Gaussian Filters for Nonlinear Filtering Problems

Transcription

1 910 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 Gaussian Filters for Nonlinear Filtering Problems Kazufumi Ito Kaiqi Xiong Abstract In this paper we develop analyze real-time accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters develop efficient accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical testings demonstrate that new filters significantly improve the extended Kalman filter with no additional cost the new Gaussian sum filter has a nearly optimal performance. Index Terms Author: please supply index terms. keywords@ieee.org for information. I. INTRODUCTION THE NONLINEAR filtering problem consists of estimating the state of a nonlinear stochastic system from noisy observation data. The problem has been the subject of considerable research interest during the past several decades because it has many significant applications in science engineering such as navigational guidance systems, radar tracking, sonar ranging, satellite airplane orbit determination [11], [14], [15]. As is well-known, the most widely used filter is the extended Kalman filter for nonlinear filtering problems. It is derived from the Kalman filter based on the successive linearization of the signal process the observation map. The extended Kalman filter has been successfully applied to numerous nonlinear filtering problems. If nonlinearities are significant, however, its performance can be substantially improved. Such efforts have also been reported in [1], [5], [7], [12], [13], [20]. In this paper our objective is to develop analyze real-time accurate filters for nonlinear filtering algorithms based on Gaussian distributions. We present the systematic formulation of Gaussian filters mixed Gaussian filters. We first develop the Gaussian filter. The proposed filter is based on the steps: 1) we assume the conditional probability density to be a Gaussian distribution (i.e., assumed density) 2) we obtain the Gaussian filter by equating the Bayesian formula with respect to the first moment (mean) the second moment (covariance). Our approach is based on the efficient numerical integration of the Bayesian formula for optimal recursive filtering. The direct evaluation of the Jacobian matrix associated with the extended Kalman filter is avoided, which is similar to the one recently reported in [12]. Secondly, we use Gaussian sum filters for the development of nearly optimal filters. The Gaussian sum filter has been studied in [1] [20]. However, we adapt our Gaussian filter for the update of Gaussian distributions. We also suggest some new update rules of weights for Gaussian sum filters. Through our experimental study we found that the filters developed in the paper perform better than or as good as the filter of Julier Uhlmann [12]. They have a significant improvement over the extended Kalman filter with no additional cost. An outline of the paper is as follows. In Section II we develop the Gaussian filter based on a single Gaussian distribution. In Section III we discuss the efficient numerical integration of the Gaussian filter based on quadrature rules, introduce the Gauss Hermite filter (GHF) the central difference filter (CDF). In Section IV we formulate the Gauss Hermite filter the central difference filter as the filter algorithms. In Section V we introduce the mixed Gaussian filter the new update rules of weights. In Section VI we discuss the relation of the Gaussian filter for the discrete time system to the continuous-time optimal filter governed by the Zakai equation. In Section VII we analyze the stability performance bound of the Gaussian filters the mixed Gaussian filters developed in the paper. In Section VIII we report our numerical findings comparison studies. Also, we demonstrate a nearly optimal performance of the mixed Gaussian filter. Finally, we conclude our results in Section IX. II. GAUSSIAN FILTERS We discuss the nonlinear filtering problem for the discrete-time signal system for -valued process (2.1) the observation process is given by (2.2) where are white noises with covariances respectively. We assume that the initial condition are independent rom variables. The optimal nonlinear filtering problem is to find the conditional expectation of the process given the observation data The probability density function of the conditional expectation is given by Bayes formula Manuscript received November 7, 1997; revised November 30, 1998 June 16, Recommended by Associate Editor, T. E. Duncan. This work was supported in part by the Office of Naval Research under Grant N MURI-AFOSR under Grant F The authors are with the Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC USA ( kito@eos.ncsu.edu; xiong@eos.ncsu.edu). Publisher Item Identifier S (00) (2.3) (2.4) /00$ IEEE

2 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 911 where is the one-step prediction is the probability density function of conditioned on That is, the recursive filter (2.3), (2.4) consists of the prediction step (2.3) the correction step (2.4). We consider a Gaussian approximation of this recursive formula develop Gaussian filters. We assume that is a single Gaussian distribution with mean covariance Then we construct the Gaussian approximation of in the following two steps. First, we consider the predictor step. We approximate by the Gaussian distribution that has the same mean covariance as By Fubini s theorem the mean covariance of are given by Next, we discuss the corrector step (2.6) where is the normalization constant we assume that is given by the Gaussian approximation defined by (2.5) (2.6). Define the innovation process Here we again approximate for the conditional distribution of given the observations up to time by a Gaussian. That is, we approximate by its Gaussian approximation This means that the probability density function of is given by the Gaussian distribution with mean covariance defined by (2.7) Remark 1.1: To derive the mean covariance of we also can use (2.1) the independence of For example which is precisely the same as the above, though differently expressed. Thus, if is a Gaussian with mean covariance then the Gaussian approximation of has mean covariance defined by Finally, we construct the Gaussian approximation of mean covariance defined by where the Kalman-filter gain is defined by the covariance is defined by (2.8) with (2.9) (2.10) (2.11) (2.5) (2.12)

3 912 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 In order to implement the Gaussian filter we must develop the approximation methods to evaluate integrals (2.5) (2.12). In Section III we discuss the approximation methods for the integration of the form If we assume by change the coordinate of integration then (3.2) where is a given function. Our discussions include the Gauss Hermite quadrature rule finite difference approximation. Note that a different approximation to this integration results in a variant of the Gaussian filter. For example, if we approximate the integration by linearizing at then we obtain the extended Kalman filter. In order to improve the performance of Gaussian filter we also discuss the mixed Gaussian filter in Section V. We approximate by the linear combination of multiple Gaussian distributions, i.e., with We apply the Gauss Hermite quadrature rule. The Gauss Hermite quadrature rule is given by where the equality holds for all polynomials of degree up to the quadrature points the weights are determined (e.g., see [9]) as follows. Let be the symmetric tridiagonal matrix with zero diagonals Then are the eigenvalues of equal to where is the first element of the th normalized eigenvector of Thus, is approximated by (3.3) where, is exact for all polynomials of the form with In order to evaluate we need -point function evaluations. For example we have Then, each Gaussian distribution is updated separately by the Gaussian filter (2.5) (2.12). In the corrector step we update the weights For example, we can determine the weights by the -projection, i.e., minimize requires nine-point function evaluations for In Julier Uhlmann [12] the stard normal distribution is approximated by the discrete distribution as over where is defined as in the corrector step (2.4), assuming where is a constant is th unit vector in Here, this discrete distribution has the same first, second, higher odd moments as the stard normal distribution. Julier Uhlmann developed a Gaussian filter based on the following quadrature rule for (3.1): (3.4) III. QUADRATURE RULES In this section we discuss the approximation methods for the integral of the form The JU-rule requires -point function evaluation is exact for all quadratic polynomials. If we set then the JU-rule coincides with the Gauss Hermite rule Finally we consider the polynomial interpolation methods. We approximate by the quadratic function that satisfies (3.1)

4 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 913 at the points in consisting of where is the stepsize. That is, is given by (3.5) where is the th coordinate of a point the vector in the symmetric matrix are defined by (3.10) If we remove the second-order correction term in (3.9) (3.10), then this coincides with the extended Kalman filter with the central difference approximation of the Jacobian of IV. FILTER ALGORITHMS In this section we describe the filter algorithms that are based on the Gaussian filter quadrature methods described in Section III. We first consider the Gauss Hermite filter based on the Gaussian Hermite quadrature rule. Let be the quadrature rule for Thus, we approximate (3.1) by the integral by (3.6) (4.1) Let be the starting values for the mean covariance of the rom variable set In Gaussian Hermite filter we apply the following predictor corrector steps recursively. A. Gauss Hermite Filter Predictor Step: Compute the factorization set Update by where is the quadratic approximation of for Next, we consider the approximation of defined by (3.7) Corrector Step: Compute the factorization set Update by where (3.8) which only bases on the values uses the diagonal second order correction of the central difference approximation of in (3.5). Then the integrals are approximated by (3.9) We use the Choresky decomposition for finding [6] in our calculations. One of the advantages of the quadrature

5 914 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 based filter is that we are not required to have the derivative of Secondly, we present the filter algorithm based on the central difference approximation with second order diagonal correction. We summarize the filter algorithm of the central difference approximation with the second-order diagonal correction as follows. B. Central Difference Filter Predictor Step: Compute the factorization set approximated by Update by where are the central difference approximations of Corrector Step: Compute the factorization set approximated by Update by V. MIXED GAUSSIAN FILTER In this section we discuss the mixed Gaussian filter. We approximate the conditional probability density by the linear combination of multiple Gaussian distributions, i.e., (5.1) Here we apply the Gaussian filter (2.5) (2.12) to each Gaussian distribution obtain the update Each update is independent from the others can be performed in a parallel manner. Next, we update the weights for the new update at the end of corrector step. We discuss three update formulas in what follows. First, by equating the zero moment of each Gaussian distribution we obtain where where are the central difference approximations of Here we approximate the right-h side by the Gaussian distribution as in (2.5) (2.6) obtain In the algorithm we avoid calculating the derivatives of Instead, 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 -point Gauss Hermite rule then the algorithm requires function evaluations. If we use the Julier Uhlmann discrete Gaussian rule then the algorithm requires function evaluations. If we use the central difference algorithm, the function evaluations are required. As we will see, our numerical testings indicate that the algorithm based on the Gauss Hermite rule performs better than the others. If we use the three-point rule, then we require 729 function evaluations for the case which can be done in real-time. (5.2) where are defined by (2.7) (2.8), which is the update formula discussed in [1]. Next, we apply the collocation condition at

6 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 915 to obtain the update where is chosen so that the singularity of the matrix is avoided the matrices are defined by (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 Thus, we solve (5.6) to obtain the weights at each corrector step by using the existing numerical optimization method (e.g., see [6]). The theoretical foundation of the Gaussian sum approximation as above is that any probability density function can be approximated as closely as desired by a Gaussian sum. More precisely, we state the following error estimate, the proof of which is found in [17]. Theorem 5.1: Let be a nonnegative integer For any there must exist a mask such that for any density function all the estimate In order to perform the minimization (5.4) we need to evaluate the integral of the form is a linear combination of Gaussian distri- holds, where butions given by (5.7) (5.8) it is relatively expensive. Hence we propose the minimization of the sum of collocation distances with is a constant independent of Here are defined by (5.5) over satisfying A positive constant is chosen so that the likelihood of each Gaussian distribution is nonzero [e.g., ]. Problem (5.5) is formulated as the quadratic programming subject to (5.6) respectively. Roughly speaking, the estimate (5.7) shows that any probability density function can be

7 916 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 approximated by a sum of Gaussian distributions each of whose components is given by (6.3). We approximate the process by the discrete-time process by with order for (5.9) with (6.6) VI. RELATION TO CONTINUOUS-TIME FILTER In this section let us discuss the nonlinear filtering problem for the continuous-time signal process in generated by (6.1) where is the stard Brownian motion, i.e., is the diffusion process [3], [11], [19]. So, (6.1) holds in the sense of Ito satisfies Here the interval is subdivided into the subintervals where So, denotes the approximation of the process at denotes the discretization of the ordinary differential equation for all Now we consider the continuous observation process We employ a family of Runge Kutta methods in our calculation, including the Euler approximation (6.2) where is the stard Brownian process that is independent of Or, we consider the discrete observation process (6.3) where is the stepsize is white noise with covariance We assume that the initial condition 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], [15], the discrete-time filter (2.3), (2.4) applied to the time-discretized signal system of (6.1), (6.2) The predictor step (2.5), (2.6) is applied successively to obtain i.e., times (6.7) (6.5) provides an approximation method for the continuous-time optimal filter to (6.1), (6.2). Note that denotes the finite difference approximation with (6.8) for the ordinary differential equation The proposed Gaussian filter does not converge to the optimal filter (6.4) as unless is Gaussian. However, it converges to a Gaussian filter, which approximates (6.4) will be discussed elsewhere. Next, we discuss the discrete-time filter for the continuous-time process (6.1) with the discrete-time observation for Hence, the mean covariance of are given 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 performance bound of the Gaussian filters (2.5) (2.12) for the signal process

8 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 917 (6.5) with i.e., a discrete approximation of the continuous signal system (6.1) can be given by by assumption (7.1). That is (7.5) where is the white noise with covariance We assume that for (7.1) Then we have the following theorem. The theorem shows stability of the proposed Gaussian filter independent of the fineness of on a fixed bounded time interval. Theorem 7.1: The solution to (2.5) (2.12) has the estimate Note that for any the covariance is nonegative satisfies 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 approximation method discussed in Sections III 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) Note that (7.2) Proof: From (2.5), (2.6) the Gaussian approximation of has the mean covariance defined by (7.3) (7.4) where the th Gaussian distribution is computed by the Gaussian filters described in Sections III IV in a parallel manner. The weights are determined at the end of the corrector step based on the likelihood of each Gaussian distribution. Under the assumption (7.1) each Gaussian filter in the Gaussian sum (7.6) is stable in the sense of (7.2). Next we discuss the performance bound. In general, let the operators denote the one-step update of conditional probability density functions that are based on the exact filter (2.3), (2.4) the mixed Gaussian filter discussed in Section V, respectively. We estimate the difference between the exact filter the mixed Gaussian filter Thus by the Cauchy Schwarz inequality we have is the probability density function of rom vari- where able (7.7) where Hence it follows from (7.4) that is the mixed Gaussian approximation of Let Then we have the following result. Theorem 7.2: Assume that for any probability density function Then the error estimate (7.8) (7.9) holds, where is the initial error defined by

9 918 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 the error Proof: Note that is given by (7.10) In this section we demonstrate the feasibility of our proposed filter algorithms, Gauss Hermite filter (GHF), central difference filter (CDF), using the three test examples as in Examples We compare the performance of our proposed filters against that of the extended Kalman filter (EKF) the filter of Julier Uhlmann (JUF) [12]. We use the average root mean square error for our comparison of the methods. The average root mean square error is defined by (8.1) Thus we have which leads to (7.9). Remark 7.2: Suppose is a mixed Gaussian distribution we set Also assume that are affine functions. Then we can show that i.e., the mixed Gaussian filter is exact, since the Gaussian filter is exact for the Gaussian distribution in this case. To underst the validity usage of error formula (7.9) we adopt the following filter (which is different from the one we implemented), i.e., the mixed Gaussian filter with fixed means covariances. For simplicity of our discussion we assume Let us consider the fixed Gaussian elements with nodal points covariance Then we define the subspace of the probability distributions on by Define is the projection of onto We consider the mixed Gaussian filter means covariances, i.e., where with fixed Thus is the fixed operator in this case. Hence can be estimated based on the error estimate formula (7.9), assuming the smoothness of VIII. NUMERICAL RESULTS AND COMPARISONS which is based on different simulation runs. Here subscript denotes the th component of the vector its corresponding estimate the superscript denotes the th simulation run. Note that for a one-dimensional system the filter of Julier Uhlmann [12] coincides with the three-point Gauss Hermite filter when Also when the stepsize is the central difference filter has the same estimate algorithm as the filter of Julier Uhlmann except for the estimation Hence, in this section the stepsize is always chosen as for our study. We also note that our proposed filter algorithms maintain the nonnegative definiteness of the covariance update. We carried out our comparison study using various examples different starting states different problem data, but we only present the selected results in what follows. Example 8.1: We consider the one-dimensional signal process the observation process where (8.2) (8.3) It is motivated from the continuous time systems (6.1) (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 1, one unstable equilibrium, 0. So, we observed that the signal process is distributed around one of stable equilibria. The observation function is the square of shifted distance from the origin distinguishes the two stable equilibria marginally. The performance of the filter is possibly tested because of the minor margin of detecting measurement function of the signal process. We use the following problem data: the time elapse the initial condition i.e., the starting point the system

10 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 919 Fig. 1. The average of root mean square error. parameters We consider the time from 0to4. is chosen as the best value, i.e., (see [12]). Fig. 1 shows the average root mean square errors committed by each filter across a simulation consisting of 50 runs. From Fig. 1 we see that the five-point Gauss Hermite filter performs better than others. The central difference filter has similar performance like the filter of Julier Uhlmann [12] in this example. All the three filters perform superior to the extended Kalman filter. Example 8.2: In this example, we consider the Lorenz system where is the -valued process. The -valued processes are white noises with the same covariance Here is a constant vector is a constant scalar. Also (8.6) The observation function is chosen as the shifted distance from the origin given by It is motivated from the Lorenz stochastic differential system of the form (8.4) as described in Section VI where The three-dimensional deterministic equations the system. Here, we choose which is a mathematically very interesting case, because in the case there are three unstable equilibria a strange attractor for the equations. We use the following problem data: the system parameters are chosen as the initial condition is i.e., We consider the time from 0 to 4. The initial estimate is with covariance In Figs. 2 4 we show the average of root mean square errors for each component of system state committed by the algorithms of the Gauss Hermite quadrature rule, the central difference approximation, the JU-rule, respectively, across a simulation consisting of 50 runs. As shown in Figs. 2 4, the GHF the CDF have smaller errors than the JUF the EKF. From these figures we conclude that in this example the GHF has a substantial improved performance, the GHF performs a little better than the CDF. Example 8.3: We further discuss the three-dimensional continuous signal process the discrete observation process (8.8) (8.9) where are zero mean uncorrelated noises with covariances given by respectively. The function is given by (8.7) is the Lorenz equation in which are positive parameters. The three parameters have a great deal of impact on The above system states represent altitude (in feet), velocity (in feet per second), constant ballistic

11 920 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 Fig. 2. The average of root mean square error. Fig. 3. The average of root mean square error. coefficient (in per second), respectively. The detailed physical meaning of the system its parameters can be found in [2]. We chose this example as a benchmark because it contains significant nonlinearities in the signal observation processes had been discussed widely in the literature. In the previous literature, a fourth-order Runge Kutta scheme with 64 steps between each observation is employed for numerical integration of (8.8) in order to deal with the significant nonlinearity of signal process [2], [12]. That is, in the predicator steps of EKF JUF, the means are calculated using the numerical scheme. Then their covariance are propagated from the th to th step using

12 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 921 Fig. 4. The average of root mean square error. Fig. 5. The absolute value of average altitude error. with where was evaluated at Thus, In the comparison, we implement the proposed approximation 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 Euler approximation of signal process (8.8) based on 32 or 64 steps (i.e., 32 or 64) between each observation (they are shortly denoted by GHF32 or GHF64, respectively). However, in NGHF the signal process system (8.8) is rewritten as the discrete form

13 922 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 Fig. 6. The absolute value of average velocity error. Fig. 7. The absolute value of average x -error. where is the approximation of the process at function with We use the following data: the system parameters are chosen as the initial condition is We consider the time from 0 to 30. The initial estimate is with covariance which is the fourth-order Runge Kutta scheme. Specifically, in NGHF64.

14 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 923 Fig. 8. The comparison of signal; estimates of two mixed Gaussian filter optimal filters. Fig. 9. The weights of two mixed Gaussian filters. We also choose the optimization number in JUF for the three-dimensional system. In Figs. 5 7 we show the absolute value of average error for each component committed by the algorithms of the Gauss Hermite quadrature rule, the JU-rule, the EKF, respectively, across a simulation consisting of 50 runs. These figures show that in this example the GHF has superior performance than the JUF the EKF. A. Mixed Gaussian Example In this section we first demonstrate a nearly optimal performance 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

15 924 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 Fig. 10. The comparison of the probability density functions of mixed Gaussian filter optimal filter at t =1: Fig. 11. The comparison of the probability density functions of mixed Gaussian filter optimal filter at t =2:75: respectively, the initial weights are In order to demonstrate the effectiveness of the proposed update (5.5), (5.6) we jumped the process at [which corresponds to in the continuous time process (8.4)] to It may represent an impulse force at We compare the performance of the mixed Gaussian filter against the one of the optimal Zakai filter (6.4) with in Figs Here GHF1 (GHF2) represents the three-point Gauss Hermite filter which is the first (second) component of the two mixed Gaussian filter with Weight 1 (Weight 2). We approximate the Zakai equation by the operator-splitting method as described in [10]. We observed that the mixed Gaussian filter performs nearly optimally up to We also note that the Zakai filter is no longer optimal after because of the jump. As seen as Figs. 8 9, the update formula (5.5),

16 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 925 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 steady point to the other The observations are also supported by the comparison in Figs Example 8.3 (Revisited): Next, we also apply our mixed Gaussian method to Example 8.3. We employ the two Gaussian distribution starting from with the same covariance

17 926 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 Fig. 14. The comparison of x errors of each Gaussian filter. Fig. 15. The weights of two mixed Gaussian filters. respectively, the initial weights are We demonstrate the performance of the mixed Gaussian filter with in Figs We see that the mixed Gaussian filter based on the update formula (5.5), (5.6) captures the signal process very well. That is, the weights of the two mixed Gaussian filters shown in Fig. 15 effectively pick up the combined performance of the Gaussian filters: GHF1 GHF2. More detailed discussions numerical testings of the mixed Gaussian filter will be presented in the forthcoming paper. IX. CONCLUSIONS The paper presents the systematic formulation of Gaussian filters mixed Gaussian filters based on the efficient numerical integration of the Bayesian formula for optimal recursive filter. Based on our formulation we develop the two filter algorithms, namely, the Gauss Hermite filter (GHF) the central difference filter (CDF). We demonstrated the feasibility of our proposed filter algorithms for testing nonlinear filtering prob-

18 ITO AND XIONG: GAUSSIAN FILTERS FOR NONLINEAR FILTERING PROBLEMS 927 lems. Our numerical results indicate that both the Gauss Hermite filter the central difference filter have superior performance to the filter of Julier Uhlmann the extended Kalman filter. We also proposed the new update rules for the Gauss sum filters show that they can perform near optimally. ACKNOWLEDGMENT The authors thank the referees for their careful reviews helpful suggestions. REFERENCES [1] D. L. Alspace H. W. Sorenson, Nonlinear Bayesian estimation using Gaussian sum approximation, IEEE Trans. Automat. Contr., vol. 17, pp , [2] M. Athans, R. P. Wishner, A. Bertolini, Suboptimal state estimation for continuous-time nonlinear systems from discrete noise measurements, IEEE Trans. Automat. Contr., vol. 13, pp , [3] A. Bensoussan, Nonlinear filtering theory, in Progress in Automation Information Systems, Recent Advances in Stochastic Calculus,J.S. Baras V. Mirelli, Eds. New York: Springer-Verlag, [4] A. Bensoussan, R. Glowinski, A. Rascanu, Approximation of the Zakai equation by the splitting up method, SIAM J. Contr. Optim., vol. 28, pp , [5] Y. Bar-Shalom X. R. Liu, Estimation Tracking: Principles, Techniques, Software. New York: Artech, [6] J. E. Dennis R. B. Schnabel, Numerical Methods for Unconstrained Optimization Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, [7] C. P. Fang, New algorithms of Gaussian assumed density filter a convergence result, IEEE Trans. Automat. Contr.. [8] P. Florchinger F. Le Gl, Time-discretization of the Zakai equation for diffusion process observed in correlated noise, in Proc. 9th Conf. Analysis Optimization of Systems, A. Bensoussan J. L. Lions, Eds. New York: Springer-Verlag, 1990, vol (Lecture Notes on in Control Inform. Sci.). [9] G. H. Golub, Some modified matrix eigenvalue problems, SIAM Rev., vol. 15, pp , [10] K. Ito, Approximation of the Zakai equation for nonlinear filtering, SIAM J. Contr. Optim., vol. 34, pp , [11] A. H. Jazwinski, Stochastic Process Filtering Theory. New York: Academic, [12] S. J. Julier J. K. Uhlmann. (1994) A general method for approximating nonlinear transformations of probability distributions. [Online]. Available: www [13] S. J. Julier, J. K. Uhlmann, H. F. Durrant-Whyte, A new approach for filtering nonlinear systems, in Proc. Amer. Contr. Conf., Seattle, WA, June 1995, pp [14] R. E. Kalman, A new approach to linear filtering prediction problems, Trans. ASME, J. Basic Eng., vol. 82D, pp , Mar [15] H. J. Kushner, Approximation to optimal nonlinear filters, IEEE Trans. Automat. Contr., vol. 12, pp , [16], private communication. [17] V. Mazya G. Schmidt, On approximate approximations using gaussian kernels, IMA J. Numer. Anal., vol. 16, pp , [18] M. Piccioni, Convergence of implicit discretization schemes for linear differential equations eith application to filtering, in Stochastic Partial Differential Equations Application, G. Da Prato L. Tabbaro, Eds. Trento, 1985, vol (Lecture Note in Mathematics). [19] B. L. Rozovskii, Stochastic Evolution Systems, Linear Theory Application to Nonlinear Filtering. Norwell, MA: Kluwer, [20] H. W. Sorenson D. L. Alspace, Recursive Bayesian estimation using Gaussian sums, Automatica, vol. 7, pp , Kazufumi Ito received the Ph.D. degree in system science mathematics from Washington University in Subsequently, he held the positions in ICASE, Brown University University of Southern California before joining the Department of Mathematics, North Carolina State University, where he is currently a Professor. His research interests include nonlinear filtering, stochastic process, control analysis of partial differential equations. He co-organized an AMS-IMS-SIAM Joint Summer Research Conference on Identification Control in Systems governed by PDE s, 1992 the workshop on Stochastic Control Nonlinear filtering, at NC State University, Kaigqi Xiong received the M.S. Ph.D. degrees in applied mathematics from the Claremont Graduate School in , respectively. In , he was also a Researcher of Electrical Engineering at the University of California, Riverside. From , he was a Visiting Assistant Professor in the Center for Research in Scientific Computation at the North Carolina State University. As a Researcher, he is visiting the Department of Mechanical Aerospace Engineering at the University of California, Irvine. His research interests include nonlinear systems, mathematical control theory, filtering algorithms, parameter estimation, flight control, neural networks applications.