Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6969453, 9 pages

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

## Mixed-Degree Spherical Simplex-Radial Cubature Kalman Filter

^{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; nc.ude.uws@ksnaud

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

Academic Editor: Bo Shen

Copyright © 2017 Shiyuan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Conventional low degree spherical simplex-radial cubature Kalman filters often generate low filtering accuracy or even diverge for handling highly nonlinear systems. The high-degree Kalman filters can improve filtering accuracy at the cost of increasing computational complexity; nevertheless their stability will be influenced by the negative weights existing in the high-dimensional systems. To efficiently improve filtering accuracy and stability, a novel* mixed-degree spherical simplex-radial cubature Kalman filter* (MSSRCKF) is proposed in this paper. The accuracy analysis shows that the true posterior mean and covariance calculated by the proposed MSSRCKF can agree accurately with the third-order moment and the second-order moment, respectively. Simulation results show that, in comparison with the conventional spherical simplex-radial cubature Kalman filters that are based on the same degrees, the proposed MSSRCKF can perform superior results from the aspects of filtering accuracy and computational complexity.

#### 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* ×* 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 aforementioned 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 aswhere denotes the state; denotes the measurement; and are both Gaussian noises with zero means and covariances and , respectively.

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

*Prediction*where represents the Gaussian distribution with the mean and the covariance .

*Correction*where

From (2) to (4), the Bayesian filter is generalized as the calculation of the following integral:where 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,where is the total number of points; represents the quadrature point and is the corresponding weight.

##### 2.1. Spherical-Radial Transformation

For simplicity, let denote the numerical integration. can be regarded as a* d*th-degree rule of , if it is exact for with and is not exact for at least one polynomial of degree [4, 19, 22].

Consider the following integral:

Let with , where . 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.where and 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 and the radial integral . Assume that and can be approximated by the following numerical integration:where and denote the quadrature point and the corresponding weight of ; and denote the quadrature point and the corresponding weight of ; and are the number of the points, respectively.

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

##### 2.2. Spherical Simplex Rule

The aforementioned spherical integration in (9) can be efficiently calculated using the transformation group of the regular -simplex [21, 22] with the vector , as follows:

The projection from the midpoints of the vector on the spherical sphere leads to the following points [21, 22]:For example, when the dimension , we can get that and .

Employing the central symmetry of the cubature formula and just treating the points and of the cubature points as in (9), we can obtain the third-degree spherical simplex rule with points as [22]and the fifth-degree spherical simplex rule with points as [22]where denotes the surface area of the sphere; the Gamma function is defined as with the properties of and .

##### 2.3. Radial Rule

The radial integral in (10) can be approximated bySubstituting with the th-degree monomial , we obtainwhere 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

Similarly, the fifth-degree radial rule [22] with can be obtained by

##### 2.4. Conventional Spherical Simplex-Radial Rule

Based on (11), the spherical simplex-radial rule for the standard Gaussian distribution can be obtained as

For the calculation of the multidimensional Gaussian distribution , a linear transformation of is required [22], where is the final quadrature point combining the spherical simplex rule with the radial rule, and denotes the square root matrix of ; that is, .

Conventional spherical simplex-radial cubature Kalman filters are all based on the same th-degree spherical simplex rule and radial rule, respectively [22]. The degree for the spherical simplex-radial rule is thus up to the th-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 th-degree can also be used to achieve the th-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

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

Substituting the numerical integration of (21) into the Bayesian filter framework from (2) to (4) generates the mixed-degree spherical simplex-radial cubature Kalman filter (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 and the covariance .*(II) Prediction*

(1) Evaluate the sampling points and by (22), where , and .

(2) Calculate the points propagating through the state process equation(3) Estimate the predicted state value and the predicted error covariance matrix*(III) Correction*

(4) Regenerate the points and by (22), where , and .

(5) Calculate the propagated points(6) Estimate the predicted measurement value(7) Estimate the cross covariance matrix and the innovation covariance matrix(8) Calculate the Kalman gain(9) Update the state estimate and the corresponding error covariance matrix

*Remark 2. *In comparison with the 3SSRCKF [22] with points, the proposed MSSRCKF with 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 requires less computational complexity than the 5SSRCKF with points.