WP7 - 3:OO. TWO LOWER BOUNDS ON THE COVARIANCE. FOR NONLINEAR FILTERING PROBLEMS*. C. B. Chang. Lexington, Massachusetts. 02173.
-
WP7 3:OO TWO LOWER BOUNDS ON THE COVARIANCE
FOR NONLINEAR FILTERING PROBLEMS*
C. B . Chang Lincoln Laboratory, Lexington, Massachusetts
-Abstract Two covariancelower bounds f o r n o n l i n e a r f i l t e r ingproblemsarepresentedinthispaper. Thesebounds a r e basedupon the Cramer-Rao bound f o r t r e a t i n g n u i sanceparameters. The t i g h t n e s so ft h e s e bounds a r e examined using a nonlinearsystem where t h e r e c u r s i v e equation for covariance computation can be obtained. These r e s u l t s a r e a l s o compared with the bound o f Bobrovsky and Zakai
.
1.
lnroduction
The i n t r a c t a b l e n a t u r e o f t h e o p t i m a l n o n l i n e a r f i l t e r i n g problem means that approximations andad-hoc techniquesmust be employed t o c o n s t r u c t p r a c t i c a l f i l t e r sf o rn o n l i n e a r systems. Thus numerous i n v e s t i gations havebeen m o t i v a t e d t o seekboundson theestiso as t o mationerrorof an o p t i m a l n o n l i n e a r f i l t e r provide abenchmark t h a t a p r a c t i c a l f i l t e r d e s i g n can be compared t o [1:-[61.
ti.I .T. 02173
o ft h e Cramer-Raobound. I t hasbeen show, by means o f Monte C a r l os i m u l a t i o nt h a tt h e Cramer-Raobound f o r see [ 8 1 ) I. n d e t e r m i n i s t i c systems i s t i g h t (e.g., 111, [21 a l l t h i s case our boundsand t h e boundof converge t ot h e Cramer-Raobound. E v a l u a t i o no ft i g h t ness f o r a g e n e r a l s i t u a t i o n i s d i f f i c u l t s i n c e t h e covariancecomputationrequiressolving a setofinfiIt i s demonn i t e number o fd i f f e r e n t i a le q u a t i o n s . s t r a t e di n [91however, t h a tt h e r ee x i s t sc e r t a i nc l a s s o f n o n l i n e a r systemswheretheoptimalestimatorfor these systems i sf i n i t ed i m e n s i o n a l . We t h e r e f o r e use onesuchsystemasanexample toevaluatethetightness ofour boundsand the bound o f BobrovskyandZakai.
[ l o ] . I nt h a t Thispaper i s anextended v e r s i o no f paper,the Cramer-Raobound withnuisanceparameters was revi.ewedand t h e r e s u l t s f o r d i s c r e t e systemstated. we recapture some h i g h l i g h t s o f [ l o ] I nt h i sp a p e r , and p r e s e n tt h er e s u l t sf o rc o n t i n u o u s systems. We then applyour bounds t o a nonlinearsystemdescribedin[9] andcompare r e s u l t s w i t h BobrovskyandZakaiboundand thetruecovariance.
Among t h ee s t i m a t i o nt h e o r e t i c a l bounds known, the 2. Development o f Bound Equations Cramer-Rao lower bound i s n o t a b l e f o r i t s s i m p l i c i t y o f c a l c u l a t i o n and t i g h t n e s s i n t h e i m p o r t a n t case o f 2 . 1 . D i s c r e t e Systems h i g hs i g n a l - t o - n o i s er a t i o s U . n f o r t u n a t e l ya , ttempts toextendthe bound t o r e c u r s i v e e s t i m a t i o n p r o b l e m s Consider the following discrete system andmeasurewithprocessnoise have been o n l y p a r t i a l l y s u c c e s s f u l . ment equations. The major d i f f i c u l t y l i e s i n t h e f a c t t h a t n o n l i n e a r [l], e s t i m a t i o ni s an i n f i n i t e dimensionalproblem.In [ 2 ] , BobrovskyandZakaipresentedaversionof CramerRao bound f o r n o n l i n e a r systems w i t h a d d i t i v e Gaussian process andmeasurement noise.This bound i so b t a i n e d as t h e c o v a r i a n c e o f e a t i r n a t i n g t h e s t a t e s o f an aresystem andmeasurement n o i s ep r o where u andv "equivalent"linear systemwhere t h e s y s t e m m a t r i x i s v i s an uncorrelated c e s s e s ; k r e s p e d i v e l y . Assume t h a t + defined as t h e e x p e c t a t i o n o f t h e J a c o b i a n m a t r i x o f Gaussianrandomsequence w i t h z e r o mean and covariance thenonlinear system. Rk and + u i s a d i s c r e t e random process with an a r b i t r a r yb u t known d i s t r i b u t i o n . One wishes t oe s t i m a t e x I n t h i s paper, we present two v e r s i o n s o f t h e -k k.In basedupon t h e s e t o f measurements y., j = l , Crmer-Rao bound f o r n o n l i n e a r systems w i t h a d d i t i v e o r d e rt o estimate one x must j o i n t j y estimate Gaussianmeasurement noise. The system noise does n o t x x x Cramer-Rao bound on the covariance need t o beGaussian o ra d d i t i v e . Our bound, which i s x+d;imates i s obtajned by e v a l u a t i n g a o b t a i n e d w i t h a completelydifferentapproachfromthat 4 d e s c r i b e di n 111 and [ 2 ] , i s basedon t h em o d i f i e d matrix with the CCk+lln x ( k + l ) n )F i s h e r ' si n f o r m a t i o n Cramer-Rao bound d e r i v e d i n [7] f o rt r e a t i n gp a r a m e t e r s . x n ) ) being ( i , j ) t h submatrix(dimension(n The e s s e n t i a li d e ai st ot r e a tt h e unknown process Thus the noise componentsas nuisanceparameters. Cramer-Rao bound f o r systems w i t h o u tp r o c e s sn o i s ei s obtained as a f u n c t i o n o f t h e p a r t i c u l a rp r o c e s sn o i s e sequence r e a l i z e d , and t h i s f u n c t i o n i s averaged w i t h r e s p e c tt ot h ed i s t r i b u t i o no ft h ep r o c e s sn o i s et o wheren i st h ed i m e n s i o no ft h es t a t ev e c t o r It i s o b t a i nt h ef i n a l bound. (2.3) o b v i o u st h a t even i f i t i sp o s s i b l et oe v a l u a t e
...,
,..., 8'cl...,
6;
&.
A strongpointoftheapproach i s thatstandard software for Monte-Carlo evaluation of the performance oftheextended Kalman f i l t e r s can be employed t o generatethe bound. Thus computationofthe bound i s v e r ya t t r a c t i v ei np r a c t i c a la p p l i c a t i o n s . Sincethe bound o f [l], 121 and t h i s paperare versionsoflower bounds t o t h e Cramer-Rao bound, the t i g h t n e s so ft h e s e bounds i s t h e r e f o r e l i m i t e d by t h a t *Thiswork was sponsoredbytheDepartment o f t h e Army. The viewsandconclusionscontainedinthis document a r et h o s eo ft h ec o n t r a c t o r and shouldnot be i n t e r preted as n e c e s s a r i l y r e p r e s e n t i n g t h e o f f i c i a l p o l i c i e s ,e i t h e re x p r e s s e do ri m p l i e do ft h eU n i t e d States Government.
i t maybecome large.
intractableratherquickly
ask
becomes
It i s w e l l - k n o w nt h a tt h ee s t i m a t i o no ft h es t a t e sequence, isequivalenttotheestimationof and theprocessnoise sequence %, ,u enables one troe p l a cjeo i ndte n s i t y yk; % ,&). used in?3!31 by p{xl I n f i l t e r i n g problemsone'sgoal istoestimate x 4' can t h e r e f o r e the process noise sequence %, be t r e a t e d as a s e t of nuisance,parameters. We can thenapplythe Cramer-Rao bound f o r t r e a t i n g n u i s a n c e parameters t o t h i s problem. Two suchbounds f o rp a r a We now extend meterestimation were discussed i n [ 7 ] . them t os t a t ee s t i m a t i o nb e l o w .
...
,...,
,...
...,&-,
s,...
Assume t h a t a set of i sg i v e nt,h e n (2.1) becomes a d e t e r m i n i s t l c system. Further assume
3 74 0191-2216/80/000C+0374$00.75
@
1980 IEEE
that + v i s a Gaussian w h i t e n o i s e sequence w i t h z e r o i s Gaussianindependent mean andcovariance R and o f v w i t h mean II anb covarlance Po, then -k 4
x+
where u i s a s p e c i f i e d random process and + v is a #Mi te*Gaussianrandomprocess w i t h z e r o mean and coThe i n i t i a ls t a t ev e c t o r x i s Gaussian, variance R and w i t h mean4&, and c o v a r i independent o f and ance P
.
s,
0'
Forcontinuoussystems, we f i r s t o b t a i n t h e l i k e l i hood r a t i oo fs i g n a lp r e s e n tv e r s u ss i g n a la b s e n to f
YG{h
A ( Y t ;%/Ut)
The F i s h e r ' si n f o r m a t i o nm a t r i x is
--
; Oactland
conditioned on U tA =$
;
05sl.
(&-h(xT)) TRT -1 ('&-h-(&))dT
= cexp
-f(y&,)
on e s t i m a t i n g x given -k
%, . ..
%
T Po - 1 (x+-$, (2.11)
F(Qu~-~)=
C GTH~~ TR
~ - ~ H+ ~G~GTp0 ~ - 1 G~
(2.5)
i=l
The i n f o r m a t i o n m a t r i x o f x given U t can be ob-t t a i n e dw i t hs t r a i g h tf o r w a r dm a n l p u l a t i o n s .
where
-1
G i + l ; i = k-I,. . . , I
Gi
=
Oi
Gk
=
t (an i d meant tr i txy)
O
=
(2.12)
j Hi Uk-l
(2.6)
J a c o b i a nm a t r i xo f ( L , E j w ) i t hr e s p e c t o and x e v a l u a t eaxdt.
-
where
fFSds
-J
=
J a c o b i a nm a t r i xo fh ( L )e v a l u a t e da t
=
t h ec o l l e c t i o no fa l l
O(t,r)
3 F
= eT
c(x,;)
= theJacobianmatrixof and e v a l u a t e da t xand
w i t hr e s p e c tt o
. h(x) e v a l u a t e d a t
+ x
u
- 5 -
U s i n gt h eF i s h e r ' si n f o r m a t i o nm a t r i xd e f i n e di n (2.5) two lower boundson t h ec o v a r i a n c eo fe s t i m a t i n g 5 canbecomputed 171, [lo]. They a r e
HT = the Jacobian matrix of
S i m i l a r l y , we o b t a i n twobounds
based upon F(x+/U
t
).
B1 (x+) = E I F - ' ( x + / U t ) ] Ut'
(2.13)
k
(2.14)
and B2(x+) =
t[F(x+/Utl~-l
U where Ee [.I denotestheexpectationoftheenclosed 8. N o t i c et h a tt h ed i f f e r e n c e q u a n t i t yw i t hr e s p e c tt o between thesetwobounds i s i n whetheronetakesthe (E2) o r a f t e r (Bl) t h ei n v e r s i o no f expectationbefore t h eF i s h e r ' si n f o r m a t l o nm a t r i x .
One can e a s i l y show t h a t t h e F ( x + / t ) abovecan be canputedusingthematrixRiccattiequavionforthe f i l t e r i n g p r o b l e m o f a d e t e r m i n i s t i c systemevaluated usingJacobianmatricesabout a g i v e ns t a t et r a j e c t o r y
b&, 05s).
2.3. /U ) i st h e We n o t et h a t h ei n v e r s eo fF ( x solutionofthematrixRiccatti equ&iook-#or t h e f i l t e r ing problem of a deterministic system with noisy measurements evaluatedusingJacobianmatricesfor a t r u es t a t et r a j e c t o r y .
N o t i c et h a tt h e bound computationrequirestaking expectationwithrespecttotheprocessnoise.Since t h i s e x p e c t a t i o n can not be o b t a i n e d a n a l y t i c a l l y i n general, we suggestthefollowingprocedurefor bound computation.
Sincetheestimateof xdepends o ne s t i m a t e so f w h i l teh ae b o v e * o n u l a t i o n does n o t e x p l i c i 3 b ; ' i n c I u d e suchdependency,one therefore e x p e c t s n e i t h e r of theabove bounds t o be as t i g h t as theexact Cramer-Rao bound, equation ( 2 . 3 ) . These bounds a r e n e v e r t h e l e s s a t t r a c t i v e because of t h e i r s i m p l i c i t y and t h e i n t r a c t a b l e n a t u r e o f t h e e x a c t bound equation.
3 ,...,
2.2.
+ u Draw asample processnoisetimehistory, whichcan be obtainedfrom an a r b i t r a r y b u t specifieddistribution.
or
4,
Compute t h e t r u e s t a t e t r a j e c t o r y u s i n g t h i s p r o c e s sn o i s eh i s t o r y . O b t a i nc o r r e s p o n d i n gs o l u t i o no ft h em a t r i x R i c c a t t ie q u a t i o n ,i . e . ,t h ec o v a r i a n c em a t r i x e v a l u a t e du s i n gt r u es t a t e s .I t si n v e r s e informationmatrix.
Continuous Systems
We usetheseformalequationstodenote tinuous stochastic- with state vector measurement vector
Discussion
acon-
~r, and
Repeat ( 1 ) - ( 3 ) many times;then o ft h ea v e r a g e di n f o r m a t i o nm a t r i x averagedcovariancematrix.
&*
i s the
B2 i st h ei n v e r s e and Bl i s t h e
We have already comnented t h a t n e i t h e r 6. nor Bi s as t i g h t as t h e& a c t Cramer-Raobound. These td bounds areneverthelessusefulbecause the exact bound 375
cannotbe c q p u t e d i n g e n e r a l and f u r t h e r y o r e ,o u r systems. bounds a p p l y t o a v e r y g e n e r a l c l a s s o f
B1 and B i su n f o r The c a n p a r i s o no ft i g h t n e s so f i f o8e a p p l i e s t u n a t e l ya l s on o n t r i v i a l . Forexample, Jensen’sinequality onconvex/concave f u n c t i o n s t o comt o be inconpare B1 and B2, t h e r e s u l t can beshown clusive.Numerical examples o f 171 showed t h a t B 1 i s H e u r i s t i c argumentswere a l s o used i n t i g h t e rt h a n B 171 t os p e c u l a t et h a t B i st i g h t e rt h a n B2. I t was basedupon t h e f a c t t h a l B 1 a p p l i e s t o amore s t r i c t B2. Inthenumericalexample c l a s so fe s t i m a t o r st h a n o f t h i s paper, B i sa g a i n seen t i g h t e rt h a n B2 although i t i s d i f f i c u l t lo p r o v e a n a l y t i c a l l y .
.
3.
Example N o t i c et h a t ill and a r es o l u t i o n sotfh e Kalman-Bucy covarianceequation at% d e t e r m i n i s t i c , t h e y a r e t h e r e f o r e r e p l a c e dw i t h P1, and P 2 . The c o r r e l a t i o n o f jil and Q 2 , P12, i sz e r o i f I t si n i t l a 1c o n d i t i o ni sz e r o .
We now useasystem discussed i n 191 and [Ill t o evaluate the t i g h t n e s s o f our bounds and the bound o f I t i s shown i n [9] t h a tt h e r e BobrovskyandZakai. eixsts a certain class of nonlinear systemswhere t h e optimalestimatorsforthese systems a r e r e c u r s i v e and f i n i t ed i m e n s i o n a l . Thesesystems arerepresented by c e r t a i n V o l t e r r a s e r i e s e x p a n s i o n s o r by b i l i n e a r systems w i t hn i l p o t e n tL i ea l g e b r a . One o f suchsystems can be represented by these following equations.
Ue n o t e t h a t t h e above r e s u l t s d i s a g r e e w i t h t h a t o f [111 i n which S1 and S2 were s e tt oz e r o .
Let E let
yand
enntethe B 1 boundon estimatingxl, then 2 i st h es o l u t i o no f
dE(i),
E =
Ff
- FHTHt:
+ ;FT
where
z are Where xl,x2 andy a r es t a t ev a r i a b l e s , zland 2 measurement v a r i a b l e s , a , 6, and y a r e p o s i t i v e conmean, stants, w 1, w2, vl, and v2areindependent,zero ( O ) , x ( O ) , and u n i t v a r i a n c e Wienerprocesses,andx1 y(0)areindpendentGaussian random v a r i a b l e 3 w i t h known means and variances.
i t here.
B, Bound
3.2. The
x2,and (3.12)
Noticethatthedifference betweenthe above equation and t h e l i n e a r i z e d R i c c a t t i e q u a t i o n i s t h a t t h e ex l i c i t dependence o f E on theprocessnoisevariance The 2 i s s t i l l i m p l i c i t l y q 7s n o t shown i n (3.12). i n f l u e n c e by theprocessnoise dl! since it i s a f u n c t i o n o ft h es t a t e sx 1 andx2. R e w r i t i n g (3.12) i n terms o f thecanponents o f .f y i e l d s
N o t i c e t h a t x (t) and x 2 ( t ) a r e s t a t e s o f a comp l e t e l ys p e c i f i e dl i n e a r GuassianMarkovprocess, their o p t i m a le s t i m a t e sa r et h es o l u t i o no ft h e Kalman-Bucy f i l t e r . The s t a t ey ( t )r e p r e s e n t s however,a nonlinear system. The o p t i m a l( f i n i t ed i m e n s i o n a l )e s t i m a t o rf o r y ( t ) was s p e c i f i e di n [9]. The e r r o rc o v a r i a n c e on t h e was s t a t e di n [Ill.* We b r i e f l y y ( t )e s t i m a t e s ,F ( t ) . d e r i v et h eo p t i m a le r r o rc o v a r i a n c ee q u a t i o n s and the bound equation on 9 ( t ) below.
3.1.
(3.7) was
The e v a l u a t i o n o f t h e l a s t t e r m o f d e t a i l e d i n [ill, we will notrepeat
The OptimalCovarianceEquation
A
y(t)/Zt) denote ihg error L e t P = c o v ( x l ( t ) , x,(t), covariance on estimating x and y ( t ) and Z = 0 3 5 1 . U s i n ge q u a t i k A t t j . 1 ) - ( 3 . 3 ) and [131, one o b t a i n s t h e v a r i a n c e on e s t i m a t i n g y ( t ) ,
%,
N o t i c et h a t
we have replaced
Ell
€(Ez3), 2
RB
S!
2
3.1.
376
then the B1
f o l l o w st h ef o l l o w
33
ZR;
+
ZR:
= -2(a+y+ E +sll 1
b
+ z ~ b, R ~
i33 = +
ii
-
by E and C Z 2 211 b A g2E(e13), S2 =
E(EZ3xl),
X
boundon t h ev a r i a n c e so fq ( t ) , s e t o f d i f f e r e n t i ael q u a t i o n s .
where E denotes the conditional expectation with respect t o Z and PI1 and PZ2 f o l l o wt h e Kalman-Bucy covariance equation.
o b t a i n e ds l i g h t l yd i f f e r e n tr e s u l t s .S e e ‘ S e c t i o n
R1b A = E(Zl3x2), and
2
and
s i n c et h e ya r ed e t e r m i n i s t i c .L e t
r-’(~f+
$1
(3.18) (3.19)
k;
= -(a+P-y+T)Rl E11
b
+
T h i si si d e n t i c a lt ot h e
EllQ2
B2 bound o f t h i s
paper.
We would l i k e t o emphasize t h a t t h e above r e s u l t does n o t c h a r a c t e r i z e t h e p r o p e r t y o f t h e B o b r o v s k y and i f t h e r ei s Zakaibound i n general.Forinstance a d d i t i v e GaussianWhi.teprocessnoise i n Eq. (3.21, t h i s bound becomes n o n t r i v i a l w h i l e o u r bounds a r e unchanged.
i,
= -2m2
+q
3.5.
We compare thesethree bounds and t h e t r u e e r r o r c o v a r i a n c ef o r a=B=y=.g. The i n i t i a le r r o rc o v a r i a n c e i s chosen t o be an i d e n t i t ym a t r i x .I nF i g u r e 3.1, we show t h e s er e s u l t sa s a functionfortimeforq=r=l. A s d i s c u s s e db e f o r e ,s i n c ea l lt h r e e boundsconverge t o zero,they become t r i v i a l bounds f o rs t e a d ys t a t e . B1 bound i s t h et i g h t e s t . D u r i n gt r a n s i e n tp e r i o d ,t h e Sincethe f i l t e r p e r f o r m a n c ed u r i n gt r a n s i e n tp e r i o d can be i m p o r t a n t f o r many a p p l i c a t i o n s , we explorethese we show t h e B r e s u l t sf u r t h e r .I nF i g u r e s 3.2and3.2 1 and B2 bound ( i t i s equal t o t h e Bobrovsky and Zakai bound f o r t h i s example) normalized by t h e t r u e e r r o r covarianceas a f u n c t i o n o f r and q , r e s p e c t i v e l y , a t one second a f t e rf i l t e ri n i t i a t i o n . These r e s u l t s i n d i c a t et h a ta l l bounds a r e t i g h t f o r h i g h s i g n a l t o n o i s e r a t i o s (low r and q values) and loose f o r low r and q v a l u e s )w h i l et h e s i g n a lt on o i s er a t i o s( h i g h B, bound i s a l w a y st i g h t e rt h a nt h e B2 bound.
where and O2 a r et h eu n c o n d i t i o n e dv a r i a n c e so xf 1 andx2. N o t i c et h a t Bl bound e q u a t i o n sa r ev e r ys i m t l a r (3.7) (3.11). t ot h et r u eo p t i m a lc o v a r i a n c ee q u a t i o n s The m a j o rd i f f e r e n c ei st h a t Sb S k , R t , R k a r ed r i v e n z e r o w h i l e S , S 2 , Rl, by E l l and X Z 2 whichconverge and P22 whichconverge a R a r ed r i v e n by P o f r and q. The Bl sgeady s t a t es o l u t l h na sf u n t i o n bound i st h e r e f o r et r i v i a lf o rs t e a d ys t a t e . I t should be t i g h t however, d u r i n gt r a n s i e n tp e r i o d .
-
1;
3.3.
tb
The B2 Bound L e t D d e n o t et h ei n f o r m a t i o nm a t r i xa sd e f i n e di n then D i s t h e s o l u t i o n o f t h e f o l l o w i n g e q u a t i o n .
(2.12),
=
- 6 ~-
FTa +
1 HTH
(3.23)
A
(3.12). Let D E(E) then where F and H a r ed e f i n e di n one c a n r e a d i l y show t h a t t h e components o f D f o l l o w t h e f o l l o w i n gs e to fd i f f e r e n t i a le q u a t i o n s .
DZ2 = 2BDZ2 DII
= 2aD,1
2AZ1
+
r
2A12
+
r
Althoughour bounds a r e t r i v i a l f o r s t e a d y s t a t e f o r t h i s example, we have a p p l i e d them t o o t h e r nonl i n e a r problemsand t h e Bl bounddoes approach a nonsome problems. z e r os t e a d ys t a t ef o r
4.
(3.25)
(3.27)
= (Y+B-a)AZ1
-Q
~
D
~
~
where and 0 a r de e f i n e iden q u a t i o n s (3.23), (3.24). The a b o v e r e s u f t s a r e o b t a i n e d w i t h t h e i n i t i a l c o n d i t i o n o f D b e i n g a diagonalmatrix. It i s seen t h a t The ( 3 . 3 ) t ht e r mo f t h em a t r i x D ( t ) remainsdiagonal. of v ( t ) . t h ei n v e r s e of D i s t h e B2 bound onthevariance L e t t h i s bound be denoted by Ej3,oneobtains
The above r e s u l tr e p r e s e n t s a t r i v i . a l boundbecause i t converges t o z e r o and i s independent o fn o i s ep a r a m e t e r s r ,q.
3.4.
The Bobrovsky and Zakai
Bound
T h i s bound was a l s o usedby 1111 forcomparison. L e t EBzdenote t h i s bound, i t i st h es o l u t i o nf o rt h e f o l l o w i n g R i c c a t t i equation.
iBz = 9” + EBzFT where
+TH
+ qE Bz GG Z Bz
The featureofour bounds i s t h e i r s i m p l i c i t y i n c o m p u t a t i o na n dt i g h t n e s sf o rh i g hs i g n a lt on o i s e ratios.Furthermore,theycan be a p p l i e dt o a v e r y generalclass of nonlinearproblems. The weakness ofour bounds i s t h a t t h e p r o c e s s noisecovariance does not alwaysappear e x p l i c i t l y i n t h e bound equation. To a l l e v i a t et h i sp r o b l e mr e q u i r e s sane v e r yt e d i o u sm a n i p u l a t i o n s .T h i ss u b j e c ti s c u r r e n t l yb e i n gi n v e s t i g a t e d . Our example showed t h a t one o f o u r bounds i s tighterthantheBobrovsky and Zakaiboundandtheother not one i s t h e same. T h i sc o n c l u s i o n however,does h o l d i n g e n e r a l . We cannot a n a l y t i c a l l yp r o v ew h i c h one of o u r bounds i s t i g h t e r a l t h o u g h we have c o n s i s t e n t l yo b s e r v e dn u m e r i c a l l yt h a tt h e bound u s i n g t h e average o f t h e i n v e r s e o f t h e i n f o r m a t i o n m a t r i x i s an a p p r o p r i a t e t i g h t e r (see a l s o [ 7 ] 1 , We f e e lt h a t c a n b i n a t i o n of a l l t h r e e bounds will g i v e t h e t i g h t e s t lower bound f o r n o n l i n e a r f i l t e r i n g problems t o d a t e . Acknowledgment
(3.30)
The a u t h o r w o u l d l i k e t o e x p r e s s h i s a p p r e c i a t i o n The o r i g i n a l t o Drs. K. P. Dunn and N. R. S a n d e l l ,J r . i d e a o f t h i s paper came o u t o f d i s c u s s i o n s among us. EspeciallyDr. Dunn, w h o has p r o v i d e d s t i m u l a t i n g d i s c u s s i o n st h r o u g h o u tt h ec o u r s eo ft h i sr e s e a r c h .
-
F = E(F)
GT =
[b
3
One can r e a d i l y shcw thattheBobrovsky onthevarianceof q(t) satisfies
S u m r y and Discussion
We have presentedtwolower bounds f o r n o n l i n e a r f i l t e r i n g problems. They a r eo b t a i n e du s i n gt h e CramerRao boundby treatingtheprocessnoiseprocessas a setofnuisanceparameters.
D33 = 2YD33
A,,
Numerical Results
and Zakai bound
377
------ -
aptimal Variance B1 Bound
B 2 Bound
r =q- L
1
0
3
2
4
5
Time F i g . 1.
Bound and variancevs.time.
L-P .9
.a
9 ' 2
-
.7 '
P
g
.b
BIBoundl@timum Variance
-
m
-mii
.5-
J
.4
L
-7
0
c
-0
-
.3 '
E
B2Boundl(PtimumVariance
.2.l' 0
I
.Ol
I
I
I
wv-1
.os
I
. l
I
I
.5
-
..'I
Measurement Noise Variance Fig. 2.
s
B
z B
7
m
Bound and
v a r i a n c er a t i ov s .
I
1.
I
1
5.
-r
measurement noisevariance.
References
[l] B. Z. Bobrovskyand
M. Zakai, "A lower bound onthe e s t i m a t i o ne r r o rf o r Markovprocesses", IEEE Trans. Automat. Contr., Vol. AC-20, 785-788, (December 1975)
.
[ 2 ] 8. Z. Bobrovskyand M. Zakai, "A lower bound on the estimation error for certain diffusion processes", I E E E Trans.Inform.Theory, Vol. IT-22, 45-52, (January1976). ~~
~~
~
and I . B. Rhodes, " F i l t e r i n g and [ 3 ] 0. L. Snyder controlperformance bounds w i t h i m l i c a t i o n on Vol. 8, a s y m p t o t i cs e p a r a t i o n " ,A u t m t i c a , 747-753, (November 1972).
I . 8. Rhodes, "Cone-bounded non[ 4 ] A. S . Gilmanand l i n e a r i t i e s andmean-square bounds-estimation Vol. ACupper bound", 1 8, 260-265,(June 1973). [ S I I . B. Rhodes and A. S . Gilman, "Cone-bounded n o n l i n e a r i t i e s andmean-square bounds-estimation lower bound", IEEE Trans. Automat. Contr., Vol AC-20, 632-642, (October 1975). . H. T a,v l o r .. "The Cramer-Rao e s t i m a t i o ne r r o r _161_ Jlower bound c o m p u t a t i o nf o rd e t e r m i n i s t i cn o n l i n e a r systems", I E E E Trans.Automat.Contr.,Vol. 343-344,(Apr i 1 1979).
AC-24,
[ 7 ] R. W. M i l l e r and C. B. Chang, "A m o d i f i e d CramerIEEE Trans. Rao boundand i t sa p p l i c a t i o n s " , V o l . IT-24, 398-400 (Hay 1978). Inform.Theory, [ 8 ] C. 8 . Chang, " B a l l i s t i ct r a j e c t o r ye s t i m a t i o nw i t h angle-only measurements", IEEE Trans.Automat. Contr.,Vol. AC-24, 474-480(June1980).
[9] S . I . Marcusand
A. S . Willsky,"Algebraicstruct u r e and f i n i t e dimensionalnonlinearestimation", S l A H J. Math,Anal.,Vol. 9, 312-327, ( A p r i l 1978).
[lo]
C . B. Chang, K. P. Ounn, and N. R . Sandell,Jr., "A Cramer-Rao bound f o r n o n l i n e a r f i l t e r i n g p r o b l e m s w i t h a d d i t i v e GaussianMeasurementsNoise". I E E E Conf.onDecision Proceedings o ft h e1 8 t h and C o n t r o l , 511-512, (December 1979).
[ll] C. H. L i u and S. I . Marcus, "Estimatorperformance foraclassofnonlinearestimation problems", Proceedings o f t h e 1 8 t h IEEE ConferenceonDecision and C o n t r o l , 518-523, (December 19791, a l s o IEEE Trans.Automat.Contr., Vol. AC-25, 299-302 G i l 1980)
.
[12] H. L. Van Trees,Detection,Estimation, and Vol. I , Wiley, New York, 1968. ModulationTheory, [13] A. H. Jazwinski,StochasticProcessesandFiltering Theory, Academic Press, New York, 1970.
379
