Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 8052967, 12 pages https://doi.org/10.1155/2018/8052967
Research Article Strong Tracking Filter for Nonlinear Systems with Randomly Delayed Measurements and Correlated Noises Hongtao Yang ,1 Xinxin Meng,1 Hui Li ,1,2 and Xiulan Li3 1
College of Electrical and Electronic Engineering, Changchun University of Technology, Changchun 130012, China Automotive Engineering Research Institute, Changchun University of Technology, Changchun 130012, China 3 Engineering Training Center, Changchun University of Technology, Changchun 130012, China 2
Correspondence should be addressed to Hongtao Yang; hongtao
[email protected] Received 16 September 2017; Accepted 8 January 2018; Published 7 February 2018 Academic Editor: Xuejun Xie Copyright ยฉ 2018 Hongtao Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper proposes a novel strong tracking filter (STF), which is suitable for dealing with the filtering problem of nonlinear systems when the following cases occur: that is, the constructed model does not match the actual system, the measurements have the onestep random delay, and the process and measurement noises are correlated at the same epoch. Firstly, a framework of decoupling filter (DF) based on equivalent model transformation is derived. Further, according to the framework of DF, a new extended Kalman filtering (EKF) algorithm via using first-order linearization approximation is developed. Secondly, the computational process of the suboptimal fading factor is derived on the basis of the extended orthogonality principle (EOP). Thirdly, the ultimate form of the proposed STF is obtained by introducing the suboptimal fading factor into the above EKF algorithm. The proposed STF can automatically tune the suboptimal fading factor on the basis of the residuals between available and predicted measurements and further the gain matrices of the proposed STF tune online to improve the filtering performance. Finally, the effectiveness of the proposed STF has been proved through numerical simulation experiments.
1. Introduction Over the past decades, the filtering problem of nonlinear systems has been an active field of research on account of its widespread applications, for example, dynamic target tracking [1, 2], signal processing [3], and integrated navigation [4]. As a general rule, in view of the fact that the Bayesian estimator of nonlinear systems in minimum mean square error (MMSE) sense is usually faced with intractable computation [5], consequently using approximation methods to design cost-efficient estimators has received much attention. One approximation method is to use piecewise and time-varying linear functions to approximate nonlinear functions, for instance, the extended Kalman filter (EKF) [5] derived from Taylor expansion, the interpolation-based central difference Kalman filter (CDKF) [6], and the divided difference filter (DDF) [7] by using the polynomial interpolation of Stirling. These methods usually have better computational efficiency but are sensitive to linearization errors or differential operations. Another approximation method is to use Gaussian or
Gaussian mixture distribution to represent the conditional state probability density function [8, 9]. By utilizing different numerical integration technologies with cost-effective and acceptable accuracy, this approximation method yields the framework of different Gaussian approximation filters (GAFs), such as unscented transform- (UT-) based unscented Kalman filter (UKF) [10], the Gauss-Hermit filter (GHF) [11] based on the rule of Gaussian-Hermite quadrature, the Gaussian sum-quadrature Kalman filter (GS-QKF) [12] based on statistical linear regression and Gauss-Hermite quadrature, the square-root quadrature Kalman filter (SRQKF) [13] on the basis of triangular decomposition of matrix, the cubature Kalman filter (CKF) [14] according to the rule of third-degree spherical radial cubature, the high-degree CKFs [2] based on the rule of arbitrary-degree spherical radial cubature, and the sparse-grid quadrature filter (SGQF) [15] derived from sparse-grid theory. In general, all the above approximation methods can obtain better filtering accuracy in case the models of statespace and measurement have sufficient accuracy. However,
2 in practical applications, the above two models may have model uncertainties; that is, the model does not match the actual system, and the main reasons for the model uncertainties are as follows: model simplification, inaccurate description of noise characteristic, original condition, or system parameter. For the sake of improving the filtering performance of nonlinear systems with model uncertainties, the STF, which can timely alter the matrices of predicted state error covariance and gain by introducing the timevarying suboptimal fading factors, was first proposed in [16]. Subsequently, the numerous transformations of STF have been presented. In [17], the sampling strong tracking nonlinear UKF was proposed for use in eye tracking. An adaptive UKF [18] based on STF and wavelet transform was presented to further enhance the tracking performance and robustness of standard UKF. The authors in [19] combine particle filter (PF) with the idea of STF and proposed an adaptive PF with strong tracking ability in the case of particle degeneracy and target state mutation. For the sake of tracking maneuvering target, a strong tracking spherical simplexradial CKF has been designed in [20]. All the aforementioned strong tracking filtering methods are formulated in case the measurements can be reached on time and the process noise is not correlated with the measurement noise. But the following cases may occur in practical situations; that is, the measurements received may be affected by the random delay, or the process noise may be correlated with the measurement noise. For the first case, literatures [21, 22] have, respectively, proposed the networked STF and STF with randomly delayed measurements (STF/RDM). For the second case, a novel nonlinear filter derived from square-root CKF and the idea of STF was proposed in [23]. However, when the above two cases exist simultaneously, these existing STFs [17โ23] are not suitable for dealing with the filtering problem in the above two coupled cases, and little attention has been paid to the study of deriving the corresponding STF. Consequently, there is a great demand to further improve the STF for the nonlinear discrete-time stochastic dynamic systems with randomly delayed measurements and correlated noises, which motivate this study. In this paper, a novel filter, which we have called the strong tracking filter with randomly delayed measurements and correlated noises (STF/RDMCN), is proposed. Basically, the novel contributions of the paper are composed of the following. Under the idea of eliminating correlated noises at the same epoch, a general decoupling filter (DF) is derived through an equivalent transformation of the system model. The implementation of DF can thus be transformed into calculating the Gaussian-weighted integrals, which is achieved through the application of the first-order linearization approximation method to develop a new EKF algorithm. In the sense of extended orthogonality principle (EOP), the adaptive adjustment formula of the suboptimal fading factor is derived, introducing the suboptimal fading factor into the new EKF algorithm to make EKF adjusting the gain matrices in real time. This results in the ultimate form of STF/RDMCN. Numerical simulation experiments with nonlinear state estimation illustrate the effectiveness of the proposed algorithm.
Mathematical Problems in Engineering The rest of this paper is arranged as follows. Section 2 gives the problem that needs to be investigated. Then, in Section 3, the general DF with one-step randomly delayed measurement is designed, and the EKF based on the firstorder linearization approximation method is developed as an implementation of the proposed DF. Thereafter, in Section 4, according to the EOP, the formula of the suboptimal fading factor is derived, and it is combined with the above EKF to form the STF/RDMCN. In Section 5, the analysis of numerical simulation experiments is provided. Finally, some conclusions are supplied in Section 6.
2. Problem Formulation Firstly, the nonlinear system model, which has one-step randomly delayed measurements and correlated noises, is formulated. Secondly, a new system model transformed from the former is given. Consider the discrete-time, nonlinear stochastic system ๐ฅ๐+1 = ๐๐ (๐ฅ๐ ) + ๐ค๐ , ๐ โฅ 0, ๐ง๐ = โ๐ (๐ฅ๐ ) + V๐ , ๐ โฅ 1, {(1 โ ๐พ๐ ) ๐ง๐ + ๐พ๐ ๐ง๐โ1 , ๐ > 1, ๐ฆ๐ = { ๐ง, ๐ = 1, { ๐
(1)
where {๐ฅ๐ ; ๐ โฅ 0} represents the ๐ร1 state vector, {๐ง๐ ; ๐ โฅ 1} represents the ๐ ร 1 real measurement vector, {๐ฆ๐ ; ๐ โฅ 1} represents the ๐ ร 1 available measurement vector, nonlinear mappings ๐๐ (โ) and โ๐ (โ) are infinitely continuously differentiable, {๐ค๐ ; ๐ โฅ 0} and {V๐ ; ๐ โฅ 1} are sequences of correlated zero-mean Gaussian white noises with covariance matrices ๐ธ[๐ค๐ ๐ค๐๐ ] = ๐๐ ๐ฟ๐๐ , ๐ธ[V๐ V๐๐ ] = ๐
๐ ๐ฟ๐๐ , and ๐ธ[๐ค๐ V๐๐ ] = ๐๐ ๐ฟ๐๐ , with ๐ฟ๐๐ representing the Kronecker delta function, the initial state ๐ฅ0 , which is independent of {๐ค๐ ; ๐ โฅ 0} and {V๐ ; ๐ โฅ 1}, denotes a random Gaussian variable having mean ๐ธ[๐ฅ0 ] = ๐ฅฬ0|0 and covariance ๐ธ[๐ฅ0 ๐ฅ0๐ ] = ๐0|0 , and {๐พ๐ ; ๐ > 1} represents a sequence of uncorrelated Bernoulli random variables that can take the value 0 or 1 with ๐ (๐พ๐ = 1) = ๐ธ [๐พ๐ ] = ๐๐ , ๐ (๐พ๐ = 0) = 1 โ ๐ธ [๐พ๐ ] = 1 โ ๐๐ , 2
๐ธ [(๐พ๐ โ ๐๐ ) ] = (1 โ ๐๐ ) ๐๐ ,
(2)
๐ธ [(๐พ๐ โ ๐๐ )] = 0, where ๐๐ denotes the delay probability. Remark 1. In fact, the Bernoulli random variable {๐พ๐ ; ๐ > 1} describes the random delay characteristic of the available measurement vector; that is, when ๐พ๐ = 0, therefore ๐ฆ๐ = ๐ง๐ which indicates that the available measurement vector is not affected by the random delay and is updated based on probability 1 โ ๐๐ ; when ๐พ๐ = 1, therefore ๐ฆ๐ = ๐ง๐โ1 which indicates that the available measurement vector randomly delays one sampling time based on probability ๐๐ .
Mathematical Problems in Engineering
3
Remark 2. Without considering the model mismatch, a general framework of GF applied in the system shown in (1) has been proposed in [24]. Here, the two-step predictive probability density function (PDF) ๐(๐ฅ๐+1 | ๐๐โ1 ) of the state is assumed to be Gaussian, and this assumption faces complicated computation procedures in the state prediction phase. According to the literature [25], we can reconstruct the nonlinear state function in (1) for decoupling the correlation between process noise and measurement noise. Afterwards, the DF can be obtained by applying the general framework of GF with one-step random delay measurements in [26], which can surmount the defect of complicated computation procedures in the state prediction phase for the method mentioned above. This means that the derivation of the STF/RDMCN is based on DF.
3. DF with One-Step Randomly Delayed Measurements
At the same epoch, for decoupling the correlation of the process and measurement noises, the literature [27] introduced a positive definite matrix
According to (9), we can find that the first two moments of ๐(๐ฅ๐+1 | ๐๐+1 ) and ๐(V๐+1 | ๐๐+1 ) in the MMSE sense need to be obtained in deducing the framework of DP. Thus, it is necessary to define an augmented state vector as follows:
๐ผ โ๐๐ ๐
๐โ1 ]. ๐๐ = [ 0 ๐ผ
๐๐ ๐
๐โ1 V๐
๐ค๐ โ ๐ผ โ๐๐ ๐
๐โ1 ๐ค๐ ]=[ ][ ] = [ V๐ V๐ 0 ๐ผ V๐ [
๐ค๐
] ]
(4)
where ๐ค๐ is pseudo-process noise satisfying ๐ธ[๐ค๐ ] = 0 and ๐ธ[๐ค๐ ๐ค๐๐ ] = (๐๐ โ ๐๐ ๐
๐โ1 ๐๐๐ )๐ฟ๐๐ . It is easy to find that the pseudo-process noise and measurement noise are uncorrelated with each other because ๐ธ[๐ค๐ V๐๐ ] = ๐ธ[๐ค๐ V๐๐ ] โ ๐๐ ๐
๐โ1 ๐ธ[V๐ V๐๐ ] = 0. Putting equation ๐ค๐ = ๐ค๐ + ๐๐ ๐
๐โ1 V๐ into the expression of ๐ฅ๐+1 in (1) yields ๐ฅ๐+1 = ๐๐ (๐ฅ๐ ) +
+ ๐ค๐ .
(5)
Define ๐ฝ๐ = ๐๐ ๐
๐โ1 , ๐น๐ (๐ฅ๐ ) = ๐๐ (๐ฅ๐ ) + ๐ฝ๐ V๐ . Then, the discretetime nonlinear stochastic system shown by (1) is transformed into the following form: ๐ฅ๐+1 = ๐น๐ (๐ฅ๐ ) + ๐ค๐ , ๐ โฅ 1, ๐ง๐ = โ๐ (๐ฅ๐ ) + V๐ ,
๐ฆ๐+1 = (1 โ ๐พ๐+1 ) [โ๐+1 (๐ฅ๐+1 ) + V๐+1 ] + ๐พ๐+1 [โ๐ (๐ฅ๐ ) + V๐ ] .
๐ฅ๐+1 ๐ =[ ], ๐ฅ๐+1 V๐+1
๐ โฅ 1,
{(1 โ ๐พ๐ ) ๐ง๐ + ๐พ๐ ๐ง๐โ1 , ๐ > 1, ๐ฆ๐ = { ๐ = 1, {๐ง๐ ,
(6) (7) (8)
where ๐ฅ0 , {๐ค๐ ; ๐ โฅ 1}, {V๐ ; ๐ โฅ 1}, and {๐พ๐ ; ๐ > 1} are mutually independent.
(9)
(10)
๐ | ๐๐+1 ) in the MMSE where the first two moments of ๐(๐ฅ๐+1 sense have the following expression: ๐ ๐ฅฬ๐+1|๐+1 =[
๐ฅฬ๐+1|๐+1 ฬV๐+1|๐+1
],
๐๐+1|๐+1
๐ค๐ = [ ], V๐
๐๐ ๐
๐โ1 V๐
3.1. The Framework of DF. Continue to consider the nonlinear system model as shown in (6)โ(8). Substituting (7) into (8), we have
(3)
Here, ๐ผ represents the unit matrix and ๐
๐ and ๐๐ denote the covariance of the measurement noise V๐ and the crosscovariance of the process noise ๐ค๐ and the measurement noise V๐ , respectively. Thus, we get ๐๐ [
Firstly, considering literature [26], the framework of DF is given, due to the fact that nonlinear system model as shown in (6)โ(8) satisfies the condition; that is, the process noise and measurement noise are uncorrelated. Secondly, a new EKF algorithm with first-order linear approximations is developed on the basis of this framework.
๐ฅV ๐๐+1|๐+1
(11)
๐ ]. ๐๐+1|๐+1 = [ ๐ฅV ๐ VV (๐๐+1|๐+1 ) ๐๐+1|๐+1 [ ] ๐ ๐ In (11), (๐ฅฬ๐+1|๐+1 , ๐๐+1|๐+1 ), (๐ฅฬ๐+1|๐+1 , ๐๐+1|๐+1 ), and (ฬV๐+1|๐+1 , VV ) are the filtering estimation and the covariance at ๐๐+1|๐+1 time ๐ + 1 of the augmented state, the state, and the mea๐ฅV is the cross-covariance surement noise, respectively; ๐๐+1|๐+1 at time ๐ + 1 of the state and the measurement noise. Considering the independence of V๐+1 with ๐ฆ๐ and ๐ฅ๐+1 , the augmented state prediction and the covariance are ๐ =[ ๐ฅฬ๐+1|๐
๐ฅฬ๐+1|๐ 0๐ร1
],
๐๐+1|๐ 0๐ร๐ ๐ ๐๐+1|๐ =[ ], 0๐ร๐ ๐
๐+1 ๐ โฅ 0. Define the mean, covariance, and cross-covariance ๐ฅฬ๐+1|๐ = ๐ธ [๐ฅ๐+1 | ๐๐ ] , ๐ | ๐๐ ] , ๐๐+1|๐ = ๐ธ [๐ฅฬ๐+1|๐ ๐ฅฬ๐+1|๐
๐ฆฬ๐+1|๐ = ๐ธ [๐ฆ๐+1 | ๐๐ ] , ๐ฆ๐ฆ
๐ | ๐๐ ] , ๐๐+1|๐ = ๐ธ [๐ฆฬ๐+1|๐ ๐ฆฬ๐+1|๐
๐งฬ๐+1|๐ = ๐ธ [๐ง๐+1 | ๐๐ ] ,
(12)
4
Mathematical Problems in Engineering ๐ง๐ง ๐ ๐๐+1|๐ = ๐ธ [ฬ๐ง๐+1|๐ ๐งฬ๐+1|๐ | ๐๐ ] ,
๐พ๐ = (
๐งฬ๐|๐ = ๐ธ [๐ง๐ | ๐๐ ] , ๐ง๐ง ๐๐|๐
=
๐ ๐ธ [ฬ๐ง๐|๐ ๐งฬ๐|๐
| ๐๐ ] ,
=(
๐ฅ๐ฆ
=
๐ ๐ธ [ฬV๐+1|๐ ๐ฆฬ๐+1|๐
=
๐ ๐ธ [๐ฅฬ๐+1|๐ ๐งฬ๐|๐
| ๐๐ ] ,
| ๐๐ ] ,
๐ฅ๐ฆ
๐ฅฬ๐+1|๐ = ๐ธ [(๐น๐ (๐ฅ๐ ) + ๐ค๐ ) | ๐๐ ] , (14)
๐ ๐ , ๐๐|๐ ), considering that ๐ค๐ Under the known (๐ฅฬ๐|๐
dent of V๐ and ๐๐ , we have
is indepen-
(22)
๐ง๐ง ๐ ๐๐+1|๐ = โซ โ๐+1 (๐ฅ๐+1 ) โ๐+1 (๐ฅ๐+1 )
(23)
+ ๐
๐+1 ,
๐ง๐ง ๐๐|๐ = โซ [โ๐ (๐ฅ๐ ) + V๐ ] [โ๐ (๐ฅ๐ ) + V๐ ] ๐ ๐ ๐ (๐ฅ๐๐ ; ๐ฅฬ๐|๐ , ๐๐|๐ ) ๐๐ฅ๐๐
โ
(24)
๐
(25)
๐ ๐งฬ๐|๐ ๐งฬ๐|๐ ,
(26)
โ
๐ (๐ฅ๐+1 ; ๐ฅฬ๐+1|๐ , ๐๐+1|๐ ) ๐๐ฅ๐+1
๐
(15)
๐ ๐ฅฬ๐+1|๐ ๐ฅฬ๐+1|๐
๐ โ ๐ฅฬ๐+1|๐ ๐งฬ๐+1|๐ , ๐ฅ๐ง = โซ [๐๐ (๐ฅ๐ ) + ๐ฝ๐ V๐ ] [โ๐ (๐ฅ๐ ) + V๐ ] ๐๐+1,๐|๐
+ ๐๐ โ ๐ฝ๐ ๐
๐ ๐ฝ๐๐ . ๐ ๐ Putting (15) into (12), the predictive estimates (๐ฅฬ๐+1|๐ , ๐๐+1|๐ ) of the augmented state can be obtained. ๐ ๐ , ๐๐|๐ ) and Step 2 (state correction). Under the known (๐ฅฬ๐|๐ ๐ ๐ (๐ฅฬ๐+1|๐ , ๐๐+1|๐ ), considering that ๐ค๐ , V๐ , ๐พ๐ , and ๐๐ are mutually independent, we have
๐ฆ๐ฆ
๐งฬ๐+1|๐ = โซ โ๐+1 (๐ฅ๐+1 ) ๐ (๐ฅ๐+1 ; ๐ฅฬ๐+1|๐ , ๐๐+1|๐ ) ๐๐ฅ๐+1 ,
๐ฅ๐ง ๐ ๐๐+1|๐ = โซ ๐ฅ๐+1 โ๐+1 (๐ฅ๐+1 )
๐ ๐ , ๐๐|๐ ) ๐๐ฅ๐๐ , ๐ฅฬ๐+1|๐ = โซ [๐๐ (๐ฅ๐ ) + ๐ฝ๐ V๐ ] ๐ (๐ฅ๐๐ ; ๐ฅฬ๐|๐
๐ ๐ ๐ฅฬ๐+1|๐+1 = ๐ฅฬ๐+1|๐ + ๐พ๐ (๐ฆ๐+1 โ ๐ฆฬ๐+1|๐ ) ,
),
where ๐พ๐ is the gain matrix of the augmented state and
โ
โ
V๐ง V๐ง (1 โ ๐๐+1 ) ๐๐+1|๐ + ๐๐+1 ๐๐+1,๐|๐
(21)
๐ ๐ , ๐๐|๐ ) ๐๐ฅ๐๐ , ๐งฬ๐|๐ = โซ [โ๐ (๐ฅ๐ ) + V๐ ] ๐ (๐ฅ๐๐ ; ๐ฅฬ๐|๐
๐ . ๐ฅฬ๐+1|๐ ๐ฅฬ๐+1|๐
๐ ๐ ๐๐+1|๐+1 = ๐๐+1|๐ โ ๐พ๐ ๐๐+1|๐ ๐พ๐๐ ,
=(
๐ฅ๐ง ๐ฅ๐ง (1 โ ๐๐+1 ) ๐๐+1|๐ + ๐๐+1 ๐๐+1,๐|๐
๐ โ
๐ (๐ฅ๐+1 ; ๐ฅฬ๐+1|๐ , ๐๐+1|๐ ) ๐๐ฅ๐+1 โ ๐งฬ๐+1|๐ ๐งฬ๐+1|๐
๐ ๐ฅฬ๐+1|๐ ๐ฅฬ๐+1|๐
= ๐ธ [๐น๐ (๐ฅ๐ ) ๐น๐๐ (๐ฅ๐ ) | ๐๐ ] + ๐ธ [๐ค๐ ๐ค๐๐ | ๐๐ ]
ร
(20)
๐๐+1|๐ ๐ฅ๐ ๐ฆ ๐๐+1|๐ = ( V๐ฆ ) ๐๐+1|๐
Step 1 (state prediction). Putting (6) into the expression of ๐ฅฬ๐+1|๐ and ๐๐+1|๐ in (13) yields
๐ ๐ ๐ (๐ฅ๐๐ ; ๐ฅฬ๐|๐ , ๐๐|๐ ) ๐๐ฅ๐๐
(19)
๐
where ๐๐ = is the set of the available measurements in (8). According to literature [26], the equations describing the framework of DF are as follows.
๐๐+1|๐ = โซ [๐๐ (๐ฅ๐ ) + ๐ฝ๐ V๐ ] [๐๐ (๐ฅ๐ ) + ๐ฝ๐ V๐ ]
,
+ (1 โ ๐๐+1 ) ๐๐+1 (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) ,
{๐ฆ๐ }๐๐=1
โ
V๐ฆ
๐๐+1|๐
(18) โ1 ๐ฆ๐ฆ ) (๐๐+1|๐ )
๐ง๐ง ๐ง๐ง = (1 โ ๐๐+1 ) ๐๐+1|๐ + ๐๐+1 ๐๐|๐
(13)
| ๐๐ ] โ
๐๐+1|๐
๐ฆ๐ฆ
V๐ง ๐ ๐๐+1,๐|๐ = ๐ธ [ฬV๐+1|๐ ๐งฬ๐|๐ | ๐๐ ] ,
๐๐+1|๐ =
โ1
๐ฆ๐ฆ
๐๐+1|๐
V๐ง ๐ = ๐ธ [ฬV๐+1|๐ ๐งฬ๐+1|๐ | ๐๐ ] , ๐๐+1|๐
๐ ๐ธ [๐ฅ๐+1 ๐ฅ๐+1
๐ฅ๐ ๐ฆ
) = ๐๐+1|๐ (๐๐+1|๐ )
๐ฆฬ๐+1|๐ = (1 โ ๐๐+1 ) ๐งฬ๐+1|๐ + ๐๐+1 ๐งฬ๐|๐ ,
๐ฅ๐ง ๐ ๐๐+1|๐ = ๐ธ [๐ฅฬ๐+1|๐ ๐งฬ๐+1|๐ | ๐๐ ] , ๐ฅ๐ง ๐๐+1,๐|๐
๐พ๐V ๐ฅ๐ฆ
๐ ๐๐+1|๐ = ๐ธ [๐ฅฬ๐+1|๐ ๐ฆฬ๐+1|๐ | ๐๐ ] , V๐ฆ ๐๐+1|๐
๐พ๐๐ฅ
(16) (17)
ร
๐ ๐ ๐ (๐ฅ๐๐ ; ๐ฅฬ๐|๐ , ๐๐|๐ ) ๐๐ฅ๐๐
V๐ง ๐๐+1|๐ = ๐
๐+1 , V๐ง = 0. ๐๐+1,๐|๐
โ
๐
(27)
๐ ๐ฅฬ๐+1|๐ ๐งฬ๐|๐ ,
(28)
3.2. Implementation of the DF. The cruces of implementing the DF in (14) and (16)โ(21) are to calculate the Gaussianweighted integrals in (15) and (22)โ(27). Owing to the nonlinearity of ๐๐ (โ) and โ๐ (โ), the analytical calculation of the above integrals is intractable and infeasible. Therefore,
Mathematical Problems in Engineering
5
some technologies of numerical approximation are needed, for example, the first-order linear approximations. Here, we use the EKF with one-step randomly delayed measurements based on the first-order linearization to implement the DF in (14) and (16)โ(21). ๐ ๐ Given the filtering estimates (๐ฅฬ๐|๐ , ๐๐|๐ ), ๐๐ (๐ฅ๐ ) and โ๐ (๐ฅ๐ ) are linearized about ๐ฅ๐ = ๐ฅฬ๐|๐ ; that is, ๐ฅ๐+1 โ ๐๐ (๐ฅฬ๐|๐ ) + ๐น๐ (๐ฅ๐ โ ๐ฅฬ๐|๐ ) + ๐ฝ๐ V๐ + ๐ค๐ , ๐ง๐ โ โ๐ (๐ฅฬ๐|๐ ) + ๐ป๐ (๐ฅ๐ โ ๐ฅฬ๐|๐ ) + V๐ ,
(29) (30)
where ๐น๐ = ๐๐๐ (๐ฅ๐ )/๐๐ฅ๐ |๐ฅ๐ =๐ฅฬ๐|๐ and ๐ป๐ = ๐โ๐ (๐ฅ๐ )/๐๐ฅ๐ |๐ฅ๐ =๐ฅฬ๐|๐ . Equations (15) are approximated as follows: ๐ฅฬ๐+1|๐ = ๐๐ (๐ฅฬ๐|๐ ) + ๐ฝ๐ ฬV๐|๐ , ๐๐+1|๐ =
๐ ๐น๐ ๐๐|๐ ๐น๐
+
๐ฅV ๐ ๐น๐ ๐๐|๐ ๐ฝ๐
(31) +
๐ฅV ๐ ๐ (๐น๐ ๐๐|๐ ๐ฝ๐ )
VV ๐ + ๐ฝ๐ ๐๐|๐ ๐ฝ๐ + ๐๐ โ ๐ฝ๐ ๐
๐ ๐ฝ๐๐ .
(32)
๐ ๐ Further, the predictive estimates (๐ฅฬ๐+1|๐ , ๐๐+1|๐ ) can be calculated by putting (31)-(32) into (12). ๐ ๐ , ๐๐+1|๐ ), we linearize Given the predictive estimates (๐ฅฬ๐+1|๐ โ๐+1 (๐ฅ๐+1 ) about ๐ฅ๐+1 = ๐ฅฬ๐+1|๐ ; that is,
๐ง๐+1 โ โ๐+1 (๐ฅฬ๐+1|๐ ) + ๐ป๐+1 (๐ฅ๐+1 โ ๐ฅฬ๐+1|๐ ) + V๐+1 ,
(33)
where ๐ป๐+1 = ๐โ๐+1 (๐ฅ๐+1 )/๐๐ฅ๐+1 |๐ฅ๐+1 =๐ฅฬ๐+1|๐ . Then, (22)โ(27) are approximated as follows: ๐งฬ๐+1|๐ = โ๐+1 (๐ฅฬ๐+1|๐ ) ,
(34)
๐ง๐ง ๐ = ๐ป๐+1 ๐๐+1|๐ ๐ป๐+1 + ๐
๐+1 , ๐๐+1|๐
(35)
๐งฬ๐|๐ = โ๐ (๐ฅฬ๐|๐ ) + ฬV๐|๐ ,
(36) ๐
๐ง๐ง ๐ฅV ๐ฅV VV = ๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐ + (๐ป๐ ๐๐|๐ ) + ๐๐|๐ , ๐๐|๐ ๐ฅ๐ง ๐๐+1|๐
=
๐ ๐๐+1|๐ ๐ป๐+1 ,
๐ฅ๐ง ๐ฅV V๐ฅ ๐ VV = ๐น๐ ๐๐|๐ ๐ป๐๐ + ๐น๐ ๐๐|๐ + ๐ฝ๐ ๐๐|๐ ๐ป๐ + ๐ฝ๐ ๐๐|๐ . ๐๐+1,๐|๐
(37) (38) (39)
Putting (34)โ(39) into (16)โ(21), the filtering estimates ๐ ๐ , ๐๐+1|๐+1 ) of the augmented state can be calculated. (๐ฅฬ๐+1|๐+1
4. Derivation of the STF/RDMCN In [16], the standard STF for nonlinear systems is proposed, and it has the following advantages: (1) when the model is uncertain due to the simplification of the system model, the uncertainty of the noise characteristics and initial conditions, or variation of system parameters, it has strong robustness, (2) it has an outstanding ability to track the state, regardless of the sudden or slow change in the state and even the system achieving a stable state or not, and (3) it adds a small amount of data overhead, and the computation complexity does not increase significantly. As a result, we consider that STF is especially suitable for the nonlinear state estimation in
these cases, namely, model uncertainties, randomly delayed measurements, and correlated noises. However, the above STF cannot be straightforwardly employed in the nonlinear system shown in (1), owing to the fact that the discretionarily chosen pairs of residuals according to the orthogonality principle are computed on the basis of measurements without random delay. Therefore, an EOP, which is applied in the nonlinear system shown in (1), is given. ๐
๐ ๐ ๐ ๐ โ ๐ฅ๐+1|๐+1 โ ๐ฅ๐+1|๐+1 ๐ธ {(๐ฅ๐+1 ) (๐ฅ๐+1 ) } = min, ๐ } = 0, ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐
๐ = 0, 1, 2, . . . ; ๐ = 1, 2, . . . .
(40)
(41)
Equation (40) is the performance index of the proposed EKF, and the corresponding derivation process can refer to [22]. Equation (41) means that the discretionarily chosen pairs of residuals which are calculated based on (9) and (19) are mutually orthogonal. 4.1. Derivation of the Suboptimal Fading Factor. It is not difficult to find that the proposed EKF offers a suboptimal estimation of the augmented state by using the given available measurements ๐๐ = {๐ฆ๐ }๐๐=1 when the system model is exact. However, when the model with uncertainty is developed, the estimation performance of the EKF will be poor or even divergent. The fundamental problem is that the gain matrix shown in (18) is not able to adapt to the change of the residuals between the available measurements and predicted measurements. In order to overcome this problem and make the proposed EKF have the excellent characteristics of STF, a natural idea is to combine the EOP with the proposed EKF to derive an STF/RDMCN by introducing a suboptimal fading ๐ of the augmented state. The factor into filtering estimate ๐๐|๐ ๐,๐ modified filtering estimate ๐๐|๐ is as follows: ๐ฅV ๐๐|๐ ๐๐|๐ ๐ ๐+1 0 ๐,๐ =[ ] [ ๐ฅV ๐ VV ] ๐๐|๐ 0 ๐ ๐+1 (๐๐|๐ ) ๐๐|๐ [ ] ๐ ๐๐|๐
๐,๐ฅV ๐๐|๐
(42)
= [ ๐,๐ฅV ๐ ๐,VV ] , (๐ ) ๐๐|๐ [ ๐|๐ ] where ๐ ๐+1 (๐ ๐+1 โฅ 1) denotes the suboptimal fading factor. Remark 3. Substituting (42) into (32), we can find that the predictive estimate ๐๐+1|๐ of the state is also modified by the same suboptimal fading factor. Continuing to consider (35), (37)โ(39), and (18), we also find that the proposed STF/RDMCN can undermine the impact of the insignificant past information by utilizing the time-varying suboptimal fading factor and adjust the gain matrix of the augmented state in real time with the aim of improving the tracking performance of the filter. Then, the next work is to determine the suboptimal fading factor ๐ ๐+1 according to the EOP.
6
Mathematical Problems in Engineering Considering (9), (19), (30), (33), (34), and (36), we get
Considering (16), (11), and (12), we have
๐ฆฬ๐+1|๐ = (1 โ ๐พ๐+1 ) (๐ป๐+1 ๐ฅฬ๐+1|๐ + V๐+1 ) + ๐พ๐+1 (๐ป๐ ๐ฅฬ๐|๐ + ฬV๐|๐ )
๐ฅ ๐ฆฬ๐+๐|๐+๐โ1 , ๐ฅฬ๐+๐|๐+๐ = ๐ฅฬ๐+๐|๐+๐โ1 โ ๐พ๐+๐โ1
+ (๐พ๐+1 โ ๐๐+1 ) (ฬ๐ง๐|๐ โ ๐งฬ๐+1|๐ ) , where ๐ฆฬ๐+1|๐ = ๐ฆ๐+1 โ ๐ฆฬ๐+1|๐ , ๐ฅฬ๐+1|๐ = ๐ฅ๐+1 โ ๐ฅฬ๐+1|๐ , ๐ฅฬ๐|๐ = ๐ฅ๐ โ ๐ฅฬ๐|๐ , ฬV๐|๐ = V๐ โ ฬV๐|๐ . Using (29) minus (31) yields ๐ฅฬ๐+1|๐ = ๐น๐ ๐ฅฬ๐|๐ + ๐ฝ๐ ฬV๐|๐ + ๐ค๐ .
V ฬV๐+๐|๐+๐ = V๐+๐ โ ๐พ๐+๐โ1 ๐ฆฬ๐+๐|๐+๐โ1 .
(43)
๐ ๐ Putting (50) into ๐ธ{๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } and ๐ธ{ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } in (48), we have ๐ ๐ธ {๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } ๐ฅ ๐ ๐ฆฬ๐+๐|๐+๐โ1 ) ๐ฆฬ๐+1|๐ }, = ๐ธ {(๐ฅฬ๐+๐|๐+๐โ1 โ ๐พ๐+๐โ1
(44)
Putting (44) into (43) yields
๐ } ๐ธ {ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐
๐ฆฬ๐+1|๐ = [(1 โ ๐พ๐+1 ) ๐ป๐+1 ๐น๐ + ๐พ๐+1 ๐ป๐ ] ๐ฅฬ๐|๐ + [(1 โ ๐พ๐+1 ) ๐ป๐+1 ๐ฝ๐ + ๐พ๐+1 ] ฬV๐|๐ + (1 โ ๐พ๐+1 ) (๐ป๐+1 ๐ค๐ + V๐+1 )
(45)
Based on (44), (46), and using a similar simplification ๐ ๐ method in (47), ๐ธ{๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } and ๐ธ{ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } can be simplified to ๐ ๐ ๐ธ {๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } = ๐ผ๐+๐โ1 ๐ธ {๐ฅฬ๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ }
Using a similar derivation procedure, we have
๐ + ๐ฝ๐+๐โ1 ๐ธ {ฬV๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ },
๐ฆฬ๐+๐+1|๐+๐
๐ ๐ } = ๐๐+๐โ1 ๐ธ {๐ฅฬ๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ } ๐ธ {ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐
= [(1 โ ๐พ๐+๐+1 ) ๐ป๐+๐+1 ๐น๐+๐ + ๐พ๐+๐+1 ๐ป๐+๐ ] ๐ฅฬ๐+๐|๐+๐
where ๐ฅ ๐ผ๐+๐โ1 = ๐น๐+๐โ1 โ ๐พ๐+๐โ1 ๐๐+๐โ1 ,
+ (๐พ๐+๐+1 โ ๐๐+๐+1 ) (ฬ๐ง๐+๐|๐+๐ โ ๐งฬ๐+๐+1|๐+๐ ) .
๐ฅ ๐ฝ๐+๐โ1 = ๐ฝ๐+๐โ1 โ ๐พ๐+๐โ1 ๐๐+๐โ1 ,
Putting (46) into (41) yields
V ๐๐+๐โ1 = โ๐พ๐+๐โ1 ๐๐+๐โ1 ,
๐ ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐ }
(47)
Putting (52) into (48), rearranging (48) yields ๐ ๐ } = ๐๐+๐โ1 ๐ธ {๐ฅฬ๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ } ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐
โ
ฬV๐+๐|๐+๐ + (1 โ ๐พ๐+๐+1 ) (๐ป๐+๐+1 ๐ค๐+๐ + V๐+๐+1 ) ๐ }. + (๐พ๐+๐+1 โ ๐๐+๐+1 ) (ฬ๐ง๐+๐|๐+๐ โ ๐งฬ๐+๐+1|๐+๐ )] ๐ฆฬ๐+1|๐
๐ + ๐๐+๐โ1 ๐ธ {ฬV๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ },
Considering that ๐ฅ0 , {๐ค๐ ; ๐ โฅ 1}, {V๐ ; ๐ โฅ 1}, {๐พ๐ ; ๐ > 1}, and ๐๐ are mutually independent and combining (2), (47) is simplified to the following form: that is, =
๐ ๐๐+๐ ๐ธ {๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } ๐ }, + ๐๐+๐ ๐ธ {ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐
(48)
๐๐+๐ = (1 โ ๐๐+๐+1 ) ๐ป๐+๐+1 ๐ฝ๐+๐ + ๐๐+๐+1 .
(54)
where ๐๐+๐โ1 = ๐๐+๐ ๐ผ๐+๐โ1 + ๐๐+๐ ๐๐+๐โ1 , ๐๐+๐โ1 = ๐๐+๐ ๐ฝ๐+๐โ1 + ๐๐+๐ ๐ฟ๐+๐โ1 .
(55)
According to (48) and (54), we can get the following form of ๐ ๐ธ{๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐ } by using the iterative operation: that is,
where ๐๐+๐ = (1 โ ๐๐+๐+1 ) ๐ป๐+๐+1 ๐น๐+๐ + ๐๐+๐+1 ๐ป๐+๐ ,
(53)
V ๐ฟ๐+๐โ1 = โ๐พ๐+๐โ1 ๐๐+๐โ1 .
= ๐ธ {[[(1 โ ๐พ๐+๐+1 ) ๐ป๐+๐+1 ๐น๐+๐ + ๐พ๐+๐+1 ๐ป๐+๐ ]
๐ } ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐
(52)
๐ + ๐ฟ๐+๐โ1 ๐ธ {ฬV๐+๐โ1|๐+๐โ1 ๐ฆฬ๐+1|๐ },
(46)
+ (1 โ ๐พ๐+๐+1 ) (๐ป๐+๐+1 ๐ค๐+๐ + V๐+๐+1 )
โ
๐ฅฬ๐+๐|๐+๐ + [(1 โ ๐พ๐+๐+1 ) ๐ป๐+๐+1 ๐ฝ๐+๐ + ๐พ๐+๐+1 ]
(51)
V ๐ ๐ฆฬ๐+๐|๐+๐โ1 ) ๐ฆฬ๐+1|๐ }. = ๐ธ {(V๐+๐ โ ๐พ๐+๐โ1
+ (๐พ๐+1 โ ๐๐+1 ) (ฬ๐ง๐|๐ โ ๐งฬ๐+1|๐ ) .
+ [(1 โ ๐พ๐+๐+1 ) ๐ป๐+๐+1 ๐ฝ๐+๐ + ๐พ๐+๐+1 ] ฬV๐+๐|๐+๐
(50)
(49)
๐ ๐ ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐ } = ๐๐+๐ ๐ธ {๐ฅฬ๐+๐|๐+๐ ๐ฆฬ๐+1|๐ } ๐ + ๐๐+๐ ๐ธ {ฬV๐+๐|๐+๐ ๐ฆฬ๐+1|๐ },
(56)
Mathematical Problems in Engineering
7
where
then the EOP will be satisfied. Putting (20), (35), and (37) into (61), rearranging (61) yields
๐๐+๐ = ๐๐+๐ ,
๐ (1 โ ๐๐+1 ) ๐ป๐+1 ๐๐+1|๐ ๐ป๐+1
๐๐+๐ = ๐๐+๐ ,
๐
๐ฅV ๐ฅV VV + ๐๐+1 (๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐ + (๐ป๐ ๐๐|๐ ) + ๐๐|๐ )
๐ = ๐, ๐๐+๐ = ๐๐+๐+1 ๐ผ๐+๐ + ๐๐+๐+1 ๐๐+๐ ,
0 = ๐๐+1 โ (1 โ ๐๐+1 )
๐๐+๐ = ๐๐+๐+1 ๐ฝ๐+๐ + ๐๐+๐+1 ๐ฟ๐+๐ ,
๐
โ
[๐๐+1 (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) + ๐
๐+1 ] .
๐ฅ ๐ผ๐+๐ = ๐น๐+๐ โ ๐พ๐+๐ ๐๐+๐ ,
๐ฝ๐+๐ = ๐ฝ๐+๐ โ ๐๐+๐ =
Putting (32) and (42) into (62), rearranging (62) yields
๐ฅ ๐พ๐+๐ ๐๐+๐ ,
(57)
V โ๐พ๐+๐ ๐๐+๐ ,
๐
๐ ๐ ๐+1 [(1 โ ๐๐+1 ) (๐ป๐+1 ๐น๐ ๐๐|๐ ๐น๐ ๐ป๐+1 ๐
๐ฅV ๐ ๐ ๐ฅV ๐ ๐ + ๐ป๐+1 ๐น๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 + ๐ป๐+1 (๐น๐ ๐๐|๐ ๐ฝ๐ ) ๐ป๐+1
V ๐ฟ๐+๐ = โ๐พ๐+๐ ๐๐+๐ ,
VV ๐ ๐ ๐ฅV + ๐ป๐+1 ๐ฝ๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 ) + ๐๐+1 (๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐
๐ = ๐ โ 1, ๐ โ 2, . . . , 1, ๐๐+๐ = (1 โ ๐๐+๐+1 ) ๐ป๐+๐+1 ๐น๐+๐ + ๐๐+๐+1 ๐ป๐+๐ ,
+
๐๐+๐ = (1 โ ๐๐+๐+1 ) ๐ป๐+๐+1 ๐ฝ๐+๐ + ๐๐+๐+1 ,
๐ ๐ ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐ } = ๐๐+1 ๐ธ {๐ฅฬ๐+1|๐+1 ๐ฆฬ๐+1|๐ } ๐ + ๐๐+1 ๐ธ {ฬV๐+1|๐+1 ๐ฆฬ๐+1|๐ }. ๐ฅ๐ฆ
(58)
=
0 ๐๐+1
V๐ฆ
๐ } ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐
โ (1 โ ๐๐+1 )
๐ ๐ + ๐ป๐+1 ๐๐ ๐ป๐+1 โ ๐ป๐+1 ๐ฝ๐ ๐
๐ ๐ฝ๐๐ ๐ป๐+1 ].
Similar to the idea of the literature [22], for obtaining the suboptimal fading factor ๐ ๐+1 , the trace operation is introduced into both sides of (63) as follows: ๐ tr [๐ ๐+1 [(1 โ ๐๐+1 ) (๐ป๐+1 ๐น๐ ๐๐|๐ ๐น๐ ๐ป๐+1 ๐
๐ฅV ๐ ๐ ๐ฅV ๐ ๐ + ๐ป๐+1 ๐น๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 + ๐ป๐+1 (๐น๐ ๐๐|๐ ๐ฝ๐ ) ๐ป๐+1 VV ๐ ๐ ๐ฅV + ๐ป๐+1 ๐ฝ๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 ) + ๐๐+1 (๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐
๐ = ๐๐+1 ๐ธ {(๐ฅฬ๐+1|๐ โ ๐พ๐๐ฅ ๐ฆฬ๐+1|๐ ) ๐ฆฬ๐+1|๐ }
+ ๐๐+1 ๐ธ {(V๐+1 โ
๐ ๐พ๐V ๐ฆฬ๐+1|๐ ) ๐ฆฬ๐+1|๐ }
+ (59)
๐ฅ๐ฆ
0 = ๐๐+1 (๐๐+1|๐ โ ๐พ๐๐ฅ ๐๐+1 )
๐ฅV ๐ (๐ป๐ ๐๐|๐ )
+
VV ๐๐|๐ )]]
=
0 tr [๐๐+1
V๐ฆ
โ (1 โ ๐๐+1 ) ๐
โ
[๐๐+1 (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) + ๐
๐+1
Define
0 ๐ where ๐๐+1 โ ๐ธ{๐ฆฬ๐+1|๐ ๐ฆฬ๐+1|๐ } is the covariance of the residuals. Putting (18) into (59), we obtain
๐
๐ ๐๐+1 โ (1 โ ๐๐+1 ) (๐ป๐+1 ๐น๐ ๐๐|๐ ๐น๐ ๐ป๐+1 ๐
๐ฅV ๐ ๐ ๐ฅV ๐ ๐ + ๐ป๐+1 ๐น๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 + ๐ป๐+1 (๐น๐ ๐๐|๐ ๐ฝ๐ ) ๐ป๐+1
๐ ๐ธ {๐ฆฬ๐+๐+1|๐+๐ ๐ฆฬ๐+1|๐ } ๐ฆ๐ฆ
โ1
0 = ๐๐+1 ๐๐+1|๐ (๐ผ โ (๐๐+1|๐ ) ๐๐+1 )
(๐ผ โ
โ1 ๐ฆ๐ฆ (๐๐+1|๐ )
(60)
โ1
VV ๐ ๐ ๐ฅV + ๐ป๐+1 ๐ฝ๐ ๐๐|๐ ๐ฝ๐ ๐ป๐+1 ) + ๐๐+1 (๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐
(65)
๐
๐ฅV VV + (๐ป๐ ๐๐|๐ ) + ๐๐|๐ ),
0 ๐๐+1 ).
According to (60), we can find that if a suitable suboptimal fading factor ๐ ๐+1 in (42) is chosen to ensure 0 = 0, ๐ผ โ (๐๐+1|๐ ) ๐๐+1
(64)
๐ ๐ + ๐ป๐+1 ๐๐ ๐ป๐+1 โ ๐ป๐+1 ๐ฝ๐ ๐
๐ ๐ฝ๐๐ ๐ป๐+1 ]] .
0 + ๐๐+1 (๐๐+1|๐ โ ๐พ๐V ๐๐+1 ),
๐ฅ๐ฆ
(63)
๐
Using (50) about ๐ = 1 and the expression of ๐๐+1|๐ and ๐๐+1|๐ in (13), we can get the following form of (58): that is,
๐ฆ๐ฆ
+
VV ๐๐|๐ )]
๐
For ๐ = 1, the following form of (56) can be obtained: that is,
V๐ฆ ๐๐+1 ๐๐+1|๐
๐ฅV ๐ (๐ป๐ ๐๐|๐ )
โ
[๐๐+1 (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) + ๐
๐+1
๐ = ๐, ๐ โ 1, . . . , 1.
+
(62)
(61)
0 ๐๐+1 โ ๐๐+1 โ (1 โ ๐๐+1 ) ๐
โ
[๐๐+1 (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) (ฬ๐ง๐+1|๐ โ ๐งฬ๐|๐ ) + ๐
๐+1 ๐ ๐ + ๐ป๐+1 ๐๐ ๐ป๐+1 โ ๐ป๐+1 ๐ฝ๐ ๐
๐ ๐ฝ๐๐ ๐ป๐+1 ].
(66)
8
Mathematical Problems in Engineering ๐ฆ๐ฆ
Hence, (66) can be simplified to tr [๐ ๐+1 ๐๐+1 ] = tr [๐๐+1 ] .
(67)
Then, the suboptimal fading factor ๐ ๐+1 is described as follows: ๐ ๐+1 =
tr [๐๐+1 ] . tr [๐๐+1 ]
(68)
0 Nevertheless, the covariance of the residuals ๐๐+1 in (66) is unknown, which can be determined by the following rough method: ๐
0 ๐๐+1
๐ฆฬ1|0 ๐ฆฬ1|0 { { { = { ๐๐0 + ๐ฆฬ๐+1|๐ ๐ฆฬ๐ ๐+1|๐ { ๐ { 1 + ๐ {
๐=0 ๐ โฅ 1,
(69)
where ๐ (0 < ๐ โค 1) is a forgetting factor which is often selected as ๐ = 0.95 according to [22]. For ๐ ๐+1 โฅ 1, the suboptimal fading factor ๐ ๐+1 can take effect, so ๐ ๐+1 can be ultimately calculated via ๐ ๐+1
tr [๐๐+1 ] = max {1, }. tr [๐๐+1 ]
๐ ๐0|0
๐ฅฬ0|0 0
๐ฅ๐ ๐ฆ ๐1|0
(73)
(3) For ๐ > 1, one has the following. Step 1 (calculation and introduction of suboptimal fading fac๐ ๐ , ๐๐|๐ ) tor). Assume that, at time ๐, the filtering estimates (๐ฅฬ๐|๐ and the covariance of the residuals ๐๐0 are all known. For time 0 , ๐๐+1 , and ๐๐+1 can be calculated by ๐ + 1, ๐ฅฬ๐+1|๐ , ๐ฆฬ๐+1|๐ , ๐๐+1 (31), (19), (69), (65), and (66), respectively. Putting ๐๐+1 and ๐๐+1 into (70) obtains ๐ ๐+1 . Then, introducing ๐ ๐+1 into (42) ๐,๐ . obtains ๐๐|๐ Step 2 (state prediction). ๐๐+1|๐ can be calculated by ๐
๐
๐,VV ๐ + ๐ฝ๐ ๐๐|๐ ๐ฝ๐ + ๐๐ โ ๐ฝ๐ ๐
๐ ๐ฝ๐๐ .
๐ง๐ง ๐ฅ๐ง Step 3 (state correction). ๐๐+1|๐ and ๐๐+1|๐ can be calculated ๐ง๐ง ๐ฅ๐ง by (35) and (38). ๐๐|๐ and ๐๐+1,๐|๐ can be calculated as follows: ๐ง๐ง ๐ ๐,๐ฅV ๐,๐ฅV = ๐ป๐ ๐๐|๐ ๐ป๐๐ + ๐ป๐ ๐๐|๐ + (๐ป๐ ๐๐|๐ ) ๐๐|๐ ๐,VV + ๐๐|๐ ,
(71)
(74)
๐ ๐ , ๐๐+1|๐ ) can be calculated by The predictive estimates (๐ฅฬ๐+1|๐ putting ๐ฅฬ๐+1|๐ and ๐๐+1|๐ into (12).
],
๐0|0 0 ]. =[ 0 0
๐ฅ๐ฆ
๐ฅ๐ง ๐1|0 ๐1|0 ๐1|0 ๐ป1๐ = ( V๐ฆ ) = ( V๐ง ) = ( ). ๐1|0 ๐1|0 ๐
1
๐ ๐ , ๐1|1 ) can be calculated by putting The filtering estimates (๐ฅฬ1|1 ๐ฆฬ1|0 and (73) into (16)โ(18).
(70)
๐ฅ๐ง ๐๐+1,๐|๐
=
๐ ๐น๐ ๐๐|๐ ๐ป๐๐
+
๐,๐ฅV ๐น๐ ๐๐|๐
+
๐,V๐ฅ ๐ ๐ฝ๐ ๐๐|๐ ๐ป๐
๐
(75)
๐,VV + ๐ฝ๐ ๐๐|๐ .
(2) For ๐ = 1, one has the following. Step 1 (calculation and introduction of suboptimal fading factor). ๐ฅฬ1|0 ,๐ฆฬ1|0 , ๐1 , and ๐1 can be calculated as follows: ๐ฅฬ1|0 = ๐0 (๐ฅฬ0|0 ) ,
๐ Once we obtain a new measurement ๐ฆ๐+1 , putting ๐ฅฬ๐+1|๐ , ๐ ๐ง๐ง ๐ฅ๐ง ฬ ๐๐+1|๐ , ๐ฆ๐+1|๐ , ๐๐+1|๐ , ๐๐+1|๐ , and (75) into (16)โ(21) can ๐ ๐ , ๐๐+1|๐+1 ) at time ๐ + calculate the filtering estimates (๐ฅฬ๐+1|๐+1 1.
5. Simulation
๐ฆฬ1|0 = ๐งฬ1|0 = โ1 (๐ฅฬ1|0 ) , ๐
๐1 โ ๐ป1 ๐น0 ๐0|0 ๐น0 ๐ป1๐ ,
(72)
๐1 โ ๐10 โ ๐
1 + ๐ป1 ๐0 ๐ป1๐ , where ๐10 can be calculated by (69). Putting ๐1 and ๐1 into ๐,๐ . (70) obtains ๐ 1 . Then, introducing ๐ 1 into (42) obtains ๐0|0 Step 2 (state prediction). ๐1|0 can be calculated by ๐1|0 = ๐
๐ฆ๐ฆ
๐ง๐ง ๐1|0 = ๐1|0 = ๐ป1 ๐1|0 ๐ป1๐ + ๐
1 ,
๐ ๐,๐ฅV ๐ ๐,๐ฅV ๐ ๐๐+1|๐ = ๐น๐ ๐๐|๐ ๐น๐ + ๐น๐ ๐๐|๐ ๐ฝ๐ + (๐น๐ ๐๐|๐ ๐ฝ๐ )
4.2. Computational Procedure of the STF/RDMCN. Now, we apply the first-order linear approximation method to compute these integrals in (15) and (22)โ(27) and develop a new STF. Further, the proposed STF/RDMCN for nonlinear system model (1) is summarized as follows. (1) Initialization (๐ = 0) is as follows: ๐ =[ ๐ฅฬ0|0
๐ฅ๐ ๐ฆ
Step 3 (state correction). ๐1|0 and ๐1|0 can be calculated as follows:
๐ ๐ ๐ ๐น0 ๐0|0 ๐น0 + ๐0 . The predictive estimates (๐ฅฬ1|0 , ๐1|0 ) can be calculated by putting ๐ฅฬ1|0 and ๐1|0 into (12).
To validate the effectiveness of the STF/RDM in nonlinear state estimation, the universal nonstationary growth model is used in the numerical simulation experiments. Meanwhile, we compare the performance of the three different filters, that is, the proposed filter, the STF/RDM in [22], and the existing EKF in the Appendix. The nonlinear systems model is as follows: ๐ฅ ๐ฅ๐+1 = 0.5๐ฅ๐ + 25 ๐ 2 + 8 cos (1.2๐) + ๐ค๐ , ๐ โฅ 0, 1 + ๐ฅ๐ (76) ๐ฅ๐2 ๐ง๐ = + V๐ , ๐ โฅ 1, 20
Mathematical Problems in Engineering
9
Table 1: Mean of RMSE๐ for ๐๐ = 0. One-step delay probability ๐๐ = 0.2
STF/RDMCN 14.2186
Table 2: Mean of RMSE๐ for ๐๐ = 0.5. STF/RDM 14.2186
One-step delay probability
STF/RDMCN
STF/RDM
Existing EKF
12.9925
13.8617
19.4514
๐๐ = 0.5
30 40 25
35 30 RMSE
RMSE
20
15
25 20 15
10
10 5
0
10
20
30
40
50
Time, k
5
0
10
20
30
40
50
Time, k
STF/RDM STF/RDMCN
Existing EKF STF/RDM STF/RDMCN
Figure 1: RMSE curves for ๐๐ = 0.2 and ๐๐ = 0.
Figure 2: RMSE curves for ๐๐ = 0.5 and ๐๐ = 0.5.
where the true value of the initial state, ๐ฅ0 , is zero, but, in simulation, the initial state estimate ๐ฅฬ0|0 is a random Gaussian variable with zero-mean and covariance which is a random number between zero and one, ๐ค๐ and V๐ are zero-mean Gaussian white noises satisfying ๐๐ = 10 and ๐
๐ = 1, and ๐ค๐ and V๐ are correlated with cross-covariance ๐๐ . Assuming that the available measurements are one-step randomly delayed, then ๐ฆ๐ = (1 โ ๐พ๐ ) ๐ง๐ + ๐พ๐ ๐ง๐โ1 ,
๐ > 1; ๐ฆ1 = ๐ง1 ,
(77)
where ๐พ๐ represents a sequence of uncorrelated Bernoulli random variables with ๐(๐พ๐ = 1) = ๐ for all ๐. The root mean square error (RMSE) is used as performance index for various nonlinear filters. The RMSE at time ๐ is defined as RMSE๐ = (
2 1 MC (๐ ) โ (๐ฅ๐ โ ๐ฅฬ๐(๐ ) ) ) MC ๐ =1
1/2
, 1 โค ๐ โค 50, (78)
where MC = 1000 represents the total number of the independent numerical simulation experiments and ๐ฅ๐(๐ ) and ๐ฅฬ๐(๐ ) , respectively, denote the true and estimated states at the ๐ th numerical simulation experiment. In case I, ๐๐ = 0.2 and ๐๐ = 0. Figure 1 shows the RMSE results of the proposed STF/RDMCN and the STF/RDM. The mean of RMSE๐ from the two filters is computed in Table 1. As can be seen from Figure 1 and Table 1, the proposed STF/RDMCN estimate performs exactly the same as the STF/RDM. The reason for this is that ๐๐ = 0 means that ๐ค๐ is not correlated with V๐ , and the proposed STF/RDMCN can
degrade to the STF/RDM. That is to say, regardless of whether the noises are correlated or not, the proposed STF/RDMCN can solve the filtering problem in these two cases; therefore it has a wider range of applications than the STF/RDM. In case II, ๐๐ = 0.5 and ๐๐ = 0.5. The RMSE results of the proposed STF/RDMCN, the STF/RDM, and the existing EKF are shown in Figure 2, the mean of RMSE๐ about the above three filters is shown in Table 2, and the mean of suboptimal fading factors derived from the STF/RDMCN and STF/RDM is given in Figure 3. According to Table 2 and Figures 2 and 3, the STF/RDMCN and STF/RDM outperform the existing EKF in estimation accuracy. This is due to the fact that the STF/RDMCN and STF/RDM can seasonably find out the increase of residuals and enhance the estimation precision via the suboptimal fading factors adaptively increasing while the existing EKF does not adapt to the increase of residuals. Moreover, the mean of RMSE๐ and that of the suboptimal fading factor of STF/RDMCN are smaller than the STF/RDM. Compared with the STF/RDM, the proposed STF/RDMCN can reflect the change of the residuals by lesser adjusting of the suboptimal fading factors. This means that, unlike the STF/RDM, the proposed STF/RDMCN can weaken the effect of the accumulative estimation error through the smaller suboptimal fading factor to ensure better tracking accuracy. In case III, ๐๐ = 0.1, 0.2, . . . , 0.9 and ๐๐ = 0.5. Figure 4 gives the mean of RMSE๐ calculated by utilizing the proposed STF/RDMCN, the STF/RDM, and the existing EKF. As the value of ๐ increases, the mean of the existing EKF is increased, and those of the proposed STF/RDMCN and STF/RDM
10
Mathematical Problems in Engineering 400
21 20
350
19
300 20
The mean of RMS๏ผ
k
The mean of fading factor
25
250 15
200 150
10
100
5
50
0
0
0
18 17 16 15 14
10
20
10
20
30
40
30
13
50 40
50
Time, k
12 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Sk
STF/RDM STF/RDMCN
Figure 3: The mean of fading factor for ๐๐ = 0.5 and ๐๐ = 0.5.
Existing EKF STF/RDM STF/RDMCN
Figure 5: The mean of RMSE๐ for ๐๐ = 0.5 and ๐๐ = 0.1, 0.2, . . . , 0.9. 21 20
6. Conclusion
The mean of RMS๏ผ
k
19 18 17 16 15 14 13 12 11 0.1
0.2
0.3
0.4
0.5 pk
0.6
0.7
0.8
0.9
Existing EKF STF/RDM STF/RDMCN
Figure 4: The mean of RMSE๐ for ๐๐ = 0.1, 0.2, . . . , 0.9 and ๐๐ = 0.5.
are decreased. The average of the proposed STF/RDMCN is smaller than that of the STF/RDM and the existing EKF. This figure indicates that the proposed STF/RDMCN has the best filtering performance compared to the other two filters, in case the delay probability ๐ is chosen from a greater value. In case IV, ๐๐ = 0.5 and ๐๐ = 0.1, 0.2, . . . , 0.9. Figure 5 shows the mean of RMSE๐ about the three filters. As can be seen from Figure 5, the mean of existing EKF is the worst, meaning that the proposed STF/RDMCN and the STF/RDM have better tracking accuracy. Furthermore, the proposed STF/RDMCN is more promising, regardless of whether correlation parameter ๐๐ has a large range of change or not.
In this paper, we have presented a strong tracking filter with randomly delayed measurements and correlated noises (STF/RDMCN). By reconstructing an equivalent nonlinear state function, the framework of decoupling filter (DF) is derived, which can eliminate the correlation between the process and measurement noises. Then, a new EKF with onestep randomly delayed measurements is developed by using first-order linearization approximation for calculating the Gaussian-weighted integrals in the DF framework. Further, in order to make the above EKF have a strong tracking ability, the suboptimal fading factor, which is derived in the sense of the extended orthogonality principle (EOP), is introduced. Finally, the STF/RDMCN, which can seasonably find the alteration of residuals between the available measurements and predicted measurements and keep welltracking performance via changing the suboptimal fading factor in real time, is formed. The numerical experiment outcomes confirmed that, under the condition of selecting different delay probabilities and correlation parameters, the proposed STF/RDMCN exceeds the STF/RDM and the existing EKF in tracking accuracy. It is also demonstrated that the previously mentioned STF/RDM is the special case of the proposed method because the STF/RDMCN can degrade to the STF/RDM via setting the correlation parameter to zero.
Appendix According to the literature [24], the existing EKF can be obtained by using first-order linearization approximation. Step 1 (state prediction). One has ๐ฅฬ๐+1|๐ = ๐ฅฬ๐+1|๐โ1 + ๐๐ (๐ฆ๐ โ ๐ฆฬ๐|๐โ1 ) ,
Mathematical Problems in Engineering
11
๐ฆ๐ฆ
๐๐+1|๐ = ๐๐+1|๐โ1 โ ๐๐ ๐๐|๐โ1 ๐๐๐ , ๐ฅ๐ฆ
๐พ๐+1 = [
โ1
๐ฆ๐ฆ
๐๐ = ๐๐+1,๐|๐โ1 (๐๐|๐โ1 ) ,
๐ฅ ๐พ๐+1 V ๐พ๐+1
๐ฅ๐ ๐ฆ
๐ฆ๐ฆ
โ1
] = ๐๐+1|๐ (๐๐+1|๐ )
๐ฅ๐ฆ
=[
๐ฆฬ๐|๐โ1 = (1 โ ๐๐ ) ๐งฬ๐|๐โ1 + ๐๐ ๐งฬ๐โ1|๐โ1 , ๐ฆ๐ฆ
๐ง๐ง ๐ง๐ง ๐๐|๐โ1 = (1 โ ๐๐ ) ๐๐|๐โ1 + ๐๐ ๐๐โ1|๐โ1 + (1 โ ๐๐ )
๐ฅ๐ ๐ฆ ๐๐+1|๐
๐
โ
๐๐ (ฬ๐ง๐|๐โ1 โ ๐งฬ๐โ1|๐โ1 ) (ฬ๐ง๐|๐โ1 โ ๐งฬ๐โ1|๐โ1 ) ,
=[
=[
(A.1)
โ1 ๐ฆ๐ฆ V๐ฆ ] (๐๐+1|๐ ) ๐๐+1|๐
,
๐ฅ๐ฆ
๐ฅ๐ฆ
๐ฅ๐ง ๐ฅ๐ง ๐๐+1,๐|๐โ1 = (1 โ ๐๐ ) ๐๐+1,๐|๐โ1 + ๐๐ ๐๐+1,๐โ1|๐โ1 ,
๐๐+1|๐ ๐๐+1|๐ V๐ฆ
๐๐+1|๐
]
๐ฅ๐ง ๐ฅ๐ง (1 โ ๐๐+1 ) ๐๐+1|๐ + ๐๐+1 ๐๐+1,๐|๐ V๐ง V๐ง (1 โ ๐๐+1 ) ๐๐+1|๐ + ๐๐+1 ๐๐+1,๐|๐
(A.3)
where ๐๐ represents the gain matrix and where ๐พ๐ represents the gain matrix and
๐ฅฬ๐+1|๐โ1 = ๐๐ (๐ฅฬ๐|๐โ1 ) ,
๐ฅ๐ง ๐ ๐๐+1|๐ = ๐๐+1|๐ ๐ป๐+1 ,
๐๐+1|๐โ1 = ๐น๐ ๐๐|๐โ1 ๐น๐๐ + ๐๐ , ๐๐ (๐ฅ ) ๓ตจ๓ตจ๓ตจ , ๐น๐ = ๐ ๐ ๓ตจ๓ตจ๓ตจ๓ตจ ๐๐ฅ๐ ๓ตจ๓ตจ๐ฅ๐ =๐ฅฬ๐|๐โ1
๐ป๐+1 =
๐ง๐ง ๐๐|๐โ1 = ๐ป๐ ๐๐|๐โ1 ๐ป๐๐ + ๐
๐ ,
๐งฬ๐โ1|๐โ1 = โ๐โ1 (๐ฅฬ๐โ1|๐โ1 ) + ฬV๐โ1|๐โ1 , ๐ง๐ง ๐ ๐ฅV ๐๐โ1|๐โ1 = ๐ป๐โ1 ๐๐โ1|๐โ1 ๐ป๐โ1 + ๐ป๐โ1 ๐๐โ1|๐โ1
๐ฅ๐ง ๐๐+1,๐|๐โ1 ๐ฅ๐ง ๐๐+1,๐โ1|๐โ1
๐ฅV (๐ป๐โ1 ๐๐โ1|๐โ1 )
=
๐น๐ ๐๐|๐โ1 ๐ป๐๐
=
๐ฅ๐ฅ ๐ ๐น๐ ๐๐,๐โ1|๐โ1 ๐ป๐โ1 ,
๐ป๐โ1 =
๐
VV + ๐ฝ๐ ๐๐|๐ ,
๐โ๐ (๐ฅ๐ ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ , ๐๐ฅ๐ ๓ตจ๓ตจ๓ตจ๐ฅ๐ =๐ฅฬ๐|๐โ1
+
๐โ๐+1 (๐ฅ๐+1 ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ , ๐๐ฅ๐+1 ๓ตจ๓ตจ๓ตจ๐ฅ๐+1 =๐ฅฬ๐+1|๐
๐ฅ๐ง ๐ฅV ๐ฅV = ๐น๐ ๐๐|๐ ๐ป๐๐ + ๐น๐ ๐๐|๐ + ๐ฝ๐ (๐ป๐ ๐๐|๐ ) ๐๐+1,๐|๐
๐งฬ๐|๐โ1 = โ๐ (๐ฅฬ๐|๐โ1 ) ,
๐ป๐ =
],
๐
+
(A.2)
VV ๐๐โ1|๐โ1 ,
๐น๐ =
๐๐๐ (๐ฅ๐ ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ , ๐๐ฅ๐ ๓ตจ๓ตจ๓ตจ๐ฅ๐ =๐ฅฬ๐|๐
๐ป๐ =
๐โ๐ (๐ฅ๐ ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ , ๐๐ฅ๐ ๓ตจ๓ตจ๓ตจ๐ฅ๐ =๐ฅฬ๐|๐
(A.4)
V๐ง = ๐
๐+1 , ๐๐+1|๐ V๐ง = 0. ๐๐+1,๐|๐
+ ๐๐ ,
๐ฆ๐ฆ ๐ฅ๐ฆ ๐ง๐ง ๐ฅ๐ง For ๐ = 0, ๐ฆ1 = ๐ง1 , ๐ฆฬ1|0 = ๐งฬ1|0 , ๐1|0 = ๐1|0 , ๐1|0 = ๐1|0 , V๐ฆ V๐ง ๐1|0 = ๐1|0 . Once ๐ฆ๐+1 are obtained, the filtering estimates ๐ ๐ and ๐๐+1|๐+1 at time ๐ + 1 of the augmented state can ๐ฅฬ๐+1|๐+1 be calculated.
๐โ๐โ1 (๐ฅ๐โ1 ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ , ๐๐ฅ๐โ1 ๓ตจ๓ตจ๓ตจ๐ฅ๐โ1 =๐ฅฬ๐โ1|๐โ1
๐ฅ๐ฅ ๐ฅ๐ฅ ๐ฅ ๐๐,๐โ1|๐โ1 = ๐๐,๐โ1|๐โ2 โ ๐๐โ1 ๐๐โ1|๐โ2 (๐พ๐โ1 ) ,
Conflicts of Interest
๐ฅ๐ฅ = ๐น๐โ1 ๐๐โ1|๐โ2 , ๐๐,๐โ1|๐โ2
The authors declare that there are no conflicts of interest regarding the publication of this paper.
๐ฆ๐ฆ
๐น๐โ1 =
๐
๐๐๐โ1 (๐ฅ๐โ1 ) ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ . ๐๐ฅ๐โ1 ๓ตจ๓ตจ๓ตจ๐ฅ๐โ1 =๐ฅฬ๐โ1|๐โ2
For ๐ = 0, ๐ฅฬ1|0 = ๐0 (๐ฅฬ0|0 ) and ๐1|0 = ๐น0 ๐0|0 ๐น0๐ + ๐0 . For ๐ฆ๐ฆ ๐ฅ๐ฆ ๐ง๐ง ๐ฅ๐ง , ๐2,1|0 = ๐2,1|0 . The ๐ = 1, ๐ฆ1 = ๐ง1 , ๐ฆฬ1|0 = ๐งฬ1|0 , ๐1|0 = ๐1|0 ๐ ๐ and ๐๐+1|๐ of the augmented state predictive estimates ๐ฅฬ๐+1|๐ can be calculated by putting ๐ฅฬ1|0 and ๐1|0 into (12). Step 2 (state correction). One has ๐ ๐ ๐ฅฬ๐+1|๐+1 = ๐ฅฬ๐+1|๐ + ๐พ๐+1 (๐ฆ๐+1 โ ๐ฆฬ๐+1|๐ ) , ๐ฆ๐ฆ
๐ ๐ ๐ = ๐๐+1|๐ โ ๐พ๐+1 ๐๐+1|๐ ๐พ๐+1 , ๐๐+1|๐+1
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