A Novel Step‐by‐Step High‐Order Extended Kalman Filter Design for a Class of Complex Systems with Multiple Basic Multipliers

Chinese Journal of ElectronicsVolume 30, Issue 2 p. 313-321 Original Research PaperFree Access A Novel Step-by-Step High-Order Extended Kalman Filter Design for a Class of Complex Systems with Multiple Basic Multipliers 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 March 2021 https://doi.org/10.1049/cje.2021.02.005 This work is supported by the National Natural Science Foundation of China (No.61751304, No.61806064, No.6193301, No.61703385, 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 step-by-step linearization high-order Extended Kalman filter SH-EKF is designed for a class of nonlinear systems composed of linear functions and the product of several separable basic functions. The basic functions in the state and measurement models are defined as latent variables; the state and measurement models are equivalently formulated into pseudo-linear models based on the combination of the original variable and the latent variables; latent variables are regarded as new variables, and a dynamic linear model between each latent variable and other latent variables with original state is established; the measurement model is rewritten into the first-order linear product form between the current state and each latent variable; latent variables are solved by Kalman filter step by step, and a stepwise linearized high-order extended Kalman filter is designed. Illustration examples are presented to demonstrate the effectiveness of the new algorithm. I. Introduction State estimation theory has been widely used in various fields, such as space monitoring, wireless communication, defense tracking, etc. and plays an important role[1-5]. Filter design, the core of state estimation theory, has undergone a continuous improvement since it was proposed. In 1942, Weiner et al. based on the minimum mean square error, proposed Wiener Filter[6]. However, Wiener filter is only applicable for stationary random processes, which makes it difficult to be widely applied in existing non-stationary random systems[7]. In 1960, Kalman et al. proposed Kalman filter (KF)[8]. KF is suitable for non-stationary random processes, and has excellent characteristics of real-time recursion. Therefore, it has been rapidly promoted and applied in various fields, especially in national defense. KF is an optimal filter designed with the minimum mean square error as the performance indicators under the premise that the model is linear and the error is white noise[9]. In 1969, Bucy developed a filter suitable for nonlinear systems: Extended Kalman filtering (EKF) by Taylor expansion[10]. EKF is the first-order Taylor expansion of the nonlinear state and measurement models under the current estimated point, so that the nonlinear function is approximately converted into a linear problem. However, EKF can only achieve first-order linear approximation, and will increase the rounding error as the model's nonlinearity increases. When dealing with strong non-linear problems, it will result in decline of filtering performance and even the loss of tracking target[11]. In order to improve the accuracy of the nonlinear model, UKF and CKF have been successively designed[12-15]. A large number of simulations show that either UKF or EKF can at most achieve the second-order polynomial approximation. 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[16]. PEKF employs polynomial approximation to deal with the non-linearity, which is a further improvement of EKF[16-22]. That is because, comparing with traditional EKF which is only concerned about the linear term and simply ignores the linearization error, PEKF in Ref.[16] employed all higher-order terms from 1 to u order of Carleman approximation for a nonlinear system. When μ=1, PEKF degenerated into conventional EKF. In Ref.[17], another PEKF was designed to cope with an augmented state which is made of Kronecker powers of the original state, so it achieved better filtering performance than that in Ref.[16]. Furthermore, Wang designed one more PEKF for fault diagnosis by analyzing and employing more information from rounding error in Ref.[18], so it obtained better capability than that in Ref.[17]. With the continuous improvement of PEKF performance, they used more and more nonlinear information, many corresponding results have been reported in Refs.[19-22]. Although PEKF has made a breakthrough, the expansion form of the nonlinear function is difficult to understand. Taylor expansion is required at each step, which causes the coefficients to be recalculated. As the dimensionality of the system gets higher and higher, so does the complexity, and this complexity is independent of the nonlinearity of the system. 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. Whether it is EKF or PEKF, it is centered on the latest state estimate, and polynomial series are employed to approximate the original model of the state and measurement near the expansion point. The advantage is that the format is consistent and easy to understand. However, there are still shortcomings in practical applications. At every moment, the estimated value must be centered to rewrite the polynomials of all orders in the expansion as well as the recalculation of coefficients. But the complex systems encountered in reality, such as Ref.[18], state and measurement models are all nonlinear functions composed of linear functions, nonlinear functions, or products of linear functions and nonlinear basic functions, etc. in the form of summation. And each part owns its specific physical significance, especially each basic function contains more profound physical significance. The existing approximation in the form of locally expanded polynomial series will inevitably destroy the original physical significance contained in the basic function. Combining with rounding error generated during the expansion of the basic function, it will reduce the filtering performance and even diverge as the degree of nonlinearity increases. A filter with better estimation performance will be designed if the integrity of basic functions can be maintained. Without loss of generality, a novel high order Extended Kalman filter is designed for a class of nonlinear system composed of linear functions and the product of several separable basic functions, such as f(0)(x) * f(1)(x) * f(2)(x) * … * f(r)(x). The main contribution of the paper can be highlighted as follows: 1) r basic functions f(i)(x) are defined as latent variables, that is α(i)(k) := f(i)(x(k)), i = 0, 1, …, r. If α(i)(k) is regarded as a time-varying parameter, the system models can be simplified formally; 2) assuming that the estimated values of α(1)α(2), …, α(i–1) at time k and α(i), α(+1), …, α(r), x at time k – 1 are obtained, the state and measurement models employed to estimate α(i)(k) can be transformed to be linear; 3) when the estimated value of α(1), α(2), …, α (r) at time k are obtained, the original state and measurement models employed to estimate k are approximately rewritten into the linear form of the state variable, and finally real-time update of x is realized by KF. The rest of the paper is organized as follows. In Section II, a description of nonlinear system is given. In Section III, a simplified description based on latent variable functions is presented. Section IV establishes a linear dynamic model description of latent variables. Section V designs a step-by-step linearization high-order extended filter. The effectiveness of the algorithm is illustrated through simulation examples in Section VI. Conclusions are given in Section VII. II. Description of Nonlinear System Consider the following class of stochastic discrete-time nonlinear system: x i ( k + 1 ) = f i ( x ( k ) , k ) + w i ( k ) = ∑ j = 1 n a i j x j ( k ) ∏ l = 1 r f j ( l ) ( x ( k ) ) + w i ( k ) ( 1 ) (1) y i ( k + 1 ) = h i ( x ( k + 1 ) , k + 1 ) + v i ( k + 1 ) = ∑ j = 1 n h i j x j ( k + 1 ) ∏ l = 1 r f j ( l ) ( x ( k + 1 ) ) + v i ( k + 1 ) ( 2 ) (2) where x(k) ∈ Rn is state vector; y(k) ∈ Rm is measurement vector; xi(k) is the ith component of state vector x(k) f j ( l ) ( x ( k ) ) , i , j = 1 , 2 , ⋯ , n ; l = 1 , 2 , ⋯ , r is the lth nonlinear function of the ith component of x(k); w(k) and v(k + 1) are Gaussian white noise sequences, E{w(k)} = 0, E{w(k)wT(j)} = Q(k)δkj; E{v(k + 1)} = 0, E{v(k + 1)vT(j + 1)} = R(k + 1)δkj. Q(k) is a positive semidefinite matrix, and R(k + 1) is a positive definite matrix; if k = j, δkj = 1, else δkj = 0. aij and hij are system coefficients. III. Simplified Description Based on Latent Variable Functions Due to the complexity of the problem, in order to facilitate the description, this and subsequent sections will take m = n = r = 2 as an example for an exemplary description. In this section, latent variables will be introduced to describe the state model Eq.(1) and measurement model Eq.(2) in pseudo-linear form. Let α j ( l ) ( k ) = f j ( l ) ( x ( k ) ) , j = 1 , 2 ; l = 1 , 2 ( 3 ) (3) where α j ( l ) ( k ) is a latent function relative to the original variable x(k). Then, Eq.(1) follows directly that x ( k + 1 ) = A ( α ( 1 ) ( k ) , α ( 2 ) ( k ) , x ( k ) ) + w ( 0 ) ( k ) ( 4 ) (4) where x ( k ) = x 1 ( k ) x 2 ( k ) , w ( 0 ) ( k ) = w 1 ( k ) w 2 ( k ) α ( l ) ( k ) = α 1 ( l ) ( k ) α 2 ( l ) ( k ) and A ( α ( 1 ) ( k ) , α ( 2 ) ( k ) , x ( k ) ) = a 11 α 1 ( 1 ) ( k ) α 1 ( 2 ) ( k ) a 12 α 2 ( 1 ) ( k ) α 2 ( 2 ) ( k ) a 21 α 1 ( 1 ) ( k ) α 1 ( 2 ) ( k ) a 22 α 2 ( 1 ) ( k ) α 2 ( 2 ) ( k ) x 1 ( k ) x 2 ( k ) Corresponding to Eqs.(3) and (4), Eq.(2) follows directly that y ( k + 1 ) = H ( α ( 1 ) ( k + 1 ) , α ( 2 ) ( k + 1 ) ) x ( k + 1 ) + v ( k + 1 ) ( 5 ) (5) where y ( k + 1 ) = y 1 ( k + 1 ) y 2 ( k + 1 ) , x ( k + 1 ) = x 1 ( k + 1 ) x 2 ( k + 1 ) , v ( k + 1 ) = v 1 ( k + 1 ) v 2 ( k + 1 ) , and H ( α ( 1 ) ( k + 1 ) , α ( 2 ) ( k + 1 ) , x ( k + 1 ) ) = h 11 α 1 ( 1 ) ( k + 1 ) α 1 ( 2 ) ( k + 1 ) h 21 α 1 ( 1 ) ( k + 1 ) α 1 ( 2 ) ( k + 1 ) h 12 α 2 ( 1 ) ( k + 1 ) α 2 ( 2 ) ( k + 1 ) h 22 α 2 ( 1 ) ( k + 1 ) α 2 ( 2 ) ( k + 1 ) × x 1 ( k + 1 ) x 2 ( k + 1 ) Note 1: In this section, a complex basic function is defined as a simple latent variable parameter, so that the pseudo-linear description of the original model is realized. IV. Linear Dynamic Model Description of Latent Variables By regarding latent variables as system parameter variables, a dynamic linear model between each latent variable and the remaining latent variables (including state variables) is established. We have α ( l ) ( k + 1 ) = S ( l 0 ) ( k ) x ( k ) + S ( l 1 ) ( k ) α ( 1 ) ( k ) + S ( l 2 ) ( k ) α ( 2 ) ( k ) + w ( l ) ( k ) = S ( l 0 ) ( k ) x ( k ) + ∑ u = 1 2 S ( l u ) ( k ) α ( u ) ( k ) + w ( l ) ( k ) ( 6 ) (6) where w ( l ) ( k ) = w 1 ( l ) ( k ) w 2 ( l ) ( k ) S ( l u ) ( k ) = s 11 ( l u ) s 12 ( l u ) s 21 ( l u ) s 22 ( l u ) s i j ( l u ) in the correlation matrix S(lu)(k) represents the coupling coefficient between α(l)(k + 1) and α(u)(k). Without any prior information, s i j ( l u ) obeys the following rules. s i j ( l u ) ( k ) = I , l = u 0 , l ≠ u ( 7 ) (7) Furthermore, it can be seen from Eqs.(4) and (6) that the joint linearized state representations of the original state variable x(k) and all latent variables α(l)(k) is realized. Meanwhile, the linearized measurement model only for the original state variable x(k) is given in Eq.(5). For the latent variables α(l)(k + 1), Eq.(5) can be equivalently formulated into Eq.(8) by combining the original state variable x(k + 1) and other latent variables α(l)(k + 1), l = 1, 2 as the linearized representation model of the observation matrix parameter. y ( k + 1 ) = H α ( 1 ) ( k + 1 ) , α ( 2 ) ( k + 1 ) x ( k + 1 ) + v ( k + 1 ) = H x ( k + 1 ) , α ( 1 ) ( k + 1 ) α ( 2 ) ( k + 1 ) + v ( k + 1 ) = H x ( k + 1 ) , α ( 2 ) ( k + 1 ) α ( 1 ) ( k + 1 ) + v ( k + 1 ) ( 8 ) (8) Therefore, based on Eqs.(4) and (7), a step-by-step linear observation can be realized on state variable x(k + 1) and latent variables α(l)(k + 1), l = 1, 2. Note 2: Step-by-step linearization means that, due to the separability of the basic function, as each latent variable is determined, the degree of non-linearity of the system gradually decreases until all α(l) is determined; then the original state and the measurement function are transformed into linear forms. V. Step-by-Step Linearization Design of High-Order Extended Kalman Filter Suppose that the observation value y(1), y(2), …, y(k), the estimated value α ^ ( k | k ) , ◯(k|k), the estimated error covariance matrix Pα(k|k) and Px(k|k) at time step k have been obtained, the three-stage step by step Kalman filter is designed as follows. 1) Stage 1: Kalman filter design for α(1)(k + 1) Assuming that the estimated values α ^ ( 1 ) ( k | k ) , α ^ ( 2 ) ( k | k ) and ◯(k|k), the estimated error covariance matrix P α ( 1 ) ( k | k ) , P α ( 2 ) ( k | k ) and Px(k|k) are known, the linear state and measurement model with α(1)(k + 1) as the state variable are respectively as follows. α ( 1 ) ( k + 1 ) = S ( 11 ) ( k ) α ( 1 ) ( k ) + S ( 10 ) ( k ) x ^ ( k | k ) + S ( 12 ) ( k ) α ^ ( 2 ) ( k | k ) + w ( 1 ) ( k ) ( 9 ) (9) y α ( 1 ) ( k + 1 ) = H α ( 1 ) ( x ^ ( k + 1 | k ) , α ^ ( 2 ) ( k + 1 | k ) ) α ( 1 ) ( k + 1 ) + v ( k + 1 ) ( 10 ) (10) Let H α ( 1 ) ( x ^ , α ^ ( 2 ) ) : = H α ( 1 ) ( x ^ ( k + 1 | k ) , α ^ ( 2 ) ( k + 1 | k ) ) , then H α ( 1 ) ( x ^ , α ^ ( 2 ) ) = h 11 x ^ 1 α ^ ( 2 ) h 12 x ^ 2 α ^ ( 2 ) h 21 x ^ 1 α ^ ( 2 ) h 22 x ^ 2 α ^ ( 2 ) The purpose of stage 1 is to obtain the estimated state α ^ ( 1 ) ( k + 1 k + 1 ) and the estimated error covariance matrix P α ( 1 ) ( k + 1 | k + 1 ) . Step 1: For Eqs.(10)–(11), the filter to be designed is of the following form. α ^ ( 1 ) ( k + 1 | k + 1 ) : = E α ( 1 ) ( k + 1 ) | x ^ 0 , y ( 1 ) , y ( 2 ) , ⋯ , y ( k ) , y ( k + 1 ) = E α ( 1 ) ( k + 1 ) | α ^ ( 1 ) ( k | k ) , α ^ ( 2 ) ( k | k ) , x ^ ( k | k ) , y ( k + 1 ) = α ^ ( 1 ) ( k + 1 | k ) + K α ( 1 ) ( k + 1 ) y α ( 1 ) ( k + 1 ) - y ^ α ( 1 ) ( k + 1 | k ) ( 11 ) (11) Step 2: State prediction and prediction error covariance α ^ ( 1 ) ( k + 1 | k ) = S ( 11 ) ( k ) α ^ ( 1 ) ( k | k ) + S ( 10 ) ( k ) x ^ ( k | k ) + S ( 12 ) ( k ) α ^ ( 2 ) ( k | k ) ( 12 ) (12) α ˜ ( 1 ) ( k + 1 | k ) = α ( 1 ) ( k + 1 ) - α ^ ( 1 ) ( k + 1 | k ) = S ( 11 ) ( k ) α ˜ ( 1 ) ( k | k ) + w ( 1 ) ( k ) ( 13 ) (13) P α ( 1 ) ( k + 1 | k ) = E α ( 1 ) ( k + 1 ) - α ^ ( 1 ) ( k + 1 | k ) × α ( 1 ) ( k + 1 ) - α ^ ( 1 ) ( k + 1 | k ) T = S ( 11 ) ( k ) P α ( 1 ) ( k | k ) ( S ( 11 ) ( k ) ) T + Q w ( 1 ) ( k ) ( 14 ) (14) Step 3: Measurement prediction and measurement error y ^ ( k + 1 | k ) = H α ( 1 ) ( x ^ ( k + 1 | k ) , α ^ ( 2 ) ( k + 1 | k ) ) α ^ ( 1 ) ( k + 1 | k ) ( 15 ) (15) y ˜ ( k + 1 | k ) = H α ( 1 ) ( x ^ ( k + 1 | k ) , α ^ ( 2 ) ( k + 1 | k ) ) × [ α ( 1 ) ( k + 1 ) - α ^ ( 1 ) ( k + 1 | k ) ] + v ( k + 1 ) = H α ( 1 ) ( x ^ ( k + 1 | k ) , α ^ ( 2 ) ( k + 1 | k ) ) × α ˜ ( 1 ) ( k + 1 | k ) + v ( k + 1 ) ( 16 ) (16) Step 4: Filter gain K α ( 1 ) ( k + 1 ) = P α ( 1 ) ( k + 1 | k ) H α ( 1 ) ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k ) , α ^ ( 2 ) ( k | k ) ) T × ( H α ( 1 ) ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k ) , α ^ ( 2 ) ( k | k ) ) P α ( 1 ) ( k + 1 | k ) × H α ( 1 ) ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k ) , α ^ ( 2 ) ( k | k ) ) T + R α ( 1 ) ( k + 1 ) - 1 ( 17 ) (17) Step 5: Estimation error covariance P α ( 1 ) ( k + 1 | k + 1 ) = E { α ( 1 ) ( k + 1 ) - α ( 1 ) ( k + 1 | k + 1 ) × α ( 1 ) ( k + 1 ) - α ( 1 ) ( k + 1 | k + 1 ) T } = [ I - K α ( 1 ) ( k + 1 ) H α ( 1 ) ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k | k ) ) ] × P α ( 1 ) ( k + 1 | k ) ( 18 ) (18) 2) Stage 2: Kalman filter design for α(2)(k + 1) Assuming that α ^ ( 1 ) ( k + 1 | k + 1 ) and P α ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k | k ) , x ^ ( k | k ) , P α ( 2 ) ( k | k ) and Px(k|k) are known, the linear state and measurement model with α(2)(k + 1) as the state variable are respectively as follows. α ( 2 ) ( k + 1 ) = S ( 22 ) ( k ) α ( 2 ) ( k ) + S ( 20 ) ( k ) x ^ ( k | k ) + S ( 21 ) ( k ) α ^ ( 1 ) ( k | k ) + w ( 2 ) ( k ) ( 19 ) (19) y ( k + 1 ) = H α ( 2 ) ( x ^ ( k + 1 | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) ) α ( 2 ) ( k + 1 ) + v ( k + 1 ) ( 20 ) (20) Let H α ( 2 ) ( x ^ , α ( 1 ) ) : = H α ( 2 ) ( x ^ ( k + 1 | k ) , α ( 1 ) ( k + 1 | k + 1 ) ) then H α ( 2 ) ( x ^ , α ( 1 ) ) = h 11 x ^ 1 α 1 ( 1 ) h 12 x ^ 2 α 2 ( 1 ) h 21 x ^ 1 α 1 ( 1 ) h 22 x ^ 2 α 2 ( 1 ) The purpose of stage 2 is to obtain the estimated state α ^ ( 2 ) ( k + 1 k + 1 ) and the estimated error covariance matrix P α ( 2 ) ( k + 1 | k + 1 ) . Step 1: For Eqs.(20)–(21), the filter to be designed is of the following form. α ^ ( 2 ) ( k + 1 | k + 1 ) : = E α ( 2 ) ( k + 1 ) | x ^ 0 , y ( 1 ) , y ( 2 ) , ⋯ , y ( k ) , y ( k + 1 ) = E α ( 2 ) ( k + 1 ) | α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k | k ) , x ^ ( k | k ) , y ( k + 1 ) = α ^ ( 2 ) ( k + 1 | k ) + K α 2 ( k + 1 ) y ( k + 1 ) - y ^ α ( 2 ) ( k + 1 | k ) ( 21 ) (21) Step 2: State prediction and prediction error covariance α ^ ( 2 ) ( k + 1 | k ) = S ( 22 ) ( k ) α ^ ( 2 ) ( k | k ) + S ( 20 ) ( k ) x ^ ( k | k ) + S ( 21 ) ( k ) α ^ ( 1 ) ( k + 1 | k + 1 ) ( 22 ) (22) α ˜ ( 2 ) ( k + 1 | k ) = α ( 2 ) ( k + 1 ) - α ^ ( 2 ) ( k + 1 | k ) = S ( 22 ) ( k ) α ˜ ( 2 ) ( k | k ) + w ( 2 ) ( k ) ( 23 ) (23) Step 3: Measurement prediction and measurement error y ^ ( k + 1 | k ) = H α ( 2 ) ( ( x ^ ( k + 1 | k ) , α ( 1 ) ( k + 1 | k + 1 ) , α ( 2 ) ( k + 1 | k ) ) ( 24 ) (24) y ˜ ( k + 1 | k ) = H α ( 2 ) ( ( x ^ ( k + 1 | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k ) ) × [ α ( 2 ) ( k + 1 ) - α ^ ( 2 ) ( k + 1 | k ) ] + v ( k + 1 ) = H α ( 2 ) ( ( x ^ ( k + 1 | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k ) ) × α ˜ ( 2 ) ( k + 1 | k ) + v ( k + 1 ) ( 25 ) (25) Step 4: Filter gain K α ( 2 ) ( k + 1 ) = P α ( 2 ) ( k + 1 | k ) H α ( 2 ) T ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) ) × H α ( 2 ) ( ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) ) P α ( 2 ) ( k + 1 | k ) × H α ( 2 ) T ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) ) + R α ( 2 ) - 1 ( 26 ) (26) Step 5: Estimation error covariance P α 2 ( k + 1 | k + 1 ) = E α ( 2 ) ( k + 1 ) - α ^ ( 2 ) ( k + 1 | k + 1 ) (1) × α ( 2 ) ( k + 1 ) - α ^ ( 2 ) ( k + 1 | k + 1 ) T = I - K α 2 H α ( 2 ) ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) ) · P α 2 ( k + 1 | k ) ( 27 ) (27) 3) Stage 3: Kalman filter design for x(k + 1) Assuming that α ^ ( l ) ( k + 1 | k + 1 ) , and P α ( l ) ( k + 1 | k + 1 ) , l = 1 , 2 , ◯(k|k) and [Px(k|k) are known, the linear state and measurement model with x(k + 1) as the state variable are respectively as follows. x ( k + 1 ) = A ( α ^ ( 1 ) ( k | k ) , α ^ ( 2 ) ( k | k ) ) x ( k ) + w ( 0 ) ( k ) ( 28 ) (28) y ( k + 1 ) = H x ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) × x ( k + 1 ) + v 0 ( k + 1 ) ( 29 ) (29) Let H x ( α ( 1 ) , α ( 2 ) ) : = H x ( α ( 1 ) ( k + 1 ) , α ( 2 ) ( k + 1 ) ) then H x ( α ( 1 ) , α ( 2 ) ) = h 11 α 1 ( 1 ) α 1 ( 2 ) h 12 α 2 ( 1 ) α 2 ( 2 ) h 21 α 1 ( 1 ) α 1 ( 2 ) h 22 α 2 ( 1 ) α 2 ( 2 ) Note 3: in Eq.(29), A ( α ( 1 ) ( k ) , α ( 2 ) ( k ) ) = A ^ ( α ^ ( 1 ) ( k | k ) , α ^ ( 2 ) ( k | k ) ) + Δ A ˜ ( α ˜ ( 1 ) ( k | k ) , α ˜ ( 2 ) ( k | k ) ) w ( 0 ) ( k ) = Δ A ˜ ( α ˜ ( 1 ) ( k | k ) , α ˜ ( 2 ) ( k | k ) ) Δ w + w ( k ) are state errors. For understanding, let α ( 1 ) ( k ) ≈ α ^ ( 1 ) ( k | k ) , α ( 2 ) ( k ) ≈ α ^ ( 2 ) ( k | k ) Note 4: in Eq.(30), H ( α ( 1 ) ( k + 1 ) , α ( 2 ) ( k + 1 ) ) = H ^ x ( ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) + Δ H ˜ x ( α ˜ ( 1 ) ( k + 1 | k + 1 ) , α ˜ ( 2 ) ( k + 1 | k + 1 ) ) and v ( 0 ) ( k + 1 ) = Δ H ˜ ( α ˜ ( 1 ) ( k + 1 | k + 1 ) , α ˜ ( 2 ) ( k + 1 | k + 1 ) ) Δ v + v ( k + 1 ) are measurement errors. For understanding, let α ( 1 ) ( k + 1 ) ≈ α ^ ( 1 ) ( k + 1 | k + 1 ) , α ( 2 ) ( k + 1 ) ≈ α ^ ( 2 ) ( k + 1 | k + 1 ) . The purpose of stage 3 is to obtain ◯(k + 1 k + 1) and P(k + 1|k + 1). Step 1: For Eqs.(29)–(30), the filter to be designed is of the following form. x ^ ( k + 1 | k + 1 ) : = E x ( k + 1 ) | x ^ 0 , y ( 1 ) , y ( 2 ) , ⋯ , y ( k ) , y ( k + 1 ) = E x ( k + 1 ) | α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) , x ^ ( k | k ) , y ( k + 1 ) = x ^ ( k + 1 | k ) + K x ( k + 1 ) y ( k + 1 ) - y ^ x ( k + 1 | k ) ( 30 ) (30) Step 2: State prediction and prediction error covariance x ^ ( k + 1 | k ) = A ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) x ^ ( k | k ) ( 31 ) (31) P x ( k + 1 | k ) = E x ( k + 1 ) - x ^ ( k + 1 | k ) x ( k + 1 ) - x ^ ( k + 1 | k ) T = A ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) P x ( k | k ) × A T ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) + Q w ( 0 ) ( k ) ( 32 ) (32) Step 3: Measurement prediction and measurement error y ^ ( k + 1 | k ) = H x ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) x ^ ( k + 1 | k ) ( 33 ) (33) y ˜ ( k + 1 | k ) = H x ( α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) × x ˜ ( k + 1 | k ) + v ( k + 1 ) ( 34 ) (34) Step 4: Filter gain Let H x ( x ^ , α ^ ( 1 ) , α ^ ( 2 ) ) : = H x ( x ^ ( k | k ) , α ^ ( 1 ) ( k + 1 | k + 1 ) , α ^ ( 2 ) ( k + 1 | k + 1 ) ) , then K x ( k + 1 ) = P x ( k + 1 | k ) H x T ( x ^ , α ^ ( 1 ) , α ^ ( 2 ) ) × H x ( x ^ , α ^ ( 1 ) , α ^ ( 2 ) ) P x ( k + 1 | k ) H x T ( x ^ , α ^ ( 1 ) , α ^ ( 2 ) ) + R x ( k + 1 ) - 1 ( 35 ) (35) Step 5: Estimation error covariance P x ( k + 1 | k + 1 ) = E { [ x ( k + 1 ) - x ^ ( k + 1 | k + 1 ) ] × [ x ( k + 1 ) - x ^ ( k + 1 | k + 1 ) ] T } = I - K x ( k + 1 ) H x ( x ^ , α ^ ( 1 ) , α ^ ( 2 ) ) · P x ( k + 1 | k ) ( 36 ) (36) VI. Illustrative Examples In this section, three cases are given to illustrate the effectiveness of the proposed method. 1) Case 1 Consider the following nonlinear discrete time system: state model is strongly nonlinear and measurement model is linear. x 1 ( k + 1 ) = 0 . 85 x 1 ( k ) + 0 . 5 x 2 ( k ) sin ( x 1 ( k ) ) + w 1 ( k ) x 2 ( k + 1 ) = - 0 . 5 x 1 ( k ) sin ( x 2 ( k ) ) + w 2 ( k ) z ( k + 1 ) = x 1 ( k + 1 ) + x 2 ( k + 1 ) + v ( k + 1 ) ( 37 ) (37) where, state noise and measurement noise obey the following distributions: w1(k) ~ N(0, 0.1), w2(k) ~ N(0, 0.1), w1(k) and w2(k) are uncorrelated, and v(k + 1) ~ N(0, 0.1). Figs.1 and 2 show the actual states and the estimates of x1 and x2 under EKF and SH-EKF. Fig.3 shows the estimated error, and Table 1 shows the mean squared error of x1 and x2. F