This paper evaluates the state estimation performance for processing nonlinear/non-Gaussian systems using the cubature particle filter (CPF), which is an estimation algorithm that combines the cubature Kalman filter (CKF) and the particle filter (PF). The CPF is essentially a realization of PF where the third-degree cubature rule based on numerical integration method is adopted to approximate the proposal distribution. It is beneficial where the CKF is used to generate the importance density function in the PF framework for effectively resolving the nonlinear/non-Gaussian problems. Based on the spherical-radial transformation to generate an even number of equally weighted cubature points, the CKF uses cubature points with the same weights through the spherical-radial integration rule and employs an analytical probability density function (pdf) to capture the mean and covariance of the posterior distribution using the total probability theorem and subsequently uses the measurement to update with Bayes’ rule. It is capable of acquiring a maximum a posteriori probability estimate of the nonlinear system, and thus the importance density function can be used to approximate the true posterior density distribution. In Bayesian filtering, the nonlinear filter performs well when all conditional densities are assumed Gaussian. When applied to the nonlinear/non-Gaussian distribution systems, the CPF algorithm can remarkably improve the estimation accuracy as compared to the other particle filter-based approaches, such as the extended particle filter (EPF), and unscented particle filter (UPF), and also the Kalman filter (KF)-type approaches, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF) and CKF. Two illustrative examples are presented showing that the CPF achieves better performance as compared to the other approaches.

State estimation for the dynamic system [

The nonlinear state-space method is convenient for handing multivariate data and nonlinear/non-Gaussian processes, and it provides a significant advantage over time-series approach for state estimation problems [

One example of the several approximate methods that have been proposed is the extended Kamlan filter (EKF) [

Nevertheless, the UKF-calculated estimation covariance matrix is not always guaranteed to be positive definite, and thus decomposition of the covariance matrix is sometimes unavailable. The UKF is likely to become unstable due to the possible negative weights on the center point for high-dimensional nonlinear systems. To overcome these limitations, a nonlinear filter based on the Bayesian framework, commonly referred to as the cubature Kalman filter (CKF) proposed by Arasaratnam et al. [

When the non-linearity and non-Gaussianity are highly prominent, the Kalman filter (KF)-type approaches (e.g., EKF, UKF and CKF discussed in this paper) assume the noise to be Gaussian distribution, which does not provide a good approximation to the posterior distribution. Proposed to approximate the posterior distribution of states through sequential importance sampling (SIS), the particle filter (PF) is a non-parametric filter and hence can easily deal with nonlinear and/or non-Gaussian state estimation [

This paper is organized as follows. In Section 2, preliminary background on the Kalman filter-type approaches is briefly reviewed and successively presents the Bayesian solution for nonlinear/non-Gaussian state estimation problems. The various types of particle filters including the EPF, UPF and CPF are discussed in Section 3. In Section 4, two illustrative examples are presented for assessment of nonlinear/non-Gaussian state estimation capabilities using the CPF algorithm in comparison to those by to the relatively conventional approaches. Conclusions are given in Section 5.

The well-known KF is an optimal closed-form solution in linear systems with Gaussian process and measurement noises. In nonlinear systems, the optimal estimation solution to the recursive Bayesian filtering problem is infinite dimensional and computationally intractable. The most widely used approximate nonlinear filter is the EKF.

The nonlinear system governed by the stochastic difference equations can be written as:

where the state vector

where

From the Bayesian perspective, the state estimation problem is required to construct the probability density function

These equations can be derived using the Markov property and Bayes’ rule from probability theory.

The EKF is an approximate nonlinear filter which linearizes the dynamic system and measurement equations about a single sample point with the assumption that the a priori distributions are Gaussian. The state distribution of the EKF is approximated by a Gaussian random variable (GRV), which is then propagated analytically through the linearization of the nonlinear system. The EKF might suffer from the performance degradation and divergence problem due to the linearization process for the system nonlinearity. To better treat the nonlinearity, other filters such as the UKF were proposed. Unlike the EKF with first-order accuracy where the linearization process using the Jacobian matrices is involved, the UKF employs a minimal set of sigma points (weighted samples) by deterministic sampling approach and at least the second order accuracy of the posterior mean and covariance can be captured.

Consider an

where

The first step in the UKF is to sample the prior state distribution by generating the sigma points through the UT. A set of weighted samples (sigma points) are deterministically chosen to adequately capture the true mean and covariance of the random variable. The basic premise is that to approximate a probability distribution is easier than to approximate an arbitrary nonlinear transformation. The samples are propagated through true nonlinear equations, and the linearization of the model is not required. The UKF requires less computational cost due to deterministic sampling of the sigma points as opposed to the randomly generated particles in the particle filter.

Not guaranteed always to be positive definite, the decomposition of the covariance matrix in the UKF is sometimes unavailable. Proposed by Arasaratnam et al. [

From the perspective of numerical analysis, the third-degree spherical-radial cubature rule can be viewed as an UT of special form with better numerical stability. The CKF is known as the approximate filter in the sense of completely preserving second-order information due to the maximum entropy principle and thus provides an efficient solution even for high-dimensional nonlinear filtering problems. For improving numerical accuracy in nonlinear system, the CKF is reformulated to propagate the square roots of the error-covariance matrices, and hence it avoids computing numerically sensitive matrix calculations. In contrast to UKF, the CKF follows directly from the cubature rule for numerically computing Gaussian-weighted integrals whose important property is that it does not entail any free parameters, whereas the UKF introduces a nonzero scaling parameter.

The CKF algorithm involves the following stages: Firstly, it approximates the mean and variance of the probability distribution through a set of 2_{i}

where

The CKF also involves a two stage procedure comprising of prediction step and update step. Under the assumption that the posterior density at time

In contrast to KF-type approaches, the PF was presented for handling multimodal probability density functions and solving nonlinear non-Gaussian problems [

The sequential importance sampling (SIS) is one of the methods which enable the Bayesian estimation by Monte Carlo simulation. The principle of SIS uses the samples with weights to approximate the posterior

Let

The recursive estimate for the importance weights of particle

where

the estimated state vector can then be approximated by

In general, the PF relies on the sequential importance sampling and requires the design of proposal distributions used to approximate the posterior distribution reasonably well. As can be seen from

The importance density function is used in the SIS scheme, where the transition prior does not take into consideration the most recent measurement data

where

It should be noticed that the EKF tends to underestimates the true covariance of the state in highly nonlinear systems. This violates the distribution support requirement for the proposal distribution and may lead to poor performance and even filter divergence. It provides a good alternative for propagating the mean and covariance of the Gaussian approximation to the state distribution.

The CPF introduces the CKF into the PF framework for generating the importance density function, so as to closely match the true posterior density by integrating the latest observation information. In other words, the CPF considers the recent measurement, such as the recent capacity degradation data, to iteratively update the weights of the random particles used in the PF framework.

To construct the proposed distribution function of the PF, the CPF uses the current measurement information and cubature points to calculate the mean and the variance of the nonlinear random function directly by setting a defined group of sample points and corresponding weights. Theoretically, The CPF requires fewer cubature points than the UPF when generating the importance proposal distribution, thus requires less computational overheads. Furthermore, the CPF uses the square root of the error covariance for iterating and possesses better stability and accuracy performance. Implementation algorithm for the PF-based approaches: EPF, UPF, and CPF, is provided in

To assess the efficiency of the state estimation using the CPF algorithm in comparison with those of UPF, EPF and KF-type approaches, two illustrative examples are adopted for demonstrating the effectiveness under nonlinear/non-Gaussian environments. The two examples presented includes the univariate nonstationary growth model (UNGM) and the ballistic target tracking. Both have significant nonlinearity and have been extensively investigated in the literature as the benchmark problems [

The UNGM is important in econometrics and has been used as a benchmark for validating and comparing nonlinear filters. Its high nonlinearity and bimodality makes filtering a difficult task. The UNGM dynamic process model is given by

and the measurement equation is

where the process noise _{k −1} is a zero mean Gaussian random variable with variance _{k}_{k}_{k}

or equivalently,

where _{0} = 0.1 and the total number of measurements

Due to the errors induced by arithmetic operations performed on finite word-length computers, the basic properties of an error covariance matrix of UT, i.e., symmetry and positive definiteness, are not always guaranteed to hold. However, the cubature-based nonlinear filter essentially propagates the square-root of the predictive and posterior error covariance, which possesses the ability to preserve symmetry and positive definite, and thereby improves the numerical accuracy and stability. The result shows that the performance of CPF is superior to UPF solutions when same numbers of particles are used, since the proposal distribution based on CKF taken into approximate the true posterior distribution is more precise than UKF. The CKF is developed using the spherical-radial rule, which is more accurate than the Gaussian quadrature rule involved in UT.

In the second example, application of the nonlinear filters to the target tracking problem using the range measurement is performed. The altitude, velocity and constant ballistic coefficient of a vertically falling body are estimated. The geometry for the ballistic target tracking using ground radar for the benchmark problem is illustrated in

The dynamic process model of this nonlinear system is given by

where the sequence

where

The results are shown from

Filter | Execution time (s) |
---|---|

EKF | 0.01033 |

UKF | 0.07916 |

CKF | 0.04517 |

EPF | 4.09327 |

UPF | 12.77373 |

CPF | 10.30431 |

The CKF demonstrates noticeable improvement over the EKF and UKF while the EPF slightly outperforms the CKF. The particles degeneracy can be attributed to the measurement of highly non-Gaussian noise. The measurement noise should have a relatively heavy tail so that it is insensitive to the outliers. To determine the number of particles are important for the sequential importance sampling, which depends on the importance distribution of particles. Therefore, an importance density tuned for a particular problem will yield an appropriate trade-off between the number of particles and the estimation accuracy.

The UKF introduces a non-zero scaling parameter, which defines the non-zero center point and is often associated with a set of weighted samples higher than that of the minimal set of sigma points. The CKF follows directly from the spherical-radial cubature rule for numerically computing Gaussian-weighted integrals with the property without entailing free parameters. Although additional tuning on parameters in UKF provides flexibility, one can fix them as their default values or just exclude the center point and the CKF is automatically obtained, if bothering to tune them. The CKF in this work is based on the third-degree spherical-radial cubature rule to propagate the cubature points through the nonlinear functions, so as to solve the integration in Bayesian filtering problem for numerically computing the multivariate moment integrals, which are numerically computed by the spherical cubature rule and the Gaussian quadrature rule, respectively.

The filtering performance of the PF-based approaches (namely CPF, UPF and EPF) outperforms the KF-type approaches (namely CKF, UKF, and EKF) due to consideration of the latest observations.

This paper provides profound insight into the estimation performance of CPF for nonlinear/non-Gaussian processes. Assessment of the nonlinear filtering approaches to the nonlinear/non-Gaussian state estimation performance has been carried out. The CPF algorithm possesses the merits of the PF framework to handle non-Gaussian errors and the CKF can deal with the nonlinearity with better numerical stability to improve the estimation accuracy. In a CPF, the CKF is used to generate the importance proposal distribution of the PF. The CKF employs third-degree spherical-radial cubature rule to solve the integration in Bayesian filtering problem for numerically computing the multivariate moment integrals encountered. By integrating the latest observation information and approximating the posterior distribution, the CKF performance is improved. Furthermore, the CKF will facilitate selection of importance sampling in practice that is useful to effectively alleviate the degeneracy and impoverishment problems in the PF.

To improve the stability of the nonlinear filter, the CKF can effectively avoid round-off errors of numerical computation, and possesses better stability than the UKF and EKF. The spherical-radial cubature rule employed in CKF is a special case of the quadrature rules involved in UKF. Namely, if the parameters of the UPF are well-tuned, the estimation performance of the UPF and the CPF will be similar or identical. Although the CKF can be treated as a special case of the UKF, CKF is, in general, considered to be more accurate and stable than the UKF in nonlinear filtering realization without additional tuning on parameters as in the UKF. The result shows that the performance of CPF is superior to UPF solutions when same numbers of particles are used, since the proposal distribution based on CKF taken into approximate the true posterior distribution is more precise than UKF.

To assess the performance of various estimation algorithms, two illustrative examples are presented, especially for the cases under nonlinear/non-Gaussian environments. Performance comparisons on the KF-type approaches: EKF, UKF, CKF and PF-based approaches: EPF, UPF, CPF have been presented. Simulation results show that the CPF algorithm possesses superior performance than the KF-type approaches and other PF-based approaches, among which the CPF algorithm shows superior estimation accuracy with less computational cost than the UPF. The performance in terms of estimation accuracy, numerical stability and computational costs can be improved. Developed to deal with nonlinear and/or non-Gaussian distribution assumptions, the CPF algorithm possesses good potential as the alternative for the nonlinear and/or non-Gaussian state estimation.