Gaussian filters for nonlinear filtering problems...

910 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000

Gaussian Filters for Nonlinear Filtering ProblemsKazufumi Ito and Kaiqi Xiong

Abstract—In this paper we develop and analyze real-time andaccurate filters for nonlinear filtering problems based on theGaussian distributions. We present the systematic formulationof Gaussian filters and develop efficient and accurate numericalintegration of the optimal filter. We also discuss the mixedGaussian filters in which the conditional probability densityis approximated by the sum of Gaussian distributions. A newupdate rule of weights for Gaussian sum filters is proposed.Our numerical testings demonstrate that new filters significantlyimprove the extended Kalman filter with no additional cost andthe new Gaussian sum filter has a nearly optimal performance.

Index Terms—Author: please supply index terms. E-mail: [email protected] for information.

I. INTRODUCTION

T HE NONLINEAR filtering problem consists of estimatingthe state of a nonlinear stochastic system from noisy ob-

servation data. The problem has been the subject of consider-able research interest during the past several decades becauseit has many significant applications in science and engineeringsuch as navigational and guidance systems, radar tracking, sonarranging, and satellite and airplane orbit determination [11], [14],[15]. As is well-known, the most widely used filter is the ex-tended Kalman filter for nonlinear filtering problems. It is de-rived from the Kalman filter based on the successive lineariza-tion of the signal process and the observation map. The extendedKalman filter has been successfully applied to numerous non-linear filtering problems. If nonlinearities are significant, how-ever, its performance can be substantially improved. Such ef-forts have also been reported in [1], [5], [7], [12], [13], and [20].In this paper our objective is to develop and analyze real-timeand accurate filters for nonlinear filtering algorithms based onGaussian distributions. We present the systematic formulationof Gaussian filters and mixed Gaussian filters.

We first develop the Gaussian filter. The proposed filter isbased on the steps: 1) we assume the conditional probabilitydensity to be a Gaussian distribution (i.e., assumed density)and 2) we obtain the Gaussian filter by equating the Bayesianformula with respect to the first moment (mean) and the secondmoment (covariance). Our approach is based on the efficientnumerical integration of the Bayesian formula for optimalrecursive filtering. The direct evaluation of the Jacobian matrixassociated with the extended Kalman filter is avoided, whichis similar to the one recently reported in [12]. Secondly, we

Manuscript received November 7, 1997; revised November 30, 1998 and June16, 1999. Recommended by Associate Editor, T. E. Duncan. This work was sup-ported in part by the Office of Naval Research under Grant N00014-96-1-0265and MURI-AFOSR under Grant F49620-95-1-0447.

The authors are with the Center for Research in Scientific Computation,North Carolina State University, Raleigh, NC 27695-8205 USA (e-mail:[email protected]; [email protected]).

Publisher Item Identifier S 0018-9286(00)04155-6.

use Gaussian sum filters for the development of nearly optimalfilters. The Gaussian sum filter has been studied in [1] and[20]. However, we adapt our Gaussian filter for the update ofGaussian distributions. We also suggest some new update rulesof weights for Gaussian sum filters. Through our experimentalstudy we found that the filters developed in the paper performbetter than or as good as the filter of Julier–Uhlmann [12]. Theyhave a significant improvement over the extended Kalman filterwith no additional cost.

An outline of the paper is as follows. In Section II we de-velop the Gaussian filter based on a single Gaussian distribu-tion. In Section III we discuss the efficient numerical integrationof the Gaussian filter based on quadrature rules, and introducethe Gauss–Hermite filter (GHF) and the central difference filter(CDF). In Section IV we formulate the Gauss–Hermite filter andthe central difference filter as the filter algorithms. In Section Vwe introduce the mixed Gaussian filter and the new update rulesof weights. In Section VI we discuss the relation of the Gaussianfilter for the discrete time system to the continuous-time optimalfilter governed by the Zakai equation. In Section VII we ana-lyze the stability and performance bound of the Gaussian fil-ters and the mixed Gaussian filters developed in the paper. InSection VIII we report our numerical findings and comparisonstudies. Also, we demonstrate a nearly optimal performance ofthe mixed Gaussian filter. Finally, we conclude our results inSection IX.

II. GAUSSIAN FILTERS

We discuss the nonlinear filtering problem for the dis-crete-time signal system for -valued process

(2.1)

and the observation process is given by

(2.2)

where and are white noises with covariancesandrespectively. We assume that the initial condition and

are independent random variables. The optimal non-linear filtering problem is to find the conditional expectation

of the process given the observation dataThe probability density function of the

conditional expectation is given by Bayes’ formula

(2.3)

and

(2.4)

0018–9286/00$10.00 © 2000 IEEE

ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 911

where is the one-step prediction and is the probabilitydensity function of conditioned on That is, the re-cursive filter (2.3), (2.4) consists of the prediction step (2.3)and the correction step (2.4). We consider a Gaussian approx-imation of this recursive formula and develop Gaussian filters.We assume that is a single Gaussian distribution withmean and covariance Then we constructthe Gaussian approximation of in the following two steps.First, we consider the predictor step. We approximateby the Gaussian distribution that has the same mean and covari-ance as By Fubini’s theorem the mean and covarianceof are given by

and

Remark 1.1:To derive the mean and covariance ofwe also can use (2.1) and the independence of and

For example

which is precisely the same as the above, though differently ex-pressed.

Thus, if is a Gaussian with mean and co-variance then the Gaussian approximation ofhas mean and covariance defined by

(2.5)

and

(2.6)

Next, we discuss the corrector step

where is the normalization constant and we assume thatis given by the Gaussian approximation defined by (2.5) and(2.6). Define the innovation process

Here we again approximate for the conditional distributionof given the observations up to time bya Gaussian. That is, we approximate by itsGaussian approximation This means that the probabilitydensity function of is given by the Gaussian distribution withmean and covariance defined by

(2.7)

and

(2.8)

Finally, we construct the Gaussian approximation of withmean and covariance defined by

(2.9)

and

(2.10)

where the Kalman-filter gain is defined by

(2.11)

and the covariance is defined by

(2.12)


In order to implement the Gaussian filter we must developthe approximation methods to evaluate integrals (2.5)–(2.12).In Section III we discuss the approximation methods for theintegration of the form

where is a given function. Our discussions include theGauss–Hermite quadrature rule and finite difference approxi-mation. Note that a different approximation to this integrationresults in a variant of the Gaussian filter. For example, if we ap-proximate the integration by linearizing at then we obtainthe extended Kalman filter.

In order to improve the performance of Gaussian filter wealso discuss the mixed Gaussian filter in Section V. We approx-imate by the linear combination of multiple Gaussiandistributions, i.e.,

Then, each Gaussian distribution is updated separately by theGaussian filter (2.5)–(2.12). In the corrector step we update theweights For example, we can determine the weightsby the -projection, i.e., minimize

over where is defined as in the corrector step (2.4),assuming

III. QUADRATURE RULES

In this section we discuss the approximation methods for theintegral of the form

(3.1)

If we assume and change the coordinate of integrationby then

(3.2)

with We apply the Gauss–Hermite quadra-ture rule. The Gauss–Hermite quadrature rule is given by

where the equality holds for all polynomials of degree up toand the quadrature points and the weights are de-

termined (e.g., see [9]) as follows. Letbe the symmetric tridi-agonal matrix with zero diagonals and

Then are the eigenvalues of and equalto where is the first element of theth normalizedeigenvector of Thus, is approximated by

(3.3)

where , and is exact for all polyno-mials of the form with In orderto evaluate we need -point function evaluations. For ex-ample we have

and

and requires nine-point function evaluations forIn Julier–Uhlmann [12] the standard normal distribution is

approximated by the discrete distribution as

and

where is a constant and is th unit vector inHere, this discrete distribution has the same first, second,and higher odd moments as the standard normal distribution.Julier–Uhlmann developed a Gaussian filter based on thefollowing quadrature rule for (3.1):

(3.4)

The JU-rule requires -point function evaluation and isexact for all quadratic polynomials. If we set andthen the JU-rule coincides with the Gauss–Hermite rule

Finally we consider the polynomial interpolation methods.We approximate by the quadratic function that satisfies


at the points in consisting of

and

where is the stepsize. That is, is given by

(3.5)

where is the th coordinate of a point and the vectorin and the symmetric matrix are

defined by

Thus, we approximate (3.1) by

(3.6)

and the integral

by

(3.7)

where is the quadratic approximation of forNext, we consider the approximation of defined by

(3.8)

which only bases on the valuesand and uses the diagonal second order correc-

tion of the central difference approximation ofin (3.5). Then the integralsand are approximated by

(3.9)

and

(3.10)

If we remove the second-order correction term in (3.9) and(3.10), then this coincides with the extended Kalman filter withthe central difference approximation of the Jacobian of

IV. FILTER ALGORITHMS

In this section we describe the filter algorithms that are basedon the Gaussian filter and quadrature methods described in Sec-tion III. We first consider the Gauss–Hermite filter based on theGaussian–Hermite quadrature rule. Let be the quadra-ture rule for

(4.1)

Let and be the starting values for the mean and co-variance of the random variable and set and

In Gaussian–Hermite filter we apply the followingpredictor and corrector steps recursively.

A. Gauss–Hermite Filter

Predictor Step: Compute the factorizationand set Update

by

Corrector Step: Compute the factorizationand set Update by

where

We use the Choresky decomposition for findingand [6]in our calculations. One of the advantages of the quadrature


based filter is that we are not required to have the derivative ofandSecondly, we present the filter algorithm based on the cen-

tral difference approximation with second order diagonal cor-rection. We summarize the filter algorithm of the central differ-ence approximation with the second-order diagonal correctionas follows.

B. Central Difference Filter

Predictor Step: Compute the factorizationand set approx-

imated by Updateby

where are the central difference approximations of

Corrector Step: Compute the factorizationand set approximated by

Update by

where

where are the central difference approximations of

In the algorithm we avoid calculating the derivatives ofandInstead, we use the central difference.The following are the operation counts in terms of function

evaluations.

• If we use the quadrature rule based on the-pointGauss–Hermite rule then the algorithm requiresfunction evaluations.

• If we use the Julier–Uhlmann discrete Gaussian rule thenthe algorithm requires function evalua-tions.

• If we use the central difference algorithm, thefunction evaluations are required.

As we will see, our numerical testings indicate that the algo-rithm based on the Gauss–Hermite rule performs better than theothers. If we use the three-point rule, then we require 729function evaluations for the case which can be done inreal-time.

V. MIXED GAUSSIAN FILTER

In this section we discuss the mixed Gaussian filter. We ap-proximate the conditional probability density by thelinear combination of multiple Gaussian distributions, i.e.,

(5.1)

Here we apply the Gaussian filter (2.5)–(2.12) to each Gaussiandistribution and obtain the update

Each update is independent from the others andcan be performed in a parallel manner.

Next, we update the weights for the new updateat the end of corrector step. We discuss three update formulasin what follows.

First, by equating the zero moment of each Gaussian distri-bution we obtain

Here we approximate the right-hand side by the Gaussian dis-tribution as in (2.5) and (2.6) and obtain

(5.2)

where are defined by (2.7) and (2.8), which is the updateformula discussed in [1].

Next, we apply the collocation condition at


to obtain the update

(5.3)

Finally we discuss the simultaneous update of the weights.We determine the weights by the -projection, i.e.,

minimize

(5.4)

over where is defined as in the corrector step (2.4)with

In order to perform the minimization (5.4) we need to evaluatethe integral of the form

and it is relatively expensive. Hence we propose the minimiza-tion of the sum of collocation distances

(5.5)

over satisfying A positive constantis chosen so that the likelihood of each Gaussian distribution isnonzero [e.g., ]. Problem (5.5) isformulated as the quadratic programming

subject to (5.6)

where is chosen so that the singularity of the matrixis avoided and the matrices are defined by

Thus, we solve (5.6) to obtain the weights at each correctorstep by using the existing numerical optimization method (e.g.,see [6]).

The theoretical foundation of the Gaussian sum approxima-tion as above is that any probability density function can be ap-proximated as closely as desired by a Gaussian sum. More pre-cisely, we state the following error estimate, the proof of whichis found in [17].

Theorem 5.1:Let be a nonnegative integer andFor any there must exist and a mask

such that for any density functionand all the estimate

(5.7)

holds, where is a linear combination of Gaussian distri-butions given by

(5.8)

with

and is a constant independent of Here andare defined by

and

respectively.Roughly speaking, the estimate (5.7) shows that any proba-

bility density function can be


approximated by a sum of Gaussian distributions each of whosecomponents is given by

(5.9)

with order for

VI. RELATION TO CONTINUOUS-TIME FILTER

In this section let us discuss the nonlinear filtering problemfor the continuous-time signal process in generated by

(6.1)

where is the standard Brownian motion, i.e., is thediffusion process [3], [11], [19]. So, (6.1) holds in the sense ofIto and satisfies

for allNow we consider the continuous observation process

(6.2)

where is the standard Brownian process that is independentof and

Or, we consider the discrete observation process

(6.3)

where is the stepsize and is white noise withcovariance We assume that the initial condition and

are independent.Then the conditional probability density

satisfies the Zakai equation

(6.4)

where is the Fokker–Planck operator [3], [19].As shown in [8], [10], and [15], the discrete-time filter (2.3),

(2.4) applied to the time-discretized signal system of (6.1), (6.2)

(6.5)

provides an approximation method for the continuous-time op-timal filter to (6.1), (6.2). Note that denotes the finite dif-ference approximation

for the ordinary differential equation

The proposed Gaussian filter does not converge to the optimalfilter (6.4) as unless is Gaussian. However, itconverges to a Gaussian filter, which approximates (6.4) andwill be discussed elsewhere.

Next, we discuss the discrete-time filter for the contin-uous-time process (6.1) with the discrete-time observation

(6.3). We approximate the process by the discrete-timeprocess by

(6.6)

with

Here the interval is subdivided into thesubintervals

where andSo, denotes the approximation of the process at

and denotes the discretization of the ordi-nary differential equation

We employ a family of Runge–Kutta methods in our calculation,including the Euler approximation

The predictor step (2.5), (2.6) is applied successivelytimesto obtain i.e.,

(6.7)

and

(6.8)

with

for Hence, the mean and covariance of aregiven by

(6.9)

Then the corrector step (2.7)–(2.12) is applied as it is to obtain

VII. STABILITY ANALYSIS OF FILTER ALGORITHMS

In this section we analyze the stability and performancebound of the Gaussian filters (2.5)–(2.12) for the signal process


(6.5) with i.e., a discrete approximationof the continuous signal system (6.1) can be given by

where is the white noise with covarianceWe assume that

for (7.1)

Then we have the following theorem. The theorem shows sta-bility of the proposed Gaussian filter independent of the finenessof on a fixed bounded time interval.

Theorem 7.1:The solution to (2.5)–(2.12) has the esti-mate

(7.2)

Proof: From (2.5), (2.6) the Gaussian approximation ofhas the mean and covariance defined by

(7.3)

and

(7.4)

Note that

Thus by the Cauchy and Schwarz inequality we have

where

Hence it follows from (7.4) that

by assumption (7.1). That is

(7.5)

Note that for any the covariance is nonegative and satis-fies the following property:

Thus by (7.5) we obtain the estimate (7.2).Remark 7.1:Using the similar arguments that lead to (7.2),

we can prove that the same result holds for the approximationmethod discussed in Sections III and IV. That is, the estimate(7.2) is still valid for the Gaussian–Hermite filter, provided that

Next, we discuss the stability of the mixed Gaussian filter.The mixed Gaussian filter assumes the form

(7.6)

where the th Gaussian distribution is com-puted by the Gaussian filters described in Sections III and IVin a parallel manner. The weights are deter-mined at the end of the corrector step based on the likelihoodof each Gaussian distribution. Under the assumption (7.1) eachGaussian filter in the Gaussian sum (7.6) is stable in the senseof (7.2).

Next we discuss the performance bound. In general, let theoperators and denote the one-step updateof conditional probability density functions that are based on theexact filter (2.3), (2.4) and the mixed Gaussian filter discussedin Section V, respectively. We estimate the difference betweenthe exact filter and the mixed Gaussian filter

(7.7)

where is the probability density function of random vari-able and

is the mixed Gaussian approximation of LetThen we have the following result.

Theorem 7.2:Assume that for any probability density func-tion and

(7.8)

Then the error estimate

(7.9)

holds, where is the initial error defined by


and the error is given by

(7.10)

Proof: Note that

Thus we have

which leads to (7.9).Remark 7.2:Suppose is a mixed Gaussian distribution

and we set Also assume that andare affine functions. Then we can show that i.e., themixed Gaussian filter is exact, since the Gaussian filter is exactfor the Gaussian distribution in this case.

To understand the validity and usage of error formula (7.9)we adopt the following filter (which is different from the one weimplemented), i.e., the mixed Gaussian filter with fixed meansand covariances. For simplicity of our discussion we assume

Let us consider the fixed Gaussian elements

with nodal points and covarianceThen we define the subspace of the probability distribu-

tions on by

Define is the projection of ontoWe consider the mixed Gaussian filter with fixed

means and covariances, i.e.,

where

Thus is the fixed operator in this case. Hencecan be estimated based on the error estimate formula (7.9),

assuming the smoothness of

VIII. N UMERICAL RESULTS AND COMPARISONS

In this section we demonstrate the feasibility of our proposedfilter algorithms, Gauss–Hermite filter (GHF), and central dif-ference filter (CDF), using the three test examples as in Exam-ples 8.1–8.3. We compare the performance of our proposed fil-

ters against that of the extended Kalman filter (EKF) and thefilter of Julier–Uhlmann (JUF) [12].

We use the average root mean square error for our comparisonof the methods. The average root mean square error is definedby

(8.1)

which is based on different simulation runs. Here subscriptdenotes the th component of the vector and its corre-

sponding estimate and the superscript denotes thethsimulation run.

Note that for a one-dimensional system the filter ofJulier–Uhlmann [12] coincides with the three-point Gauss–Her-mite filter when Also when the stepsize is thecentral difference filter has the same estimate algorithm as thefilter of Julier–Uhlmann except for the estimation Hence,in this section the stepsize is always chosen asfor our study.

We also note that our proposed filter algorithms maintain thenonnegative definiteness of the covariance update.

We carried out our comparison study using various examplesand different starting states and different problem data, butwe only present the selected results in what follows.

Example 8.1:We consider the one-dimensional signalprocess

(8.2)

and the observation process

(8.3)

where

and

It is motivated from the continuous time systems (6.1) and (6.2)of the form

(8.4)

as described in Section VI.This example is used as a benchmark because of the following

reasons. The deterministic equation

(8.5)

has two stable equilibria, 1 and 1, and one unstable equi-librium, 0. So, we observed that the signal processis dis-tributed around one of stable equilibria. The observation func-tion is the square of shifted distance fromthe origin and distinguishes the two stable equilibria marginally.The performance of the filter is possibly tested because of theminor margin of detecting measurement function of thesignal process.

We use the following problem data: the time elapsethe initial condition i.e., the

starting point and the system


Fig. 1. The average of root mean square error.

parameters and We consider the time from0 to 4. is chosen as the best value, i.e., (see [12]).

Fig. 1 shows the average root mean square errors committedby each filter across a simulation consisting of 50 runs. FromFig. 1 we see that the five-point Gauss–Hermite filter performsbetter than others. The central difference filter has similar per-formance like the filter of Julier–Uhlmann [12] in this example.All the three filters perform superior to the extended Kalmanfilter.

Example 8.2: In this example, we consider the Lorenzsystem

where is the -valued process. The-valued processes and are white noises with the

same covariance Here is a constant vector andis a constant scalar. Also

(8.6)

The observation function is chosen as the shifted distance fromthe origin given by

It is motivated from the Lorenz stochastic differential systemof the form (8.4) as described in Section VI where

The three-dimensional deterministic equations

(8.7)

is the Lorenz equation in which are positiveparameters. The three parameters have a great deal of impact on

the system. Here, we choose andwhich is a mathematically very interesting case, because in thecase there are three unstable equilibria and a strange attractorfor the equations.

We use the following problem data: the system parameters arechosen as and the initial condition is

i.e.,and We consider the time from 0 to 4. The initialestimate is with covariance

In Figs. 2–4 we show the average of root mean square errorsfor each component of system state committed by the algorithmsof the Gauss–Hermite quadrature rule, the central difference ap-proximation, and the JU-rule, respectively, across a simulationconsisting of 50 runs. As shown in Figs. 2–4, the GHF and theCDF have smaller errors than the JUF and the EKF. From thesefigures we conclude that in this example the GHF has a substan-tial improved performance, and the GHF performs a little betterthan the CDF.

Example 8.3:We further discuss the three-dimensional con-tinuous signal process

(8.8)

and the discrete observation process

(8.9)

where and are zero mean and uncorrelatednoises with covariances given by andrespectively. The function is given by

The above system states and represent altitude(in feet), velocity (in feet per second), and constant ballistic




coefficient (in per second), respectively. The detailed physicalmeaning of the system and its parameters can be found in [2].

We chose this example as a benchmark because it containssignificant nonlinearities in the signal and observation processesand had been discussed widely in the literature.

In the previous literature, a fourth-order Runge–Kutta schemewith 64 steps between each observation is employed for numer-ical integration of (8.8) in order to deal with the significant non-

linearity of signal process [2], [12]. That is, in the predicatorsteps of EKF and JUF, the means are calculated using the nu-merical scheme. Then their covariance are propagated from theth to th step using



Fig. 5. The absolute value of average altitude error.

with

where was evaluated at and Thus,

In the comparison, we implement the proposed approxima-tion schemes (6.7)–(6.9) to solve the filtering problem of (8.8),

(8.9). In the predictor step of GHF, we directly use the Eulerapproximation of signal process (8.8) based on 32 or 64 steps(i.e., 32 or 64) between each observation (they are shortlydenoted by GHF32 or GHF64, respectively).

However, in NGHF the signal process system (8.8) isrewritten as the discrete form


Fig. 6. The absolute value of average velocity error.

Fig. 7. The absolute value of averagex -error.

where is the approximation of the process atand function with

which is the fourth-order Runge–Kutta scheme. Specifically,in NGHF64.

We use the following data: the system parameters are chosenas and

the initial condition isand We consider the time from 0 to 30. The

initial estimate is withcovariance


Fig. 8. The comparison of signal; estimates of two mixed Gaussian filter and optimal filters.

Fig. 9. The weights of two mixed Gaussian filters.

We also choose the optimization number in JUF for thethree-dimensional system.

In Figs. 5–7 we show the absolute value of average errorfor each component committed by the algorithms of theGauss–Hermite quadrature rule, the JU-rule, and the EKF,respectively, across a simulation consisting of 50 runs. Thesefigures show that in this example the GHF has superior perfor-mance than the JUF and the EKF.

A. Mixed Gaussian Example

In this section we first demonstrate a nearly optimal perfor-mance of the mixed Gaussian method described in Section V.We consider the same example as in Example 8.1 with

We employ the two Gaussian distributions starting from


Fig. 10. The comparison of the probability density functions of mixed Gaussian filter and optimal filter att = 1:

Fig. 11. The comparison of the probability density functions of mixed Gaussian filter and optimal filter att = 2:75:

and

respectively, and the initial weights are Inorder to demonstrate the effectiveness of the proposed update(5.5), (5.6) we jumped the process at [whichcorresponds to in the continuous time process (8.4)]to It may represent an impulse force atWe compare the performance of the mixed Gaussian filter

against the one of the optimal Zakai filter (6.4) within Figs. 8–11. Here GHF1 (GHF2) represents the three-pointGauss–Hermite filter which is the first (second) component ofthe two mixed Gaussian filter with Weight 1 (Weight 2). We ap-proximate the Zakai equation by the operator-splitting methodas described in [10]. We observed that the mixed Gaussianfilter performs nearly optimally up to We also notethat the Zakai filter is no longer optimal after becauseof the jump. As seen as Figs. 8 and 9, the update formula (5.5),


Fig. 12. The comparison of altitude errors of each Gaussian filter.

Fig. 13. The comparison of velocity errors of each Gaussian filter.

(5.6) quickly captures the phase change from the one steadypoint to the other The observations are alsosupported by the comparison in Figs. 10 and 11.

Example 8.3 (Revisited):Next, we also apply our mixedGaussian method to Example 8.3. We employ the two Gaussiandistribution starting from

and

with the same covariance


Fig. 14. The comparison ofx errors of each Gaussian filter.

Fig. 15. The weights of two mixed Gaussian filters.

respectively, and the initial weights areWe demonstrate the performance of the mixed Gaussian filterwith in Figs. 12–15. We see that the mixed Gaussianfilter based on the update formula (5.5), (5.6) captures thesignal process very well. That is, the weights of the two mixedGaussian filters shown in Fig. 15 effectively pick up the com-bined performance of the Gaussian filters: GHF1 and GHF2.More detailed discussions and numerical testings of the mixedGaussian filter will be presented in the forthcoming paper.

IX. CONCLUSIONS

The paper presents the systematic formulation of Gaussianfilters and mixed Gaussian filters based on the efficient numer-ical integration of the Bayesian formula for optimal recursivefilter. Based on our formulation we develop the two filter algo-rithms, namely, the Gauss–Hermite filter (GHF) and the centraldifference filter (CDF). We demonstrated the feasibility of ourproposed filter algorithms for testing nonlinear filtering prob-


lems. Our numerical results indicate that both the Gauss–Her-mite filter and the central difference filter have superior perfor-mance to the filter of Julier–Uhlmann and the extended Kalmanfilter. We also proposed the new update rules for the Gauss sumfilters and show that they can perform near optimally.

ACKNOWLEDGMENT

The authors thank the referees for their careful reviews andhelpful suggestions.

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Kazufumi Ito received the Ph.D. degree in systemscience and mathematics from Washington Univer-sity in 1981. Subsequently, he held the positionsin ICASE, Brown University and University ofSouthern California before joining the Departmentof Mathematics, North Carolina State University,where he is currently a Professor. His research inter-ests include nonlinear filtering, stochastic process,control and analysis of partial differential equations.He co-organized an AMS-IMS-SIAM Joint SummerResearch Conference on Identification and Control

in Systems governed by PDE’s, 1992 and the workshop on Stochastic Controland Nonlinear filtering, at NC State University, 1996.

Kaigqi Xiong received the M.S. and Ph.D. de-grees in applied mathematics from the ClaremontGraduate School in 1995 and 1996, respectively. In1995–1996, he was also a Researcher of ElectricalEngineering at the University of California, River-side. From 1996–1999, he was a Visiting AssistantProfessor in the Center for Research in ScientificComputation at the North Carolina State University.As a Researcher, he is visiting the Department ofMechanical and Aerospace Engineering at the Uni-versity of California, Irvine. His research interests

include nonlinear systems, mathematical control theory, filtering algorithms,parameter estimation, flight control, neural networks and applications.

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