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SUN Xiaohui, WEN Chenglin, WEN Tao. Maximum Correntropy High-Order Extended Kalman Filter[J]. Chinese Journal of Electronics, 2022, 31(1): 190-198. DOI: 10.1049/cje.2020.00.334
Citation: SUN Xiaohui, WEN Chenglin, WEN Tao. Maximum Correntropy High-Order Extended Kalman Filter[J]. Chinese Journal of Electronics, 2022, 31(1): 190-198. DOI: 10.1049/cje.2020.00.334

Maximum Correntropy High-Order Extended Kalman Filter

Funds: This work was supported by the National Natural Science Foundation of China (61751304, 61806064, 61933013, 61703385, U1664264) and Science and Technology Project of China Electric Power Research Institute (SGHB0000KXJS1800375)
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  • Author Bio:

    SUN Xiaohui: was born in 1993. She received B.E. degree from Harbin University of Science and Technology. She studied for a master’s degree and a Ph.D. in Hangzhou Dianzi University in 2016 and 2018, respectively, and is currently a Ph.D. candidate. Her research interest is filter design. (Email: sun_xh1993@163.com)

    WEN Chenglin: was born in 1963. He graduated from Henan University in 1986, graduated from Zhengzhou University with a master degree in 1993 and received a Ph.D. from Northwestern Polytechnical University 1999. He went out of the Postdoctoral Mobile Station of Control Science and Engineering of Tsinghua University in 2002. He is a Professor of Hangzhou Dianzi University and Guangdong University of Petrochemical Technology. His research interests include information fusion and target detection, fault diagnosis and active security control, deep learning and optimization decision-making systems, cyberspace security and attack detection and positioning. (Email: wencl@hdu.edu.cn)

    WEN Tao: (corresponding author) received the B.Eng. degree in computer science from Hangzhou Dianzi University in 2011, the M.Sc. degree from the University of Bristol, Bristol, U.K. in 2013, and the Ph.D. degree from the Birmingham Centre for Railway Research and Education, University of Birmingham, Birmingham, U.K. in 2018. He currently works in the School of Electronic and Information Engineering, Beijing Jiaotong University. His research interests include CBTC system optimization, railway signaling simulation, railway depend-ability improvement, wireless signal processing, and digital filter research (Email: wentao@bjtu.edu.cn)

  • Received Date: October 09, 2020
  • Accepted Date: December 02, 2020
  • Available Online: September 21, 2021
  • Published Date: January 04, 2022
  • In this paper, a novel maximum correntropy high-order extended Kalman filter (H-MCEKF) is proposed for a class of nonlinear non-Gaussian systems presented by polynomial form. All high-order polynomial terms in the state model are defined as implicit variables and regarded as parameter variables; the original state model is equivalently formulated into a pseudo-linear form with original variables and parameter variables; the dynamic relationship between each implicit variable and all variables is modeled, then an augmented linear state model appears by combing with pseudo-linear state model; similarly, the nonlinear measurement model can be equivalently rewritten into linear form; once again, the statistical characteristics of non-Gaussian modeling error are described by mean value and variance based on their finite samples; combing original measurement model with predicted value regarded as added state measurement, a cost function to solve the state estimation based on maximum correntropy criterion (MCC) is constructed; on the basis of this cost function, the state estimation problem can be equivalently converted into a recursive solution problem in the form of Kalman filter, in which the filter gain matrix is solved by numerical iteration though its fixed-point equation; illustration examples are presented to demonstrate the effectiveness of the new algorithm.
  • Filter has been applied in various fields, including navigation, national defense construction, neural network training and so on[1-6]. For general nominal linear systems, Kalman filter (KF) provides an optimal solution step-by-step[7]. For nonlinear problems, extended Kalman filter is the most typical method, which approximates the nonlinear system by its first order linearization[8]. Another nonlinear filters are based on sigma sampling, such as unscented Kalman filter (UKF), cubature Kalman filter (CKF), and so on[9-12], which have better filtering performance compared with EKF.

    KF and its extensions in general have been established to meet the conditions of the standard KF[13-15]. Nevertheless, their performances may get worse when employed to non-Gaussian situations. This is because their objective function and criterion depend on large outliers, they are not suitable for non-Gaussian environments[16].

    For non-Gaussian systems, Chen proposed maximum correntropy KF (MCKF)[17]. MCKF only needs a limited number of realizations for modeling error to establish a mean estimator of correntropy. Lately, the Maximum correentropy extended KF (MCEKF) that can solve nonlinear non-Gaussian noise systems have appeared based on EKF[16]. However, similar to EKF, MCEKF can realize the approximation of the nonlinear function not higher than the second order.

    In the paper, we develop a novel high-order extended KF, called the maximum correntropy high-order KF (H-MCEKF) by combining MCC with a fixed-point iterative algorithm. Similar to EKF, the H-MCEKF not only retains the propagation process of state, but that of covariance matrix. Therefore, the new algorithm has two significant advantages: real-time and recursion. Unlike the first-order Taylor expansion of MCEKF, HMCEKF uses polynomials to have a full description for nonlinear functions. It should be emphasized that it is not difficult to convert a nonlinear function into a high-order polynomial by employing Taylor expansion or multi-dimensional Taylor network[17].

    There are two contributions that can be shown in this paper. 1) The idea of converting a nonlinear function to a linear form is given: a) all high-order polynomials in the system are defined as implicit variables and treated as parameter variables; b) the original state model is equivalently formulated into a pseudo-linear form; c) an augmented linear state model is established by combing with pseudo-linear state model; d) the original measurement model is rewritten into linear form. 2) The solution of high-order polynomials are converted into KF.

    This paper will be divided into several sections to describe and derive the design process of the new filter: Brief introduction in Section II; modeling the nonlinear non-Gaussian systems in Section III; linearizing the systems in Section IV; derivation process of H-MCEKF in Section V; simulations in Section VI and conclusions and future in Section VII finally.

    For one-dimensional random variable X,Y with FXY(x,y), then the definition of correntropy is[17]

    V(X,Y)=E[Υ(XY)]=Υ(XY)dFXY(x,y) (1)

    where, E is expectation, and Υ(XY) is Mercer Kernel. For example, Gaussian kernel is as follows.

    Υ(XY)=Gσ(e)=exp(e22σ2) (2)

    where, e is the difference between x and y, σ is their covariance.

    Perform Taylor expansion on Eq.(2),

    Υ(XY)=p=0(1)p2nσ2np!E{(xy)2p} (3)

    then, the correntropy of Eq.(1) can be formulated

    V(X,Y)=E[Υ(XY)]=p=0(1)p2pσ2pp!(xy)2pdFXY(x,y)=p=0(1)p2pσ2pp!E{(XY)2p} (4)

    where, E{(XY)2p}=(xy)2pdFXY(x,y) is 2p order moment of the random variable X,YR

    In general, random variable pairs may be relatively easy to obtain, and FXY is difficult to obtain. In this case, we can get the correntropy by Eq.(5)

    E{(XY)2p}=1n(ni=1(x(i)y(i))2p) (5)

    then, the correntrpy of random variable pairs (X,Y) under limited data is

    ˆV(X,Y)=E[Υ(XY)]=p=0(1)p2pσ2pp!(xy)2pdFXY(x,y)=p=0(1)p2pσ2pp!1N(Nj=1(x(j)y(j))2p)=1Np=0Gσ(e(j)) (6)

    When X,YRn and the components in e=XY are independent of each other, the multi-dimensional correntropy can be obtained by Eq.(7)

    ˆV(X,Y)=E{Υ(XY)}=1nNni=1Nj=1Gσ(e(j)i) (7)

    where, i denotes ith component of e; j denotes jth sampler.

    Remark 1 The larger the σ is , the greater the influence of the second moment on the correntropy is.

    Given a class of state models and observation models with strong nonlinear characteristics

    x(k+1)=f(x(k))+w(k) (8)
    y(k+1)=h(x(k+1))+v(k+1) (9)

    where, x(u) and y(u) are n-dimensional state variable and m-dimensional measurement variable, respectively; state modeling errors ω(u) and measurement modeling error ν(u) are both with non-Gaussian systems; f(x(u)) is nonlinear state function and h(x(u+1)) represents nonlinear function of x(u+1).

    For ease of understanding, we simplify the representation of the high-dimensional linearization process based on Eq.(8) and Eq.(9).

    For facilitate understanding, we only describe and establish the filter for the two-dimensional systems and suppose.

    fi(x(u))=l1+l2=ll1,l2lai,l1,l2xl11(u)xl22(u) (10)

    where, ai,l1,l2 is the corresponding weight value for l1+l2=ll1,l2lxl11(u)xl22(u).

    Definition 1 x(l)(u):={xl11(u)xl22(u),l1+l2=l;0ljl;l=0,1,,r} is a set of hidden variables with sequential order.

    Definition 2 a(l)i is weight vector for x(l)(u) where

    a(l)i:=[a(l)i;1,a(l)i;2,,a(l)i;nl]=[ai;l,0,ai;l1,1,,ai;0,l]i=1,2

    On the basis of Definition 1, 2 and Eq.(10), we have

    [x(1)1(u+1)x(1)2(u+1)]=[a(1)1a(2)1a(l)1a(r)1a(1)2a(2)2a(l)2a(r)2]×[x(1)(u)x(2)(u)x(l)(u)x(r)(u)]+[w(1)1(u)w(1)2(u)] (11)

    Let

    x(u):=x(1)(u)=[x(1)1(u)x(1)2(u)],A(l):=[a(l)1a(l)2],w(u):=w(1)(u)=[x(1)1(u)x(1)2(u)]

    then,

    x(1)(u+1)=A(1)x(1)(u)+rl=2A(l)x(l)(u)+w(1)(u) (12)

    Similarly, suppose the measurement function in Eq.(9) is shown as follows

    hi(x(1)(u+1))=r1+r2=rr1,r2rhi,r1,r2xr11(u+1)xr22(u+1) (13)

    Similar to Definition 1 and Definition 2, Eq.(10) and Eq.(11), Eq.(13) has the matrix form as following.

    y(1)(u+1)=H(1)x(1)(u+1)+rl=2H(l)x(l)(u+1)+v(1)(u+1) (14)

    In order to linearize the nonlinear functions, we build the dynamic model as follows

    x(l)(u+1)=ru=1A(q)l(u)x(q)(u) (15)

    where, A(q)l(u) can be solved by given prior information. Specially, we may suppose that

    A(q)l(u)={I,l=q0,lq (16)

    Combining Definition 1 and Definition 2, Eq.(11) and Eq.(12), state model Eq.(8) has further linear form

    [x(1)(u+1)x(2)(u+1)x(l)(u+1)x(r)(u+1)]=[A(1)1(u)A(2)1(u)A(u)1(u)A(r)1(u)A(1)2(u)A(2)2(u)A(u)2(u)A(r)2(u)A(1)l(u)A(2)l(u)A(u)l(u)A(r)l(u)A(1)r(u)A(2)r(u)A(u)r(u)A(r)r(u)][x(1)(u)x(2)(u)x(l)(u)x(r)(u)]+[w(1)(u)w(2)(u)w(l)(u)w(r)(u)] (17)

    Let

    X(u)=[(x(1)(u))T(x(2)(u))T(x(r)(u))T]TA(u+1,u)=[A(1)1(u)A(2)1(u)A(r)1(u)A(1)2(u)A(2)2(u)A(r)2(u)A(1)r(u)A(2)r(u)A(r)r(u)]W(u)=[w(1)(u)w(2)(u)w(r)(u)]T

    Eq.(17) is equivalently rewritten as follows

    X_(u+1)=A_(u+1,u)X_(u)+W_(u) (18)

    where, W_(u) denotes the modeling uncertainty of the augmented linearized state.

    Similarly, the linear matrix form of the measurement model (9) is

    [y1(u+1)y2(u+1)]=[h(1)1(u+1)h(2)1h(r)1h(1)2(u+1)h(2)2h(r)2]×[x(1)(u+1)x(2)(u+1)x(r)(u+1)]+[v1(u+1)v2(u+1)] (19)

    On the basis of Eq.(19), we can obtain the linearization of Eq.(2).

    Y_(u+1)=H_(u+1)X_(u+1)+V_(u+1) (20)

    where, V(k) is the modeling error.

    For linear models Eq.(18) and Eq.(20), we have

    [ˆX_(u+1|u)Y_(u+1)]=[IH_(u+1)]X_(u+1)+Δ(u+1) (21)

    where, ˆX_(u+1|u)=A_(u+1,k)ˆX_(u|u) can be obtained by Eq.(18).

    u(u+1)=[˜X_(u+1|u))V_(u+1)] (22)

    where, ˜X_(u+1|u)=X_(u+1|u+1)ˆX_(u+1|u)

    and

    E[u(u+1)uT(u+1)]=[P_(u+1|u)00R_V(u+1)] (23)

    where, E[] can be obtained by finite sampling.

    P_(u+1|u)=A_(u+1,u)P_(u|u)A_T(u+1,u)+Q(u) (24)

    where

    Q(u)=diag{Q(1)(u)Q(r)(u)}Q(1)(u)=1NNj=1{[ω(1,j)(u)ˉω(u)][ω(1,j)(u)ˉω(u)]T}ˉω(u)=1NNj=1ω(j)(u)

    ω(1,j)(u) is the jth sampling of modeling error ω(1)(u); and ω(l)(u)N(0,Q(l)),l=2,3,,r.

    Similarly, RV(u+1) is the covariance matrix of v(u+1) in original state model (9).

    R_V(u+1)=1NNj=1{[v(j)(u+1)ˉv(u+1)][v(j)(u+1)ˉv(u+1)]T} (25)

    where ˉv(u+1)=1NNj=1v(j)(u+1), v(j)(u+1) is the jth sampling of the non-Gaussian random noise vector v(u+1).

    u in Eq.(23) is L=rl=1nl+m dimension modeling error vector, and the relationship between components are not independent. In order to employ the correntropy shown in Eq.(7), u(u+1) needs to be transformed independently. Then, Eq.(23) can be further formulated.

    E{u(u+1)uT(u+1)}=[B_X(u+1|u)B_TX(u+1|u)00B_Y(u+1)B_TY(u+1)]=B_(u+1)B_T(u+1) (26)

    where, B_X(u+1) and B_Y(u+1) can be obtained by Cholesky decomposition of P_(k+1) and R_V(u+1), respectively.

    Combing Eq.(26) and B_1(u+1), Eq.(21) can be simplified as

    D_(u+1)=S_(u+1)X_(u+1)+e(u+1) (27)

    where

    D_(u+1)=B_1(u+1)[ˆX_(u+1|u)Y_(u+1)]S_(u+1)=B_1(u+1)[IH_(u+1)]e(u+1)=B_1(u+1)u(u+1)

    with

    E{e(u)eT(u)}=E{[B_1(u+1)u(u+1)][B_1(u+1)u(u+1)]T}=B_1(u+1)E{u(u+1)uT(u+1)}(B_1(u+1))T=B_1(u+1)B_(u+1)B_T(u+1)(B_1(u+1))T=I (28)

    Therefore, the components of the obtained random variable e(u+1) become statistically independent after the non-Gaussian modeling error u(u+1) is equivalently transformed by B1(u+1).

    We propose the following objective function for solving ˆX_(u+1|u+1) by Eq.(21)

    JL(X_(u+1))=1LLi=1(1NNj=1Gσ(d(j)i(u+1)si(u+1)X_(u+1)))=1LNLi=1Nj=1Gσ(e(j)i) (29)

    where, d(j)i is the jth realization of the ith element of D(k),

    According to Eq.(30), we can get the optimal solution.

    JL(X_(u+1))X_(u+1)=0 (30)

    Further

    X_(u+1)=(Nj=1Li=1Gσ(e(j)i(u+1))sTi(u+1)si(u+1))1×(Nj=1Li=1Gσ(e(j)i(u+1))sTi(u+1)d(j)i(u+1)) (31)

    Considering that e(j)i(u+1)=d(j)i(u+1)si(u+1)X_(u+1), Eq.(31) can be written as a fixed point equation about X_(u+1)

    X_(u+1)=f(X_(u+1)) (32)

    Let

    S_(u+1)=Li=1si(u+1)D_(u+1)=Li=1di(u+1)C_(u+1)=Li=1Nj=1Gσ(e(j)i(u+1))

    then

    f(X_(u+1))=(S_T(u+1)C_(u+1)S_(u+1))1×S_T(u+1)C_(u+1)D_(u+1) (33)

    where

    S_T(u+1)C(u+1)S_(u+1)=Li=1Nj=1[sTi(u+1)Gσ(e(j)i(u+1))si(u+1)] (34)
    S_T(u+1)C_(u+1)D_(u+1)=Li=1Nj=1[sTi(u+1)Gσ(e(j)i(u+1))di(u+1)] (35)

    From Ref [14], Eq.(34) and Eq.(35), we arrive at

    [S_T(u+1)C_(u+1)S_(u+1)]1=ˉP(u+1|u)ˉP(u+1|u)H_T(u+1)×[H_(u+1)ˉP(u+1|u)H_T(u+1)ˉR(u+1)]1×H_(u+1)ˉP(u+1|u) (36)
    S_T(u+1)C_(u+1)D_(u+1)=ˉP1(u+1|u)ˆX_(u+1|u)+H_T(u+1)ˉR1(u+1)Y(u+1) (37)

    where

    ˉP(u+1|u)=BX(u+1))C1X(u+1)BTX(u+1)ˉR(u+1)=BY(u+1))C1Y(u+1)BTY(u+1)

    Further, we get

    X_(u+1)=ˆX_(u+1|u)+ˉK(u+1|u)×[Y(u+1)H_(u+1)ˆX_(u+1|u)] (38)

    where

    ˉK(u+1)=ˉP(u+1|u)H_T(u+1)×[H_(u+1)ˉP(u+1|u)H_T(u+1)+ˉR1(u+1)]1 (39)

    Thus, the equivalent conversion is completed from fixed-point equation to Kalman filter.

    With the above derivations, we summarize the proposed H-MCEKF algorithm. Select a proper σ and ε, where σ controls the local scope of the Gaussian kernel function and ε represents the estimation accuracy. Set an initial estimate ˆX_(k|k)0=ˆX_(k|k1), P(k|k), then

    ˆX_(u|u)t+1=ˆX_(u|u)t+ˉK(u)t[Y(u)H(u)ˆX_(u|u)t] (40)

    where

    ˉK(u)t=ˉP(u|u1)tH_T(u)×(H_(u)ˉP(u|u1)tH_T(u)+ˉR(u)t)1 (41)
    ˉP(u|u1)t=B_X(u|u1)C_X(u|u1)tB_TX(u|u1) (42)
    ˉR(u)t=B_Y(u)C_Y(u)tB_TY(u) (43)
    C_X(u)t=diag{G1σ(e1(u)t),,G1σ(eL1(u)t)} (44)
    C_Y(u)t=diag{G1σ(eL1+1(u)t),,G1σ(eL1+m(u)t)} (45)
    ˉei(u)t=d(j)i(u)si(u)ˆX_(u|u)t (46)

    If Eq.(47) holds,

    (47)
    \begin{split} \underline P (u|u) =& E\left\{ {\underline {\tilde X} {{(u|u)}_t}{{\underline {\tilde X} }^{\rm{T}}}{{(u|u)}_t}} \right\} \\ = &E\left\{ {[\underline X (u) - \underline {\hat X} {{(u|u)}_t}]{{[ \underline X (u) - \underline {\hat X} {{(u|u)}_t}]}^{\rm{T}}}} \right\} \\ =& \left[ {I - \bar K{{(u)}_t}H(u)} \right]\underline P {(u|u - 1)}{\left[ {I - \bar K{{(u)}_t}H(u)} \right]^{\rm{T}}} \\ &+ \bar K{(u)_t}\bar R{(u)_t}{\bar K^{\rm{T}}}{(u)_t} \end{split} (48)

    Remark 2 Without confusion, let u + 1: = u

    Remark 3 Eq.(44) and Eq.(45) contribute to the adjustment of the noise uncertainty. As the system iterates, Eq.(44) and Eq.(45) will gradually converge to 1.

    Remark 4 In the polynomial expansion method in the paper, as the dimensionality increases, higher-order terms will show sparseness and its proportion will be less and less. We can use the method of pruning to model it again, so that the dimensionality can be controlled, and at the same time, the dynamic statistical properties of high-order terms will be retained.

    This chapter verifies the performance of the proposed new filter through several simulations.

    Giving the following systems shown in Eq.(8) and Eq.(9)

    where

    \begin{split} {f_1}(x(u)) =& {x_1}(u) + {x_2}(u) - \frac{1}{6}x_1^3(u) - \frac{1}{6}x_2^3(u)\\ & + \frac{1}{{120}}x_1^5(u) + \frac{1}{{120}}x_2^5(u) \\ {f_2}(x(u)) = &{x_1}(u) - \frac{1}{2}x_1^2(u) - \frac{1}{2}x_2^2(u)\\ &+ \frac{1}{{24}}x_1^4(u) + \frac{1}{{24}}x_2^4(u)\\ z(u + 1) = &x(u + 1) + v(u + 1) \end{split}

    The simulation is performed many times from 1-500, where w(u) \sim 0.8N(0,0.01) + 0.2N(0,0.1) for the first 200 times and w(u + 1)\sim 0.8N(0,0.01) + 0.2N(0,0.1) + {[\begin{array}{*{20}{c}} {0.1}&{0.1} \end{array}]^{\rm{T}}} for the remaining; {v}(u+1) \sim N(0,0.01) ; {x^{(2)}}(u) = {\left[ {\begin{array}{*{20}{c}} {x_1^2(u)}&{x_2^2(u)} \end{array}} \right]^{\rm{T}}}, {x^{(3)}}(u) = {\left[ {\begin{array}{*{20}{c}} {x_1^3(u)}&{x_2^3(u)} \end{array}} \right]^{\rm{T}}} , {x^{(4)}}(u) = {\left[ {\begin{array}{*{20}{c}} {x_1^4(u)}&{x_2^4(u)} \end{array}} \right]^{\rm{T}}}, and {x^{(5)}}(u) = \left[ {c} {x_1^5(u)}\;\; {x_2^5(u)}\right]^{\rm{T}} are defined as implicit variables and their modeling errors satisfies that the mean is 0 and the variance is 0.01; x(0) = {[1\;\;1]^{\rm{T}}} , \hat x(0|0) = {[1\;\;1]^{\rm{T}}} + N(0,0.01) \times randn(2,1) and P(0|0) = 0.01 \times diag(1,1)

    Case 1 only considers the process noise is non-Gaussian and takes \sigma = 10,\varepsilon = {10^{ - 1}} as an example to demonstrate the estimation results of MCEKF and H-MCEKF, as shown in Fig.1(a)-(d). Table 1 summarizes the estimation errors under different \sigma and \varepsilon .

    Figure  1.  The actual state x_1 and its estimate (a) with estimate error (c); The actual state x_2 and its estimate (b) with estimate error (d)
    Table  1.  Estimation errors with different filters
    MSE of {x_{\rm{1}}}MSE of {x_{\rm{2}}}MSE
    \sigma\varepsilon MCEKFH-MCEKFImprovedMCEKFH-MCEKFImprovedMCEKFH-MCEKFImproved
    \sigma {\rm{ = }}2 \varepsilon {\rm{ = }}{10^{ - 4}} 0.0469 0.0467 0.42% 0.0226 0.0204 9.73% 0.0348 0.0335 3.74%
    \varepsilon {\rm{ = }}{10^{ - 6}} 0.0434 0.0421 3.00% 0.0344 0.0256 25.58% 0.0389 0.0338 13.11%
    \sigma {\rm{ = }}5 \varepsilon {\rm{ = }}{10^{ - 1}} 0.0205 0.0189 7.80% 0.0091 0.0089 2.20% 0.0148 0.0139 6.08%
    \varepsilon {\rm{ = }}{10^{ - 2}} 0.0107 0.0099 7.47% 0.0112 0.0083 25.89% 0.0110 0.0091 17.27%
    \varepsilon {\rm{ = }}{10^{ - 4}} 0.0201 0.0178 11.44% 0.0235 0.0195 17.02% 0.0218 0.0187 14.22%
    \varepsilon {\rm{ = }}{10^{ - 6}} 0.0203 0.0111 45.32% 0.0165 0.0148 10.30% 0.0184 0.0130 29.34%
    \sigma {\rm{ = }}10 \varepsilon {\rm{ = }}{10^{ - 1}} 0.0128 0.0105 17.96% 0.0107 0.0083 22.42% 0.0117 0.0094 19.65%
    \varepsilon {\rm{ = }}{10^{ - 2}} 0.0150 0.0140 6.67% 0.0147 0.0131 18.88% 0.0148 0.0136 8.11%
    \varepsilon {\rm{ = }}{10^{ - 4}} 0.0133 0.0092 30.82% 0.0122 0.0108 11.48% 0.0127 0.0100 21.26%
    \varepsilon {\rm{ = }}{10^{ - 6}} 0.0190 0.0169 11.05% 0.0155 0.0116 25.16% 0.0173 0.0142 17.92%
    \sigma {\rm{ = }}15 \varepsilon {\rm{ = }}{10^{ - 1}} 0.0081 0.0071 12.35% 0.0083 0.0061 26.50% 0.0082 0.0066 19.51%
    \varepsilon {\rm{ = }}{10^{ - 2}} 0.0092 0.0074 19.56% 0.0077 0.0055 28.57% 0.0084 0.0064 23.80%
    \varepsilon {\rm{ = }}{10^{ - 4}} 0.0080 0.0069 13.75% 0.0080 0.0057 28.75% 0.0080 0.0063 21.25%
    \varepsilon {\rm{ = }}{10^{ - 6}} 0.0086 0.0073 15.11% 0.0084 0.0053 36.90% 0.0085 0.0063 25.88%
     | Show Table
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    Consider the nonlinear non-Gaussian system shown in Eq.(8) and Eq.(9).where

    \begin{split} {f_1}(x(u)) =& {x_1}(u) + {x_2}(u) - \frac{1}{6}x_1^3(u) - \frac{1}{6}x_2^3(u)\\ &+ \frac{1}{{120}}x_1^5(u) + \frac{1}{{120}}x_2^5(u) \\ {f_2}(x(u)) =& {x_1}(u) - \frac{1}{2}x_1^2(u) - \frac{1}{2}x_2^2(u) + \frac{1}{{24}}x_1^4(u) \\ &+ \frac{1}{{24}}x_2^4(u)\\ {h_1}(x(u & + 1)) = {x_1}(u + 1) - x_1^3(u + 1)\\ {h_2}(x(u & + 1)) = {x_2}(u + 1) - x_2^3(u + 1) \end{split}

    The process noises are uncorrelated Gaussian white noises but the observation process noises are non-Gaussian noise with a mixed-Gaussian distribution. The simulation is performed many times for 1−500, where v(u+1) \sim 0.8N(0,0.01) + 0.2N(0,0.1) for the first 200 times and v(u + 1)\sim 0.8N (0,0.01) + 0.2N(0,0.1) + {[\begin{array}{*{20}{c}} {0.1}&{0.1} \end{array}]^{\rm{T}}} for the remaining; {w}(u) \sim N(0,0.01) ; all high-order polynomials are regarded as implicit values,and their modeling errors satisfies that the mean is 0 and the variance is 0.01; x(0) = {[1\;\;1]^{\rm{T}}} , \hat x(0|0) = {[1\;\;1]^{\rm{T}}} + N(0,0.01) \times randn(2,1) and P(0|0) = 0.01 \times diag(1,1)

    Case 2 only considers the measurement noise is non- Gaussian and takes \sigma = 10,\varepsilon = {10^{ - 1}} as an example to compare the filtering performance between MCEKF and H-MCEKF, as shown in Fig.2(a)(d)). Table 2 describes the error comparison in detail under several \sigma and \varepsilon .

    Figure  2.  The actual state x_1 and its estimate (a) with estimate error (c); The actual state x_2 and its estimate (b) with estimate error (d)
    Table  2.  Estimation errors with different filters
    MSE of {x_{\rm{1}}}MSE of {x_{\rm{2}}}MSE
    \sigma\varepsilon MCEKFH-MCEKFImprovedMCEKFH-MCEKFImprovedMCEKFH-MCEKFImproved
    \sigma {\rm{ = }}2\varepsilon {\rm{ = }}{10^{ - 1}}0.58000.135176.70% 0.05470.036333.64% 0.31730.085772.99%
    \varepsilon {\rm{ = }}{10^{ - 2}}0.52460.224657.18%0.05080.041218.89%0.28770.132953.80%
    \varepsilon {\rm{ = }}{10^{ - 4}}0.30860.253517.85%0.11150.045357.56%0.21010.149428.89%
    \varepsilon {\rm{ = }}{10^{ - 6}}0.40880.209548.75%0.07980.061722.68%0.24430.135644.49%
    \sigma {\rm{ = }}5\varepsilon {\rm{ = }}{10^{ - 1}}0.19810.19501.56%0.0299 0.02874.01%0.11400.1119 1.82%
    \varepsilon {\rm{ = }}{10^{ - 2}}0.23360.184321.10%0.03410.02710.53%0.13380.105721.00%
    \varepsilon {\rm{ = }}{10^{ - 4}}0.19970.18049.66% 0.04610.034325.59% 0.11700.11333.16%
    \varepsilon {\rm{ = }}{10^{ - 6}}0.22680.191015.78%0.04610.024546.85%0.12560.11855.65%
    \sigma {\rm{ = }}10\varepsilon {\rm{ = }}{10^{ - 1}}0.51570.179565.19%0.03550.027223.38%0.27560.103462.48%
    \varepsilon {\rm{ = }}{10^{ - 2}}0.58880.133677.31%0.02740.02730.36%0.30810.080573.87%
    \varepsilon {\rm{ = }}{10^{ - 4}}0.55170.166769.78%0.03110.026016.39%0.29140.096466.92%
    \varepsilon {\rm{ = }}{10^{ - 6}}0.58620.249757.40%0.04280.04094.44% 0.31450.145353.80%
    \sigma {\rm{ = }}15\varepsilon {\rm{ = }}{10^{ - 1}}0.53260.268149.66%0.04960.04617.06% 0.29110.157146.03%
    \varepsilon {\rm{ = }}{10^{ - 2}}0.52940.238354.98%0.03680.033110.05%0.28310.135752.06%
    \varepsilon {\rm{ = }}{10^{ - 4}}0.54610.156671.32%0.03730.03467.23%0.29170.095667.22%
    \varepsilon {\rm{ = }}{10^{ - 6}}0.51090.209259.05%0.03260.021234.96%0.27180.115257.61%
     | Show Table
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    Case 1 and Case 2 take \sigma = 10,\varepsilon = {10^{ - 1}} as examples to express the filtering performance of H-MCEKF and MCEKF, respectively. Table 1 and Table 2 summarize the corresponding MSEs of original state variables under different kernel bandwidth \sigma and threshold \varepsilon for H-MCEKF and MCEKF. In the simulation, \sigma is set to 2, 5, 10 and 15 and \varepsilon is set to {10^{ - 1}} , {10^{ - 2}} , {10^{ - 4}} , {10^{ - 6}} . The results confirms that the proposed H-MCEKF can outperform the MCEKF under the assumption that the state noise and measurement noise are non-Gaussian. Besides, we can see compared with the kernel bandwidth \varepsilon , \sigma has little effect.

    A novel maximum correntropy high-order extended Kalman filter (H-MCEKF) has been designed for nonlinear and non-Gaussian systems in this paper. 1) the nonlinear polynomials has been defined as implicit function variables, which transforms the state model into pseudo-linearization. 2) establishing the linear model between all implicit function variables; 3) then the state model have been equivalently rewritten to linear model by combing with original states and implicit variables, which is similar to the measurement model.

    Function variables with additive form can directly employ the new filter for state estimation. For general nonlinear functions, multi-dimensional Taylor network can be employed to develop into additive polynomials. But this method is not perfect, and model uncertainty still exists. Therefore, there is still a lot of work to solve nonlinear non-Gaussian systems.

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