 Intelligent Control and Automation, 2011, 2, 47-56 doi:10.4236/ica.2011.21006 Published Online February 2011 (http://www.SciRP.org/journal/ica) Copyright © 2011 SciRes. 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 E-mail: aalcorta@fcfm.uanl.mx, srostro@hotma il.com, mautor2@gmail.com 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  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 fXt 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 , regard a dynamic sys-tem where fXt is a nonlinear function, linear ob-servations and existence of parameter  multiplying the diffusion term in both equations (state and observa-tions). In  were obtained the suboptimal risk-sensitive filtering equations for polynomial systems of third de-gree and applied to the pendulum equations , in which the original system was linearized applying Taylor series around the equilibrium point. In [10,11] it is regarded fXt as nonlinear function. This paper presents an application of the equations obtained in [10,11] for sin-gular form of fXt (polynomial of second and third degree). The goal of this work is to obtain the optimal risk- sensitive filtering equations when the form of fXt 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. Copyright © 2011 SciRes. ICA 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 Xt denotes the state process, Yt denotes a continuous accumulated observations process, Xt satisfies the diffusion model given by  22dX tfX tdtdW t (1) where fXt represents the nominal dynamics, and W is a Brownian motion, and the observation process Yt satisfies the equation:   2, 00,2dY 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 1exp 0kX for some constant k. Let us consider  01log exp,,TJELXtmttdtYt (3) the exponential mean-square cost criterion to be mini-mize. In the rest of the paper the assumptions (A1)-(A4) (from ) are hold: ● (A1) ,,nfghwith ,xxfh bounded. ● (A2) 221211Dxx Dx . Here xf is the matrix of partial derivatives of f with xh defined similarly. x is a continuous, real-valued function satisfying (A2) for some positive 1D, 2D. ● (A3) ,nfh with f, h bounded and xxf, xxh bounded and globally Hölder continuous. (A function u is globally Hölder continuous if there exists 0,1, K such that  uxuyKxy for all ,xy). ● (A4) Given R, there exists RK such that RxyKxy for all , xy. Let ,qTX t denotes the unnormalized condi-tional density of XT, given accumulated observa-tions Yt for 0tT. It satisfies the Zakai stochas-tic PDE, in a sense made precise, for instance in . It is assumed that  110, exp,,,exp ,qXt XtqTXtpTX 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: *.pKLT ppT (5) where, for every ng, let  2 ,21,21 2 gXXXXXXXLtrag fgLT gLgaYThgKTXtaXtYT hYT hLYT hh  (6) L denotes the differential generator of the Markov dif-fusion Xt in (1). By assumptions (A1) and (A3) in , 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 0C 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 . Further, p is a classical solution to (5) and (4) with p continuous on 10, nT and partial derivatives ,, ,,1,,iijTX XXpp pijn continuous for 10TT [13,14]. Moreover, ,;0pTX tY. We rewrite (5) as fol-lows: 12XX XpBtr aXtpAppT (7) where   XXAfXtaXtYthxtdiv at  (8) ,BTXt M. A. ALCORTA-GARCIA ET AL. Copyright © 2011 SciRes. ICA 49 2 2 (()) ,,XXXtr aXtdivfX taXtYThXtKTXt  ,1,1,1,,iijnXijij XjnXX ijij XXdiv atajntr aa These assumptions imply uniform bounds for A and B, depending on the sup norm Y on 10,T, but not on . Taking log transform: ,log,ZTXt pTXt, which satisfies the nonlinear parabolic PDE: 1,22XXXX XZtrZA ZZZBT (9) with initial condition 0,XZXt Xt . The optimal risk-sensitive filtering problem consists in found the estimate Ct, of the state Xt through verifica-tion that    1,2 ,TZTXtXtCtQt XtCtTYThXt (10) is a viscosity solution of (9). Where ,nXt ,mwt ,Yt ,pvt ,f nh with , Xxfh bounded is assumed through- out. Here xh is the matrix of partial derivatives of h and the same form for XZ. 3. Optimal Risk-Sensitive Filtering Problem 3.1. Optimal Risk-Sensitive Filtering for Stochastic System of Second Degree Taking  12 ,TfXtAtAtXtAtXtXt  1hXtEtE tXt with ,nAt 1,nnAt M 2,nnnAt T  1,pEtE t npM where ijM denotes the field of matrixes of dimension ij and ijkT denotes the field of tensors of dimension ijk n. The following stochastic equations system is obtained:   121t ,,TdXAtAt X tAt Xt X tdB tdYtEtEtXtdB t  (11) where 22. The optimal filtering problem con-sists in to obtain the estimate of the state Xt given the observations equations, which minimizes the expo-nential mean-square cost criterion, taking ,ZTXt (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:      1111121111 2, .TTTCtAtA tCtQtQtCtQtEtdyEt QtCtAtQ tQtA tQtQtAtQtQtEtEt    (12) where Ct is the state estimate vector with initial conditions with initial condition 00CC, and Qt is a symmetric matrix negative defined, where the initial condition 00Qq is derived from initial conditions for Z. If , 0TXtXtKXtQ K . Proof: The value function is proposed    1,()2 ,TZTXtXtCtQtXtCtTYThXt (13)  0,, , , XZXtXtCt Qtt are func-tions defined on 0, ,,nTCt 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). ,XXXZZ are the partial derivatives of Z re-spect to Xt and Z is the gradient of Z. Then the partial derivatives of Z are given by:         1112 121 ,2.TTTXTXXZXtCtQXtCtXtCtQtCttYtEtE tXtZQtXtCtXtCtQtYtEtZQt    (14) Let us consider:  121 TAAtAt XtAtXt X tYTE t  (15)      221 112211 122–1 2TBAtXtAtYTEtAtAtXtA tXtXtYTEtEtEtXt   M. A. ALCORTA-GARCIA ET AL. Copyright © 2011 SciRes. ICA 50 Substituting (14) and the expressions for A, B in (9); we obtain:           11211112112102 2 11 22 11221 21 2TTTTTTXtCtQtXtCtQtCttYtEtEtXt trQtAtAtXt AtXtXtYtEtQtXt CtXt CtQtYtE tQtXtCtXtCtQt YtEtAtYtEtAt AtXtAtX      2111 .2tXt YtEtEtEtXt (16) Collecting the   ,TTTXtXt XtXtXt and   TTXtXtXtXt terms, and replacing Xt by Ct; we obtain the matrix equation for Qt. Col-lecting the Xt terms, the vectorial equations for Ct are obtained (12). 3.2. Optimal Risk-Sensitive Filtering for Stochastic System of Third Degree Taking 12TfXtAtAtXtAtXtXt    3,TTAtX tXtX t 1hXtEtE tXt with ,nAt 1,nnAt M 2,nnnAt T 3,nnnnAt T ,pEt  1npEt M where ijM denotes the field of matrixes of dimension ij, ijkT denotes the field of tensors of dimension ijk and ijklT denotes the field of tensors of dimension ijkl. The fol-lowing stochastic equations system is obtained:      12310tA ,, ,0TTTdXt AtXt AtXtXtAtX tXtX tdBtdYtEtEtXtdB tXX  (17) where 22. Theorem 2. The solution to the filtering problem, for the system (17) with criterion (3) takes the form:   11111 ,2CtAtA tCtQtQtCtQtEtdyEtQtCt   (18)      11222 TTQtAtQtQtAtAtQtCtAtQtCt AtQtCt        3333111 TTTTTTTTAtCtC tQtAtCtC tQtAtQtCtCtAtQtCtC tQtQtdivfCYtEtEtEt where Ct is the state estimate vector with initial conditions with initial condition 00CC, and Qt is a symmetric matrix negative defined, where the initial condition 00Qq is derived from initial conditions for Z. If , 0TXtXtKXtQ 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 .   111 122211222221,2.2aaaXtDXutdWtXtDXXDXdWt   (19) where 1Xt represents the unnormalized concentra-tion PoPC of a certain specie P of the reactor, 2Xt represents unnormalized concentration QPoCC 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 PFN and the nominal con-centration PoC, this is PFPouN FC, where F is the volumetric flow of alimentation on 31ms. 11aDkVF, 22aPoDkVCF where V is the volume of reactor in 3m, 1k and 2k are constants of first degree given in 1s. It can take that 1aD and 2aD are considering by 11aD and 21aD. Q is highly sour while P is neuter. Then, the following dynamical stochastic system is ob-tained:   11 1222122 222,2.2XtXutdWtXtXXXdWt  (20) M. A. ALCORTA-GARCIA ET AL. Copyright © 2011 SciRes. ICA 51Applying the equations (12) to the system (20), the equations of the optimal risk-sensitive filtering are ob-tained: 22111111 1212121111 12122222222212 12221111122 1122211 22121211 212 11 222 12111 1122122122211 221241,322 1, ,2,QQQQQQQQQQQQQ QQQQCQQCQCQQ QQQCQCYQYQCQCQCQCQQQQ Q   2 11222211 212 11 122 111112121222 . 2 CQCQQCQCYQ YQQCCQCQC (21) The initial conditions for the risk-sensitive filtering equations are: 121020, 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  and the equations of the extended Kalman-Bucy filtering , 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  are given by:   222111112 112212111221211 1212222222222122 221222221111 1112222211222212 1122 22 24,2232 ,2224 ,222,2 .PP PPPP PmPPPPPPPPmP PPmm PYmPYmmmmm PPY mPYm     (22) where the initial conditions are 1020,X 2010,X 102,Y 201,Y 101,m 11 0100,P 12 01,P 722 0110P. 4.1.2. Extended Kalman-Bucy Filtering Equations The equations of the extended Kalman-Bucy  filter-ing are given by:  2221112 112212111211 12122222222122212 22221111 11122224,223,222 ,222,PP PPPP PPPPPPPP PPmm PYmPYm    (23) 2211121122222.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: 1212020, 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 211122201, 05, 06, 00.0001, 07.CC QQQ  b) For polynomial filtering equations: 12 111272201, 05, 0100, 01,0110.mm PPP  c) For Extended Kalman-Bucy filtering equations: 1211 122201, 05, 05, 03,05.mm PPP 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 RSJ 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:  1212050, 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 211122201, 05, 07, 00.0001,07.5.CC QQQ  b) For polynomial filtering equations:  12 111272201, 05, 085, 010., 0210mm PPP  c) For Extended Kalman-Bucy filtering equations: 12 11122201, 05, 02, 05,0 10.mm PPP M. A. ALCORTA-GARCIA ET AL. Copyright © 2011 SciRes. ICA 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 RSJ 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:  121200.05, 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 211122201, 05, 06, 00.0001,07.CC QQQ  b) For polynomial filtering equations:  12 111272201, 05, 0100, 05,0110.mm PPP  c) For Extended Kalman-Bucy filtering equations:  12 11122201, 05, 01.85, 03,05.mm PPP  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 RSJ 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 1Error and 2Error which are defined as  11 1ErrorXtC t (in same form for 2Error); 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 .  RSJ PolJ KBJ 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 .  RSJ PolJ KBJ 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 .  RSJ PolJ KBJ 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 Error1 Error2 Criterion R-S 0 0.5 1 1.5 2time 201510500 0.5 1 1.5 2time 0 0.5 1 1.5 2time 05–56040200 Figure 1. Graphs of the 1Error , 2Error , 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 . 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:   2112120.5 12XtXtXtdWt  M. A. ALCORTA-GARCIA ET AL. Copyright © 2011 SciRes. ICA 53  22122, .22XtdWtYtXtdWt (24) where 1Xt represents the orientation angle of the satellite, measured with respect of a secondary axis which does not coincide with the principal one. 2Xt 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: 211121 2 1121222211221 2122211121222222212221 2221111212211 2212121122 222211 22121112122 ,0.5 ,,1 2, QQCCQCQCQQQCCQQQQQQQQQCCQCCQCQYQQ QQCQCQCQQQCQ CQC  12111212211 2212111122 22112211 2212222 .QCQ CQYQQ QQCQ CQCQQQQCQ (25) Error1 Error2 Criterion POL 0 0.5 1 1.5 2time 2012160 15105096300 0.5 1 1.5 2time 0 0.5 1 1.5 2time 120 80400 Figure 2. Graphs of the 1Error , 2Error , 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. Error1 Error2 Criterion K-B 0 0.5 1 1.5 2time 20122500 0.5 1 1.5 2time 15105096302001501005000 0.5 1 1.5 2time Figure 3. Graphs of the 1Error , 2Error , 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  are given by: 1112 1211122211122221112 2222211 1212122222 2212211122111121221133323,2220.5, ,20.5 1.5 0.52, 2. PPPP PPPmmPPmPP PPP PmmmPmmPYmPmYm   (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: 1200.09, 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. Copyright © 2011 SciRes. ICA 54 a) For risk-sensitive filtering equations: 12 11122200.92, 00.5, 0400, 0100,0 850.CCQQQ b) For polynomial filtering equations:  1211122200.92, 00.5, 010, 020,05.mmPPP 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 RSJ 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: 1200.115, 00.073,XX  1202, 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:  121112 2200.92, 00.5, 04500,0100, 08500.CCQQQ b) For polynomial filtering equations:  121112 2200.92, 00.5, 01500,0879.21, 01000.mmPPP  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 RSJ 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 , with respect to the exponential mean-square exponential criterion J. The Figures 4 and 5 show the 1Error and 2Error which are defined as  11 1ErrorXtC t (in same form for 2Error ); 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.  RSJ PolJ 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.  RSJ PolJ 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 Error1 Error2 Criterion 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time 0.50.40 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 Time0–0.5–10.50.25 0–0.25 –0.5 0.30.20.10 Figure 4. Graphs of the 1Error , 2Error , 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. Copyright © 2011 SciRes. ICA 55Error1 Error2 Criterion 0.50.4Time0–0.50.30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time0 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 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Figure 5. Graphs of the 1Error , 2Error , 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 RSJ in final time, are smaller than PolJ and KBJ for all values given to the intensity parameter . 6. References  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.  M. V. Basin and M. A. 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