High‐Order Extended Kalman Filter Design for a Class of Complex Dynamic Systems with Polynomial Nonlinearities

Chinese Journal of ElectronicsVolume 30, Issue 3 p. 508-515 Original Research PaperFree Access High-Order Extended Kalman Filter Design for a Class of Complex Dynamic Systems with Polynomial Nonlinearities SUN Xiaohui, Corresponding Author SUN Xiaohui sun_xh1993@163.com School of Automation, Hangzhou Dianzi University, Hangzhou, 310018 ChinaSearch for more papers by this authorWEN Chenglin, Corresponding Author WEN Chenglin wencl@hdu.edu.cn School of Automation, Hangzhou Dianzi University, Hangzhou, 310018 China School of Automation, Guangdong University of Petrochemical Technology, Maoming, 525000 ChinaSearch for more papers by this authorWEN Tao, Corresponding Author WEN Tao wentao@bjtu.edu.cn School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, 100044 ChinaSearch for more papers by this author SUN Xiaohui, Corresponding Author SUN Xiaohui sun_xh1993@163.com School of Automation, Hangzhou Dianzi University, Hangzhou, 310018 ChinaSearch for more papers by this authorWEN Chenglin, Corresponding Author WEN Chenglin wencl@hdu.edu.cn School of Automation, Hangzhou Dianzi University, Hangzhou, 310018 China School of Automation, Guangdong University of Petrochemical Technology, Maoming, 525000 ChinaSearch for more papers by this authorWEN Tao, Corresponding Author WEN Tao wentao@bjtu.edu.cn School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, 100044 ChinaSearch for more papers by this author First published: 01 May 2021 https://doi.org/10.1049/cje.2021.04.004 This work is supported by the National Natural Science Foundation of China (No.61751304, No.61806064, No.61933013, No.U1664264) and Science and Technology Project of China Electric Power Research Institute (No.SGHB0000KXJS1800375). AboutSectionsPDF 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 onFacebookTwitterLinked InRedditWechat Abstract A novel High-order extended Kalman Filter (HEKF) is designed for a class of complex dynamic systems with polynomial nonlinearities. The state and measurement models are represented by multi-dimensional high-order polynomials, respectively. All high-order polynomials in the state model are defined as implicit variables. By combining original variables with implicit variables, the state model is equivalently formulated to be a pseudo-linear form. By modeling dynamic relationship between implicit variables and combining original variables with all implicit variables, a new linear augmented state model is established correspondingly. The measurement model can be equivalently formulated as a linear form. On the basis of the new linear state and linear measurement models, the HEKF is designed and derived in detail. Simulation results demonstrate the effectiveness of the proposed estimator. I. Introduction The application of filters is irreplaceable in various fields at home and abroad. The development and progress of filters have played an important role in the construction of the national economy, especially in the national defense construction, such as real-time state estimation and target tracking[1–4]. In 1942, Weiner et al. proposed famous Wiener filter based on the minimum variance estimation[5]. However, due to the non-recursive characteristics, difficulty in solving and only applicability for stationary random processes, Wiener filter has not been widely used[6]. In 1960, Kalman et al. proposed famous Kalman filter (KF)[7], because it possesses good characteristics, such as real-time, recursion and optimality under the minimum variance criterion, KF has been widely used in various fields, especially in military field. However, KF will also encounter difficult problems in application, like non-linearity or non-Gaussian. In order to make KF applicable to a wider range of nonlinear systems, in 1969, Bucy used Taylor expansion to develop a filtering algorithm, named Extended Kalman filter (EKF)[8]. However, because all high-order terms in the Taylor expansion are discarded, the filtering performance will decline and even diverge while dealing with strongly nonlinear system[9]. In order to solve the problem met by EKF, Julier proposed a new filtering algorithm by combing UT transform and sigma sampling, named Unscented Kalman filter (UKF)[10]. UKF is deterministic sampling. For example, for a given n-dimensional state system, 2 n + 1 sampling points are required. Therefore, for low-dimensional systems, it is difficult to reflect superiority. Inevitably, it faces the same shortcomings as EKF, that is, it fails with the increase of nonlinearity[11]. Due to above problems, Ienkaram proposed a new filtering algorithm with the idea of integration, named Cubature Kalman filter (CKF)[12,13]. Compared with UKF, CKF uses spherical integration and radial integration to approximate the nonlinear function. The essence is to optimize the point-taking method and weight selection in UKF, and improve the stability of the filtering algorithm. A large number of simulations show that either UKF or EKF can achieve the second-order polynomial approximation at most. But as the nonlinearity increases, the same problem will be encountered: performance degradation and divergence. After CKF, the research on EKF has not made substantial progress until Polynomial extended Kalman filter (PEKF) came out in 1991[14]. PEKF employs polynomial approximation to deal with the non-linearity, which is a further improvement of EKF[14–20]. That is because, comparing with EKF which is only concerned about the linear term and simply ignores the linearization error, PEKF in Ref.[14] employed all higher-order terms from 1 to u order of Carleman approximation for a nonlinear system. When μ = 1 , PEKF degenerated into EKF. In Ref.[15], another PEKF was designed to cope with an augmented state which is made of Kronecker powers of the original state, so it achieved better filter performance than that in Ref.[14]. Furthermore, Wang designed one more PEKF for fault diagnosis by analyzing and employing more information from rounding error in Ref.[16], so it obtained better capability than that in Ref.[15]. With the continuous improvement of PEKF performance, they used more and more nonlinear information, many corresponding results have been reported in Refs.[17–20]. Although PEKF has made a breakthrough, but it is difficult for ordinary readers to read and understand, let alone reproduce and apply. The reason is that the expansion form of the nonlinear function is difficult to understand, and Taylor expansion is required at each step, which causes the coefficients to be recalculated. Moreover, the analysis shows that whether simple or complex function, complicate methods are required such as Carleman approximation or Kronecker powers, which does not consider the specific characteristics of the function. Therefore, in order to overcome the complexity, such as complex Taylor expansion and recalculations, we propose a novel HEKF under the assumption that state and measurement functions can be modeled as polynomials. It should be emphasized that it is easy to convert a general high-dimensional derivable nonlinear function into polynomials by Multi-dimensional Taylor Network[21]. Without loss of generality, a class of polynomial nonlinear dynamic systems is studied to establish a novel HEKF. The main contribution of the paper can be highlighted as follows: 1) The state model is equivalently formulated to be pseudo-linear by regarding all high-order terms in the polynomial model as the implicit variables, which avoids rounding of truncation errors; 2) By combining original states and implicit variables, a linear augmented state model is established; 3) Corresponding the new state model, the measurement model is equivalently rewritten into a linear form; 4) A new high-order extended Kalman filter is designed to realize the joint estimation of state and implicit variables; and 5) Comparing with EKF, the novel HEKF has the same properties: recursion and real-time. The rest of the paper is organized as follows. In Section II, a description of strongly nonlinear dynamic system is given. In Section III, a pseudo-linearized representation based on high-order polynomials is shown. A linearized representation based on augmented state is established in Section IV. A high-order extended filter for polynomial nonlinear systems is designed in Section V. Simulation examples are illustrated in Section VI and the paper is concluded in Section VII. II. A Description of the Strongly Nonlinear Dynamic System Consider the following class of stochastic discrete-time nonlinear systems: x ( k + 1 ) = f ( x ( k ) ) + w ( k ) (1) y ( k + 1 ) = h ( x ( k + 1 ) ) + v ( k + 1 ) (2) where, x ( k ) is the system state; y ( k ) is the measurement output; x ( k + 1 ) = x 1 ( k + 1 ) x 2 ( k + 1 ) , f ( x ( k ) ) = f 1 ( x ( k ) ) f 2 ( x ( k ) ) , y ( k + 1 ) = y 1 ( k + 1 ) y 2 ( k + 1 ) , h ( x ( k + 1 ) ) = h 1 ( x ( k + 1 ) ) h 2 ( x ( k + 1 ) ) , w ( k ) = w 1 ( k ) w 2 ( k ) , v ( k + 1 ) = v 1 ( k + 1 ) v 2 ( k + 1 ) Hypothesis 1.The state transition function f i ( x ( k ) ) , i = 1 , 2 has a continuous smooth derivative of order r for the multidimensional state variable x ( k ) ; state observation function h i ( x ( k + 1 ) ) , i = 1 , 2 has a continuous smooth derivative of order r for the multidimensional state variable x ( k + 1 ) . Hypothesis 2.The state noise w ( k ) and the output noise v ( k + 1 ) are uncorrelated zero-mean sequences. And E { w ( k ) } = 0 , E w ( k ) w T ( j ) = Q ( k ) δ k j ; E { v ( k + 1 ) } = 0 , E v ( k ) v T ( j ) = R ( k ) δ k j . Q ( k ) is a positive semi-definite matrix, and R ( k + 1 ) is a positive definite matrix. When k = j , δ k j = 1 ; else, δ k j = 0 . Hypothesis 3.The initial state x ( 0 ) is random and independent of the noises. It satisfies E { x ( 0 ) } = x 0 , E { [ x ( 0 ) − x 0 ] [ x ( 0 ) − x 0 ] T } = P 0 , P 0 is a positive definite matrix. E { w ( k ) v T ( k + 1 ) } = 0 , E { x ( 0 ) w T ( k ) } = 0 , E { x ( 0 ) v T ( k + 1 ) } = 0 . III. A Pseudo-Linearized Representation Based on High-Order Polynomials This section gives a polynomial description of the state transition function for the above strongly nonlinear system with second-order variables, we shall also give an introduction of higher-order implicit variables, and the pseudo-linearized representations of state measurement function. 1. Pseudo-linearized representation of state transition function f i ( x ( k ) ) A polynomial representation of the state transition function f i ( x ( ( k ) ) in Eq.(1) is as follows f i ( ( x ( k ) ) = a i , 0 + a i , 1 , 0 x 1 ( k ) + a i , 0 , 1 x 2 ( k ) + a i , 2 , 0 x 1 2 ( k ) + a i , 1 , 1 x 1 ( k ) x 2 ( k ) + a i , 0 , 2 x 2 2 ( k ) + a i , 3 , 0 x 1 3 ( k ) + a i , 2 , 1 x 1 2 ( k ) x 2 ( k ) + ⋯ + a i , 0 , 3 x 2 3 ( k ) + ⋯ + ∑ l 1 + l 2 = l l 1 , l 2 ≤ l a i , l 1 , l 2 x 1 l 1 ( k ) x 2 l 2 ( k ) + ⋯ + ∑ r 1 + r 2 = r r 1 , r 2 ≤ r a i , r 1 , r 2 x 1 r 1 ( k ) x 2 r 2 ( k ) (3) where, a i , 0 is a constant. ∑ l 1 + l 2 = l l 1 , l 2 ≤ l a i , l 1 , l 2 x 1 l 1 ( k ) x 2 l 2 ( k ) , 0 ≤ l 1 , l 2 ≤ l ; l = 1 , 2 , … , r (4) is the weighted sum of all l order tensor terms a i , l 1 , l 2 , l 1 + l 2 = l ; 0 ≤ l 1 , l 2 ≤ l ; l = 1 , 2 , … , r (5) is the corresponding weight value. Definition 1.Let x ( l ) ( k ) : = x 1 l 1 ( k ) x 2 l 2 ( k ) , l 1 + l 2 = l ; 0 ≤ l j ≤ l ; l = 1 , 2 , … , r (6) be a set of l order implicit variables corresponding to the original system variable x ( k ) . Definition 2.The weight vector corresponding to the implicit variables vector of l order is a i ( l ) : = a i ; 1 ( l ) , a i ; 2 ( l ) , … , a i ; n l ( l ) = a i ; l , 0 , a i ; l − 1 , 1 , … , a i ; 0 , l , i = 1 , 2 (7) According to the Definitions 1, 2 and Eq.(3), the pseudo-linearization of polynomial state model Eq.(1) is expressed as x 1 ( 1 ) ( k + 1 ) x 2 ( 1 ) ( k + 1 ) = a 1 ( 1 ) a 1 ( 2 ) ⋯ a 1 ( l ) ⋯ a 1 ( r ) a 2 ( 1 ) a 2 ( 2 ) ⋯ a 2 ( l ) ⋯ a 2 ( r ) × x ( 1 ) ( k ) x ( 2 ) ( k ) ⋮ x ( l ) ( k ) ⋮ x ( r ) ( k ) + w 1 ( 1 ) ( k ) w 2 ( 1 ) ( k ) (8) Denote x ( k ) : = x ( 1 ) ( k ) = x 1 ( 1 ) ( k ) x 2 ( 1 ) ( k ) , A ( l ) : = a 1 ( l ) a 2 ( l ) , w ( 1 ) ( k ) : w ( k ) = w 1 ( 1 ) ( k ) w 2 ( 1 ) ( k ) then the matrix form of Eq.(8) is x ( 1 ) ( k + 1 ) = [ A ( 1 ) A ( 2 ) ⋯ A ( l ) ⋯ A ( r ) ] x ( 1 ) ( k ) x ( 2 ) ( k ) ⋮ x ( l ) ( k ) ⋮ x ( r ) ( k ) + w ( 1 ) ( k ) = A ( 1 ) x ( 1 ) ( k ) + ∑ l = 2 r A ( l ) x ( l ) ( k ) + w ( 1 ) ( k ) (9) Remark 1.Compared with the original state model x ( 1 ) ( k ) = [ x 1 ( k ) x 2 ( k ) ] T , the pseudo-linear form of Eq.(9) with the higher-order implicit variable x ( l ) ( k ) , l = 2 , 3 , … , r is time-varying. There is only a change in representation form and no essential difference, and hence it is called pseudo linearization. 2. Pseudo-linearized representation of measurement functions h i ( x ( k + 1 ) ) A polynomial representation of the measurement function h i ( x ( k + 1 ) ) in Eq.(2) is as follows h i ( x ( 1 ) ( k + 1 ) ) = h i , 0 + h i , 1 , 0 x 1 ( k + 1 ) + h i , 0 , 1 x 2 ( k + 1 ) + h i , 2 , 0 x 1 2 ( k + 1 ) + ⋯ + h i , 0 , 2 x 2 2 ( k + 1 ) + h i , 3 , 0 x 1 3 ( k + 1 ) + ⋯ + h i , 0 , 3 x 2 3 ( k + 1 ) + ⋯ + ∑ l 1 + l 2 = l l 1 , l 2 ≤ l h i , l 1 , l 2 x 1 l 1 ( k + 1 ) x 2 l 2 ( k + 1 ) + ⋯ + ∑ r 1 + r 2 = r r 1 , r 2 ≤ r h i , r 1 , r 2 x 1 r 1 ( k + 1 ) x 2 r 2 ( k + 1 ) (10) According to the Definitions 1 and 2 and Eqs.(4–9), the pseudo-linearization of polynomial measurement model Eq.(2) is expressed as y 1 ( 1 ) ( k + 1 ) y 2 ( 1 ) ( k + 1 ) = h 1 ( 1 ) h 1 ( 2 ) ⋯ h 1 ( l ) ⋯ h 1 ( r ) h 2 ( 1 ) h 2 ( 2 ) ⋯ h 2 ( l ) ⋯ h 2 ( r ) × x ( 1 ) ( k + 1 ) x ( 2 ) ( k + 1 ) ⋮ x ( l ) ( k + 1 ) ⋮ x ( r ) ( k + 1 ) + v 1 ( 1 ) ( k + 1 ) v 2 ( 1 ) ( k + 1 ) (11) then the matrix form of Eq.(11) is y ( 1 ) ( k + 1 ) = [ H ( 1 ) H ( 2 ) ⋯ H ( l ) ⋯ H ( r ) ] × x ( 1 ) ( k + 1 ) x ( 2 ) ( k + 1 ) ⋮ x ( l ) ( k + 1 ) ⋮ x ( r ) ( k + 1 ) + v ( 1 ) ( k + 1 ) = H ( 1 ) x ( 1 ) ( k + 1 ) + ∑ l = 2 r H ( l ) x ( l ) ( k + 1 ) + v ( 1 ) ( k + 1 ) (12) IV. Linearized Representation of Polynomial Nonlinear Dynamic System Based on Augmented State In order to convert the pseudo-linearized model established in Section III into a true linearized form, it is necessary to expand the dimension by treating high-order implicit variables as new variables of the system. 1. Linearized representation of polynomial nonlinear state model based on augmented state To solve the problem in Remark 1, a linear coupling relationship between l order implicit variable x ( l ) ( k + 1 ) and u order implicit variable x ( u ) ( k ) is established by considering x ( l ) ( k ) as a time-varying parameter. x ( l ) ( k + 1 ) = A l ( u ) ( k ) x ( u ) ( k ) l , u = 2 , 3 , … , r (13) then, the implicit variable is used as an extension of the original state vector to achieve a linear representation of the state model. A l ( u ) ( k ) can be identified based on the input information of the original state model; when there is no priori information, it is set as follows. A l ( u ) ( k ) = I , l = u 0 , l ≠ u (14) According to Eqs.(6)–(9), the state model has the following linearized matrix form x ( 1 ) ( k + 1 ) x ( 2 ) ( k + 1 ) ⋮ x ( l ) ( k + 1 ) ⋮ x ( r ) ( k + 1 ) = A 1 ( 1 ) A 1 ( 2 ) ⋯ A 1 ( u ) ⋯ A 1 ( r ) A 2 ( 1 ) A 2 ( 2 ) ⋯ A 2 ( u ) ⋯ A 2 ( r ) ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ A l ( 1 ) A l ( 2 ) ⋯ A l ( u ) ⋯ A l ( r ) ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ A r ( 1 ) A r ( 2 ) ⋯ A r ( u ) ⋯ A r ( r ) × x ( 1 ) ( k ) x ( 2 ) ( k ) ⋮ x ( l ) ( k ) ⋮ x ( r ) ( k ) + w ( 1 ) ( k ) w ( 2 ) ( k ) ⋮ w ( l ) ( k ) ⋮ w ( r ) ( k ) (15) Denote X ( k ) = ( x ( 1 ) ( k ) ) T , ( x ( 2 ) ( k ) ) T , ⋯ , ( x ( r ) ( k ) ) T T A ( k + 1 , k ) = A 1 ( 1 ) ( k ) A 1 ( 2 ) ( k ) ⋯ A 1 ( u ) ( k ) ⋯ A 1 ( r ) ( k ) A 2 ( 1 ) ( k ) A 2 ( 2 ) ( k ) ⋯ A 2 ( u ) ( k ) ⋯ A 2 ( r ) ( k ) ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ A l ( 1 ) ( k ) A l ( 2 ) ( k ) ⋯ A l ( u ) ( k ) ⋯ A l ( r ) ( k ) ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ A r ( 1 ) ( k ) A r ( 2 ) ( k ) ⋯ A r ( u ) ( k ) ⋯ A r ( r ) ( k ) W ( k ) = w ( 1 ) ( k ) w ( 2 ) ( k ) ⋮ w ( l ) ( k ) ⋮ w ( r ) ( k ) then the linearized representation of the strongly nonlinear state model Eq. (1) is in the following form. X ( k + 1 ) = A ( k + 1 , k ) X ( k ) + W ( k ) (16) 2. Linearized representation of polynomial nonlinear measurement model based on augmented state Based on the above-mentioned augmented state variables, in order to achieve a linear representation of the measurement model, the augmented state variables are regarded as new variables measured under the new system; and the pseudo-measurement model Eq.(12) is equivalently rewritten as follows. y 1 ( k + 1 ) y 2 ( k + 1 ) = h 1 ( 1 ) h 1 ( 2 ) ⋯ h 1 ( u ) ⋯ h 1 ( r ) h 2 ( 1 ) h 2 ( 2 ) ⋯ h 2 ( u ) ⋯ h 2 ( r ) × x ( 1 ) ( k + 1 ) x ( 2 ) ( k + 1 ) ⋮ x ( l ) ( k + 1 ) ⋮ x ( r ) ( k + 1 ) + v 1 ( k + 1 ) v 2 ( k + 1 ) (17) Denote H ( k + 1 ) = h 1 ( 1 ) h 1 ( 2 ) ⋯ h 1 ( u ) ⋯ h 1 ( r ) h 2 ( 1 ) h 2 ( 2 ) ⋯ h 2 ( u ) ⋯ h 2 ( r ) = H ( 1 ) H ( 2 ) ⋯ H ( u ) ⋯ H ( r ) H ( u ) = h 1 ( u ) h 2 ( u ) , V ( k + 1 ) = v 1 ( k + 1 ) v 2 ( k + 1 ) then the linearized representation of nonlinear measurement model Eq.(2) is of the following form. Y ( k + 1 ) = H ( k + 1 ) X ( k + 1 ) + V ( k + 1 ) (18) V. High-Order Extended Kalman Filter Design for Polynomial Nonlinear Systems Given the linearized state model and measurement model shown in Eq.(16) and Eq.(18) Deduction 1. E { W ( k ) W T ( k ) } = Q W ( k ) (19) E { V ( k + 1 ) V T ( k + 1 ) } = R V ( k + 1 ) (20) E { W ( k ) V T ( k + 1 ) } = 0 (21) Step 1.Set initial value of the new system.According to the initial value x ( 0 ) of x ( k ) which satisfies the following equation. E { x ( 0 ) } = x ^ 0 = x ^ 10 , x ^ 20 T (22) E { [ x ( 0 ) − x 0 ] [ x ( 0 ) − x 0 ] T } = P 0 = d i a g p 1 , 0 , p 2 , 0 (23) then X ( 0 ) satisfies the following characteristics. X ( 0 ) = ( x ( 1 ) ( 0 ) ) T , ( x ( 2 ) ( 0 ) ) T , … , ( x ( r ) ( 0 ) ) T T (24) E { X ( 0 ) } = X ^ 0 = ( x ^ 0 ( 1 ) ) T , ( x ^ 0 ( 2 ) ) T , … , ( x ^ 0 ( r ) ) T T (25) where x ^ 0 ( l ) : = x ^ 1 , 0 l 1 x ^ 2 , 0 l 2 , l 1 + l 2 = l ; l = 1 , 2 , … , r E X ( 0 ) − X ^ 0 X ( 0 ) − X ^ 0 T = P ¯ 0 (26) where, P ¯ 0 = d i a g P 0 ( 1 ) , P 0 ( 2 ) , … , P 0 ( l ) , … , P 0 ( r ) ≥ 0 is a positive semi-definite matrix. Step 2.Recursive filtering.Assuming that y ( 1 ) , y ( 2 ) , … , y ( k ) have been obtained, that is, X ^ ( k | k ) and P ¯ ( k | k ) have been known of new system, then the new filter of r − t h order extended Kalman filter at time step k → k + 1 is shown below. X ^ ( k + 1 | k + 1 ) : = E X ( k + 1 ) | X ^ 0 , y ( 1 ) , y ( 2 ) , … , y ( k ) , y ( k + 1 ) = E X ( k + 1 ) | X ^ ( k | k ) , y ( k + 1 ) (27) then, the corresponding estimated error covariance is P ¯ ( k + 1 | k + 1 ) = E [ X ( k + 1 ) − X ^ ( k + 1 | k + 1 ) ] × [ X ( k + 1 ) − X ^ ( k + 1 | k + 1 ) ] T (28) Step 3.Time update.1) According to X ^ ( k | k ) and A ( k + 1 , k ) , X ^ ( k + 1 | k ) is initially obtained . X ^ ( k + 1 | k ) = A ( k + 1 , k ) X ^ ( k | k ) (29)2) According to P ¯ ( k | k ) and Q W ( k ) , P ¯ ( k + 1 | k ) is obtained. P ¯ ( k + 1 | k ) = A ( k + 1 , k ) P ¯ ( k | k ) A T ( k + 1 , k ) + Q W ( k ) (30) Step 4.Measurement update.1) According to P ¯ ( k + 1 | k ) and the related information of the measured values, the gain matrix K ( k + 1 ) can be given. K ( k + 1 ) = P ¯ ( k + 1 | k ) H T ( k + 1 ) × H ( k + 1 ) P ¯ ( k + 1 | k ) H T ( k + 1 ) + R V ( k ) − 1 (31) 2) According to X ^ ( k + 1 | k ) , K ( k + 1 ) and the residual between the actual and predicted observation, a novel high-order extended Kalman filter appears. X ^ ( k + 1 | k + 1 ) = X ^ ( k + 1 | k ) + K ( k + 1 ) × y ( k + 1 ) − H ( k + 1 ) X ^ ( k + 1 | k ) (32) 3) Calculate error covariance matrix P ¯ ( k + 1 | k + 1 ) . P ¯ ( k + 1 | k + 1 ) = [ I − K ( k + 1 ) H ( k + 1 ) ] P ¯ ( k + 1 | k ) (33) VI. Numerical Simulations In this section, two cases are employed to illustrate the effectiveness of the proposed method, respectively: 1) State equation is a high-order polynomial, and measurement equation is a linear model; 2) Both state and the measurement equations are high-order polynomials. 1. Case 1 Consider the following nonlinear system where state model is a high-order polynomial and measurement model is linear. x 1 ( k + 1 ) = x 1 ( k ) − x 2 ( k ) − 1 6 x 1 3 ( k ) − 1 6 x 2 3 ( k ) + 1 120 x 1 5 ( k ) + 1 120 x 2 5 ( k ) + w 1 ( k ) x 2 ( k + 1 ) = 1 − 1 2 x 1 2 ( k ) − 1 2 x 2 2 ( k ) + 1 24 x 1 4 ( k ) + 1 24 x 2 4 ( k ) + w 2 ( k ) (34) y 1 ( k + 1 ) = x 1 ( k + 1 ) + v 1 ( k + 1 ) y 2 ( k + 1 ) = x 2 ( k + 1 ) + v 2 ( k + 1 ) (35) where, w ( k ) and v ( k ) are uncorrelated zero-mean sequences and w 1 ( k ) ∼ N ( 0 , 0 . 1 ) , w 2 ( k ) ∼ N ( 0 , 0 . 1 ) , v 1 ( k + 1 ) ∼ N ( 0 , 0 . 1 ) , v 2 ( k + 1 ) ∼ N ( 0 , 0 . 1 ) . Figs.1 and 2 show the actual states and the estimates of state variables x 1 and x 2 under EKF and HEKF. Fig.3 shows the estimated error, and mean squared error of x 1 and x 2 is given in Table 1. Fig. 1Open in figure viewerPowerPoint The actual state x 1 and its estimate It can be seen from Figs.1–3 and Table 1, compared with EKF, HEKF can increase the accuracy by 12.7% and 6.9%, respectively, and entirety increases by 17.5%, which further illustrates the effectiveness of the proposed method. Fig. 2Open in figure viewerPowerPoint The actual state x 2 and its estimate Fig. 3Open in figure viewerPowerPoint Estimate error Table 1. Error comparison between HEKF and EKF Filter MSE of x 1 MSE of x 2 MSE EKF 0.0691 0.0626 0.0755 HEKF 0.0603 0.0583 0.0623 Improved 12.7% 6.9% 17.5% 2. Case 2 Consider the following nonlinear system where state and measurement models are both high-order polynomials. x 1 ( k + 1 ) = x 1 ( k ) − x 2 ( k ) − 1 6 x 1 3 ( k ) − 1 6 x 2 3 ( k ) + 1 120 x 1 5 ( k ) + 1 120 x 2 5 ( k ) + w 1 ( k ) x 2 ( k + 1 ) = 1 − 1 2 x 1 2 ( k ) − 1 2 x 2 2 ( k ) + 1 24 x 1 4 ( k ) + 1 24 x 2 4 ( k ) + w 2 ( k ) (36) y ( k + 1 ) = x 1 ( k + 1 ) + x 2 ( k + 1 ) − 1 6 x 1 3 ( k + 1 ) − 1 6 x 2 3 ( k + 1 ) − 1 2 x 1 2 ( k + 1 ) x 2 ( k + 1 ) − 1 2 x 1 ( k + 1 ) x 2 2 ( k + 1 ) + v ( k + 1 ) (37) where, w ( k ) and v ( k ) are uncorrelated zero-mean sequences and w 1 ( k ) ∼ N ( 0 , 0 . 01 ) , w 2 ( k ) ∼ N ( 0 , 0 . 01 ) , v ( k + 1 ) ∼ N ( 0 , 0 . 01 ) . Figs.4 and 5 show the actual states and the estimates of state variables x 1 and x 2 under EKF and HEKF. Fig.6 shows the estimated error, and mean squared error of x 1 and x 2 is given in Table 2. Fig. 4Open in figure viewerPowerPoint The actual state x 1 and its estimate Fig. 5Open in figure viewerPowerPoint The actual state x 2 and its estimate Fig. 6Open in figure viewerPowerPoint Estimate error Table 2. Error comparison between HEKF and EKF Filter MSE of x 1 MSE of x 2 MSE EKF 0.0557 0.0559 0.0555 HEKF 0.0413 0.0427 0.0398 Improved 25.9% 23.6% 28.3 It can be seen from Figs.4–6 and Table 2, compared with EKF, HEKF can increase the accuracy by 25.9% and 23.6%, respectively, and entirety increases by 28.3%, which further illustrates the effectiveness of the proposed method. In this section, two simulation examples are given to illustrate that HEKF has a better estimated effect and is more suitable for complex nonlinear systems. VII. Conclusions In this paper, a novel high-order extended Kalman filter is designed for a dynamic stochastic system composed of strongly nonlinear state and measurement models which are expressed in polynomial forms. First, the high-order polynomial in the state and measurement model are defined as implicit variables; second, the state and measurement model of the original system are rewritten to be pseudo-linear; again, by coupling between implicit variables and combining original variables with implicit variables, an augmented linear state model that combines original states and parameters is established and measurement model is equivalently rewritten into linear form; finally, with the help of joint estimation of state and parameters, the detailed design process of HEKF is shown, and its effectiveness is verified by simulations. For easy understanding, this paper gives the design idea of the HEKF in the form of two-dimensional variables. This idea can be generalized to general nonlinear functions, but how to represent the strong nonlinear functions with global equivalent polynomials is a problem that needs to be solved in the future. Biographies SUN Xiaohui was born in 1993. She received B.E. degree from Harbin University of Science and Technology. She is currently a Ph.D. candidate in Hangzhou Dianzi University. Her research interest is filter design. WEN Chenglin was born in 1963. He graduated from Henan University in 1986, graduated from Zhengzhou University with a M.S. degree in 1993 and received a Ph.D. degree 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. WEN Tao (corresponding author) received the B.E. degree in computer science from Hangzhou Dianzi University in 2011, the M.S. 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. References 1Z.H. Li, L. Ning and S.N. Xu, "Nonlinear non-Gaussian system filtering based on Gaussian sum and divided difference filter", Control and Decision, Vol. 27, No.1, pp. 129– 134, 2012. (in Chinese) CASGoogle Scholar 2Q.B. Ge, T.L. Xu, X.L. 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