MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 457536 10.1155/2014/457536 457536 Research Article A Robust Recursive Filter for Nonlinear Systems with Correlated Noises, Packet Losses, and Multiplicative Noises Qian Hua-Ming 1 http://orcid.org/0000-0003-1953-892X Huang Wei 1 Liu Biao 2 Shen Chen 1 Su Xiaojie 1 College of Automation Harbin Engineering University Harbin 150001 China hrbeu.edu.cn 2 Dongguan Techtotop Microelectronics Co., Ltd. Chengdu R & D Branch Chengdu 610041 China 2014 3132014 2014 02 11 2013 16 01 2014 24 01 2014 31 3 2014 2014 Copyright © 2014 Hua-Ming Qian 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.

A robust filtering problem is formulated and investigated for a class of nonlinear systems with correlated noises, packet losses, and multiplicative noises. The packet losses are assumed to be independent Bernoulli random variables. The multiplicative noises are described as random variables with bounded variance. Different from the traditional robust filter based on the assumption that the process noises are uncorrelated with the measurement noises, the objective of the addressed robust filtering problem is to design a recursive filter such that, for packet losses and multiplicative noises, the state prediction and filtering covariance matrices have the optimized upper bounds in the case that there are correlated process and measurement noises. Two examples are used to illustrate the effectiveness of the proposed filter.

1. Introduction

In recent years, the state estimation theory has received extensive attention in many fields of application, such as attitude estimation , target tracking , signal processing , and integrated navigation . State estimation refers to a methodology that is used for estimating the state of a time-varying system through noisy measurements, which are different from other methods . So far, various kinds of filtering algorithms for state estimation have been presented, for example, Kalman filter , extended Kalman filter (EKF) , unscented Kalman filter (UKF) , and so forth. As is well known, among those filters, Kalman filter is an optimal solution based on the minimum mean square error rule for linear systems and EKF is an effective way for softly nonlinear system to estimate the state by using linearization techniques. Although EKF is a popular estimating algorithm in engineering practice, its use must satisfy the following two assumptions: (1) the system model should be accurate and (2) the additive noises should be Gaussian and uncorrelated. Otherwise, the performance of EKF can be degraded severely, even unstable. Unfortunately, in real world, the model uncertainty is an unavoidable and crucial problem for nonlinear systems. Therefore it is required to develop a more general filtering algorithm. To this end, the robust filtering technique has been developed to reduce the unfavorable effect of model uncertainties by establishing an appropriate uncertain model in consideration of uncertainties. Up to now, a lot of literatures on the robust filtering problem with model uncertainties have been published, such as the H filter , set-valued nonlinear filter [17, 18], mixed H2/H filter [19, 20], and robust extended Kalman filter design [21, 22]. In these reports, the robust recursive filter design has been investigated to be available for handling the nonlinear filtering problem with model uncertainties. For instance, a discrete-time robust extended Kalman filter has been presented for uncertain systems with sum quadratic constraints in . Due to the influence of the misalignments of star sensors, by considering the model uncertainties, a nonlinear robust filter for satellite attitude determination is developed and verified in .

In literature mentioned above, however, only additive noises are considered for nonlinear systems. Actually, another important noise called multiplicative noise is often encountered in many engineering systems, such as attitude estimation systems and airborne synthetic aperture radar systems. It is coupled with the state and has an unknown noise variance, which results in a negative impact on the state estimation. Hence, the multiplicative noise is usually viewed as a model uncertainty. Currently, the nonlinear robust filtering problem with multiplicative noises has been much less researched. In [23, 24], by utilizing linear matrix inequality approach, a robust Kalman filter is derived for linear systems. Different from them, another robust Kalman filter is proposed for linear systems by finding two Riccati differential equations and determining the filter parameters in . Then,  extends the work to nonlinear systems. Apart from multiplicative noises, signal transmissions in the sensor networks are often unreliable. For example, sudden sensor failure, random communication delays, and packet losses appear in the practical system frequently . All these lead to the measurement mode uncertainty. Accordingly, the filtering problem with packet losses has stirred considerable research attention and many research results have been published recently; see, for example . In most literatures, the packet loss is described as a random variable in the distribution of Bernoulli, which may not be available because of the existence of the different transmission process in multiple sensors. In [36, 37], a diagonal matrix composed of Bernoulli random variables is introduced to the measurement equation, which means that individual sensor might have different missing rates. Meanwhile, it is not difficult to find that the most existing filtering researches concerning packet losses are subject to linear systems. However, as we all know, nonlinearity is inevitable in almost all engineering applications, which will directly degrade the quality of the filtering performance. For this purpose, a quantized recursive filtering is presented for a class of nonlinear systems with missing measurements, multiplicative noises, and quantization effects in . Though missing measurements and multiplicative noises are taken into consideration at the same time, this work endures the limitation that the measurement equation must be linear, which makes that the algorithm in  cannot be extended to solve the general nonlinear filtering problems in the case that the process and measurement model are all nonlinear. But note that the multiplicative noise case is just a special case of the stochastic nonlinearities considered in . Therefore, an explicit and systematic solution to this problem can be extended.

In addition, the correlation of additive noises is one of the key factors to the filtering algorithm. Disturbed by the complicated environment, the additive noises often show the characteristic of correlation in the practical application. Unluckily, the design procedures of all the above filters for multiplicative noises or packet losses are based on the assumption that there are uncorrelated additive noises in the system. In fact, this assumption does not always come into existence, and the process noise might be correlated with the measurement noise in real applications. In , a modified UKF for nonlinear systems with correlated additive noises is proposed. Wang et al.  extend the work to develop a Gaussian approximation recursive filter framework to deal with correlated noises. But model uncertainties are not considered in these works. To the best of the authors’ knowledge, up to the present, based on the assumption that the process noise is correlated with the measurement noise, the nonlinear robust filtering problem with multiplicative noises and packet losses has not been reported. Therefore, in order to better reflect the actual situation and consider the complex dynamical systems, there is a strong desire to develop a robust recursive filter to handle the robust filtering problem with correlated additive noises, multiplicative noises, and packet losses.

Motivated by the above discussion, we present a robust recursive filter for a class of nonlinear systems with correlated additive noises, multiplicative noises, and packet losses. In this paper, multiplicative noises are assumed as zero mean Gaussian white noises and the packet losses are modeled as independent Bernoulli random variables. Based on the structure of the extended Kalman filter with correlated noises, the proposed filter designs an optimal upper bound of the prediction error and the filtering error covariance matrices, respectively. The main contributions of the paper are as follows. (1) In the case that the process noise is correlated with the measurement noise, a recursive filter framework is established to deal with the robust filtering problem for nonlinear systems in the presence of multiplicative noises and packet losses. (2) The addressed robust recursive filter problem is new especially when the correlated additive noises appear in the system. (3) The developed robust filter is recursive, which is suitable for online applications. The remainder of the paper is organized as follows. In Section 2, the problem is formulated. In Section 3, the robust recursive filter with correlated additive noises, multiplicative noises, and packet losses is developed. In Section 4, two simulation examples are employed, and the simulation analysis is given. In Section 5, some conclusions are drawn.

2. Problem Formulation and Preliminaries

Consider a general class of discrete time-varying systems with multiplicative noises, correlated additive noises, and packet losses: (1)xk+1=f(xk)+i=1qAiksηikxk+wk,(2)yk=Σkh(xk)+i=1rCiksξikxk+vk, where xkRn is the state vector, ykRm is the measurement vector, ηik and ξik are the uncorrelated zero mean Gaussian multiplicative noises, and Aiks and Ciks are known matrices with appropriate dimension. The diagonal matrix Σk is denoted as Σk=diag{λk1,λk2,,λkm}, where λki  (i=1,2,,m) are independent Bernoulli random variables. It is assumed that λki has the probability density function p(λki) on the interval [0,1] with mean μki and covariance (σki)2. The process noise wk and the measurement noise vk are correlated zero mean Gaussian white noises, which satisfies (3)E(wk)=0,cov(wk,wjT)=QkδkjE(vk)=0,cov(vk,vjT)=Rkδkjcov(wk,vjT)=Skδkj.

The deterministic nonlinear functions f(xk):RnRn and h(xk):RnRm are known. According to the known measurement equation, we employ the assumption in  as the following form: (4)h(xk)a1xk+a2, where a1 and a2 are the nonnegative scalars.

Because of existing correlated additive noises, for system (1)-(2), a recursive filter with correlated noises to be designed is constitutive of the following two steps including the state prediction and correction:

State prediction: (5)x^k+1k-1=f(x^kk-1),(6)x^k+1k=x^k+1k-1+Lk[yk-Σ-kh(x^kk-1)],

State correction: (7)x^k+1k+1=x^k+1k+Kk+1[yk+1-Σ-k+1h(x^k+1k)],

where Σ-k=E(Σk)=diag(μk1,μk2,,μkm); x^kk-1 is the one-step state prediction at time k-1 with x^0-1=x^00; x^k+1k-1 is the two-step state prediction at time k-1; Lk and Kk+1 are the gain parameters to be determined;   x^k+1k+1 is the state estimation at time k+1.

The aim of the paper is to design a recursive filter for the structures (5)–(7), which make the filtering prediction and estimation covariance have upper bounds in the presence of multiplicative noises and packet losses. Suppose that two positive definite matrices Ξk+1k and Ξk+1k+1 satisfy (8)E[(xk+1-x^k+1k)(xk+1-x^k+1k)T]Ξk+1k,E[(xk+1-x^k+1k+1)(xk+1-x^k+1k+1)T]Ξk+1k+1.

The addressed filtering problem is that the designed filter parameters Lk and Kk+1 in (5)–(7) should minimize the upper bounds Ξk+1k and Ξk+1k+1.

Remark 1.

In engineering applications, multiplicative noises constantly existing in the systems depend on the real state value, which results in the unknown noise variance. As discussed in , it should be seen as a model uncertainty. Moreover, the definition (3) shows that the process noise is correlated with the measurement noise. This requires employing a new state prediction step in (5)-(6), which is different from the state prediction of the recursive filter form in [36, 37]. Meanwhile, the unknown prediction gain Lk does not exist in the literature [36, 37], which will lead to the different estimation results. Subsequently, the packet losses are described by utilizing the diagonal matrix Σk in (2), which indicates that the different sensors have different failure rates. Since multiplicative noises, correlated additive noises, and packet losses are taken into account, the system (1)-(2) is more generalized to describe the realistic situations in engineering.

3. Design of Robust Recursive Filter 3.1. The Error Covariance Matrix

Denote the two-step prediction error as x~k+1k-1=xk+1-x^k+1k-1 and the one-step prediction error as x~k+1k=xk+1-x^k+1k. From (1), (2), (5), and (6), they can be calculated as (9)x~k+1k-1=f(xk)-f(x^kk-1)i=1qAiksηikxk+wk,(10)x~k+1k=x~k+1k-1-Lk[Σkh(xk)+i=1rCiksξikxk+vk-Σ-kh(x^kk-1)].

The nonlinear functions f(xk) and h(xk) can be linearized by utilizing the Taylor series expansion around x^kk-1: (11)f(xk)=f(x^kk-1)+Akx~kk-1+o(|x~kk-1|),(12)h(xk)=h(x^kk-1)+Ckx~kk-1+o^(|x~kk-1|), where Ak=f(xk)/xk|xk=x^kk-1; Ck=h(xk)/xk|xk=x^kk-1; o(|x~kk-1|) and o^(|x~kk-1|) represent the high-order terms of the Taylor series expansion. According to the literature , o(|x~kk-1|) and o^(|x~kk-1|) can be expressed as (13)o(|x~kk-1|)=BkβkEkx~kk-1,o^(|x~kk-1|)=DkαkEkx~kk-1, where BkRn×n and DkRm×n are known scaling matrices, EkRn×n is a known tuning matrix, and βkRn×n and αkRn×n are unknown time-varying matrices accounting for the linearization errors of the system model that satisfies (14)βkβkTI,αkαkTI.

According to (9), (11), and (13), the two-step prediction error can be written as (15)x~k+1k-1=(Ak+BkβkEk)x~kk-1+i=1qAiksηikxk+wk.

Substituting (12), (13), and (15) into (9), the one-step prediction error can be determined as (16)x~k+1k=(Ak+BkβkEk)x~kk-1+i=1qAiksηikxk+wk-Lk[Σkh(xk)+i=1rCiksξikxk+vk-Σ-k(h(xk)-(Ck+DkαkEk)x~kk-1)i=1rCiksξikxk]=[(Ak+LkΣ-kCk)+(Bkβk+LkΣ-kDkαk)Ek]x~kk-1-Lk(Σk-Σ-k)h(xk)-Lk(i=1rCiksξikxk+vk)+i=1qAiksηikxk+wk=[(Ak+LkΣ-kCk)+HkFkJkEk]x~kk-1-Lk(Σk-Σ-k)h(xk)-Lk(i=1rCiksξikxk+vk)+i=1qAiksηikxk+wk, where it is assumed that Hk=[BkLkΣ-kDk], Fk=[βk0n×n0n×nαk], and Jk=[In×nIn×n] and from (14) we have FkFkTI.

The one-step prediction error covariance can be obtained as (17)Pk+1k=E(x~k+1kx~k+1kT)=[(Ak+LkΣ-kCk)+HkFkJkEk]Pkk-1×[(Ak+LkΣ-kCk)+HkFkJkEk]T-SkLkT-LkSkT+Lk[i=1rCiksE(xkxkT)(Ciks)TΣ˘kE(h(xk)hT(xk))+i=1rCiksE(xkxkT)(Ciks)T+Rk]LkT+i=1qAiksE(xkxkT)(Aiks)T+Qk, where Σ˘k=diag{(σk1)2,(σk2)2,,(σkm)2}.

Denote the estimation error as (18)x~k+1k+1=xk+1-x^k+1k+1.

From (1), (2), and (7), it can be rewritten as (19)x~k+1k+1=x~k+1k-Kk+1[yk+1-Σ-k+1h(x^k+1k)]=x~k+1k-Kk+1[Σk+1h(xk+1)+i=1rCik+1sξik+1xk+1+vk+1-Σ-k+1h(x^k+1k)i=1rCik+1sξik+1xk+1]=(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)x~k+1k-Kk+1(i=1rCik+1sξik+1xk+1+vk+1)-Kk+1(Σk+1-Σ-k+1)h(xk+1), where C-k+1=h(xk)/xk|xk=x^k+1k; D-k+1 is a known problem-dependent scaling matrix; E-k+1 is a known tuning matrix; α-k+1 is an unknown time-varying matrix that satisfies (20)α-k+1α-k+1TI.

Subsequently, in the light of (19), the filtering error covariance can be expressed as (21)Pk+1k+1=E(x~k+1k+1x~k+1k+1T)=(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)×Pk+1k(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)T+Kk+1[Rk+1Σ˘k+1E(h(xk+1)hT(xk+1))i=1rCik+1sE(xk+1xk+1T)(Cik+1s)T+i=1rCik+1sE(xk+1xk+1T)(Cik+1s)T+i=1rCik+1sE(xk+1xk+1T)(Cik+1s)TRk+1]Kk+1T.

Remark 2.

Since there are the high-order errors, the matrices βk, αk, and α-k+1 are unknown, which makes the fact that the prediction covariance Pk+1k and the filtering error covariance Pk+1k+1 from (17) and (21) cannot be computed directly. In order to complete the design of the filter, an effective way is to calculate the upper bounds for the Pk+1k and Pk+1k+1 and then design the prediction gain Lk and the filtering gain Kk+1 to minimize the upper bounds. Due to the influence correlated noises and unknown prediction gain Lk, there is a striking contrast between the prediction covariance Pk+1k in this paper and the counterpart in the literature [36, 37].

3.2. The Robust Recursive Filter Design

To develop the robust recursive filter, the following lemmas are given.

Lemma 3 (see [<xref ref-type="bibr" rid="B40">40</xref>]).

Let A=[aij]n×n be a real matrix and let B=diag(b1,b2,,bn) be a diagonal random matrix. Then (22)E{BABT}=[E{b12}E{b1b2}E{b1bn}E{b2b1}E{b22}E{b2bn}E{bnb1}E{bnb2}E{bn2}]A, where is the Hadamard product.

Lemma 4 (see [<xref ref-type="bibr" rid="B41">41</xref>]).

Given matrices A, H, E, and F with compatible dimensions such that FFTI, let X be a symmetric positive definite matrix and let γ be an arbitrary positive constant such that (23)γ-1I-EXET>0. Then the following matrix inequality holds: (24)(A+HFE)X(A+HFE)TA(X-1-γETE)-1AT+γ-1HHT.

Lemma 5 (see [<xref ref-type="bibr" rid="B42">42</xref>]).

For 0kn, suppose that X=XT>0, ek(X)=ekT(X)Rn×n, and gk(X)=gkT(X)Rn×n. If there exists YX such that (25)ek(Y)ek(X),gk(Y)>ek(Y), then the solutions Mk and Nk to the following difference equations, (26)Mk+1=ek(Mk),Nk+1=gk(Nk),M0=N0>0, satisfy MkNk.

According to these lemmas, the following theorem is given to obtain the main results of the robust recursive filter.

Theorem 6.

Consider the covariance matrices of the one-step prediction errorand the filtering error in (17) and (21). Assume that the conditions shown in (14) and (20) come into existence. Let λ1, λ2, ε1, and ε2 be positive scalars. If the following two discrete-time Riccati difference equations, (27)Ξk+1k=(Ak+LkΣ-kCk)(Ξkk-1-1-2λ1EkTEk)-1(Ak+LkΣ-kCk)T+λ1-1(BkBkT+LkΣ-kDkDkTΣ-kTLkT)+Qk-SkLkT-LkSkT+Lk{i=1rvΣ˘k[2(a12tr(Ωkk-1)+a22)]I+i=1rCiksΩkk-1(Ciks)T+Rk}LkT+i=1qAiksΩkk-1(Aiks)T,(28)Ξk+1k+1=(I-Kk+1Σ-k+1C-k+1)(Ξk+1k-1-λ2E-k+1TE-k+1)-1×(I-Kk+1Σ-k+1C-k+1)T+λ2-1Kk+1Σ-k+1D-k+1D-k+1TΣ-k+1TKk+1T+Kk+1{i=1rCik+1sΔk+1k(Cik+1s)TΣ˘k+1[2(a12tr(Δk+1k)+a22)]I+i=1rCik+1sΔk+1k(Cik+1s)T+Rk+1}Kk+1T with initial covariance Ξ0-1=Ξ00>0 have positive definite solution, such that for 0kN(29)λ1-1I-JkEkΞkk-1(JkEk)T>0,λ2-1I-E-k+1Ξk+1kE-k+1T>0, where (30)Ωkk-1=(1+ε1)Ξkk-1+(1+ε1-1)x^kk-1x^kk-1T,Δk+1k=(1+ε2)Ξk+1k+(1+ε2-1)x^k+1kx^k+1kT, then the prediction gain Lk and the filtering gain Kk+1 are given by (31)Lk=[Sk-Ak(Ξk|k-1-1-2λ1EkTEk)-1CkTΣ-k]×{i=1rCiksΩk|k-1(Ciks)TΣ-kCk(Ξk|k-1-1-2λ1EkTEk)-1CkTΣ-k+λ1-1Σ-kDkDkTΣ-kT+Σ˘k[2(a12tr(Ωk|k-1)+a22)]I+i=1rCiksΩk|k-1(Ciks)T+Rk}-1,(32)Kk+1=(Ξk+1|k-1-λ2E-k+1TE-k+1)-1C-k+1TΣ-k+1×{[2(a12tr(Δk+1|k)+a22)]i=1rCik+1sΔk+1|k(Cik+1s)TΣ-k+1C-k+1(Ξk+1|k-1-λ2E-k+1TE-k+1)-1C-k+1TΣ-k+1+λ2-1Σ-k+1D-k+1D-k+1TΣ-k+1T+i=1rCik+1sΔk+1|k(Cik+1s)Ti=1rCik+1sΔk+1|k(Cik+1s)T+Σ˘k+1[2(a12tr(Δk+1|k)+a22)]I+Rk+1(Ξk+1|k-1-λ2E-k+1TE-k+1)-1}-1 and the matrices Ξk+1k and Ξk+1k+1 are upper bounds for Pk+1k and Pk+1k+1; that is, (33)Pk+1kΞk+1k,Pk+1k+1Ξk+1k+1. Moreover, the prediction gain Lk given by (31) minimizes the upper bound Ξk+1k and the filtering gain Kk+1 given by (32) minimizes the upper bound Ξk+1k+1.

Proof.

According to (17) and (21), the prediction covariance Pk+1k and the filtering error covariance Pk+1k+1 can be expressed as the functions of Pkk-1 and Pk+1k: (34)Pk+1k=Pk+1k(Pkk-1),Pk+1k+1=Pk+1k+1(Pk+1k).

Assume that ε1 is a positive scalar. The matrix inequality can be obtained as (35)x~kk-1x^kk-1T+x^kk-1x~kk-1Tε1x~kk-1x~kk-1T+ε1-1x^kk-1x^kk-1T.

From (35), we have (36)E(xkxkT)E[(x~kk-1+x^kk-1)(x~kk-1+x^kk-1)T](1+ε1)Pkk-1+(1+ε1-1)x^kk-1x^kk-1T=𝚿kk-1.

According to the literature , based on (4), we obtain (37)E{h(xk)hT(xk)}E{h(xk)2}IE{(a1xk+a2)2}I(2a12E{xk2}+2a22)I=2[a12tr(E(xkxkT))+a22]I.

Substituting (36) into (37), it can be rewritten as (38)E{h(xk)hT(xk)}2[a12tr(𝚿kk-1)+a22]I.

Furthermore, inserting (36) and (38) into (17), the one-step prediction error covariance can be rearranged as (39)Pk+1|k[(Ak+LkΣ-kCk)+HkFkJkEk]Pk|k-1×[(Ak+LkΣ-kCk)+HkFkJkEk]T-SkLkT-LkSkT+Lk[i=1rCiks𝚿k|k-1(Ciks)T+RkΣ˘k2[a12tr(𝚿k|k-1)+a22]Iyyyyyyyyy+i=1rCiks𝚿k|k-1(Ciks)T+Rki=1rCiks𝚿k|k-1(Ciks)T+Rk}LkTyyyyyyyyy+i=1qAiks𝚿k|k-1(Aiks)T+Qk,

Similar to (36) and (38), let ε2 be a positive scalar; we have (40)E(xk+1xk+1T)(1+ε2)Pk+1k+(1+ε2-1)x^k+1kx^k+1kT=Λk+1k,E{h(xk+1)hT(xk+1)}2[a12tr(Λk+1k)+a22]I.

Substituting (40) into (21), we have (41)Pk+1|k+1=(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)Pk+1|k×(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)T+Kk+1{i=1rCik+1sΛk+1|k(Cik+1s)TΣ˘k+1[2(a12tr(Λk+1|k)+a22)]I+i=1rCik+1sΛk+1|k(Cik+1s)T+Rk+1}Kk+1T.

From (27) and (28), Ξk+1k and Ξk+1k+1 can be rewritten as the functions of Ξkk-1 and Ξk+1k: (42)Ξk+1k=Ξk+1k(Ξkk-1),Ξk+1k+1=Ξk+1k+1(Ξk+1k).

Assume that there exist λ1>0 and λ2>0. Let the matrices Ek and E-k+1 satisfy the inequalities (43)λ1-1I-JkEkΞkk-1(JkEk)T>0,λ2-1I-E-k+1Ξk+1kE-k+1T>0.

According to Lemma 4, we have (44)[(Ak+LkΣ-kCk)+HkFkJkEk]Ξkk-1×[(Ak+LkΣ-kCk)+HkFkJkEk]T(Ak+LkΣ-kCk)(Ξkk-1-1-2λ1EkTEk)-1×(Ak+LkΣ-kCk)T+λ1-1(BkBkT+LkΣ-kDkDkTΣ-kTLkT),(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)Ξk+1k×(I-Kk+1Σ-k+1C-k+1-Kk+1Σ-k+1D-k+1α-k+1E-k+1)T(I-Kk+1Σ-k+1C-k+1)(Ξk+1k-1-λ2E-k+1TE-k+1)-1×(I-Kk+1Σ-k+1C-k+1)T+λ2-1Kk+1Σ-k+1D-k+1D-k+1TΣ-k+1TKk+1T.

Combining (39) and (41)–(44), the condition (25) can be satisfied in Lemma 5. Thus, according to Lemmas 4 and 5, we have (45)Pk+1kΞk+1k,Pk+1k+1Ξk+1k+1.

To minimize the upper bounds, constructingthe optimized prediction gain Lk and the filtering gain Kk+1 is to minimize the upper bounds Ξk+1k and Ξk+1k+1; according to (27) and (28), we have (46)tr(Ξk+1|k)Lk=2(Ak+LkΣ-kCk)(Ξk|k-1-1-2λ1EkTEk)-1tr(Ξk+1|k)Lk=×CkTΣ-k-2Sktr(Ξk+1|k)Lk=+2Lk{i=1rCiksΩk|k-1(Ciks)Tλ1-1Σ-kDkDkTΣ-kTtr(Ξk+1|k)Lk=+Σ˘k[2(a12tr(Ωk|k-1)+a22)]Itr(Ξk+1|k)Lk=+i=1rCiksΩk|k-1(Ciks)T+Rk}=0,tr(Ξk+1|k+1)Kk+1=-2(I-Kk+1Σ-k+1C-k+1)(Ξk+1|k-1-λ2E-k+1TE-k+1)-1×C-k+1TΣ-k+1+2Kk+1{i=1rCik+1sΔk+1|k(Cik+1s)Tλ2-1Σ-k+1D-k+1D-k+1TΣ-k+1T+Σ˘k+1[2(a12tr(Δk+1|k)+a22)]I+i=1rCik+1sΔk+1|k(Cik+1s)T+Rk+1}=0, where Ωkk-1 and Δk+1k are defined in (30).

Considering (46), the optimized prediction gain Lk and the filtering gain Kk+1 can be obtained in (31) and (32). The proof is completed.

For the sake of clarity, the robust recursive filter is summarized as follows.

Step 1.

Given x^kk-1 and Ξkk-1 and from (31), the prediction gain Lk is computed. Using (5) and (31), the one-step state prediction x^kk-1 and the upper bound Ξk+1k can be obtained by (6) and (27).

Step 2.

The filtering gain Kk+1 can be given by (32). The state estimation x^k+1k+1 and the upper bound Ξk+1k+1 can be given by (7) and (28).

Step 3.

Repeat Step 1 to update the one-step state prediction and its upper bound and use Step 2 to obtain the state estimation.

Remark 7.

The robust recursive filter problem is removed by using Theorem 6 for nonlinear systems with multiplicative noises, correlated additive noises, and packet losses. Different from the most existing robust filter literature, the robust recursive filter design proposed in this paper is based on the structure including state prediction and state correction in the presence of the correlated additive noises. Note that the phenomenon of correlated additive noises, multiplicative noises, and packet losses arises in the engineering applications. In order to solve this problem, a robust recursive filter is derived to find the upper bound of the prediction error covariance and the filtering error covariance and design the filter parameters to minimize the upper bounds. It is worth mentioning that, though the correction terms in (28) and (32) are similar to the corresponding component in , there is a clear difference between the prediction terms in (27) and (31) caused by correlated additive noises and the counterpart in [36, 37], which will directly affect the estimation results. This distinguishes our work from the work in [36, 37].

4. Simulation

To show the effectiveness of the proposed robust recursive filter (RRF), it is compared with the finite-horizon extended Kalman filter (FEKF) in the literature  by employing the following examples.

Example 8.

The discretized maneuvering target tracking example in [36, 37] is presented including correlated additive noises, multiplicative noises, and packet losses: (47)xk+1=[x1,k+1x2,k+1]=[0.8x1,k+x1,kx2,k1.5x2,k-x1,kx2,k]+[0.060.080.090.12]ηk[x1,kx2,k]+[0.010.03]wk,yk=Σk×7.5sin(x2,k)+[0.150.2]ξk[x1,kx2,k]+vk, where the state xk=[x1,kTx2,kT]Trepresents the position and velocity of target; ηk and ξk are independent zero mean Gaussian white noises with covariance 1; wk and vk are correlated zero mean Gaussian noises with Qk=0.05 and Rk=0.05. Let Sk=0.02. The mean and covariance of Σk are determined as μk=0.9 and (σk)2=0.065.

The initial state and covariance are set as x^00=[1.80.2]T and Ξ00=20I2. Let ε1=0.4, ε2=0.35, λ1=λ2=0.002, Dk=D-k+1=[0.10.15]T, Ek=E-k+1=0.01I2, and Bk=diag{0.1,0.2}.

To evaluate the performance of the proposed robust recursive filter, the mean square error (MSE) is employed. And it can be expressed as (48)MSE=1Nk-1N(xk-x^kk)2, where N is the sample number.

Simulation results are shown in Figures 14. From Figures 1 and 2, it can be seen that, compared with the FEKF in , the proposed robust recursive filter performs better when the model is correlated with additive noises, multiplicative noises, and packet losses. Both true position and velocity are tracked well. This is because the effect of the proposed algorithm can compensate for the correlated noise, while the FEKF cannot. Shown in Figures 3 and 4 are the comparisons of MSE of the estimated states with the corresponding diagonal elements of the estimation error covariance. Obviously, for the proposed algorithm, the MSE of the estimated state is always lower than the upper bound. This confirms the results of Theorem 6. Meanwhile, the MSE of the RRF stays below the MSE of the FEKF, which further illustrates that the proposed algorithm has higher precision than the FEKF in presence of correlated additive noises, multiplicative noises, and packet losses.

The trajectory of the actual state x1 and its estimate.

The trajectory of the actual state x2 and its estimate.

MSE of the estimated state x1 and upper bound.

MSE of the estimated state x2 and upper bound.

Example 9.

According to the literature , the robust recursive filter is considered to handle the attitude estimation problem with correlated additive noises, multiplicative noises, and packet losses. The system process models are expressed as follows: (49)xk+1=[qk+1βk+1]=[I4×4+Δt2Ω(ω~k-βk)04×303×4I3×3][qkβk]+[-Δt2Ξ(qk)03×3]ηv+[04×1ηu]=f(xk,ω~k)+i=1sAikηikxk+[04×1σuΔtI3×1]wk, where the state xk consisted of the quaternion vector qk and the gyro bias vector βk; ω~k is the gyro measured angular rate at time k; Δt is the gyros sampling interval; ηv is the Gaussian white-noise process with zero mean and covariance σv2; ηu is the zero mean Gaussian white-noise process with covariance σu2Δt; s=3; ηik is the zero mean multiplicative noise with covariance 1; wk is the zero mean Gaussian noise with covariance 1; [ω×] is a cross-product matrix defined by (50)[ω×]=[0-ω3ω2ω30-ω1-ω2ω10],Ω(ω)=[-[ω×]ω-ωT0];

A i k are known scaling matrices with appropriate dimension, which can be expressed as (51)Aik=-Δtσv2[Aik104×303×403×3], where (52)A1k1=[000100100-100-1000],A2k1=[00-10000110000-100],A3k1=[0100-1000000100-10]. The measurement model can be described as (53)zk=[zk1zk2zk3]=Σk[A(qk)r1A(qk)r2A(qk)r3]+[vk1vk2vk3]=Σkh(xk)+σsI9×1vk, where zki  (i=1,2,3) is the measurement vector; A(qk) is the real attitude matrix at time k; ri  (i=1,2,3) is the reference vector of the star sensors; i is the number of star sensors. vki is a zero mean Gaussian white noise with covariance matrix σs2I3  ×  3; vk is the zero mean Gaussian noise with covariance 1; if q=[q1,q2,q3,q4]T=[ρT,q4]T, the attitude matrix can be written as (54)A(q)=(q42-ρTρ)I3×3+2ρρT-2q4[ρ×].

The simulation conditions are set as follows: the gyro sampling interval is Δt=0.25s; the standard deviation of gyros’ measurement noise is σv=1.45444  ×  10-6 rad/s1/2; the standard deviation of gyros’ drift noise is σu=1.3036  ×  10-9rad/s3/2; wk and vk are correlated zero mean Gaussian noises with Qk=1 and Rk=1; let Sk=0.5; the standard deviation of star sensors’ measurement noise is all σs=18′′; due to using three star sensors, the reference vectors of the star sensors are r1=T, r2=T,and r3=T; because of the high precision of the star sensors, the estimation error is rather small in the attitude estimation system. Therefore, set Bk=0, Dk=D-k+1=0. The random variables μki  (i=1,2,,9) satisfy the Bernoulli distribution with Γ-k=diag{0.8,0.8,0.8,0.9,0.9,0.9,0.95,0.95,0.95}; let ε1=ε2=0.1 and λ1=λ2=0.0001; let Cik+1=0, a1=1, and a2=0.05.

The simulated results are shown in Figures 5 and 6. In Figure 5, blue lines represent the quaternion estimation errors of the RRF, green dashed lines represent the quaternion estimation errors of the FEKF, and red dashed lines show the corresponding diagonal elements of the error covariance of the RRF. It can be seen clearly that the estimation errors of the quaternion vector part of the RRF are generally within the boundaries of the computed covariance, which indicates that the proposed filter can control correlated additive noises, multiplicative noises, and packet losses. Besides, the quaternion estimation errors of the RRF are smaller than the quaternion estimation errors of the FEKF. These show that the RRF performs better than the FEKF. Furthermore, since it is very important for attitude estimation to get the angle information, the estimated quaternion needs to be converted as the form of Euler angles. In Figure 6, root mean square error (RMSE) is employed to express the quality of the Euler angle estimation. For the RRF, the means of RMSE of the attitude angles are 3.199′′, 3.125′′, and 3.195′′, respectively, which are lower than the REKF obviously. The reason why our method has such advantages is that the effects of arising multiplicative noises, packet losses, and correlated additive noises are all compensated for without loss of generality.

The quaternion estimation errors of the proposed filter.

RMSE of attitude angles in the proposed filter.

5. Conclusion

Due to the fact that existing robust filtering algorithms are difficult in dealing with correlated additive noises, a robust recursive filter is developed in this paper for nonlinear systems with consideration of correlated additive noises, multiplicative noises, and packet losses. The proposed algorithm is designed to minimize the upper bound on the prediction covariance and the filtering covariance. Simulated results demonstrate that the proposed filter provides effective performance for controlling correlated additive noises, multiplicative noises, and packet losses.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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