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Mixed-Degree Spherical Simplex-Radial Cubature Kalman Filter.

1. Introduction

Bayes' rule has been widely applied into solution to state estimation problem for both linear and nonlinear systems [1, 2]. The application of Bayes' rule to a probabilistic state space model generates the Bayesian filter, which can be generally divided into two categories, that is, the global approach and the local approach [3, 4]. The global approach makes no assumption of probability density function (pdf) and can achieve preferable performance at the cost of enormous computational burden, for example, the adaptive grid point mass filter [5], the Gaussian mixture filter [6], and the particle filter [7]. The local approach is an approximation solution based on some assumption of pdf.

Under Gaussian assumption, the local approach can be derived using the minimum mean square error (MMSE) criterion [8] and thus be summarized as the calculation of the multidimensional integrals whose integrands bear the form nonlinear function x Gaussian pdf, which is intractable to directly solve [4]. The problem is thus transformed into the approximation of the nonlinear function or the Gaussian pdf [4]. The approximation of the nonlinear function is achieved mainly by utilizing different polynomial expansions, which yields a set of nonlinear filters, for example, the extended Kalman filter (EKF) [9-11], the divided difference filter (DDF) [12], the Chebyshev polynomial Kalman filter (CPKF) [13], and the Fourier-Hermit Kalman filter (FHKF) [14]. As the most commonly used one of this kind of nonlinear Kalman filters, the EKF performs the first-order Taylor series approximation of the nonlinear functions, which is appropriate for a "mild" nonlinear environment [9-11]. However, when the system is highly nonlinear, the truncated higher order terms of the nonlinear system may degrade the estimation accuracy [10]. An efficient approximation of the Gaussian pdf is therefore required. A series of nonlinear Kalman filters based on deterministic sampling strategy are proposed, for example, the unscented Kalman filter (UKF) [10,15,16], the cubature Kalman filter (CKF) [4,17-19], the Gauss-Hermite quadrature filter (GHQF) [3], and the sparse grid quadrature filter (SGQF) [20]. For non-Gaussian noise environment, for example, impulse noise, the maximum correntropy Kalman filter (MCKF) [8] utilizes the maximum correntropy criterion (MCC) to combat heavy-tailed noises.

From the numerical integration perspective, these afore mentioned nonlinear Kalman filters based on approximation of the Gaussian pdf are only different in numerical integration methods that are utilized to calculate the Gaussian weighted integrals in the filtering framework. An efficient numerical integral rule is thus required. Based on the spherical simplex radial cubature rule [21,22], a new class of nonlinear Kalman filters were proposed, that is, the third-degree SSRCKF and the fifth-degree SSRCKF. The SSRCKF can obtain a better filtering performance in comparison with the conventional CKF [22]. In this paper, a novel mixed-degree spherical simplex-radial cubature Kalman filter (MSSRCKF), which combines the third-degree spherical simplex rule and the fifth-degree radial rule, is proposed. The proposed MSSRCKF can improve filtering accuracy effectively at the cost of slightly increasing computational complexity.

2. Numerical Integration Based on Bayesian Filtering Framework

The general discrete-time nonlinear dynamic systems can be described as

[x.sub.k] = g ([x.sub.k-1]) + [q.sub.k-1], [z.sub.k] = h ([x.sub.k]) + [r.sub.k]>

where [x.sub.k] [member of] [R.sup.n] denotes the state; [z.sub.k] [member of] [R.sup.m] denotes the measurement; [q.sub.k-1] and [r.sub.k] are both Gaussian noises with zero means and covariances [Q.sub.k-1] and [R.sub.k], respectively.

Based on (1), under the Gaussian assumption, the Bayesian filter can be divided into the following two steps: prediction and correction.

Prediction

[mathematical expression not reproducible] (2)

where [mathematical expression not reproducible] represents the Gaussian distribution with the mean [x.sub.k-i] and the covariance [P.sub.k-1].

Correction

[mathematical expression not reproducible] (3)

where

[mathematical expression not reproducible] (4)

From (2) to (4), the Bayesian filter is generalized as the calculation of the following integral:

[mathematical expression not reproducible], (5)

where f(*) denotes arbitrary nonlinear function.

Generally, it is intractable for directly calculating the integral of (5). Hence, we can obtain an approximated expression by utilizing the numerical integration theory; that is,

[mathematical expression not reproducible] (6)

where L is the total number of points[[xi].sub.1] represents the quadrature point and [w.sub.i] is the corresponding weight.

2.1. Spherical-Radial Transformation. For simplicity, let Q(f) = ([[summation].sub.i])[w.sub.i]f([[xi].sub.i]) denote the numerical integration. Q(f) can be regarded as a dth-degree rule of 7(f), if it is exact for [mathematical expression not reproducible] d and is not exact for at least one polynomial of degree [[summation].sup.n.sub.i=1] [d.sub.i] [less than or equal to] [d.sub.i] < d +1 [4,19, 22]. Consider the following integral:

[mathematical expression not reproducible]. (7)

Let x = ry with [y.sup.T]y = 0, where r = [square root of [square root of [x.sup.T]x]. It is difficult to directly find numerical approximation of (7). Therefore, an integral transformation is required and (7) is transformed into the following spherical-radial coordinate integral.

[mathematical expression not reproducible], (8)

where [U.sub.n] = {y [member of] [R.sup.n] | [y.sup.T]y = 1} and [sigma](*) represent the spherical surface and surface measure of the sphere, respectively [4].

According to the spherical-radial transformation, (8) can be decomposed into the spherical integral [mathematical expression not reproducible] and the radial integral [mathematical expression not reproducible] Assume that S(r) and R can be approximated by the following numerical integration:

[mathematical expression not reproducible], (9)

[mathematical expression not reproducible] (10)

where y and [w.sub.s,i] denote the quadrature point and the corresponding weight of S(r); [r.sub.j] and [w.sub.r,j] denote the quadrature point and the corresponding weight of R; [L.sub.s] and [L.sub.r] are the number of the points, respectively.

Therefore, applying (9) and (10) into (8) generates the following approximation:

[mathematical expression not reproducible] (11)

2.2. Spherical Simplex Rule. The aforementioned spherical integration S(r) in (9) can be efficiently calculated using the transformation group of the regular n-simplex [21, 22] with the vector [a.sub.i] = [[[a.sub.i,1], [a.sub.i,2], ..., [a.sub.i,n]].sup.T], i = 1, 2, ..., n + 1 as follows:

[mathematical expression not reproducible] (12)

The projection from the midpoints of the vector a; on the spherical sphere leads to the following points [21, 22]:

[mathematical expression not reproducible] (13)

For example, when the dimension n = 2, we can get that [mathematical expression not reproducible].

Employing the central symmetry of the cubature formula and just treating the points [a.sub.i] and [b.sub.i] of the cubature points as [Y.sub.i] in (9), we can obtain the third-degree spherical simplex rule with 2n + 2 points as [22]

[mathematical expression not reproducible] (14)

and the fifth-degree spherical simplex rule with [n.sup.2] + 3n + 2 points as [22]

[mathematical expression not reproducible]

where [A.sub.n] = [2[GAMMA].sup.n] (1/2)/[GAMMA](n/2) = 2 [square root of [[pi].sup.n]/[GAMMA](n/2) denotes the surface area of the sphere; the Gamma function is defined as [mathematical expression not reproducible] with the properties of [GAMMA] (1/2) = [square root of [pi]] and [[GAMMA](n + 1) = n[GAMMA](n).

2.3. Radial Rule. The radial integral in (10) can be approximated by

[mathematical expression not reproducible] (16)

Substituting S(r) with the Zth-degree monomial [r.sup.l], we obtain

[mathematical expression not reproducible] (17)

where l represents an even integer.

Since the resultant spherical simplex-radial cubature rule is fully symmetric, we only need to match the even degree monomials. Matching different even-degree monomials yields different quadrature points and weights. The third degree radial rule [4,19, 22] can be expressed as

R (3) = 1/2 [GAMMA] (n/2) S ([square root of n/2]). (18)

Similarly, the fifth-degree radial rule [22] with [L.sub.r] = 2 can be obtained by

[mathematical expression not reproducible]. (19)

2.4. Conventional Spherical Simplex-Radial Rule. Based on (11), the spherical simplex-radial rule for the standard Gaussian distribution N(x; 0, [I.sub.n]) can be obtained as

[mathematical expression not reproducible] (20)

For the calculation of the multidimensional Gaussian distribution N(x; [bar.x], P), a linear transformation of [square root of P[[xi].sub.i]] + [bar.x] is required [22], where [[xi].sub.i] is the final quadrature point combining the spherical simplex rule with the radial rule, and [square root of P] denotes the square root matrix of [mathematical expression not reproducible].

Conventional spherical simplex-radial cubature Kalman filters are all based on the same dth-degree spherical simplex rule and radial rule, respectively [22]. The degree for the spherical simplex-radial rule is thus up to the dth-degree, for example, the third-degree spherical simplex-radial rule (3SSR) and the fifth-degree spherical simplex-radial rule (5SSR). However, it is interesting to note that the spherical simplex rule or radial rule with higher than dth-degree can also be used to achieve the dth-degree accuracy for integration. Hence, the degrees of the spherical simplex rule and radial rule are not necessarily required to be the same [19].

2.5. Mixed-Degree Spherical Simplex-Radial Cubature Kalman Filter. To improve filtering accuracy and reduce computational complexity, a novel mixed-degree spherical simplex radial rule (MSSR), namely, the third-degree spherical simplex rule and the fifth-degree radial rule, is proposed in this paper.

Combining (14) and (19), we can get the MSSR as

[mathematical expression not reproducible] (21)

The quadrature points and weights based on the MSSR are therefore denoted by

[mathematical expression not reproducible] (22)

Substituting the numerical integration of (21) into the Bayesian filter framework from (2) to (4) generates the mixed-degree spherical simplex-radial cubature Kalman fil ter (MSSRCKF), which is summarized in Algorithm 1.

Algorithm 1 (mixed-degree spherical simplex-radial cubature Kalman filter).

(I) Initialization. Set the initial state estimate value x0 and the covariance [P.sub.0].

(II) Prediction

(1) Evaluate the sampling points [[xi].sub.i,k-1] and [w.sub.i] by (22), where [mathematical expression not reproducible].

(2) Calculate the points propagating through the state process equation

[mathematical expression not reproducible]. (23)

(3) Estimate the predicted state value

[mathematical expression not reproducible] (24)

and the predicted error covariance matrix

[mathematical expression not reproducible] (25)

(III) Correction

(4) Regenerate the points [[xi].sub.i,k|k-1] and by (22), where [mathematical expression not reproducible].

(5) Calculate the propagated points

[mathematical expression not reproducible]. (26)

(6) Estimate the predicted measurement value

[mathematical expression not reproducible]. (27)

(7) Estimate the cross covariance matrix and the innovation covariance matrix

[mathematical expression not reproducible] (28)

(8) Calculate the Kalman gain

Gk = [P.sub.xz] [P.sup-1.sub.zz]. (29)

(9) Update the state estimate and the corresponding error covariance matrix

[mathematical expression not reproducible] (30)

Remark 2. In comparison with the 3SSRCKF [22] with 2n + 2 points, the proposed MSSRCKF with 2n + 3 points can achieve better filtering accuracy at the cost of almost the same computational complexity, which can be clearly seen in Figure 1. In comparison with the 5SSR, the MSSR with positive weights preferably achieves the "good" integral rule as summarized in [4]. Therefore, the MSSRCKF can obtain better filtering accuracy than the 5SSRCKF, which is shown in the following examples. In addition, the MSSRCKF with 2n+3 requires less computational complexity than the 5SSRCKF with [n.sup.2] + 3n +3 points.

Remark 3. The other combination of the fifth-degree spherical simplex rule and third-degree radial rule, which requires [n.sup.2] + 3n + 2 quadrature points, has a relatively higher computational complexity. Similar to the 5SSRCKF, this mixed order combination may introduce negative weights, which may lead to the filter instability in some high-dimensional systems. Hence, only the MSSRCKF that combines the third degree spherical simplex rule and fifth-degree radial rule is presented in this paper.

3. Accuracy Analysis

The filtering performance of nonlinear Kalman filter is directly decided by the accuracy of state estimates [1, 18]. In this section, we will give the accuracy analysis about the proposed MSSR from the aspects of the estimated mean and its covariance of nonlinear function, which shows the filtering performance of the MSSRCKF theoretically.

3.1. Accuracy in Estimating the Mean of a Nonlinear Function. Assume that a small disturbance [increment of x] with zero mean and covariance P is introduced around the mean x. The Taylor series of a nonlinear function y = f (x) can be expressed as [1]

[mathematical expression not reproducible] (31)

where the operator [mathematical expression not reproducible] is the differential of f(*) regarding the mean x, which can also be denoted using the Jacobian matrix of f (*); that is, [D.sub.[increment of x]]f = F[increment of x].

Thus, the kth term in the Taylor series is given as

[mathematical expression not reproducible]. (32)

Taking expectations of (31), we can get that

[mathematical expression not reproducible] (33)

Owing to the symmetry of the Gaussian distribution, all the odd terms in the above Taylor series are zero. Therefore, (33) can be rewritten by

[mathematical expression not reproducible] (34)

where the second term can be transformed as

[mathematical expression not reproducible] (35)

Further, since E[[increment of x][DELTA][x.sup.T]] = P, we have

[mathematical expression not reproducible] (36)

Substituting (36) into (34), the true mean of nonlinear function can be expressed by

[mathematical expression not reproducible] (37)

In (22), for simplicity, let [[delta].sub.i] = [[square root of Pa].sub.i]] and [[sigma].sub.i] = [square root of n/2[[delta].sub.i]. The propagation of the ith point through the nonlinear transformation as a Taylor expansion about [bar.x] can be written as

[mathematical expression not reproducible] (38)

According to (24), the mean of the estimated nonlinear transformation can be expressed by

[mathematical expression not reproducible] (39)

Since the odd moments for the points symmetrically distributed around [bar.x] are zero, we can obtain

[mathematical expression not reproducible] (40)

After denoting (12) by the matrix form of a with some mathematical operation, we have [aa.sup.T] = ((n + 1)/n)[I.sub.n]. The second-order term in (39) can be written as

[mathematical expression not reproducible] (41)

Therefore, (40) can be rewritten as

[mathematical expression not reproducible] (42)

Comparing (42) with (37), we can see that the mean estimated by the MSSRCKF agrees exactly with the true mean up to the third order with errors introduced in the fourth-order and higher order moments. Note that the lower moments are significant in practical applications, which have direct influences on filtering accuracy.

3.2. Accuracy in Estimating the Covariance of a Nonlinear Function. Given the true mean [bar.y].sub.true], the true covariance matrix can be calculated by

[mathematical expression not reproducible]. (43)

Combining (31) and (34), we can obtain that

[mathematical expression not reproducible] (44)

Substituting (44) into (43) and employing the symmetry of the vector [increment of x], the expected values of all odd ordered terms will be zero. Hence, the true covariance matrix can be expressed as

[mathematical expression not reproducible] (45)

Since [mathematical expression not reproducible], (45) can be rewritten as

[mathematical expression not reproducible] (46)

Given the mean of the estimated nonlinear transformation in (39), the estimated covariance matrix of the proposed MSSR is calculated by

[mathematical expression not reproducible] (47)

where

[mathematical expression not reproducible] (48)

Since

[mathematical expression not reproducible] (49)

the estimated covariance matrix (47) can be simplified as follows:

[mathematical expression not reproducible] (50)

Comparing (50) with the true covariance matrix (46), we can find that the estimated covariance matrix obtained by the MSSRCKF agrees exactly with the true covariance matrix up to the second order.

4. Simulations

In this section, two examples are presented to demonstrate the superiority of the proposed MSSRCKF. The simulation results were obtained in MATLAB R2011b using a computer with processor Inter(R) Core(TM)i5-4570 CPU @ 3.20 GHz. For better comparison, the following root mean square error (RMSE) is firstly defined:

[mathematical expression not reproducible] (51)

where M denotes the total Monte Carlo runs and M = 100 is used here; the subscript j represents the jth state, [x.sup.i.sub.j,k] and [mathematical expression not reproducible] are the real state and its corresponding estimated value at time instant k in the ith Monte Carlo run, respectively. The compared nonlinear Kalman filters are the 3SSRCKF and 5SSRCKF [22].

Example 1. Consider the following two-dimensional nonlin ear stochastic system:

[mathematical expression not reproducible] (52)

where [tau] = 0.001.

The real initial values and their corresponding estimates are set to [mathematical expression not reproducible], respectively. The other parameters are set as [Q.sub.k-1] = [0.003.sup.2][I.sub.2x2] and [R.sub.k] = [0.001.sup.2]. For all the filters, the initial covariance matrix is given by [P.sub.0] = [I.sub.2x2].

Table 1 shows the compared filtering accuracy of all the filters. From Table 1, we can see that the proposed MSSRCKF can obtain the best filtering performance among the three filters. In addition, it is interesting to note that the MSSR CKF with only one more point than the 3SSRCKF requires less computational burden than the 5SSRCKF. Hence, the proposed MSSRCKF with less computational complexity can implement performance improvement of the 5SSRCKF.

Example 2. The well-known vertically falling body model is utilized in this section to further demonstrate the efficiency of the proposed MSSRCKF. The complete system equations are given as follows [23]:

[mathematical expression not reproducible] (53)

where the parameter a is a known constant; [x.sub.1](f), [x.sub.2](f), and [x.sub.3](f) are the altitude, the velocity, and the ballistic coefficient in turn, all of which can be estimated using the radar range measurement. First, the fourth-order Runge-Kutta method [24] is used to discretize the continuous dynamic system (53). The discrete-time step size is 1/64 seconds, and the sampling interval is 1 second.

The discrete-time range measurement at time instant k is expressed by

[mathematical expression not reproducible] (54)

where [D.sub.1] denotes the horizontal distance between the falling target and the radar; [D.sub.2] is the altitude of the radar.

The real initial state values and the initial estimates are set As [mathematical expression not reproducible], respectively. The other parameters are given as follows: [alpha] = 5 x [10.sup.-5], [D.sub.1] = [D.sub.2] = 1 x [10.sup.5], [Q.sub.k-1] = 0, and [R.sub.k] = 1 x [10.sup.4]. For all the filters, the initial covariance matrix is given by [P.sub.0] = diag([1 x [10.sup.6],4 x [10.sup.6],1 x [10.sup.-4]]).

At the beginning, the altitude of the target is high and the change of the velocity can be ignored, which leads to [mathematical expression not reproducible]; the system can be considered as a linear one. When

the target approaches the radar, the sensibility of the radar to the observation noise becomes more and more sensitive, and the nonlinearity degree of the system becomes higher. The tracking results of the altitude based on three filters are shown in Figure 2. We can clearly see from Figure 2 that the MSSRCKF always achieves the lowest RMSE among the three filters.

To further compare the filters, we also give the mean values of RMSE and the mean consumed time in Table 2. It can be seen that the proposed MSSRCKF requiring almost the same time as 3SSRCKF achieves the smallest mean value of RMSE. The same conclusions as Example 1 can be obtained in Example 2.

5. Conclusion

In this paper, the MSSRCKF is proposed to solve nonlinear estimation problems. The accuracy analysis shows that the mean estimation obtained by the MSSRCKF is accurate to the third-order moment of the true mean. In addition, the covariance estimated by the MSSRCKF agrees accurately with the true covariance matrix up to the second-order moment. Simulations show that the proposed MSSRCKF outperforms the conventional low degree SSRCKF at the cost of slightly increasing computational burden. It is also interesting to note that the MSSRCKF can achieve almost the same or even better filtering performance than the 5SSRCKF that may have the stability problem in high-dimensional nonlinear system. This is beyond our intuition, because the numerical integral rule of the 5SSRCKF is higher than that of the MSSRCKF. The theoretical analysis on this phenomenon is our future work.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (nos. 61671389, 61672436, 61372139, and 61571372), Fundamental and Frontier Research Project of Chongqing (no. cstc2014jcyjA40020), China Postdoctoral Science Foundation Funded Project (no. 2016M590853), and Fundamental Research Funds for the Central Universities (nos. XDJK2014B001, XDJK2016E029).

https://doi.org/10.1155/2017/6969453

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Shiyuan Wang, (1,2) Yali Feng, (1,2) Shukai Duan, (1,2) and Lidan Wang (1,2)

(1) College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

(2) Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China

Correspondence should be addressed to Shukai Duan; duansk@swu.edu.cn

Received 1 September 2016; Accepted 20 February 2017; Published 19 March 2017

Caption: Figure 1: Comparison of the number of points required by different filters versus dimension.

Caption: Figure 2: RMSE of all the filters about the target falling altitude.
Table 1: Comparison of the mean values of RMSE
about state x1 among all the filters.

Filters   Mean values    Time (s)       Points
          of RMSE

3SSRCKF   0.36675522     0.12705626     2n + 2
5SSRCKF   0.36545063     0.23100751     [n.sup.2] + 3n + 3
MSSRCKF   0.36545053     0.13303506     2n + 3

Table 2: Filtering performance comparison
about the target falling altitude.

Filters   Mean values    Time (s)       Points
          of RMSE

3SSRCKF   393.56981130   0.00816492     2n + 2
5SSRCKF   358.05064678   0.01639527     [n.sup.2] + 3n + 3
MSSRCKF   336.41120229   0.00939589     2n + 3
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Title Annotation:Research Article
Author:Wang, Shiyuan; Feng, Yali; Duan, Shukai; Wang, Lidan
Publication:Mathematical Problems in Engineering
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Date:Jan 1, 2017
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