Polynomial chaos Kalman filter for target tracking applications

IET Radar, Sonar & NavigationVolume 17, Issue 2 p. 247-260 ORIGINAL RESEARCHOpen Access Polynomial chaos Kalman filter for target tracking applications Kundan Kumar, Kundan Kumar orcid.org/0000-0002-2977-8815 Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorRanjeet Kumar Tiwari, Ranjeet Kumar Tiwari Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorShovan Bhaumik, Shovan Bhaumik Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorParesh Date, Corresponding Author Paresh Date [email protected] Department of Mathematics, College of Engineering, Design and Physical Sciences, Brunel University London, London, UK Correspondence Paresh Date, Department of Mathematics, College of Engineering, Design and Physical Sciences, Brunel University London, London, UK. Email: [email protected].ac.ukSearch for more papers by this author Kundan Kumar, Kundan Kumar orcid.org/0000-0002-2977-8815 Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorRanjeet Kumar Tiwari, Ranjeet Kumar Tiwari Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorShovan Bhaumik, Shovan Bhaumik Department of Electrical Engineering, Indian Institute of Technology Patna, Patna, IndiaSearch for more papers by this authorParesh Date, Corresponding Author Paresh Date [email protected] Department of Mathematics, College of Engineering, Design and Physical Sciences, Brunel University London, London, UK Correspondence Paresh Date, Department of Mathematics, College of Engineering, Design and Physical Sciences, Brunel University London, London, UK. Email: [email protected]Search for more papers by this author First published: 17 October 2022 https://doi.org/10.1049/rsn2.12338AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this paper, an approximate Gaussian state estimator is developed based on generalised polynomial chaos expansion for target tracking applications. Motivated by the fact that calculating conditional moments in an approximate Gaussian filter involves computing integrals with respect to Gaussian density, the authors approximate the non-linear dynamics using polynomial chaos expansion. Second-order as well as third-order polynomial chaos expansions were used for approximate filtering, to derive the necessary recursive algorithm and also provide certain algebraic simplifications which reduce the computational burden without significantly affecting the filtering performance. Two comprehensive numerical experiments for multivariate systems, including one for a multi-model system, demonstrate the potential of the new algorithms. 1 INTRODUCTION Target tracking [1-4] is a state estimation problem to estimate the kinematics of a target, such as the position, the velocity and the turn rate, from noisy measurements received from the sensors. In target tracking problems, the process and measurement equations are usually non-linear, and a closed-form optimal estimate is not available. There is no single non-linear filter which clearly outperforms all other filtering algorithms for a range of tracking problems. Hence, the selection of filtering algorithm depends on a trade-off between estimation accuracy and computational complexity, which is specific to each application. Several filtering heuristics have been developed over time to address this. The first popular non-linear filter was the extended Kalman filter (EKF) [5, 6]. It is based on first-order Taylor series approximation of the process and measurement functions around the previous estimate and uses the standard recursive Kalman filter (KF) structure. However, for highly non-linear systems, the EKF performs poorly [5, 7]. Due to the limitations of the EKF, more advanced estimation techniques have been developed. In the literature, mainly two approaches are available for non-linear filtering. In the first one, the prior and the posterior probability density functions (PDFs) are approximated as Gaussian. Their mean and covariances are evaluated with a few deterministic sample points and the associated weights [8]. The cubature Kalman filter (CKF) [9], the interpolatory CKF (ICKF) [10], the embedded CKF (ECKF) [11], the unscented Kalman filter (UKF) [12, 13] and the Gauss-Hermite filter (GHF) [14], among others, are examples of such estimators. In another approach, the prior and posterior pdfs are approximated with a large number of random particles generated using importance sampling and their corresponding probability weights. It is popularly known as particle filter (PF) [15]. Although the PF and its variants can achieve high estimation accuracy, their computation demand is relatively higher than that of the first class of filters. This problem becomes worse with an increase in the system dimension. In this paper, we will employ polynomial chaos to arrive at Gaussian prior and posterior densities of the state estimate. Polynomial chaos theory was first introduced in Ref. [16], where Hermite polynomial was used to model a stochastic process with the help of Gaussian random variables. Later, the authors in Ref. [17] presented generalised chaos theory by using orthogonal polynomials for different continuous and discrete distributions. In Ref. [18], the chaos expansion is used to represent power flow response. The chaos expansion is reported to converge in L 2 ${\mathcal{L}}_{2}$ space for a general class of stochastic processes having finite second moments [17, 19]. Polynomial chaos-based square-root Kalman filter was developed in Ref. [20], where the coefficients are calculated from the ensemble Kalman filter and the filter gain is calculated from the coefficients. A few attempts have been made to develop an estimation method with the help of generalised polynomial chaos for a non-linear system adopting the EKF approach [21] and the ensemble Kalman filter approach [22]. Both the papers considered continuous stochastic processes rather than discrete time processes, which are (arguably) more realistic in target tracking applications with discretely sampled measurements. Further, for a highly non-linear system, the generalised polynomial chaos EKF (gPC-EKF) performs poorly [23]. A second-order polynomial chaos expansion of a multivariate non-linear function leads to terms of the form xixj. In addition to these terms, a third-order polynomial chaos expansion leads to terms of the form xixjxk. We call the terms in the expansion with i = j for second-order terms and i = j = k for third-order terms as 'alike terms' and any terms with i ≠ j are called as 'cross terms'. Since approximate Gaussian filters, such as the CKF, ICKF, ECKF and UKF, among others, involve computing conditional moments of the state with respect to Gaussian density, it is intuitively attractive to use a polynomial expansion of the underlying non-linear functions using Hermite polynomials (i.e. polynomial chaos expansion), which provide an optimal approximation in L 2 ${\mathcal{L}}_{2}$ sense. Recently, Xu et al. [23] used polynomial chaos expansion to approximate non-linear functions and subsequently derive an algorithm for estimating states of a discrete non-linear system. They considered only a second-order polynomial chaos expansion with alike terms. Further, they used noise covariances to distribute the collocation points (CPs) around the mean. This makes the algorithm sensitive with filter initialisation and estimation diverges in tracking the truth in many scenarios. In this paper, we also adopt the polynomial chaos expansion to approximate a non-linear function. Our specific contributions to Ref. [23] are as follows: (i) Ref. [23] considered only second-order polynomial chaos expansion with alike terms. In this paper, we develop an estimation algorithm using up to third-order polynomial chaos expansion. (ii) Further, we consider both alike terms and cross terms in both second-order and third-order expansion, and evaluate the effect of neglecting cross terms both empirically through comprehensive examples and in terms of flop counts. (iii) Ref. [23] used (2n + 1) number of CPs on the axis at a distance of 3 $\sqrt{3}$ from the origin. Instead of using such highly constrained CPs, we use generalised CPs, for example, as suggested in Refs. [24, 25]. We propose a structured procedure to choose CPs for polynomial chaos filtering and show that the proposed filters with generalised CPs achieve a high degree of accuracy and low track loss compared to existing approximate Gaussian filters. The developed filters are applied to two target tracking problems, where we have seen that the estimator derived in Ref. [23] provides a comparable results with the existing Gaussian filters such as the CKF [9], the ICKF [10] and the UKF [12, 13]. However, the proposed method provides more accurate tracking compared to Ref. [23] and other existing Gaussian filters at a modest increment in computational cost. The computational budget required for the proposed filter is studied in terms of flop counts and is compared with the existing filters. The rest of the paper is organised as follows. Section 2 presents the Gaussian approximated filter under the Bayesian framework. In Section 3, we first approximate the function using polynomial chaos expansion, and subsequently derive the polynomial chaos Kalman filter (PCKF). The computation complexity of the proposed PCKF is compared with the CKF and UKF in Section 4. The simulation results are presented in Section 5, followed by a brief discussion and conclusions. 2 BAYESIAN FRAMEWORK OF APPROXIMATE GAUSSIAN FILTERING The process and the measurement equation in discrete-time state space model are given as X k = f X k − 1 + η k − 1 , ${\mathcal{X}}_{k}=f\left({\mathcal{X}}_{k-1}\right)+{\eta }_{k-1},$ (1) Y k = h X k + ν k , ${\mathcal{Y}}_{k}=h\left({\mathcal{X}}_{k}\right)+{\nu }_{k},$ (2)where X k ∈ R n ${\mathcal{X}}_{k}\in {\mathbb{R}}^{n}$ and Y k ∈ R n y ${\mathcal{Y}}_{k}\in {\mathbb{R}}^{{n}_{y}}$ are the state and measurement vectors. f ( ⋅ ) : R n → R n $f(\cdot ):{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ and h ( ⋅ ) : R n → R n y $h(\cdot ):{\mathbb{R}}^{n}\to {\mathbb{R}}^{{n}_{y}}$ are process and measurement function, respectively. The process noise, ηk−1, and measurement noise, νk, are assumed to be Gaussian with mean zero and covariance matrices Qk−1 and Rk, respectively. It is also assumed that the initial state, X 0 ${\mathcal{X}}_{0}$ , the process and measurement noise are independent to each other. In Bayesian filtering, the state X k ${\mathcal{X}}_{k}$ is being estimated recursively using the measurement Y 1 : k ${\mathcal{Y}}_{1:k}$ in two steps: (i) prediction step (ii) update step. In the prediction step, using Chapman–Kolmogorov equation we construct the prior pdf p X k | Y 1 : k − 1 $p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right)$ from the knowledge of p X k − 1 | Y 1 : k − 1 $p\left({\mathcal{X}}_{k-1}\vert {\mathcal{Y}}_{1:k-1}\right)$ : p X k | Y 1 : k − 1 = ∫ p X k | X k − 1 p X k − 1 | Y 1 : k − 1 d X k − 1 . $p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right)=\int p\left({\mathcal{X}}_{k}\vert {\mathcal{X}}_{k-1}\right)p\left({\mathcal{X}}_{k-1}\vert {\mathcal{Y}}_{1:k-1}\right)d{\mathcal{X}}_{k-1}.$ (3) In the update step, Bayes' rule is used to construct the posterior density function p X k | Y 1 : k $p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k}\right)$ p X k | Y 1 : k = p Y k | X k p X k | Y 1 : k − 1 ∫ p Y k | X k p X k | Y 1 : k − 1 d X k . $p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k}\right)=\frac{p\left({\mathcal{Y}}_{k}\vert {\mathcal{X}}_{k}\right)p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right)}{\int p\left({\mathcal{Y}}_{k}\vert {\mathcal{X}}_{k}\right)p\left({\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right)d{\mathcal{X}}_{k}}.$ (4) For a linear system, the prior and posterior density functions (Equations 3-4) remain Gaussian, and the first two moments of the posterior density can be obtained recursively using the Kalman filter. For a non-linear system, the prior and posterior pdfs are no longer Gaussian. However, many approximate Gaussian filtering algorithms assume them to be Gaussian and calculate the mean vector and the covariance matrix recursively [6, Chap. 6, 9, 12, 14. The prior mean can be calculated as [9, 14, 26] X ˆ k | k − 1 = ∫ − ∞ ∞ f X k − 1 N X k − 1 ; X ˆ k − 1 | k − 1 , P k − 1 | k − 1 d X k − 1 . $\begin{array}{r}\hfill {\widehat{\mathcal{X}}}_{k\vert k-1}=\int \nolimits_{-\infty }^{\infty }f\left({\mathcal{X}}_{k-1}\right)\mathcal{N}\left({\mathcal{X}}_{k-1};{\widehat{\mathcal{X}}}_{k-1\vert k-1},{P}_{k-1\vert k-1}\right)d{\mathcal{X}}_{k-1}.\end{array}$ (5) The prior error covariance P k | k − 1 X X $\left({P}_{k\vert k-1}^{\mathcal{X}\mathcal{X}}\right)$ , the expected value of measurement Y ˆ k | k − 1 $\left({\widehat{\mathcal{Y}}}_{k\vert k-1}\right)$ , the covariance of measurement P k | k − 1 Y Y $\left({P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}\right)$ and the cross-covariance of state and measurement P k | k − 1 XY $\left({P}_{k\vert k-1}^{\mathcal{XY}}\right)$ can also be expressed in a similar way [9, 14]. For exact expressions, see Section 2 of Ref. [26]. After receiving the new measurement, Y k ${\mathcal{Y}}_{k}$ , we construct the posterior mean and the posterior covariance as X ˆ k | k = X ˆ k | k − 1 + K k Y k − Y ˆ k | k − 1 , ${\widehat{\mathcal{X}}}_{k\vert k}={\widehat{\mathcal{X}}}_{k\vert k-1}+{K}_{k}\left({\mathcal{Y}}_{k}-{\widehat{\mathcal{Y}}}_{k\vert k-1}\right),$ (6)and P k | k X X = P k | k − 1 X X − K k P k | k − 1 Y Y K k T , ${P}_{k\vert k}^{\mathcal{X}\mathcal{X}}={P}_{k\vert k-1}^{\mathcal{X}\mathcal{X}}-{K}_{k}{P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}{K}_{k}^{T},$ (7)where the Kalman gain (Kk) is expressed as K k = P k | k − 1 XY P k | k − 1 Y Y − 1 . ${K}_{k}={P}_{k\vert k-1}^{\mathcal{XY}}{\left({P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}\right)}^{-1}.$ (8) In general, for any arbitrary non-linear function, the integrals in estimating the conditional mean and the conditional covariance matrices are computationally intractable. In this paper, we use polynomial chaos expansion of the non-linear function to evaluate these integrals. 3 POLYNOMIAL CHAOS-BASED KALMAN FILTER The polynomial chaos expansion is a series expansion of Gaussian random variables in terms of Hermite polynomials [27, p. 44]. In the next subsection, we discuss the functional approximation using polynomial chaos expansion and develop the filtering algorithm in subsequent subsections. Approximation using polynomial chaos expansion Here, we approximate the transition function, f ( X ) $f(\mathcal{X})$ , where the state X $\mathcal{X}$ follows Gaussian distribution with mean X ˆ $\widehat{\mathcal{X}}$ and covariance P. It is important to note that the approximation is not generally on the mean value; rather, it is determined by the sample points. This point is made clearer later in the discussion, when we talk about the prior mean. At first, we transform X $\mathcal{X}$ into a standard Gaussian random variable, x, following X = X ˆ + S x $\mathcal{X}=\widehat{\mathcal{X}}+Sx$ , where P = SST. After the transformation, the function f ( X ) $f(\mathcal{X})$ becomes f X ˆ + S x $f\left(\widehat{\mathcal{X}}+Sx\right)$ , and its dth order chaos expansion can be written as [18, 28] f X ˆ + S x ≈ a 0 + ∑ i 1 = 1 n a i 1 H 1 x i 1 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 a i 1 i 2 H 2 x i 1 , x i 2 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 ∑ i 3 = 1 i 2 a i 1 i 2 i 3 H 3 x i 1 , x i 2 , x i 3 + ⋯ + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 ⋯ ∑ i d = 1 i d − 1 a i 1 i 2 ⋯ i d H d x i 1 , x i 2 , … , x i d , $\begin{array}{rl}\hfill f\left(\widehat{\mathcal{X}}+Sx\right)\approx & {a}_{0}+\sum\limits _{{i}_{1}=1}^{n}{a}_{{i}_{1}}{H}_{1}\left({x}_{{i}_{1}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}{a}_{{i}_{1}{i}_{2}}{H}_{2}\left({x}_{{i}_{1}},{x}_{{i}_{2}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}\sum\limits _{{i}_{3}=1}^{{i}_{2}}{a}_{{i}_{1}{i}_{2}{i}_{3}}\hfill \\ \hfill & {H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{2}},{x}_{{i}_{3}}\right)+\cdots +\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}\cdots \sum\limits _{{i}_{d}=1}^{{i}_{d-1}}{a}_{{i}_{1}{i}_{2}\cdots {i}_{d}}{H}_{d}\left({x}_{{i}_{1}},{x}_{{i}_{2}},\text{\ldots },{x}_{{i}_{d}}\right),\hfill \end{array}$ (9)where a i 1 i 2 ⋯ i d ${a}_{{i}_{1}{i}_{2}\cdots {i}_{d}}$ are the coefficients of the chaos expansion, x 1 x 2 ⋯ x n T = x ${\left[\begin{array}{cccc}\hfill {x}_{1}\hfill & \hfill {x}_{2}\hfill & \hfill \cdots \hfill & \hfill {x}_{n}\hfill \end{array}\right]}^{T}=x$ , and Hd(x1, x2, …, xn) is the Hermite polynomial of degree d, which is given by [17, 18] H d x 1 , x 2 , … , x n = ( − 1 ) d exp 1 2 x T x ∂ n ∂ x 1 ∂ x 2 ⋯ ∂ x n exp − 1 2 x T x . $\begin{array}{r}\hfill {H}_{d}\left({x}_{1},{x}_{2},\text{\ldots },{x}_{n}\right)={(-1)}^{d}\mathrm{exp}\left(\frac{1}{2}{x}^{T}x\right)\frac{{\partial }^{n}}{\partial {x}_{1}\partial {x}_{2}\cdots \partial {x}_{n}}\mathrm{exp}\left(-\frac{1}{2}{x}^{T}x\right).\end{array}$ (10) Hermite polynomials are orthogonal in the inner product sense [27, p. 45] that is, E[HpHq] = 0 if Hp ≠ Hq, where the expectation is taken with respect to the standard Gaussian measure. Further, polynomial chaos expansion as expressed in Equation (9) converges in the mean square sense to the true function [27, 28] and the squared error of approximation decreases with the increase in order of expansion. In Refs. [18, 27, 28], it has been mentioned that the approximation of degree more than three provides only a marginal improvement in accuracy and is overshadowed by the increase in computational burden. In this work, we confine ourselves to chaos expansion up to third order while approximating the non-linear state transition function and the measurement function. From Equation (9), we approximate the transition function with chaos expansion of third order as follows: f X ˆ + S x ≈ a 0 + ∑ i 1 = 1 n a i 1 H 1 x i 1 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 a i 1 i 2 H 2 x i 1 , x i 2 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 ∑ i 3 = 1 i 2 a i 1 i 2 i 3 H 3 x i 1 , x i 2 , x i 3 ≈ a 0 + A H ( x ) , $\begin{array}{rl}\hfill f\left(\widehat{\mathcal{X}}+Sx\right)& \approx {a}_{0}+\sum\limits _{{i}_{1}=1}^{n}{a}_{{i}_{1}}{H}_{1}\left({x}_{{i}_{1}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}{a}_{{i}_{1}{i}_{2}}{H}_{2}\left({x}_{{i}_{1}},{x}_{{i}_{2}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}\sum\limits _{{i}_{3}=1}^{{i}_{2}}{a}_{{i}_{1}{i}_{2}{i}_{3}}{H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{2}},{x}_{{i}_{3}}\right)\hfill \\ \hfill & \approx {a}_{0}+AH(x),\hfill \end{array}$ (11)where the matrix H(x) and A are given by H ( x ) ( m − 1 ) × 1 = H 1 x 1 ⋯ H 3 x n , x n , x n T , $H{(x)}_{(m-1)\times 1}={\left[\begin{array}{ccc}\hfill {H}_{1}\left({x}_{1}\right)\hfill & \hfill \cdots \hfill & \hfill {H}_{3}\left({x}_{n},{x}_{n},{x}_{n}\right)\hfill \end{array}\right]}^{T},$ (12)and A n × ( m − 1 ) = a 1 a 2 ⋯ a nnn , ${A}_{n\times (m-1)}=\left[\begin{array}{cccc}\hfill {a}_{1}\hfill & \hfill {a}_{2}\hfill & \hfill \cdots \hfill & \hfill {a}_{\mathit{nnn}}\hfill \end{array}\right],$ (13)respectively. The total number of terms in the approximated polynomial is denoted by m, and for a third-order polynomial chaos expansion, m = n + 3 3 $m=\left(\begin{matrix}{n+3}\\{3}\end{matrix}\right)$ . Note that as per our nomenclature in the Section 1, a0 and coefficient a with subscript i1 = i2 for second-order terms and subscript i1 = i2 = i3 for third-order terms in Equation (11) are alike terms and the rest of the coefficients are cross terms. From Equation (10), we know that H 1 x i 1 = x i 1 , ${H}_{1}\left({x}_{{i}_{1}}\right)={x}_{{i}_{1}},$ H 2 x i 1 , x i 1 = x i 1 2 − 1 2 , H 2 x i 1 , x i 2 = x i 1 x i 2 , ${H}_{2}\left({x}_{{i}_{1}},{x}_{{i}_{1}}\right)=\frac{\left({x}_{{i}_{1}}^{2}-1\right)}{\sqrt{2}},\,{H}_{2}\left({x}_{{i}_{1}},{x}_{{i}_{2}}\right)={x}_{{i}_{1}}{x}_{{i}_{2}},$ H 3 x i 1 , x i 1 , x i 1 = x i 1 3 − 3 x i 1 6 , H 3 x i 1 , x i 2 , x i 2 = x i 1 x i 2 2 − 1 2 , H 3 x i 1 , x i 2 , x i 3 = x i 1 x i 2 x i 3 . ${H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{1}},{x}_{{i}_{1}}\right)=\frac{\left({x}_{{i}_{1}}^{3}-3{x}_{{i}_{1}}\right)}{\sqrt{6}},\,{H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{2}},{x}_{{i}_{2}}\right)=\frac{{x}_{{i}_{1}}\left({x}_{{i}_{2}}^{2}-1\right)}{\sqrt{2}},\,{H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{2}},{x}_{{i}_{3}}\right)={x}_{{i}_{1}}{x}_{{i}_{2}}{x}_{{i}_{3}}.$ Similar to Equation (11), we approximate the measurement function (Equation 2) as h X ˆ + S x ≈ b 0 + ∑ i 1 = 1 n b i 1 H 1 x i 1 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 b i 1 i 2 H 2 x i 1 , x i 2 + ∑ i 1 = 1 n ∑ i 2 = 1 i 1 ∑ i 3 = 1 i 2 b i 1 i 2 i 3 H 3 x i 1 , x i 2 , x i 3 ≈ b 0 + B H ( x ) , $\begin{array}{rl}\hfill h\left(\widehat{\mathcal{X}}+Sx\right)& \approx {b}_{0}+\sum\limits _{{i}_{1}=1}^{n}{b}_{{i}_{1}}{H}_{1}\left({x}_{{i}_{1}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}{b}_{{i}_{1}{i}_{2}}{H}_{2}\left({x}_{{i}_{1}},{x}_{{i}_{2}}\right)+\sum\limits _{{i}_{1}=1}^{n}\sum\limits _{{i}_{2}=1}^{{i}_{1}}\sum\limits _{{i}_{3}=1}^{{i}_{2}}{b}_{{i}_{1}{i}_{2}{i}_{3}}{H}_{3}\left({x}_{{i}_{1}},{x}_{{i}_{2}},{x}_{{i}_{3}}\right)\hfill \\ \approx {b}_{0}+BH(x),\hfill & \end{array}$ (14)where B is a matrix of dimension ny × (m − 1), and can be expressed in a similar fashion as Equation (13). Polynomial chaos approximation-based filter In this subsection, we derive the filtering algorithm for the following approximate non-linear system: X k = a 0 + A H X k − 1 + η k − 1 , Y k = b 0 + B H X k + ν k , $\begin{array}{rl}\hfill {\mathcal{X}}_{k}& ={a}_{0}+AH\left({\mathcal{X}}_{k-1}\right)+{\eta }_{k-1},\hfill \\ \hfill {\mathcal{Y}}_{k}& ={b}_{0}+BH\left({\mathcal{X}}_{k}\right)+{\nu }_{k},\hfill \end{array}$ (15)where a0, b0, A and B are as defined in the previous section and X k = X ˆ k + S x ${\mathcal{X}}_{k}={\widehat{\mathcal{X}}}_{k}+Sx$ , with S being a Cholesky factor of the covariance matrix of X k ${\mathcal{X}}_{k}$ and x being a random variable with standard normal distribution. The following three lemmas allow computation of moments using polynomial chaos expansion of non-linear functions. Remark 1.For a standard Gaussian random variable x, the matrix H(x) has the following properties: E [ H ( x ) ] = 0 , $E[H(x)]=0,$ (16) E H ( x ) H ( x ) T = I . $E\left[H(x)H{(x)}^{T}\right]=I.$ (17) Lemma 1.For the system in Equation (15), the prior mean and the error covariance are given by X ˆ k | k − 1 = a 0 , ${\widehat{\mathcal{X}}}_{k\vert k-1}={a}_{0},$ (18) P k | k − 1 X X = A A T + Q k − 1 . ${P}_{k\vert k-1}^{\mathcal{X}\mathcal{X}}=A{A}^{T}+{Q}_{k-1}.$ (19) Proof.The prior mean can be calculated as X ˆ k | k − 1 = E X k | Y 1 : k − 1 . ${\widehat{\mathcal{X}}}_{k\vert k-1}=E\left[{\mathcal{X}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right].$ Now, using the affine transformation [29, pp. 36–37] X k − 1 = X ˆ k − 1 | k − 1 + S k − 1 | k − 1 x k − 1 ${\mathcal{X}}_{k-1}={\widehat{\mathcal{X}}}_{k-1\vert k-1}+{S}_{k-1\vert k-1}{x}_{k-1}$ , the above equation becomes X ˆ k | k − 1 = E f X ˆ k − 1 | k − 1 + S k − 1 | k − 1 x k − 1 + η k − 1 = E a 0 + A H x k − 1 + η k − 1 = a 0 . $\begin{array}{rl}\hfill {\widehat{\mathcal{X}}}_{k\vert k-1}& =E\left[f\left({\widehat{\mathcal{X}}}_{k-1\vert k-1}+{S}_{k-1\vert k-1}{x}_{k-1}\right)+{\eta }_{k-1}\right]\hfill \\ \hfill & =E\left[{a}_{0}+AH\left({x}_{k-1}\right)+{\eta }_{k-1}\right]\hfill \\ \hfill & ={a}_{0}.\hfill \end{array}$ The predicted estimation error is X k − X ˆ k | k − 1 = A H x k − 1 + η k − 1 . ${\mathcal{X}}_{k}-{\widehat{\mathcal{X}}}_{k\vert k-1}=AH\left({x}_{k-1}\right)+{\eta }_{k-1}.$ (20)The prior error covariance can be calculated as P k | k − 1 X X = E X k − X ˆ k | k − 1 X k − X ˆ k | k − 1 T | Y 1 : k − 1 = E A H x k − 1 + η k − 1 A H x k − 1 + η k − 1 T = A E H x k − 1 H x k − 1 T A T + Q k − 1 = A A T + Q k − 1 . ${P}_{k\vert k-1}^{\mathcal{X}\mathcal{X}}=E\left[\left({\mathcal{X}}_{k}-{\widehat{\mathcal{X}}}_{k\vert k-1}\right){\left({\mathcal{X}}_{k}-{\widehat{\mathcal{X}}}_{k\vert k-1}\right)}^{T}\vert {\mathcal{Y}}_{1:k-1}\right]=E\left[\left(AH\left({x}_{k-1}\right)+{\eta }_{k-1}\right){\left(AH\left({x}_{k-1}\right)+{\eta }_{k-1}\right)}^{T}\right]=AE\left[H\left({x}_{k-1}\right)H{\left({x}_{k-1}\right)}^{T}\right]{A}^{T}+{Q}_{k-1}=A{A}^{T}+{Q}_{k-1}.$ Lemma 2.For the system in Equation (15), the expected value of one-step-ahead measurement, Y ˆ k | k − 1 ${\widehat{\mathcal{Y}}}_{k\vert k-1}$ , the covariance of the same measurement, P k | k − 1 Y Y ${P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}$ , and the cross covariance of the state and the measurement, P k | k − 1 XY $\left({P}_{k\vert k-1}^{\mathcal{XY}}\right)$ , are, respectively, given by Y ˆ k | k − 1 = b 0 , ${\widehat{\mathcal{Y}}}_{k\vert k-1}={b}_{0},$ (21) P k | k − 1 Y Y = B B T + R k , ${P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}=B{B}^{T}+{R}_{k},$ (22) P k | k − 1 XY = A B T . ${P}_{k\vert k-1}^{\mathcal{XY}}=A{B}^{T}.$ (23) Proof.The expected value of measurement is Y ˆ k | k − 1 = E Y k | Y 1 : k − 1 . $\begin{array}{rl}\hfill {\widehat{\mathcal{Y}}}_{k\vert k-1}& =E\left[{\mathcal{Y}}_{k}\vert {\mathcal{Y}}_{1:k-1}\right].\hfill \end{array}$ Using Equations (2) and (14), the above equation can be written as Y ˆ k | k − 1 = E b 0 + B H x k + ν k | Y 1 : k − 1 = b 0 . ${\widehat{\mathcal{Y}}}_{k\vert k-1}=E\left[\left({b}_{0}+BH\left({x}_{k}\right)+{\nu }_{k}\right)\vert {\mathcal{Y}}_{1:k-1}\right]={b}_{0}.$ Now, since Y k − Y ˆ k | k − 1 = B H x k + ν k , ${\mathcal{Y}}_{k}-{\widehat{\mathcal{Y}}}_{k\vert k-1}=BH\left({x}_{k}\right)+{\nu }_{k},$ (24)the covariance of the measurement can be expressed as P k | k − 1 Y Y = E Y k − Y ˆ k | k − 1 Y k − Y ˆ k | k − 1 T | Y 1 : k − 1 = E B H x k + ν k B H x k + ν k T = B E H x k H x k T B T + R k = B B T + R k . ${P}_{k\vert k-1}^{\mathcal{Y}\mathcal{Y}}=E\left[\left({\mathcal{Y}}_{k}-{\widehat{\mathcal{Y}}}_{k\vert k-1}\right){\left({\mathcal{Y}}_{k}-{\widehat{\mathcal{Y}}}_{k\vert k-1}\right)}^{T}\vert {\mathcal{Y}}_{1:k-1}\right]=E\left[\left(BH\left({x}_{k}\right)+{\nu }_{k}\right){\left(BH\left({x}_{k}\right)+{\nu }_{k}\right)}^{T}\right]=BE\left[H\left({x}_{k}\right)H{\left({x}_{k}\right)}^{T}\right]{B}^{T}+{R}_{k}=B{B}^{T}+{R}_{k}.$ (25)Further, the cross-covariance can be calculated as P k | k − 1 XY = E X k − X ˆ k | k − 1 Y k − Y ˆ k | k − 1 T |