e a iables based on a
mul idimensional signal + noise as inpu
Kalman Fil e
Con ol
inpu s
Noise Sys em
signal + noise
Noise Measu ing
de ice
Es ima e o
sys em s a e
a iables
Kalman Fil e
Con ol
inpu s
Noise Sys em
signal + noise
Noise Measu ing
de ice
Es ima e o
sys em s a e
a iables
ISSC 2004, Bel as , June 30 - July 2
whe e is he a p io i s a e es ima e a s ep k,
κ
x
ˆ
is
he a pos e io i s a e es ima e a s ep k, P
k
-
is he a
p io i es ima e e o co a iance ma ix, P
k-1
is he a
pos e io i es ima e e o co a iance ma ix , K
n
is he
Kalman Gain ma ix.
Equa ion (3) is he S a e es ima ion ex apola ion, (4)
is he e o co a iance ex apola ion, (5) is he
Kalman Gain, (6) is he s a e es ima e obse a ional
upda e and (7) is he e o co a iance upda e. These
equa ions a e implemen ed in he Kalman Fil e
applica ion which we will now desc ibe.
III THE 2D TEACHING TOOL
The Kalman Fil e eaching ool is a Ja a applica ion
ha can be used o e i y he s uden s’ knowledge
and allow hem explo e hei own unde s anding o
he e ec s o noise on measu ing a sys em esponse.
The ini ial applica ion window o e s he use h ee
pa hs o ollow (see Figu e 3):
1. A wo-dimensional e sion wi h sc olling
g aphics and wi h pa ame e s ha can be a ied
in eal- ime;
2. A guided s ep-by-s ep e sion o inpu ing da a
ha can ex end o six inpu s and six
measu emen s;
3. A mul i-dimensional e sion so ha he use
can inpu da a in any o de . The e is an uppe
limi o six inpu s and six measu emen s.
We will now explo e each o hese e sions in mo e
de ail h ough he use o wo simple examples.
The 2D e sion is limi ed o a single measu emen o
a single a iable as a unc ion o ime. Fo example,
we migh wan o op imally es ima e he empe a u e
o a oom o e ime using a sensing de ice. The use
se s he ele an pa ame e s h ough an in ui i e
g aphical use in e ace (see Figu e 4). When all
da a is en e ed he use s a s he Kalman Fil e
p ocess and da a is plo ed o sc een. The applica ion
can be s opped and s a ed o expe imen wi h he
uning and ansien e ec s o he il e se ings. All
pa ame e s can be a ied in eal- ime and he e ec s
on he ou pu obse ed simul aneously. When he
ou pu window ills wi h da a i sc olls o
accommoda e new da a. Help bu ons a e associa ed
wi h each unc ion bu on o p o ide a sho
explana ion o each ea u e.
As an example we will imagine we ha e a bucke o
nominally 10-ohm esis o s, accu a e o 1% RMS –
see du Plessis [12]. Imagine ha we selec each
esis o in u n om he bucke and using an
ohmme e wi h an accu acy o 0.3-ohms RMS
andom e o , we andomly choose a esis o om
he bucke , measu e i s alue using he ohmme e and
a emp o es ima e i s ue alue. This example
in ol es no dynamics and no ma ix no a ion. The
p ocess ansi ion assumes ha all he esis o s ha e
he same alue and ha any a ia ion is accoun ed o
by andom Gaussian noise. The measu emen model
says ha we a e measu ing he ac ual s a e wi h he
addi ion o noise. The alues gi en a e inse ed in o
he applica ion as ollows:
P ocess T ansi ion 1.0
Meas./S a e ela ion 1.0
Measu emen noise a iance 0.3
P ocess noise a iance 0.1
T ue alue 10
The Kalman Fil e is hen s a ed and he plo
ob ained is shown in Figu e 4. In his plo , we can
see wo ansien pe iods. The p ocess noise is he
same in bo h cases. The da a be ween he i s and
second ansien a e calcula ed based on he
pa ame e s gi en abo e. Fo he second ansien ,
he measu emen noise has been inc eased so ha he
Kalman Fil e will place less us in he
measu emen model and mo e us in he p ocess
model.
IV THE MULTI-DIMENSIONAL
TEACHING TOOL
This e sion ope a es up o a maximum o six inpu s,
six s a es and six measu emen s. The ac ual
measu emen s can ei he be simula ed by he
p og am o inse ed manually by he use . The e a e
wo ways o inpu he Kalman Fil e se -up da a, by
ollowing a guided s ep-by-s ep p ocedu e o iewing
he en i e Kalman Fil e se -up sc een a once. The
choice is made om he ini ial sc een iew (see
Figu e 3). We will b ie ly desc ibe each e sion
be o e p esen ing a simple illus a i e example.
The s ep-by-s ep e sion guides he use h ough a
logical p ocess o en e ing he da a. Each s ep
p o ides an explana ion o he da a being eques ed
Figu e 3: The ini ial sc een iew o he Kalman Fil e
eaching ool showing he h ee op ions : 2D, s ep-by-
s ep
o mul iD e sions.
−
κ
x
ˆ
ISSC 2004, Bel as , June 30 - July 2
and ou lines he heo e ical p e equisi es o being
able o ully unde s and he s ep a hand. An
example o one o hese s eps is shown in Figu e 5.
When he las s ep is eached he ull mul i-
dimensional Kalman Fil e ool is launched wi h all
he alues ini ialised o hose he use inse ed.
The ull mul iD e sion can also be launched by
choosing he hi d op ion in he ini ial sc een. Now
he use can click any bu on on he sc een in any
o de o inse he il e se -up da a. The sc een
layou is designed o be in ui i e and o e lec he
ac ual Kalman equa ions. Da a is en e ed s a ing
om he le and wo king owa ds he igh , as he
dimensions o he ma ices depends on he numbe
o s a es, inpu s and measu emen s. When he da a
is inse ed he use p esses he 'S a ' bu on, he
calcula ions a e pe o med and he esul s a e
g aphically p esen ed o he use . Figu es 6a and
6b show wo examples o he main sc een o
en e ing da a. In each case he e is a schema ic
diag am o assis he unde s anding o he
pa ame e cu en ly being se . The e is a hype ex
help ile, which also explains he Kalman Fil e
heo y, bu i is indispensable ha a no ice be
guided in he use o he mul iD e sion o he
applica ion.
Now we will examine a simple illus a i e example
based on an objec mo ing in he X-Y plane a a
cons an eloci y subjec o andom a ia ions in i s
ajec o y. The mo ion is desc ibed by a ixed
mo ion model. The new posi ion (X,Y) is he old
posi ion plus he eloci y (∆X, ∆Y) plus noise w.
A any ime we he e o e ha e:
+
−∆
−∆
−
−
=
∆
∆
∆
∆
Y
X
Y
X
w
w
w
w
Y
X
Y
X
Y
X
Y
X
)1(
)1(
)1(
)1(
1000
0100
1010
0101
)(
)(
)(
)(
(8)
Figu e 4: The 2D Kalman Fil e Teaching Tool Sc een Sho . All inpu pa ame e s can be a ied in eal ime o
iew he e ec s o noise on he sys em and he measu emen . In his example he applica ion has been s opped
and es a ed wi h new pa ame e s. The pa ame e s can also be a ied in eal ime.
Sc olling Sc een
Figu e 5: S ep ou o he s ep-by-s ep da a inse ion
ool.
ISSC 2004, Bel as , June 30 - July 2
I we assume ha we can only obse e he posi ion
o he objec (no i s eloci y), hen we ge he
ollowing measu emen model equa ion:
+
∆
∆
=
Y
X
Y
X
Y
X
Y
X
Z
Z
)(
)(
)(
)(
0010
0001
)(
)(
(9)
Suppose we s a ou a posi ion (0,0) mo ing o he
igh wi h eloci y (4,3) and we sample a andom
ajec o y o leng h 50. The ini ial co a iance
ma ix is assumed o be a diagonal ma ix wi h
alues o 1000, indica ing ha we do no us ou
ini ial es ima e. The measu ed da a we use comes
om a simple model o he sys em simula ed using
ano he applica ion.
When all he measu emen da a is en e ed and he
Kalman Fil e pa ame e s a e se up, he p ocess
should be s a ed by selec ing he START bu on.
This p oduces a numbe o g aphs as ou pu . The
ou pu g aphs o he inpu da a used a e shown in
Figu es 7a and 7b.
These plo s indica e he ac ion o he Kalman
Fil e . We a e es ima ing he objec ’s posi ion and
eloci y as a unc ion o ime, gi en i s posi ion
only. Ou inpu model desc ibes he noise
associa ed wi h he measu emen s and wi h ou
model o he ac ual sys em. As we collec mo e
da a we combine hese unce ain ies o ob ain a
maximum likelihood es ima e o he objec posi ion
and eloci y. The co a iances associa ed wi h he
s a e es ima es hen dec eases exponen ially as can
be seen in Figu e 7a. In Figu e 7b we can see a
plo o he aw measu emen s (inpu X s inpu Y)
and he es ima ed s a e (S a e 1 s S a e 2). The
es ima ed s a e plo is a smoo hed es ima e o he
aw measu emen s plo .
Figu e 6a : A iew o he main applica ion da a inpu sc een.
In his case he use has selec ed o inpu he ansi ion
ma ix. They ha e al eady in
dica ed ha he e a e ou s a es.
The schema ic assis s in unde s anding he ansi ion ma ix
as pa o he sys em p ocess.
Figu e 6b : Ano he iew o he main applica ion da a inpu
sc een. In his case he use has selec ed o inpu he
measu emen ma ix. They ha e al eady indica ed ha he e
a e wo measu emen s and ou s a es. The schema ic assis s
in unde s anding he measu emen model.
Figu e 7a : Each s a e and i ’s co a iance ma ix a e
plo ed as a unc ion o ime.
Figu e 7b : Each measu emen is plo ed oge he
wi h i s inno a ion.
ISSC 2004, Bel as , June 30 - July 2
V CONCLUDING REMARKS
This pape desc ibed he heo y and ope a ion o
he Kalman Fil e . I hen de ailed an inno a i e
Ja a-based Kalman Fil e ool ha ope a es in wo
dimensional o mul i-dimensional mode. Such
compu e -assis ed lea ning applica ions a e
essen ial o allow s uden s expe imen wi h ‘wha
i ’ scena ios in he domain o signals and sys ems
and hus deepen hei awa eness o undamen al
enginee ing concep s.
Ou u u e wo k will ex end he cu en applica ion
o include signal il e ing and p edic ion and will
hen look a he ex ended Kalman il e o non-
linea sys ems.
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