Intelligent Control and Automation, 2011, 2, 47-56
doi:10.4236/ica.2011.21006 Published Online February 2011 (http://www.SciRP.org/journal/ica)
Optimal Risk-Sensitive Filtering for System Stochastic of
Second and Third Degree
Ma Aracelia Alcorta-Garcia, Sonia Gpe Anguiano Rostro, Mauricio Torres Torres
Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Cd. Universitaria,
San Nicolas, Mexico
Received October 9, 2010; revised February 8, 2011; accepted February 10, 2011
Abstract
The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-
sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity
parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-
tering equations are obtained using quadratic value functions as solutions to the corresponding Focker-
Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic
polynomial systems of second and third degree is verified in a numerical example against the optimal po-
lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through
comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-
tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.
Keywords: Optimal Nonlinear Filtering, Risk-Sensitive Filtering, Extended Kalman-Bucy Filtering
1. Introduction
Since the linear optimal filter was obtained by Kalman
and Bucy (60’s), numerous works are based on it, see for
example [1-5], of the variety of all those. The determi-
nistic filter model introduced by Mortensen [6] provides
an alternative to stochastic filtering theory. In this model,
errors in the state dynamics and the observations are
modeled as deterministic “disturbance functions”, and an
exponential mean-square cost criterion disturbance error
is to be minimized. Special conditions are given for the
existence, continuity and boundedness of

 
f
Xt in
the state equation, which is considered nonlinear, and the
linear function



hXt in the observation equation. A
concept of stochastic risk-sensitive estimator, introduced
more recently by McEneaney [7], regard a dynamic sys-
tem where
 
f
Xt is a nonlinear function, linear ob-
servations and existence of parameter

 multiplying
the diffusion term in both equations (state and observa-
tions). In [8] were obtained the suboptimal risk-sensitive
filtering equations for polynomial systems of third de-
gree and applied to the pendulum equations [9], in which
the original system was linearized applying Taylor series
around the equilibrium point. In [10,11] it is regarded
 
f
Xt as nonlinear function. This paper presents an
application of the equations obtained in [10,11] for sin-
gular form of




f
Xt (polynomial of second and
third degree).
The goal of this work is to obtain the optimal risk-
sensitive filtering equations when the form of




f
Xt
is polynomial of second and third degree and parameter

 multiplying the diffusion term in the state and obser-
vations equations. There filtering equations are obtained
taking a value function as solution of the nonlinear pa-
rabolic partial differential equation and exponential
mean-square exponential cost criterion to be minimized.
Undefined parameters in the value function are calcu-
lated through ordinary differential equations composed
by collecting terms corresponding to each power of the
state-dependent polynomial in the nonlinear parabolic
PDE equations. This procedure leads to the obtention of
the optimal risk-sensitive filtering equations.
The closed-form for risk-sensitive filtering equations
is explicitly obtained in this work. Although the diffi-
culty presented by systems of second and third degree, in
this work is shown an advantage for risk-sensitive filter-
ing equations versus extended Kalman-Bucy and poly-
nomial filtering equations under certain values of the
parameter

. This performance is shown verified in a
numerical example against the mean-square optimal for

M. A. ALCORTA-GARCIA ET AL.
48
polynomial filtering equations (and extended Kalman-
Bucy for systems of second degree), through comparing
the exponential mean-square cost criterion values in fi-
nite horizon time. The simulation results reveal strong
advantages in favor of the designed risk-sensitive filter-
ing equations for all values of the intensity parameters
(in Table 1) multiplying diffusion terms in state and ob-
servation equations. Tables of the criterion values and
graphs of the simulations are included. This exponential
mean-square cost criterion function contains the parame-
ter

which appear in the dynamic system, which leads
to a more robust solution. This work is organized as fol-
lows: filtering problem statement, optimal risk-sensitive
filtering for stochastic system of second degree, optimal
risk-sensitive filtering for stochastic system of third de-
gree, application for systems of second degree, applica-
tion for systems of third degree, conclusions and refer-
ences.
2. Filtering Problem Statement
Consider the following stochastic model (1), where


X
t
denotes the state process,


Yt denotes a continuous
accumulated observations process,

X
t satisfies the
diffusion model given by
   
2
2
dX tfX tdtdW t


 (1)
where
 
f
Xt represents the nominal dynamics, and
W is a Brownian motion, and the observation process

Yt satisfies the equation:
 

 
2, 00,
2
dY th XtdtdW tY


 
 (2)
where

is a parameter and W and W
 are independ-
ent Brownian motions, which are also independent of the
initial state

0X.

0X has probability density
 

1
exp 0kX



 for some constant k

.
Let us consider
   
0
1
log exp,,
T
J
ELXtmttdtYt 



 (3)
the exponential mean-square cost criterion to be mini-
mize. In the rest of the paper the assumptions (A1)-(A4)
(from [10]) are hold:
● (A1) ,,n
fghwith ,
x
x
fh bounded.
● (A2)



22
12
11Dxx Dx   . Here
x
f is
the matrix of partial derivatives of f with
x
h defined
similarly.

x
 is a continuous, real-valued function
satisfying (A2) for some positive 1
D, 2 D.
● (A3) ,n fh with f, h bounded and
x
x
f,
x
x
h bounded
and globally Hölder continuous. (A function u is globally
Hölder continuous if there exists


0,1, K   such
that
 
uxuyKxy

 for all ,
x
y).
● (A4) Given R

, there exists R K such that




R
x
yKxy 


for all , xy.
Let




,qTX t denotes the unnormalized condi-
tional density of


X
T, given accumulated observa-
tions


Yt for 0tT

. It satisfies the Zakai stochas-
tic PDE, in a sense made precise, for instance in [12]. It
is assumed that



 




      
1
1
0, exp,
,,exp ,
qXt Xt
qTXtpTX tYThxt








(4)
where




,pTX t is called pathwise unnormalized
filter density. p satisfies the following linear second-
order parabolic PDE with coefficients depending on


Yt:


*.
pK
LT pp
T




(5)
where, for every n
g, let

  








 
2
,
2
1
,2
1
2
gXXX
X
X
XX
Ltrag fg
LT gLgaYThg
K
TXtaXtYT hYT h
LYT hh


 



(6)
L denotes the differential generator of the Markov dif-
fusion


X
t in (1). By assumptions (A1) and (A3) in
[10], K is bounded and continuous.


*
LT

is the for-
mal adjoint of


LT

. Since

00,Y




0,pXt




0,qXt. The initial condition for (5) is (4). For
some given


00,YC T, (where 0
C denote the space
of continuous


Yt such that

00Y, with the sup
norm ). The pathwise filter density p is the unique
“strong” solution to (5) and (4) in a sense made precise
in [12]. Further, p is a classical solution to (5) and (4)
with p continuous on


1
0, n T and partial derivatives
,, ,,1,,
iij
TX XX
pp pijn

 continuous for 1 0TT


[13,14].
Moreover,




,;0pTX tY. We rewrite (5) as fol-
lows:
 

1
2XX X
pB
tr aXtpApp
T


 (7)
where


     

 
X
X
AfXtaXtYthxt
div at 
 
 (8)




,BTXt

M. A. ALCORTA-GARCIA ET AL.
49
 

    

 
2
2
(())
,,
XX
X
tr aXt
divfX taXtYThXt
KTXt




 




 
 
,1
,1
,1,,
i
ij
n
Xij
ij X
j
n
XX ij
ij XX
div atajn
tr aa







These assumptions imply uniform bounds for A and B,
depending on the sup norm Y on


1
0,T, but not on

. Taking log transform:







,log,
Z
TXt pTXt  ,
which satisfies the nonlinear parabolic PDE:

1,
22
XXXX X
ZtrZA ZZZB
T
 
 (9)
with initial condition
   
0,
X
Z
Xt Xt   . The
optimal risk-sensitive filtering problem consists in found
the estimate

Ct, of the state

X
t through verifica-
tion that
       
  
1
,2
,
T
Z
TXtXtCtQt XtCt
TYThXt 
 

(10)
is a viscosity solution of (9).
Where

,
n
Xt

,
m
wt


,Yt


,
p
vt
,
f
n h with ,
Xx
f
h bounded is assumed through-
out. Here
x
h is the matrix of partial derivatives of h and
the same form for X
Z
.
3. Optimal Risk-Sensitive Filtering Problem
3.1. Optimal Risk-Sensitive Filtering for
Stochastic System of Second Degree
Taking


 




12 ,
T
f
XtAtAtXtAtXtXt 










1
hXtEtE tXt with 

,
n
At


1, nn
At M



2, nnn
At T


 
1,
p
EtE t np
M where ij M

denotes the field of matrixes of dimension ij

and
ijk
T denotes the field of tensors of dimension
ijk n. The following stochastic equations system is
obtained:
 





 
12
1
t
,
,
T
dXAtAt X tAt Xt X t
dB t
dYtEtEtXtdB t


 

 



(11)
where

2
2.




 The optimal filtering problem con-
sists in to obtain the estimate of the state


X
t given
the observations equations, which minimizes the expo-
nential mean-square cost criterion, taking




,
Z
TXt
(10) as solution of the nonlinear parabolic partial differ-
ential Equation (9).
Theorem 1. The solution to the filtering problem, for
the system (11) with criterion (3) takes the form:










    

  
 
1
1
1
1
1
2
11
11
2,
.
TT
T
CtAtA tCtQtQtCt
QtEtdyEt QtCt
AtQ t
QtA tQtQtAtQtQt
EtEt




 
 

 



(12)
where


Ct is the state estimate vector with initial
conditions with initial condition

0
0CC, and


Qt
is a symmetric matrix negative defined, where the initial
condition


0
0Qq

is derived from initial conditions
for Z. If






, 0
T
X
tXtKXtQ K 

 .
Proof: The value function is proposed
       
  
1
,()
2
,
T
Z
TXtXtCtQtXtCt
TYThXt 
 

(13)








 

0,, , ,
X
Z
XtXtCt Qtt   are func-
tions defined on




0, ,,
n
TCt Qt is a symmetric
matrix of dimension nn

and ()t

is a scalar func-
tion) as a viscosity solution of the nonlinear parabolic
PDE (9). ,
XXX
Z
Z are the partial derivatives of Z re-
spect to


X
t and
Z

is the gradient of Z. Then the
partial derivatives of Z are given by:
 

 

 

 
  
  
    

1
1
1
2
1
2
1
,
2
.
T
T
T
X
T
XX
ZXtCtQXtCt
XtCtQtCtt
YtEtE tXt
ZQtXtCt
X
tCtQtYtEt
ZQt

 
 


 




(14)
Let us consider:






 
12
1
T
A
AtAt XtAtXt X t
YTE t
 
 (15)
   
 

 

 
2
21 1
12
2
11
1
22
–
1
2
T
BAtXtAtYTEt
A
tAtXtA tXtXt
YTEtEtEtXt
  

 

M. A. ALCORTA-GARCIA ET AL.
50
Substituting (14) and the expressions for A, B in (9);
we obtain:
      
  


 

       
  
    
   
1
12
1
1
11
2
112
1
02
2
11
22
11
22
1
2
1
2
TT
T
T
T
T
X
tCtQtXtCtQtCt
tYtEtEtXt trQt
AtAtXt AtXtXtYt
EtQtXt CtXt Ct
QtYtE tQtXtCt
XtCtQt YtEtAt
YtEtAt AtXtAtX



 
 
 





 














    
2
11
1
.
2
t
Xt YtEtEtEtXt
(16)
Collecting the
  


,
TTT
X
tXt XtXtXt and
  
TT
X
tXtXtXt terms, and replacing


X
t
by

Ct; we obtain the matrix equation for


Qt
. Col-
lecting the

X
t terms, the vectorial equations for

Ct
 are obtained (12).
3.2. Optimal Risk-Sensitive Filtering for
Stochastic System of Third Degree
Taking


 











12
T
f
XtAtAtXtAtXtXt 
  
3,
TT
A
tX tXtX t










1
hXtEtE tXt
with

,
n
At


1, nn
At M




2, nnn
At T




3, nnnn
At T



,
p
Et 

1np
Et M

 where ij M denotes the field
of matrixes of dimension ij, ijk T denotes the field
of tensors of dimension ijk and ijkl T denotes
the field of tensors of dimension ijkl. The fol-
lowing stochastic equations system is obtained:
 
  
  
12
3
10
tA
,
, ,0
T
TT
dXt AtXt AtXtXt
AtX tXtX tdBt
dYtEtEtXtdB tXX


 

 



(17)
where

2
2

 
.
Theorem 2. The solution to the filtering problem, for
the system (17) with criterion (3) takes the form:
 


 
1
1
1
1
1
,
2
CtAtA tCtQtQtCt
QtEtdyEtQtCt


 

 



(18)
 




      
112
22
T
T
QtAtQtQtAtAtQtCt
AtQtCt AtQtCt
 








 

  
  


 
3
3
3
3
1
11
T
T
T
T
T
T
T
T
AtCtC tQt
AtCtC tQt
AtQtCtCt
AtQtCtC t
QtQtdivfCYtEt
EtEt





 


where


Ct is the state estimate vector with initial
conditions with initial condition

0
0CC, and


Qt
is a symmetric matrix negative defined, where the initial
condition


0
0Qq

is derived from initial conditions
for Z. If






, 0
T
X
tXtKXtQ K 

 .
Proof: In similar form to Theorem 1.
4. Applications
4.1. Application for Systems of Second-Degree.
Optimal Risk-Sensitive Filtering Equations
Consider the following dynamical stochastic system as-
sociated to a continuous stirred tank reactor in which is a
chemical reaction occurs. This reaction is in liquid phase
and has isothermal character between multicomponents
[15].


 
 
111 1
2
2
2112222
2
1,
2
.
2
a
aa
X
tDXutdWt
X
tDXXDXdWt




 
 


(19)
where


1
X
t represents the unnormalized concentra-
tion
P
o
PC of a certain specie P of the reactor,


2
X
t
represents unnormalized concentration QPo CC of a
certain specie Q. The control variable u is defined as the
relation between the alimentation molar rate by volumet-
ric unit of P, designated by
P
F
N and the nominal con-
centration
P
o
C, this is
P
FPo
uN FC

, where F is the
volumetric flow of alimentation on 31
ms
. 11a
DkVF

,
22aPo
DkVCF

where V is the volume of reactor in
3
m, 1 k and 2 k are constants of first degree given in
1
s

. It can take that 1a D and 2a D are considering by
11 a
D

and 21
a
D

. Q is highly sour while P is neuter.
Then, the following dynamical stochastic system is ob-
tained:
 
 
11 1
2
2
2122 2
2
2,
2
.
2
XtXutdWt
X
tXXXdWt




 



(20)

M. A. ALCORTA-GARCIA ET AL.
51
Applying the equations (12) to the system (20), the
equations of the optimal risk-sensitive filtering are ob-
tained:


22
111111 12
12121111 121222
22
222212 1222
11
11122 1122
2
11 2212
1211 212 11 222 12
111 1122
12
2122
2
11 2212
41,
3
22 1,
,
2,
QQQQ
QQQQQQQ
QQ QQQ
Q
CQQCQC
QQ Q
QQCQCYQYQ
CQCQC
Q
CQQ
QQ Q
 
 








 












2 1122
2211 212 11 122 1111
12121222
.
2
CQC
QQCQCYQ YQQ
CCQCQC






 

(21)
The initial conditions for the risk-sensitive filtering
equations are:



121
020, 010,02,XXY


201,Y



101,C

205,C


11 06,Q

12 00.0001,Q

22 07Q , the final time is 2Ts. The system
formed by the equations (20) and (21), is simulated using
Simulink in MatLab7. The performance of the designed
equations is compared versus the equations of the poly-
nomial filtering [1] and the equations of the extended
Kalman-Bucy filtering [16], applied to the system (20),
that is optimal with respect to the conventional exponen-
tial mean-square cost criterion.
4.1.1. Polyn o mi al Filteri ng E qua ti ons
The corresponding equations for the polynomial filtering
[1] are given by:



 
 
2
22
111112 11
2
2
12111221211 121222
2
22
2222122 221222
2
2
1111 111222
2
211222
2
12 1122 22
2
4,
2
2
32 ,
2
224 ,
2
2
2,
2
.
PP PP
PP PmPPPPP
PPPmP PP
mm PYmPYm
mmmm P
PY mPYm

 









 
 
 

 












(22)
where the initial conditions are

1020,X


2010,X


102,Y

201,Y

101,m

11 0100,P


12 01,P



7
22 0110P.
4.1.2. Extended Kalman-Bucy Filtering Equations
The equations of the extended Kalman-Bucy [13] filter-
ing are given by:



 
2
22
1112 11
2
2
12111211 121222
2
22
22122212 22
2
2
1111 111222
2
4,
2
2
3,
2
22 ,
2
2
2,
PP PP
PP PPPPP
PPP PP
mm PYmPYm

 






 
 



  







(23)

2
21112112222
2.mmmPYmPYm 



 




1) Consider the stochastic dynamical system associ-
ated to a continuous stirred tank reactor and the follow-
ing initial conditions for the state and observations equa-
tions:






1212
020, 010, 02, 01,XXYY

 the
final time is 2Ts

. The initial conditions for the fil-
tering equations in which case are given by:
a) For risk-sensitive filtering equations:





1 21112
22
01, 05, 06, 00.0001,
07.
CC QQ
Q
 

b) For polynomial filtering equations:





12 1112
7
22
01, 05, 0100, 01,
0110.
mm PP
P

 

c) For Extended Kalman-Bucy filtering equations:





1211 12
22
01, 05, 05, 03,
05.
mm PP
P



Table 1 presents comparison between the exponential
mean square cost criterion J for the three types of filter-
ing equations; you can see that the
R
S
J
 values are the
smallest for all values of the intensity parameter

.
2) Consider the stochastic dynamical system associ-
ated to a continuous stirred tank reactor and the follow-
ing initial conditions for the state and observations equa-
tions:



 
1212
050, 01, 02, 01,XXYY

  the
final time is 2Ts

. The initial conditions for the fil-
tering equations in which case are given by:
a) For risk-sensitive filtering equations:





1 21112
22
01, 05, 07, 00.0001,
07.5.
CC QQ
Q
 

b) For polynomial filtering equations:



 

12 1112
7
22
01, 05, 085, 010
.
,
0210
mm PP
P
 

c) For Extended Kalman-Bucy filtering equations:





12 1112
22
01, 05, 02, 05,
0 10.
mm PP
P




M. A. ALCORTA-GARCIA ET AL.
52
Table 2 presents comparison between the exponential
mean square cost criterion J for the three types of filter-
ing equations; you can see that the
R
S
J
 values are the
smallest for all values of the intensity parameter

.
3) Consider the stochastic dynamical system associ-
ated to a continuous stirred tank reactor and the follow-
ing initial conditions for the state and observations equa-
tions:
 


1212
00.05, 050, 02, 01,XXYY
the final time is 2Ts. The initial conditions for the
filtering equations in which case are given by:
a) For risk-sensitive filtering equations:
 

1 21112
22
01, 05, 06, 00.0001,
07.
CC QQ
Q
 

b) For polynomial filtering equations:
 

12 1112
7
22
01, 05, 0100, 05,
0110.
mm PP
P
 

c) For Extended Kalman-Bucy filtering equations:
 

12 1112
22
01, 05, 01.85, 03,
05.
mm PP
P
 

Table 3 presents comparison between the exponential
mean square cost criterion J for the three types of filter-
ing equations; you can see that the
R
S
J
 values are the
smallest for all values of the intensity parameter

.
With these tables, showed that the filter risk-sensitive
is the best, because the values obtained are lower.
The Figures 1, 2 and 3 show the 1 Error and 2 Error
which are defined as
 
11 1
ErrorXtC t (in same
form for 2 Error); and the exponential mean-square cost
criterion values in 2Ts.
Table 1. Comparison of exponential mean-square cost cri-
terion values J(3) in T = 2s for risk-sensitive, polynomial
and extended Kalman-Bucy filteri ng equations .

R
S
J

P
ol
J
K
B
J

0.1 53.4293 69.2292 (t = 0.17s) 69.0816(t = 0.14s)
1 53.5165 145.7323 277.3136
10 53.7994 157.2172 235.5110
100 54.7621 858.7622 189.6937
1000 58.5054 58230 185.7343
Table 2. Comparison of exponential mean-square cost cri-
terion values J(3) in T = 2s for risk-sensitive, polynomial
and extended Kalman-Bucy filteri ng equations .

R
S
J

P
ol
J
K
B
J

1 505.8493 705.1152 (t = 1.64s) 686.3813 (t = 0.28s)
10 513.3591 712.2522 527.0787
100 537.8120 1430.4728 587.0328
1000 622.1946 59067 641.1202
10000 960.423 5597700 673.6555
Table 3. Comparison of exponential mean-square cost cri-
terion values J(3) in T = 2s for risk-sensitive, polynomial
and extended Kalman-Bucy filteri ng equations .

R
S
J

P
ol
J
K
B
J

0.1 42.0377 55.6339 70.3916 (t = 0.36s)
1 41.9340 56.1153 144.2845
10 41.6143 70.7942 317.9369
100 40.6851 763.7829 678.2942
1000 38.5611 57957 812.9141
10000 36.1603 55918000 859.8006
Error
1
Error
2
Criterion R-S
0 0.5 1 1.5 2
time
20
15
10
5
0
0 0.5 1 1.5 2
time
0 0.5 1 1.5 2
time
0
5
–
5
60
40
20
0
Figure 1. Graphs of the 1
E
rror , 2
E
rror , and exponential
mean square cost criterion corresponding to the risk-sensi-
tive optimal filtering equations for a continuous stirred
tank reactor for 10,




,
1020X

,
2010X


102,Y



201Y

.
4.2. Application for Polynomial System of Third
Degree
4.2.1. Optimal Risk-Sensitive Filtering Equations
The risk-sensitive control equations for third degree po-
lynomial systems will be applied to the problem of ori-
entation of a monoaxial satellite [15]. The description is
as follows: a satellite rotates around a fixed axis without
gravity. The rotation torques is produced by a system of
mini-engines through a controlled explosion of gases in
the opposite direction. The state equations for this model
are given by:
 

 
2
1121
2
0.5 12
X
tXtXtdWt


 


M. A. ALCORTA-GARCIA ET AL.
53
  
221
22
, .
22
X
tdWtYtXtdWt



 (24)
where

1
X
t represents the orientation angle of the
satellite, measured with respect of a secondary axis
which does not coincide with the principal one.


2
X
t
represents the angular velocity with respect to the prin-
cipal axis. Applying the system of equations (18) to the
system (24), the following optimal risk-sensitive filtering
equations are obtained:


2
11121 2 1121222211
221 2
122211121222
22
2212221 2
22
1111212
2
11 2212
12
1122 222
2
11 2212
111212
2
,
0.5 ,
,
1
2
,
QQCCQCQCQQ
QCC
QQQQQQ
QQQCC
Q
CCQCQY
QQ Q
QCQCQC
QQQ
CQ CQ
C


  
















 





12
111212
2
11 2212
11
1122 22112
2
11 2212
222
.
QCQ CQY
QQ Q
QCQ CQCQ
QQQ
CQ






 

(25)
Error
1
Error
2
Criterion POL
0 0.5 1 1.5 2
time
20
12
160
15
10
5
0
9
6
3
0
0 0.5 1 1.5 2
time
0 0.5 1 1.5 2
time
120
80
40
0
Figure 2. Graphs of the 1
E
rror , 2
E
rror , and exponential
mean square cost criteri on corresponding to the poly nomial
filtering equations for a continuous stirred tank reactor for
10,



,
1020X

,
2010X


102,Y


201Y

.
Error
1
Error
2
Criterion K-B
0 0.5 1 1.5
2
time
20
12
250
0 0.5 1 1.5 2
time
15
10
5
0
9
6
3
0
200
150
100
50
00 0.5
1 1.5 2
time
Figure 3. Graphs of the 1
E
rror , 2
E
rror , and exponential
mean square cost criterion corresponding to the extended
Kalman-Bucy filtering equations for a continuous stirred
tank reactor for 10,




,
1020X


,
2010X


102,Y



201Y

.
The initial conditions are:

10 0.115,X


20 0.073,X


100,Y



100.92,C and


200.5,C

11 0 4500,Q


12 0100,Q


22 08500,Q 1.8Ts.
4.2.2. Polyn o mi al Filteri ng E qua ti ons
The corresponding equations for the polynomial filter [1]
are given by:


1112 121112221112
22
211
12 22
222
11 1212
122222 2
2
1221112
2
11
11
2
12
211
333
2
3,
2
22
0.5, ,
2
0.5 1.5 0.5
2,
2.
PPPP PPPmm
P
Pm
PP P
PP P
mmmPmm
PYm
P
mYm
 


 
 




 

 
 








(26)
1) Consider the stochastic dynamical system associ-
ated to a problem of orientation of a monoaxial satellite
and the following initial conditions for the state and ob-
servations equations:




12
00.09, 00.65,XX


102,Y



201Y

, the final time is 1Ts. The initial condi-
tions for the filtering equations in which case are given
by:

M. A. ALCORTA-GARCIA ET AL.
54
a) For risk-sensitive filtering equations:









12 1112
22
00.92, 00.5, 0400, 0100,
0 850.
CCQQ
Q


b) For polynomial filtering equations:
 

121112
22
00.92, 00.5, 010, 020,
05.
mmPP
P


Table 4 presents comparison between the exponential
mean square cost criterion J for the two types of filtering
equations; it can be saw, that the
R
S
J
 values are the
smallest for all values of the intensity parameter

.
2) Consider the stochastic dynamical system associ-
ated to a problem of orientation of a monoaxial satellite
and the following initial conditions for the state and ob-
servations equations:




12
00.115, 00.073,XX
 
12
02, 01YY, the final time is 1Ts. The initial
conditions for the filtering equations in which case are
given by:
a) For risk-sensitive filtering equations:






 
1211
12 22
00.92, 00.5, 04500,
0100, 08500.
CCQ
QQ


b) For polynomial filtering equations:

 
1211
12 22
00.92, 00.5, 01500,
0879.21, 01000.
mmP
PP

 
Table 5 presents comparison between the exponential
mean square cost criterion J for the two types of filtering
equations; it can be saw, that the
R
S
J
 values are the
smallest for all values of the intensity parameter

.
The system Equations (24), (25) and (26) is simulated
using Simulink in MatLab7. The performance of the de-
signed equations is compared versus the equations of the
polynomial filter [1], with respect to the exponential
mean-square exponential criterion J.
The Figures 4 and 5 show the 1
Error and 2 Error
which are defined as
 
11 1
ErrorXtC t (in same
form for 2 Error ); and the exponential mean- square cost
criterion values.
Table 4. Comparison of mean-square exponential criterion
J(3) for r-s filtering equations and polynomial filtering eq-
uations.

R
S
J

P
ol
J
0.01 0.3239 2.0321
0.1 0.3232 1.1319
1 0.3198 0.6655
10 0.3063 0.3319
100 0.2800 26.8974
Table 5. Comparison of mean-square exponential criterion
J(3) for r-s filtering equations and polynomial filtering eq-
uations.

R
S
J

P
ol
J
0.01 0.3842 0.5895
0.1 0.3835 0.4287
1 0.3691 0.4013
10 0.2841 0.3054
100 0.1454 0.2454
Error
1
Erro
r
2
Criterion
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time
0.5
0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time
0
–
0.5
–
1
0.5
0.25
0
–
0.25
–
0.5
0.3
0.2
0.1
0
Figure 4. Graphs of the 1
E
rror , 2
E
rror , and exponential mean square cost criterion corresponding to the risk-sensitive op-
timal filtering equations for satellite monoaxial for 10,




,
10 0.115X


,
20 0.073X


102,Y


201Y.

M. A. ALCORTA-GARCIA ET AL.
55
Error
1
Erro
r
2
Criterion
0.5
0.4
Time
0
–
0.5
0.3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time
–
1
–
1
–
2
0
1
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Figure 5. Graphs of the 1
E
rror , 2
E
rror , and exponential mean square cost criterion c orresponding to the polynomi al filter-
ing equations for satellite monoaxial for 10,




,
10 0.115X


,
20 0.073X


102,Y



201Y.
5. Conclusions
In this paper the equations have been obtained for the
optimal risk-sensitive filtering problem, when the system
is polynomial of second and third degree, with presence
of Gaussian white noise, exponential mean-square cost
criterion to be minimized, with parameter

multiply-
ing the Gaussian white noise in the state and observa-
tions equations, and taking into account a value function
as a viscosity solution of the nonlinear parabolic PDE.
Numerical application is solved for risk-sensitive and
polynomial filtering equations for system of second and
third degree (and Kalman-Bucy for system of second
degree) for some values of parameter

. The perform-
ance for optimal risk-sensitive filtering equations is veri-
fied through of the comparison between the values of the
exponential mean-square cost criterion J for polynomial
and extended Kalman Bucy filtering equations.
It can be seen that the values of the mean square cost
criterion
R
S
J
 in final time, are smaller than
P
ol
J
and
K
B
J for all values given to the intensity parameter

.
6. References
[1] M. V. Basin and M. A. Alcorta-García, “Optimal Filter-
ing and Control for Third Degree Polynomial Systems,”
Dynamics of Continuous Discrete and Impulsive Systems,
Vol. 10, 2003, pp. 663-680.
[2] M. V. Basin and M. A. Alcorta-García, “Optimal Filter-
ing for Bilinear Systems and Its Application to Terpo-
lymerization Process State,” Proceedings of IFAC 13th
Symposium on System Identification, 27-29 August 2003,
Rotterdam, 2003, pp. 467-472.
[3] F. L. Lewis, “Applied Optimal Control and Estimation,”
Prentice Hall PTR, Upper Saddle River, 1992.
[4] V. S. Pugachev and I. N. Sinitsyn, “Stochastic Systems
Theory and Applications,” World Scientific, Singapore,
2001.
[5] S. S.-T. Yau, “Finite-Dimensional Filters with Nonlinear
Drift. I: A Class of Filters including both Kalman-Bucy
and Benes Filters,” Journal of Mathematical Systems, Es-
timation, and Control, Vol. 4, 1994, pp. 181-203.
[6] R. E. Mortensen, “Maximum Likelihood Recursive Non-
linear Filtering,” Journal of Optimization Theory and Ap-
plications, Vol. 2, No. 6, 1968, pp. 386-394. doi:10.1007/
BF00925744
[7] W. M. McEneaney, “Robust H∞ Filtering for Nonlinear
Systems,” Systems and Control Letters, Vol. 33, No. 5,
1998, pp. 315-325. doi:10.1016/S0167-6911(97)00124-2
[8] M. A. Alcorta-García, M. V. Basin, S. G. Anguiano and J.
J. Maldonado, “Sub-optimal Risk-Sensitive Filtering for
Third Degree Polynomial Stochastic Systems,” IEEE
Control Applications & Intelligent Control, St. Petersburg,
8-10 July 2009, pp. 285-289.
[9] H. K. Khalil, “Nonlinear Systems,” 3rd Edition, Prentice
Hall, Upper Saddle River, 2002.
[10] W. H. Fleming and W. M. McEneaney, “Robust Limits
of Risk Sensitive Nonlinear Filters,” Mathematics of
Control, Signals, and Systems, Vol. 14, No. 1, 2001, pp.
109-142. doi:10.1007/PL00009879
[11] W. M. McEneaney, “Max-Plus Eigenvector Representa-
tions for Solution of Nonlinear H1 Problems: Error Ana-
lysis,” SIAM Journal on Control and Optimization, Vol.
43, 2004, pp. 379-412. doi:10.1137/S0363012902414688

M. A. ALCORTA-GARCIA ET AL.
56
[12] D. L. Lukes, “Optimal Regulation of Nonlinear Dynamic
Systems,” SIAM Journal on Control and Optimization,
Vol. 7, No. 1, 1969, pp. 75-100. doi:10.1137/0307007
[13] T. Yoshida and K. Loparo, “Quadratic Regulatory Theory
for Analitic Nonlinear Systems with Additive Controls,”
Automatica, Vol. 25, No. 4, 1989, pp. 531-544. doi:10.10
16/0005-1098(89)90096-4
[14] N. N. Ural’ceva, O. A. Ladyzenskaja and V. A. Solon-
nikov, “Newblock Linear and Quasi-Linear Equations for
Parabolic Type,” American Mathematical Society, Provi-
dence, 1968.
[15] H. Sira-Ramírez, R. Márquez, F. Rivas-Echeverría and O.
Llanes-Santiago, “Automática y Robótica: Control de
Sistemas no Lineales,” Pearson, Madrid, 2005.
[16] A. H. Jazwinski, “Stochastic Processes and Filtering The-
ory,” Dover Publications, Inc., New York, 2007.