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The extended Kalman filter as a parameter estimator with application to a pharmacokinetic example

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Content THE EXTENDED KALM AN FILTER AS A PARAMETER
ESTIMATOR WITH APPLICATION TO A PHARMACOKINETIC
EXAMPLE
Mary Joan Gennuso
A T h esis P resen ted to th e
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In P a r t i a l F u lf il lm e n t of the
Requirements fo r the Degree
MASTER OF SCIENCE
(A pplied Mathematics)
August 1986
UMI Number: EP54441
All rights reserved
INFORMATION TO ALL USERS
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In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
OIssartaîion F^Kiisf»ng
UMI EP54441
Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author.
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789 East Eisenhower Parkway
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UNIVERSITY O F SO U T H E R N CALIFORNIA
TH E GRADUATE SC H O O L
U N IV ER SITY PARK
LO S A N G ELES. C A L IFO R N IA 9 0 0 0 7
This thesis, written by
Mary Joan Gennuso
under the direction of h.e.ic....Thesis Committee,
and approved by all its members, has been pre
sented to and accepted by the Dean of The
Graduate School, in partial fulfillment of the
requirements for the degree of
M aster o f S c ie n c e
Dean
D ate JylX.15^.„1986.
THESIS COMMITTEE
C^nrman *
LL &
DEDICATION
To My Family and F rien d s
11
ACKNOWLEDGEMENTS
I s i n c e r e l y thank my a d v i s o r , P ro fe sso r Alan Schumitzky fo r
a l l h i s h e lp throughout my e n ro llm en t in the Applied M athematics
program and e s p e c i a l l y fo r h is a id in t h i s t h e s i s t o p i c .
1 a l s o am g r e a t f u l to the o th e r members o f my t h e s i s
com m ittee, P ro fe s so r Proskurow ski and P ro fe s so r G o ld ste in fo r
t h e i r su p p o rt.
I l l
TABLE OF CONTENTS
PAGE
DEDICATION...................................................................................................................................i i
ACKNOWLEDGEMENTS..................................................................................................................i i i
LIST OF FIGURES.........................................................................................................................v
CHAPTER
1 BACKGROUND..................................................... 1
2 LINEAR DISCRETE KALM AN FILTER.................................................................4
3 EXTENDED DISCRETE KALMAN FILTER............................................................9
4 DERIVATION OF THE EXTENDED KALMAN FILTER...................................12
5 THE EXTENDED KALMAN FILTER AS A PARAMETER ESTIMATOR. . . 1 7
6 PHARMACOKINETIC EXAMPLE AND CONCLUSIONS......................................21
REFERENCES...................................................................................................................................28
IV
LIST OF FIGURES
PAGE
Figure
2.1 D is c r e te L in ear Kalman F i l t e r ....................................................................8
3.1 D is c re te Extended Kalman F i l t e r ............................................................ 11
6.1 Summary of Computer Generated E s ti m a t io n s ...................................26
CHAPTER 1
BACKGROUND
The h i s t o r y o f the Kalman f i l t e r can be tra c e d back to the
m athem atician Gauss who invented d e t e r m i n i s t i c l e a s t sq u a re s fo r
use in an o r b i t a l measurement problem in 1795 ( a t the age of
e i g h te e n ) . The l e a s t sq u ares a lg o rith m produces e s ti m a t e s in
which th e sum o f the squares o f th e e r r o r s i s minimum. While
working on p r o b a b i l i t y d e n s it y f u n c tio n s , c i r c a 1910, F is h e r
d ev ised the maximum l i k e li h o o d e s tim a tio n . This te c h n iq u e u ses an
e s tim a te t h a t maximizes th e p r o b a b i l i t y of th e a c t u a l measurements
w hile ta k in g i n t o account i t s known s t a t i s t i c a l p r o p e r t i e s . The
n ex t development o ccu rred w ith Wiener c i r c a 1940. He designed a
p ro ced u re f o r th e frequency domain o f optim al f i l t e r s . This was
done through h i s work with random p ro c e s s th e o ry . W iener’s
te c h n iq u e a d d ressed only the c o n tin u o u s time problem . The optim al
e s tim a te s i t produced were lim ite d to the stead y s t a t e c a s e .
Meanwhile, Kolmogorov was working on th e d i s c r e t e - t i m e c a s e . I t
was during th e 1960’s t h a t Kalman and o th e rs expounded on o p tim al
r e c u r s i v e f i l t e r te c h n iq u e s . T heir work was based on s t a t e space
time domain fo rm u la tio n s . This approach has come to be known a s
th e Kalman f i l t e r and is w ell s u i t e d fo r d i g i t a l com puters. [3]
F i l t e r i n g i s t h a t ty p e o f e s tim a tio n problem which d e a ls w ith
e s tim a tin g the c u r r e n t system s t a t e based on a l l p r i o r
measurements. The o th e r two ty p e s o f e s tim a tio n problem s a re
sm oothing and p r e d i c t i o n . Smoothing d e a ls w ith th e d e s ir e d
1
e s tim a te s f o r th e s t a t e o f a system in th e p a s t tim e span o f
a v a i l a b l e d a ta and in p r e d i c t i o n problems the d e s ir e d tim e is in
th e f u t u r e .
The Kalman f i l t e r is a r e c u r s iv e f i l t e r t h a t e s tim a te s th e
c u r r e n t s t a t e v e c to r based on n o isy samples o f p re v io u s
m easurements. The measurement v e c to r i s an incom plete and noisy
fu n c tio n o f the s t a t e v e c t o r . [9]
A c r i t i c a l component in d eterm in in g th e o p ti m a l ity o f th e
Kalman f i l t e r i s the s p e c i f i c a t i o n o f the gain m a trix . Each
column o f th e e r r o r co v a ria n c e m atrix is m u l tip l ie d by th e
a p p r o p r ia te in v e rs e o f mean sq are measurement n o is e to produce the
Kalman gain m a trix . The elem ents o f th e g a in m a trix r e p r e s e n t th e
r a t i o between s t a t i s t i c a l measures o f th e u n c e r t a in t y in th e s t a t e
e s tim a te and th e u n c e r t a in t y in a measurement. T h e re fo re , th e
gain m a trix i s p r o p o r ti o n a l to th e u n c e r t a i n t y in the e s tim a te and
in v e rs e ly p r o p o r ti o n a l to th e measruement n o is e . [3]
A problem t h a t must be faced in Kalman f i l t e r i n g i s
d iv e rg e n c e . As L jung’s work d em o n strates, many assum ptions have
been in a p p r o p r i a t e l y made a s to th e p r o p e r t i e s o f co nvergent and
d iv e rg e n t e s ti m a t io n s . These p r o p e r t i e s have been s y s t e m a tic a lly
an aly zed by Ljung. [8]
The te c h n iq u e o r i g i n a l l y developed by Kalman was in ten d ed for
l i n e a r system s. This was l a t e r extended fo r the n o n lin e a r
c a s e s . The b a s ic in p u ts fo r th e f i l t e r a r e : system model, model
measurements, system and measurement e r r o r s and i n i t i a l
c o n d i tio n s . The f i l t e r then p ro c e s s e s th e measurement data and
g iv e s an e s tim a te o f th e s t a t e and i t s c o v a ria n c e e s ti m a t e .
Since i t s in c e p tio n , th e Kalman f i l t e r h as been u t i l i z e d in
e s tim a tio n s of such d iv e r s e a p p l i c a t i o n s a s : s a t e l l i t e s o r b i t s ,
ro c k e t guidance, n u c le a r r e a c t o r s , r i v e r flow , v e c to r cardiogram
c l a s s i f i c a t i o n , t r a f f i c d e n s i t i e s and p h a rm a c o k in e tic s . [ 3 ] I t i s
to t h i s l a s t a p p l i c a t i o n t h a t s p e c i f i c a t t e n t i o n w i l l be given in
t h i s t h e s i s .
A b r i e f o u t l i n e o f th e t h e s i s fo llo w s :
In Chapter 2, th e e q u a tio n s fo r th e l i n e a r d i s c r e t e Kalman
f i l t e r a re given as w ell a s e x p la n a tio n s o f a l l v a r i a b l e s .
Chapter 3 g iv e s th e e q u a tio n s and re a s o n s fo r th e d i s r e t e
extended Kalman f i l t e r .
Chapter 4 d e t a i l s the d e r i v a t i o n o f th e extended d i s c r e t e
Kalman f i l t e r .
Chapter 5 summarizes th e param eter e s tim a to r problem
a cco rd in g to th e work o f Ljung.
Chapter 6 d e t a i l s th e p h aram cokinetic example and red u ce s i t
to a d is c re te -d y n a m ic d i s c r e t e - t i m e c a s e . A summary o f computer
r e s u l t s and c o n c lu sio n s a re a ls o given.
CHAPTER 2
LINEAR DISCRETE KALMAN FILTER
A com plete d e r i v a t i o n o f the Kalman f i l t e r can be found in
r e f e r e n c e [ 3 ] . Here, only th e model and f i l t e r e q u a tio n s w i l l be
d e fin e d .
C onsider th e l i n e a r d i s c r e t e - t i m e system w ith dynamics given
by:
X(k + 1) = F(k) X(k) + G(k) W(k)
where :
k = 1, 2...
X = S ta t e Vector
F = System T r a n s it io n M atrix
G = Noise T r a n s it io n M atrix
W = Noise Vector
(dim ension n)
(n X n)
(n X p)
(dim ension p) — random fo rc in g
fu n c tio n o f te n r e f e r r e d to as
th e p l a n t n o is e .
This e q u a tio n c o rresp o n d s to th e d i f f e r e n t i a l e q u atio n
X(k) = $(k) X(k) + r(k ) W(k)
Let th e measurement e q u a tio n s be given by:
Z(k + 1) = H(k + 1) X(k + 1) + V(k + 1)
where :
Z = O b se rv atio n Vector
H = O bservation M atrix
V = Measurement Noise Vector
(dim ension m)
(n X m)
(dim ension m)
The fo llo w in g assum ptions a re made:
E[X (0)] = X(0)
E[W(k)] = 0 fo r a l l k
E[V(k)] = 0 f o r a l l k
X (0), W(k), V(k) a r e p a irw is e u n c o r r e la te d
Cov[X(0), X(0)] = P (0 ), P is th e system e r r o r e s tim a tio n
Cov[W(k), W (j)] = Q(k)6%j
Cov[V(k), V ( j) ] = R(k)6%j
= Kronecker d e l t a
= ( 1 i f k = j
\ 0 o th e rw ise
Q = V ariance o f th e S ta te Noise
R = V ariance o f the O b serv atio n Noise
Q, R a re n o n -n e g a tiv e d e f i n i t e
Cov[W(k), V ( j) ] = 0 fo r a l l k, j
Cov[W(k), X (0)] = 0 fo r a l l k
Cov[V(k), X (0)] = 0 fo r a l l k
' N(0, Q(k))
V ,^ - N(0, R(k))
X(0) r N(0, P (0)) where N(A,B) r e p r e s e n t s th e
m u l t i v a r i a t e normal d i s t r i b u t i o n w ith
mean v e c to r A and c o v a ria n c e m atrix
B.
A A
Now d e fin e X(k/k) and X(k + 1/k) to be th e b e s t l i n e a r unbiased
minimum square e s ti m a t e s of X(k) and X(k + 1) r e s p e c t i v e l y , given
th e s e t o f o b s e r v a tio n s (Z (1 ), Z (2), . . , Z (k )). Then th e
A A
q u a tio n s fo r X(k/k) and X(k + 1/k) a re given by th e fo llo w in g
F igure 2 .1 .
Model
Dynamics;
X(k + 1) = F(k) X(k) + G(k) W(k)
Measurements ;
Z(k + 1) = H(k + 1) X(k + 1) + V(k + 1)
F i l t e r E quations
S t a t e E x tra p o la te :
X(k + 1/k) = F(k) X(k/k)
Covariance E x tra p o la te :
P(k + 1/k) = F(k) X(k/k) P (k /k ) F^(k) + G(k) Q(k) G(k)^
Kalman Gain:
K(k + 1) = P(k + 1/k + 1) H?(k + 1) (k + 1)
S ta t e Update:
X(k + 1/k + 1) = X(k + 1/k) + K(k + 1) [Z(k + 1) - H(k + 1)
* X(k + 1 /k )]
Figure 2.1
Discrete Linear Kalman Filter
C ovariance Update;
P(k + 1/k + 1) = [H^fk + 1) R"1(k + 1) H(k + 1) + P"1(k + 1/k)]"^
I n i t i a l C o n d itio n s:
X (0/-1) = E[X(0)]
P (0 /-1 ) = Cov[X(0), X (0)] = P(0)
Figure 2.1 (Continued)
Discrete Linear Kalman Filter
CHAPTER 3
EXTENDED DISCRETE KALMAN FILTER
The extended Kalman f i l t e r i s used when the model and
measurement e q u a tio n s a re n o n l i n e a r . The major d i f f e r e n c e between
th e l i n e a r and extended Kalman f i l t e r i s t h a t in th e l i n e a r case
th e g a in s can be p r e c a l c u l a t e d . The extended f i l t e r u s e s th e
c u r r e n t e s tim a te s o f th e s t a t e as a r e f e r e n c e f o r each s ta g e in
th e l i n e a r i z a t i o n p ro c e s s . T h e re fo re , th e g a in s may n ot be
p re d e te rm in e d . In the extended f i l t e r the l i n e a r i z e d s t a t e
t r a n s i t i o n m atrix and th e l i n e a r i z e d o b s e rv a tio n m a trix are
computed by u sin g th e p a r t i a l d e r i v a t i v e s .
The d i f f e r e n c e s in th e extended Kalman f i l t e r w i l l become
more a p p a re n t in th e p h arm aco k in etic example t h a t fo llo w s in
C hapter 6. The fo llo w in g F igure 3.1 l i s t s th e e q u a tio n s f o r th e
extended Kalman f i l t e r .
Model
Dynamics :
X(k + 1) = f[X (k ), k] + G[X(k), k] W(k)
Measurements :
Z(k + 1) = h[X(k + 1), k + 1] + V(k + 1)
Where :
f(X ,k ) and h(X ,k) are n o n lin e a r v e c to r valued f u n c ti o n s o f
dim ensions n and m r e s p e c t i v e l y .
Assumptions on W , V and X(0):
Same as fo r d i s c r e t e l i n e a r Kalman f i l t e r .
I n i t i a l C o n d itio n s:
Same a s fo r d i s c r e t e l i n e a r Kalman f i l t e r
F i l t e r E quations
S ta te E x tr a p o la te :
X(k + 1/k) = f [ X (k /k ) , k]
Figure 3.1
Discrete Extended Kalman Filter
10
Covariance E x tr a p o la te :
P(k + 1/k) = F [X (k/k), K] P (k /k ) F ^[X (k /k ), k]
+ G [X (k/k), k] Q(k) G ^[X (k/k), k]
C ovariance Update:
P(k + 1/k + 1) = {üF[X(k + 1 /k ), k + 1] R“ ^(k + 1)
* H[X(k + 1 /k ) , k + 1] + P’ ^ (k + 1 /k ) } “
Kalman Gain:
K(k + 1) = P(k + 1/k+ 1) H^[X(k + 1 /k ), k + 1] R’ ^(k + 1)
S ta t e Update:
X(k + 1/k + 1) = X(k + 1/k) + K(k + 1) {z(k + 1)
- h[X(k + 1 /k ), k + 1]}
Where :
F [X (k /k ), k] =
3f(X ,k)
3X
X = X(k/k)
H[X(k + 1 /k ) , k] =
3h(X,k)
3X
X = X(k + 1/k)
Figure 3-1 (Continued)
Discrete Extended Kalman Filter
11
CHAPTER 4
DERIVATION OF THE EXTENDED KALMAN FILTER
An e x te n s iv e p ro o f o f th e fo llo w in g d e r i v a t i o n can be found
in r e f e r e n c e , [10] Here only th e d e t a i l s a re sk e tc h e d .
The model i s given as:
X(k + 1) = f[ X (k ) ,k ] + G [X (k),k] W(k)
Z(k + 1) = h[X(k + 1), k +1] + V(k + 1)
The r e c u r s i v e f i l t e r can be Implemented by u sin g a feedback system
and l i n e a r i z i n g th e n o n lin e a r problem . A model i s b u i l t o f the
n o n lin e a r dynamic sytem. Output from th e model i s used to p r e d i c t
what th e n e x t o b s e rv a tio n w i l l b e . The d i f f e r e n c e between the
model p r e d i c t i o n and th e a c t u a l o b s e rv a tio n i s used to c o r r e c t the
model. The Kalman g ain must be c a r e f u l l y chosen. Gains too low
w i l l d e c re a se c o n fid e n c e in th e system . Gains to o high may cause
the system to o v e rs h o o t, o s c i l l a t e and f a i l to tra c k X(k) c lo s e
enough.
To l i n e a r i z e th e n o n lin e a r problem , th e fo llo w in g methodology
is employed:
12
Choose a nominal t r a j e c t o r y :
%nom(k + 1) = (%)'
k]
X (0): is s p e c i f i e d ,
nom
D efine :
dxCk) = X(k) - X^^^Ck)
« ] -
= % n om (k )
\
= % nom (k)
d^Ck) = Z(k) - h[Xnom(k). k ] .
A Taylor s e r i e s expansion o f th e n o n lin e a r model y i e l d s an update
in the model e q u a tio n s ;
Model
d%(k + 1) - FCXnom^k), k] d^Ck) + GCX^^^^k). k] W(k) + . . .
d^(k) = H[X^Q^(k), k] d%(k) + V(k) +
13
The above e q u a tio n s a re approxim ated by dropping th e h ig h e r o rd e r
term s in d%(k). The r e s u l t i n g l i n e a r system s d^, d^ s a t i s f y the
assum ptions o f th e l i n e a r Kalman f i l t e r g iven in Figure 2 .1 . This
le a d s to th e fo llo w in g f i l t e r e q u a tio n s :
X(k + 1/k + 1) = d (k + 1/k + 1) + X ^(k + 1)
d_(k + 1/k + 1) = F[X (k ), k] d _(k/k)
A nom A
+ K(k+1) {Z(k + D - h ( k +1 ) , k + 1 ]
nom
- k] FCX^^^(k). k] d^ (k/k)}
where :
^nom
P(k + 1/k) = F[X _ ( k ), k] P (k/k) F^[x (k ), k]
nom nom
' G[%nom(k). k] Q(k) G^[X^^^(k). k]
P(k + 1/k + 1 ) = {H^[X (k + 1), k + 1] R“ ^(k + 1)
nom
* * U , k + 1] + P” k ( X + D / k ] } " ’
14
K(k + 1) = P(k + 1/k + 1) H^[X (k + 1), k + 1] R” \ k + 1 ).
nom
The Kalman g a in , K(k +1), depends only on X^^^Ck). The
d is a d v a n ta g e o f l i n e a r i z a t i o n about a nominal t r a j e c t o r y i s t h a t
X(k) - X^Q^(k) can become too la rg e even i f X(0) - ^nom^^^ i s
s m a ll. This can occur due to th e ap p ro x im atio n s made. [11]
The e q u a tio n s can be m odified by making th e fo llo w in g
s u b s t i t u t i o n s : r e p la c e X^^^Ck) and X^^^^k + 1) w ith
X (k/k) and X(k + 1/k) = f[X (k /k ), k ] . This changes l i n e a r i z a t i o n
from a b o u t th e nominal t r a j e c t o r y to ab o u t th e old e s tim a te .
Making the above s u b s t i t u t i o n s th e e q u a tio n s become th o se of
Figure 3 .1 .
S ta t e E x tr a p o la te :
X(k + 1/k) = f[ X (k /k ) , k]
Covariance E x tra p o la te :
P(k + 1/k) = F [X (k /k ), k] P (k /k ) F ^ [X (k /k ), k]
+ G [X (k/k), k] Q(k) G ^[X (k/k), k]
C ovariance Update:
P(k + 1/k + 1) = {H^[X(k + 1 /k ) , k+1] r" ^ ( k + 1)
* H[X(k + 1 /k ) , k + 1] + P " \ k + 1 /k ) } ” ^
15
Kalman Gain:
K(k + 1) = P(k + 1/k + 1) HF[X(k + 1 /k ), k + 1] R ~ h k + 1)
S ta t e Update:
X(k + 1/k + 1) = X(k + 1/k) + K(k + 1) {Z(k + 1)
-H[X(k + 1 /k ), k + 1]}
16
CHAPTER 5
THE EXTENDED KALMAN FILTER AS A PARAMETER ESTIMATOR
In t h i s c h a p te r, we i l l u s t r a t e how the extended Kalman f i l t e r
(EKF) can be used in e s tim a tin g unknown p aram eters o f a l i n e a r
dynamical system . The id e a s behind t h i s a p p l i c a t i o n a re due to
Ljung. [8]
C onsider th e l i n e a r d i s c r e t e tim e system (analogous to t h a t
d e s c rib e d in C hapter 2) g iv e n by:
q(k + 1) = A(0) q(k) + B(0) u(k)
Z(k + 1) = C(e) q(k + 1) + V(k + 1)
q(0) ~ N(qQ,P^)
Here q, u, Z and V a re the s t a t e , in p u t, measurement and n o is e
v e c to r s r e s p e c t i v e l y .
F u rth e r th e system m a tric e s A(9), B (9), C(9) are assumed to
be independent of k but to be dependent on an unknown param eter
v e c to r 9. For s i m p l i c i t y we a l s o assume th e input u i s
d e t e r m i n i s t i c , i . e . th e re i s no p ro c e s s n o is e .
The problem i s to e s tim a te th e tr u e v alu e of 9, say 9*, given
the N measurements Z (1 ), . . . , Z(N). The EKF approach to t h i s
problem i s now d e s c rib e d .
Extend th e s t a t e v e c to r q by a d jo in in g a new " s t a t e " 9.
Since 9 i s c o n s ta n t, i t s a t i s f i e s th e "dynam ical" e q u a tio n s :
17
8(k + 1) = e(k)
e (j) ~ n( 0q, Pg)
The extended s t a t e v e c to r X becomes
X(k)T = (q (k )T . e (k )T ).
The model e q u a tio n s fo r X are th en :
X(k + 1) = f(X (k ), K)
Z(k + 1) = h(X(k + 1), k + 1) + V(k + 1)
(5.1 )
(5 .2 )
where f*^ = (f^ ^ \ fg^) is such t h a t :
f-|(X,k) = A(0) q(k) + B(0) u(k)
f2 (X ,k ) = 0(k)
and where
h(X,k) = C(0) q(k + 1).
The i n i t i a l c o n d itio n s fo r ( 5 . 1 ) - ( 5 . 2 ) a re :
X(0) = X
0
Xq ~ N (0), P (0))
P(0) =
E quations (5.1 ) - ( 5 . 2 ) a re e x a c tly of th e form g iv e n by th e
dynamics and measurement e q u a tio n s o f the EKF d e fin e d in
F igure 3 .1 . The EKF can th en be used to c a l c u l a t e th e e s tim a te s
X (k/k)^ = (q (k /k )^ , 8 ( k / k ) ^ ) . A fte r N m easurements, 6(N/N) i s
then th e e s tim a te o f 0*.
To s im p lif y th e s e c a l c u l a t i o n s , th e p a r t i a l d e r i v a t i v e
m a tric e s F and H in F ig u re 3.1 can be e x p l i c i t l y ex p ressed in th e
p a r t i t i o n e d form;
F(X, k) = I I (X,k)
A(0)
0
H(X, k) = I I (X,k)
30
I
(A(0)q + B(0) u (k ))
C(0) |g ( C ( 0 ) q )
th e Kalman g a in K(k) and c o v a ria n c e e s ti m a t e s P (k /k ) can a l s o be
" p a r t i t i o n e d " as above.
In g e n e r a l, th e EKF may "d iv e rg e " a s k tends to «. Many ad-
hoc rem edies f o r " c o r r e c t in g " t h i s b e h a v io r have been suggested
(see [ 5 ] ) . A r ig o r o u s a n a l y s i s o f t h i s phenomena was f i r s t given
by Ljung. [8] For the above model, s u f f i c i e n t c o n d itio n s fo r
"convergence" have been e s t a b l i s h e d . These c o n d itio n s in c lu d e ,
fo r example, th e assum ption t h a t A(0) i s " s t a b le " fo r a l l
" f e a s i b l e " 0. For com plete d e t a i l s , th e i n t e r e s t e d r e a d e r should
r e f e r to Ljung [8 , Theorem 6 . 1 ] .
19
In p a r t i c u l a r , th e s e c o n d itio n s in s u re t h a t th e e s tim a te s
9 (k /k ) converge to 0* a s k te n d s to This l a t t e r r e s u l t w i l l be
" e x p e rim e n ta lly " v e r i f i e d in th e n e x t c h a p te r.
20
CHAPTER 6
PHARMACOKINETIC EXAMPLE
T his problem in v o lv e s the a d m i n i s t r a t i o n and o b s e rv a tio n o f a
drug given to a p a t i e n t . The o b s e rv a tio n s o f th e drug amounts are
taken a t d i f f e r e n t tim e s . In th e s im p l e s t c a s e , i t i s assumed
t h a t th e drug beh av io r can be r e p re s e n te d by a "one-compartment"
model. This model in v o lv e s : one s t a t e v a r i a b l e , X^, i s the
amount of drug in th e compartment, and one unknown param eter A,
where A i s the e l im i n a tio n r a t e c o n s ta n t.
The model can be d e s c rib e d a s fo llo w s :
Dynamics :
X ^(t) = -A X ^(t) + u ( t )
Measurements :
Z(k) = X^(t%) + V ( k ) .
The t^ are th e tim e p o in t s a t which measurements a re ta k en ; and
u ( t ) i s a known p ie c e w ise c o n s ta n t fu n c tio n , i . e . no p ro c e s s e s
n o is e . To c o n v e rt t h i s model to a d i s c r e t e - t i m e system , th e above
dynamical e q u a tio n s a re solved e x p l i c i t l y .
The s o l u t i o n is g iv en by;
X, (t) = e - 3 ) u( 8)d 8 ,
21
Now assume:
u(k) = u. = c o n s ta n t , t . < t < t . + 1.
1 1 - - 1
Then
J e u ,d s = u
-A(t - tg)
so t h a t th e s o l u t i o n can be r e w r i t t e n a s
% l( t l + l ) - s
1 - e
- A ( tj . 1 - t . )
Making th e s u b s t i t u t i o n s
X ,(n) = X ,( t^ )
u(n) = u ,
we can r e w r i t e th e l a s t e q u a tio n as;
X^(k + 1) = A(k,A) X(k) + B(k,A) u(k)
where :
A(k,A)
B(k,A)
+ 1 -
-A(t
1 - e
k + 1
22
Now apply th e methodology o f Chapter 5. Since A i s an
unknown c o n s ta n t , r e w r i t e A a s an o th e r " s t a t e " ;
A = XgCk).
Since A i s a c o n s ta n t :
XgCk + 1) = X^Ck).
I t fo llo w s t h a t th e model dynamics a re given by
-X (k)Ak
Xi(k + 1) = e X^(k) + U j^ %_(k)
1 - e
-Xg(k)Ak
X^(k + 1) = X^Ck)
where
Ak - tk + 1 - tk=
and th e measurement e q u a tio n is given by:
Z(k + 1) = X^(k + 1) + V(k + 1).
The extended Kalman f i l t e r can now be a p p lie d to t h i s system , as
d e s c rib e d in Chapter 5.
W e have:
f = (f^i , f 2 ) where
23
-X Ak
(X,k) = e X^ +
fgCX.h) = Xg
1 - e
-XgAk
h(X ,k) = X
To s a t i s f y th e "convergence" assum ptions o f Chapter 5, we assume
t h a t th e t^ a re evenly spaced, say Ak = 1 . For s i m p l i c i t y we a ls o
assume th e u = 0. I f we f u r t h e r r e s t r i c t X to be p o s i t i v e , then
A(X) i s " s t a b l e " . I t fo llo w s :
F =
9f^(X ,k)
ax.
afgCx.k)
air
a f ^ ( x ,k )
W Z
af2(X ,k)
axl
For our computer s im u la tio n s we ta k e th e t r u e v a lu e s to be
X^ (0) = 100
XgOO) = .2.
The i n i t i a l c o n d i tio n s a re a s fo llo w s :
X^(0 /0 ) = 100
X2(0/0) = .4
P (0/0) = 0 0
0 .16
24
Q = V ariance o f th e s t a t e n o ise = 0
R = V ariance o f th e observed n o is e which w i l l vary from .1
to 1000.
The r e s u l t s a re d is p la y e d in F igure 6 .1 . The t a b l e d is p la y s
the e s tim a te d v a lu e s o f X -| and X2 fo r the v a rio u s n o is e v alu es
du ring th e span o f 10 i t e r a t i o n s . The t r u e v a lu e o f X ^ a t s ta g e
10 is 13. 53353. The tr u e value o f X2 a t s ta g e 10 i s , o f c o u rs e ,
.2. The "convergence" o f t h i s f i l t e r is rem arkably good
c o n s id e rin g th e s i g n a l - t o - n o i se r a t i o when th e v a ria n c e of R i s
1000. (A c tu a lly s i m i l a r r e s u l t s were o b ta in e d fo r v a ria n c e v a lu e s
up to 100,000.)
25
s ta g e
Noise
50
1000
1 82.478472 86.036895 93.609768 98.649225
.169566 .1 16480 .003506
-0.071673
2 67.637488 69.215519 72.214748 81 .008854
Ï2
.185404 .170526 .135115 .068617
3 55.355571 55.763109 56.337032 54.452961
Î2
.190905 .185559 .168387 .1 56217
4 45.436890 46.067791 47.506695 50.156372
Î2
.192846 .186885 .1 68842
.142095
5
! i
37.236537 37.785180
39.134257 41.533241
Ï2
.1 94482 .189097 .173252 .14961 1
6
30.422367 30.529197 30.775993
27.682378
Î2
.196360 .193008 .182867 .180390
7
?1
24.985191 25.426614 26.529652 27.933481
?2
.196485 .191590 .178598 .160494
8
! l
20.549542 21.305448 23.224760 29.015522
Î2
.196255 .189735 .173365 .137792
9
Î1
16.697247 16.962944 17.502995 16.166625
Î2
.19861 1 .194006 .184291 .178166
10
Î1
13.645430 13.856386 14.227091 12.380683
% 2
.199238 .194833 .186251 .183836
F igure 6.1
Summary of Computer G enerated E stim a tio n s
26
CONCLUSIONS
For t h i s p a r t i c u l a r p h arm acokinetic example th e Kalman f i l t e r
proved to be extrem ely s t a b l e as p re d ic te d by the Ljung th e o ry .
As e x p ec ted , th e b e s t e s tim a tio n s o ccu rred w ith th e l e a s t amount
of n o is e . For t h i s c a s e , th e convergence o f % 2 was reached
r a p id l y and a t th e 10th sta g e a ls o approached th e tr u e v a lu e .
Although n o is e v a ria n c e o f 1,000 made th e e s tim a tio n s worse i t did
not cause f i l t e r d iv e rg e n c e .
REFERENCES
[1] Anderson, B rian, Optimal F i l t e r i n g . New J e rs e y :
P r e n ti c e H a ll, 1979.
[2] Bozic, S. M., D i g i t a l and Kalman F i l t e r i n g . New York:
John Wiley and Sons, 1979
[ 3] A. Gelb, e d .. A pplied Optimal E s tim a tio n , T ec h n ical S t a f f o f
th e A n a ly tic a l S c ie n c e s C o rp o ra tio n . M assa ch u setts:
M .I.T . P re s s , 1974.
[4] H ayashi, H. S ., The D is c re te Kalman F i l t e r With A p p lic a tio n
to T arg e t Motion A n aly sis (T ec h n ical Report) Naval
Undersea Research and Development C enter, Pasadena,
C a l i f o r n i a , June 1970.
[5] Ja z w in sk i, A. H ., S to c h a s tic p ro c e s s e s and F i l t e r i n g
T heory. New York: Academio P re s s , 1973.
[6] Leigh, J . R., A pplied C ontrol T heory. England: S hort Run
P re s s , 1982.
[ 7 ] Leondes, C. Y ., e d .. C o n tro l and Dynamic System s.
New York: Academic P re s s , 1981 .
[8 ] Ljung, L . , "Asymptotic Behavior o f th e Extended Kalman
f i l t e r as a Param eter E stim ato r fo r L inear system s", IEEE
T rans. Automat. C o n tr., Volume 24, pp. 36-50, F ebruary
1979.
[ 9 ] L o b d ill, J . , "Kalman M ileage P r e d ic t o r M onitor", B y te ,
Volume 6, pp. 230-248, J u ly 1981,
[10] L uenberger, D. G ., O p tim izatio n by Vector Space M ethods.
New York: John Wiley and Sons, 1969.
[11] Schweppe, F . C ., U ncertain Dynamic System s. New J e r s e y :
P r e n tic H a ll, 1973.
[12] Sorenson, H. W., " L e a st-S q u a re s E stim a tio n : From Gauss to
Kalman", IEEE Spectrum, Volume 7, pp. 63- 6 8, J u ly 1970.
28

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Creator Gennuso, Mary Joan (author)

Core Title The extended Kalman filter as a parameter estimator with application to a pharmacokinetic example

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