Recursive state estimation for linear systems with lossy measurements under time-correlated multiplicative noises

Recursive State Estimation for Linear Systems with Lossy Measurements under Time-Correlated Multiplicative Noises

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Recursive State Estimation for Linear Systems with Lossy Measurements under Time-Correlated Multiplicative Noises Shaoying Wang, Zidong Wang, Hongli Dong, Fuad E. Alsaadi PII: DOI: Reference:

S0016-0032(19)30816-6 https://doi.org/10.1016/j.jfranklin.2019.11.031 FI 4266

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

1 August 2019 18 October 2019 10 November 2019

Please cite this article as: Shaoying Wang, Zidong Wang, Hongli Dong, Fuad E. Alsaadi, Recursive State Estimation for Linear Systems with Lossy Measurements under Time-Correlated Multiplicative Noises, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.031

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Recursive State Estimation for Linear Systems with Lossy Measurements under Time-Correlated Multiplicative Noises Shaoying Wanga,b , Zidong Wangc , Hongli Dongd,e,∗ and Fuad E. Alsaadif

Abstract This paper is concerned with the recursive state estimation problem for a class of linear discrete-time systems with lossy measurements and time-correlated multiplicative noises (TCMNs). The lossy measurements result from one-step transmission delays and packet dropouts. Different from the traditional white multiplicative noises, TCMNs are included in the measurement model in order to reflect engineering practice. Utilizing the state augmentation approach, the system under investigation is first converted into a stochastic parameter system, and some new recursive terms (including the estimation for the product of state and multiplicative noises) are introduced to handle the difficulties caused by the TCMNs. Then, by the well-known projection theorem, recursive state estimation algorithms are developed in the sense of minimum mean-square error, which facilitate the design of the filter, the multi-step predictor and the smoothers. The proposed algorithms are explicitly dependant on the key system parameters including the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. Finally, simulation results illustrate the effectiveness of the presented estimation algorithms. Index Terms State estimation; Discrete system; Time-correlated multiplicative noises; Transmission delay; Packet dropout

I. I NTRODUCTION State estimation problem has received increasing attention in recent years because of its wide application potentials in a variety of engineering problems such as target tracking, robot navigation, signal processing, fault detection and so on [6], [11], [15], [17], [19], [28], [31], [37]–[39]. Among others, the H∞ estimation [27] and the Kalman filtering [3], [24] are arguably two of the most popular estimation methods. To be specific, in case that the noise signal is energy bounded, the former method would ensure a prescribed This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF323-135-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support. This work was also supported by the National Natural Science Foundation of China under grants 61703050, 61873148, 61873058 and 61933007, the Natural Science Foundation of Shandong Province of China under grants ZR2016FQ16, ZR2018MF015 and ZR2018MF023, and the Youth Project of Binzhou University of China under grants BZXYL1604, BZXYL1505 and BZXYL2016Y27. a Institute of Applied Technologies, Northeast Petroleum University, Daqing 163318, China. b College of Science, Binzhou University, Shandong 256603, China. c Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom. d Institute of Complex Systems and Advanced Control, Northeast Petroleum University, Daqing 163318, China. e Heilongjiang Provincial Key Laboratory of Networking and Intelligent Control, Northeast Petroleum University, Daqing 163318, China. f Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. ∗ Corresponding author at: Institute of Complex Systems and Advanced Control, Northeast Petroleum University, Daqing 163318, China. E-mail addresses: [email protected] (S. Wang), [email protected] (Z. Wang), [email protected] (H. Dong)

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bound on the worst-case estimation error by using H∞ norm as a performance index. When the system model and the noise statistics are exactly known, the latter method provides an optimal filter for linear systems in the minimum variance sense. Unfortunately, noises in a real world are often non-Gaussian and the system models are unavoidably subject to parameter uncertainties. In this case, the classic Kalman filtering technique becomes inapplicable and this impels us to study new estimation algorithms. Multiplicative noises, which behave as a common kind of stochastic uncertainties, are frequently encountered in practical engineering such as signal processing, biological movement and aerospace systems [1], [2]. To date, much research effort has been devoted to the investigation on the state estimation problem with multiplicative noises, and a large body of results has been available on distributed filter fusion [21], [29], Tobit Kalman filter [10], robust state estimation [16] and optimal filtering problems [35], etc. Recently, instead of the traditional white multiplicative noises, the so-called time-correlated multiplicative noises (TCMNs) have been discussed in [18], where a new filtering algorithm has been developed for discrete-time linear systems under the TCMNs. In particular, some new terms have been introduced in [18] to cope with the difficulties caused by the TCMNs without augmenting system’s noises. It should be pointed out that, in most existing literature, the state-dependent multiplicative noises are conventionally assumed to be zero-mean white ones, which can then be usually merged into the process and observation noises, thereby forming new correlated measurement/process noises or Gaussian noises with unknown covariance. Note that the TCMNs in [18] can also be described by a linear model with white noises where the dimensions of corresponding matrices increase as the number of multiplicative noises increases. The estimation method developed in [18] has proven to be useful in reducing the computational load. Nevertheless, for linear discrete-time systems under TCMNs, there is still much room to improve the state estimation algorithms by reducing the possible conservatism through, for example, introducing new recursive terms for estimating the product of state and multiplicative noises. In a networked control system (NCS), the network-induced phenomena (e.g. packet dropout and transmission delay) are known to be almost inevitable in data transmissions via the unreliable communication channel. Such an imperfection, if not given due consideration, would deteriorate the estimator performance and it becomes vitally important to adequately model the network-induced phenomena which are mostly randomly occurring. So far, in the context of NCS, some typical models have been proposed with examples including time-delay model, packet dropouts model, missing measurements model, quantization model and fading channel model [5], [30], [36]. During the past few years, a great deal of research attention has been paid to the study of two or more network-induced phenomena. For example, by introducing some Bernoulli distribution stochastic variable, multi-step transmission delays and multiple packet dropouts have been considered in [25] and, later on, in order to avoid complete loss of packets, the latest measurement transmitted successfully has been used in [26] as a compensation for the estimation. For the cases of random observation losses, a unified model has been proposed in [4] to account for the network-induced phenomena (e.g. stochastic sensor gain degradation, missing measurements, random parameter matrices). In recent years, the model reflecting one-step delay and packet dropout has been put forward in [22], [34], indicating that packets at the sensor side are only transmitted once, and one or two data packets or nothing can be received at each time point. If nothing can be received, then the latest measurement received is used to compensate. It is worth noting that, the estimation problem regarding networked-induced uncertainties has recently

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become a research topic attracting growing interest [7]–[9], [20], [23], [32], [33]. To mention a few, a filter has been constructed in [23] for a class of discrete nonlinear stochastic systems subject to quantization effects over a finite horizon. By means of the recursive matrix inequality, sufficient conditions have been obtained for the existence of the desired filters. Utilizing the predictor of lost observation to compensate the packet loss, the centralized/distributed fusion estimators have been presented in [7] in the sense of linear unbiased minimum variance. For networked multi-rate systems, finite-time filtering problem and varianceconstrained state estimation problem have been addressed, respectively, in [32], [33]. Unfortunately, to the best of the authors’ knowledge, the state estimation problem for linear discrete-time systems with one-step delays and packet dropouts has not been properly investigated yet, not to mention the case where the TCMNs are also involved. It is, therefore, the main purpose of this paper to shorten such a gap. Motivated by the above analysis, we will investigate the optimal estimation problem for a class of linear discrete-time systems with transmission delays, packet dropouts and TCMNs. Inspired by [22], [34], the model of one-step delay and packet dropout is adopted here to describe the network-induced incomplete information. Moreover, the TCMNs are considered in the measurement equation where the vector comprising all the TCMNs could be described by a linear system model with white noises. To overcome the difficulties brought by TCMNs and incomplete information, we first use state augmentation approach to transform the original system into an equivalent stochastic parameter one. Then, based on the projection theory, the optimal estimators (including filter, multi-step predictors and smoothers) are proposed for the parameterized system. Particularly, some recursive terms (e.g. the estimates of the product of multiplicative measurement noise and state) are introduced to deal with the time-correlated noises. Moreover, the corresponding error covariances are derived by sophisticated mathematical calculations. The main contributions of this paper are highlighted as follows: 1) unlike the commonly investigated white multiplicative noises, the multiple TCMNs are considered in the measurement equation and networkinduced phenomena (e.g. one-step random delay and packet dropouts) are simultaneously taken into account in the estimator design so as to better reflect the reality; 2) different from most existing results, some new recursive terms (including the estimates of the product of multiplicative measurement noise and state) are introduced to simplify the derivation of recursive estimators while reducing the conservatism; 3) utilizing the innovation technique, not only the filter but also multi-step predictor and smoother are proposed, which are all expressed in a recursive form and therefore suitable for online implementation. The rest of this paper is organized as follows. The problem under investigation is formulated in Section II. Three lemmas and the main theorems on the design of the recursive estimators including the filter, multistep predictor and smoother are provided in Section III. Meanwhile, the corresponding error covariances are derived in this part. In Section IV, a numerical example is given to show the effectiveness of the proposed estimators, which is then followed by some conclusions in Section V. Proofs of the obtained results in Section III are provided in the Appendix. Notation. Throughout the paper, Rn is the n-dimensional Euclidean space. 0 and Im represent, respectively, the zero matrix and identity matrix with compatible dimensions. Prob(∗) means the occurrence probability of the event ∗. E(x) stands for the expectation of x. sym{∗} denotes ∗ + ∗T . All matrices are assumed to be of suitable dimensions if not explicitly stated.

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II. P ROBLEM F ORMULATION Consider the following linear discrete-time system with TCMNs: xk+1 = Ak xk + wk s X
y k = Ck + C˜τ,k ητ,k xk + vk

(1) (2)

τ =1

ητ,k+1 =

s X i=1

(τ )

hi,k ηi,k + βτ,k , k = 0, 1, 2, · · ·

(3)

where xk ∈ R is the unknown system state with given initial state x0 , yk ∈ Rm is the measurement output, ητ,k ∈ R denotes the multiplicative noise satisfying E[ητ,0 ] = 0 and E[ητ,0 ηλ,0 ] = ϕτ,λ (λ = 1, 2, · · · , s), (τ ) hi,k ∈ R is a constant at time k, wk ∈ Rn and vk ∈ Rm are zero-mean process noise and measurement noise with variances Qk and Rk , respectively, βτ,k is a zero-mean random variable, and Ak , Ck , C˜τ,k are known matrices with compatible dimensions. Throughout this paper, we give the following assumptions. Assumption 1: The initial state x0 has mean x¯0 and variance P0 , which is independent of wk , vk , βτ,k and ητ,0 . wk , vi and βτ,i (i = 0, 1, · · · , k) are mutually uncorrelated with each other. (τ,λ) Assumption 2: βτ,k is uncorrelated with βλ,j (j = 0, 1, · · · , k − 1), E[βτ,k ] = 0 and E[βτ,k βλ,k ] = σk (λ = 1, 2, · · · , s). Remark 1: It should be pointed out that, if we were to define (η1,k , η2,k , · · · , ηs,k )T as ηk , then ηk+1 can be expressed as a linear system model with the white noise ηk+1 = Hk ηk + βk , where    (1) (1) (1)  h1,k h2,k · · · hs,k β1,k (2)   β   (2) (2) h · · · hs,k   2,k   h Hk =  1,k 2,k . , βk =   ···   ··· ··· ··· ···  (s) (s) (s) βs,k h1,k h2,k · · · hs,k
Thus, the equation (2) can be transformed into yk = Ck + C˘k ηk xk + vk , C˘k = (C˜1,k , C˜2,k , · · · , C˜s,k ). Obviously, the multiplicative noises considered in this paper are time-correlated, which is different from the white multiplicative noises in [16], [21], [26]. As is well known, the phenomena of one (or two or no) measurement packet arriving at the estimator at the same moment are the consequences of finite bandwidth for data transmission over networks. Adopted from [22], [34], zk received by the estimator is modeled as follows: " # rk yk zk = (1 − rk−1 )ξk yk−1 n

+ (1 − rk )[1 − (1 − rk−1 )ξk ]zk−1

(4)

where rk and ξk are uncorrelated random variables satisfying, respectively, Prob{rk = 1} = r¯k , Prob{rk = 0} = 1−¯ rk and Prob{ξk = 1} = ξ¯k , Prob{ξk = 0} = 1−ξ¯k . Also, rk and ξk are assumed to be uncorrelated with x0 , wk , vk , βτ,k and ητ,0 . From model (4), we can easily observe that the measurement yk can be received at time k when rk = 1, rk−1 = 1 or rk = 1, ξk = 0, i.e., zk = [ykT , 0]T ; the measurement yk and yk−1 can be simultaneously T received at time k when rk = 1, rk−1 = 0, ξk = 1, i.e., zk = [ykT , yk−1 ]T ; the measurement yk−1 can be

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T received at time k when rk = 0, rk−1 = 0, ξk = 1, i.e., zk = [0, yk−1 ]T ; the packet zk−1 can be received at time k as a compensation when rk = 0, rk−1 = 1 or rk = 0, ξk = 0, i.e., zk = zk−1 . Let

(5)

θk = (1 − rk )ξk+1 , Uk = θk yk , Vk = (1 − θk )zk . Then, system (1)-(4) can be rewritten as the following form: ek Xk + B ek Wk Xk+1 =A ek Xk + D e k vk zk =C

(6) (7)

T  where Xk+1 , xTk+1 , UkT , VkT , Wk , [wkT , vkT ]T , 

 ek ,  A  

ek , C

and

Ak 0 0 0 ˆ θk Ck 0 0 0 ˆ rk C k 0 αk Im 0 0 (1 − θk )Im 0 αk Im

ek , B "

"

In 0 0 0 0 θk Im rk Im 0

#T



  , 

rk Cˆk 0 (1 − rk )Im 0 0 Im 0 (1 − rk )Im ek , D

"

rk Im 0

(8)

, #

,

#

,

s P C˜τ,k ητ,k and αk , 1 − rk − θk . with Cˆk , Ck + τ =1

ek , B ek , C ek can be rearranged as To simplify the presentation, A

ek = A(0) + θk (A(1) + A(3) ) A k k k (2)

(4)

+rk (Ak + Ak ),

ek = B (0) + θk B (1) + rk B (2) , B k k k (0) (1) (2) e Ck = C + rk C + rk C , k

where (0) Ak



  , 

e k = rk D(0) , D k

Ak 0 0 0 0 0 0 0 0 0 Im 0 0 Im 0 Im

k

k



  , 

(9)

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

 0 0 0 0  C 0 0 0   k  (1) Ak ,  ,  0 0 −Im 0  0 −Im 0 −Im   0 0 0 0  0 0 0 0    (2) Ak ,    Ck 0 −Im 0  0 0 0 −Im   0 0 0 0  C˜   τ,k 0 0 0  (1) Qτ,k ,  ,  0 0 0 0  0 0 0 0   0 0 0 0  0 0 0 0    (2) Qτ,k ,  ˜ ,  Cτ,k 0 0 0  0 0 0 0 s s X (1) X (2) (3) (4) Qτ,k ητ,k , Ak , Qτ,k ητ,k , Ak , τ =1

(0) Bk



  , 



In 0 0 0

0 0 0 0



τ =1



    (1) , Bk ,   

0 0 0 Im 0 0 0 0







  , 

T 0 0   0 I      m  (2) (0) Bk ,   , , Ck ,    Im 0   0 Im  T T  T Ck 0 C˜τ,k 0  0  0 0 0     (3) (1) Ck ,   , Qτ,k ,   −Im  0 0 0  0 −Im 0 0 " # s X Im (2) (3) (0) Ck , Qτ,k ητ,k , Dk , . 0 τ =1 0 0 0 0 0 Im 0 0

Before proceeding further, let us introduce the following notations:

ek ] = B k , E[C ek ] = C k , ek ] = Ak , E[B E[A e k ] = Dk , E[rk ] = rk , E[θk ] = θk , E[D ek − Ak , ∆Bk = B ek − B k , ∆Ak = A

ek − C k , ∆Dk = D e k − Dk , ∆Ck = C ∆rk = rk − rk , ∆θk = θk − θk ,

(10)

T

   , 

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(1)

(2)

Πτ,k = θk Qτ,k + rk Qτ,k , and thus ek − Ak ∆Ak =A (1)

(2)

(3)

(4)

=∆θk Ak + ∆rk Ak + θk Ak + rk Ak , ek − B k = ∆θk B (1) + ∆rk B (2) , ∆Bk =B k k (1) (2) e ∆Ck =Ck − C k = ∆rk C + rk C , k

(11)

k

e k − Dk = ∆rk D(0) . ∆Dk =D k

According to the definition of random variables rk and θk , it is easy to derive that E[rk2 ] = rk , E[θk2 ] = θk , E[rk θk ] = 0, E[(1 − θk )rk ] = E[rk ] = rk ,

E[(1 − θk )(1 − rk )] = 1 − rk − θk ,

(12)

E[(θk − θk ) ] = θk (1 − θk ) , ϕk , 2

E[(rk − rk )2 ] = rk (1 − rk ) , φk .

Also, from (6)-(7), the statistical properties of Wk and vk are given by " # 0 E[Wk vkT ] = , Sk , Rk " # Q 0 k E[Wk WkT ] = , Nk . 0 Rk

(13)

Considering the relationship between xk and Xk , the first goal of this paper is to obtain an optimal estimation of Xk based on the measurement sequence Zk = [z0T , z1T , · · · , zkT ]T , which includes the filter bk|k , multi-step predictor X bk+h|k (h > 0) and smoother X bk+h|k (h < 0). Thus, the optimal estimators X bk+h|k for the original system (1)-(4). x bk+h|k are obtained by x bk+h|k = [In , 0, · · · , 0]X III. M AIN R ESULTS

In this section, an optimal filter of the state Xk in the sense of linear minimum mean-square error will bk+h|k (h > 0) and be designed by the projection theory. Then, the corresponding multi-step predictor X bk+h|k (h < 0) will be also developed for the system (1)-(4). smoother X To begin with, some lemmas are introduced as follows. (τ,λ) T Lemma 1: Defining ρk+1 = E[ητ,k+1 ηλ,k+1 ], we have (τ,λ)

ρk+1 =

s X s X

(τ ) (i,j) (λ)T

i=1 j=1

Proof: See Appendix A for the proof. Lemma 2: Defining qk = E[Xk XkT ], we have (3)

(3)T

∆A(3) (qk ) =E[Ak Xk XkT Ak k

(τ,λ)

hi,k ρk hj,k + σk

]

(14)

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=

s X s X

(i,j)

(1)

(1)T

(1)

(15)

(2)T

(2)

(16)

(3)T

(3)

(17)

ρk Qi,k qk Qj,k , Γk ,

i=1 j=1 (4)

(4)T

∆A(4) (qk ) =E[Ak Xk XkT Ak k

=

s X s X

(i,j)

]

(2)

ρk Qi,k qk Qj,k , Γk ,

i=1 j=1 (2)

(2)T

∆C (2) (qk ) =E[Ck Xk XkT Ck k

=

s X s X

(i,j)

]

(3)

ρk Qi,k qk Qj,k , Γk ,

i=1 j=1

∆∆Ak (qk ) =E[∆Ak Xk XkT ∆ATk ] (1)

(1)T

(2)

(2)T

=ϕk Ak qk Ak + φk Ak qk Ak (1) (2)T (2) (1)T − θ¯k r¯k A qk A − θ¯k r¯k A qk A k

+

(1) θ¯k Γk

+

k (2) r¯k Γk

k

k

, Φk ,

(18)

where qk will be calculated in Lemma 3. Proof: See Appendix B for the proof. Lemma 3: The state covariance matrix qk = E[Xk XkT ] of the system (6) is computed recursively by (0)

(0)T

qk+1 =Ak qk Ak

(2)

(1)

(1)T

+ θ k Ak q k Ak (2)T

+ rk Ak qk Ak

(0)

(1) (2) + Ψk + θ¯k Γk + r¯k Γk (1)T

+ sym{θk Ak qk Ak

(0)

(2)T

+ rk Ak qk Ak

}

(19)

ek Wk W T B e T ] will be calculated by (50) with the initial value q0 = diag{P0 + x¯0 x¯T0 , 0}. where Ψk = E[B k k Proof: See Appendix C for the proof. To facilitate subsequent discussion, we define the following notations: bk|k , E[Xk |Zk ], X ˜ k|k , Xk − X bk|k , X bk+1|k , E[Xk+1 |Zk ], X ˜ k+1|k , Xk+1 − X bk+1|k , X

b (τ ) , E[ητ,k Xk |Zk ], X ˜ (τ ) , ητ,k Xk − X b (τ ) , X k|k k|k k|k

b (τ ) , E[ητ,k+1 Xk+1 |Zk ], zbk|k−1 , E[zk |Zk−1 ], X k+1|k ˜ (τ ) , ητ,k+1 Xk+1 − X b (τ ) , ek , zk − zbk|k−1 , X k+1|k k+1|k T T ˜ k|k−1 X ˜ k|k−1 ˜ k|k X ˜ k|k ], Mk , E[X Pk , E[X ], (τ )

Pk

˜ k|k X ˜ (τ )T ], M (τ ) , E[X ˜ k|k−1 X ˜ (τ )T ], , E[X k k|k k|k−1

(τ,λ)

Pk

(τ,λ) ˜ (τ ) X ˜ (λ)T ˜ (τ ) X ˜ (λ)T , E[X , E[X k|k k|k ], Mk k|k−1 k|k−1 ].

bk+h|k denotes the estimation of Xk+h based on observations Zk = To be more specific, for h = 0, 1, X ek+h|k = Xk+h − X bk+h|k means the corresponding estimation errors with covariance [z0T , z1T , · · · , zkT ]T , X ˜ k|k X ˜ T ] and Mk+1 = E[X ˜ k+1|k X ˜ T ], and the same are true for X b (τ ) , X b (τ ) , P (τ ) matrices Pk = E[X k k|k k+1|k k|k k+1|k (τ ) (τ,λ) (τ,λ) and Mk . In particular, Pk , Mk are called the filtering and prediction error covariance matrices ˜ (τ ) and X ˜ (λ) , respectively. In addition, the estimation of zk based on Zk−1 is also defined between X k|k−h k|k−h as zbk|k−1 and the innovation zk − zbk|k−1 is given as ek .

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Based on the above lemmas and the well-known projection theorem, the optimal linear filter, predictor and smoother will be provided in Theorems 1-3. Theorem 1: For system (6)-(7), the optimal linear filter and one-step predictor are recursively calculated by bk|k =X bk|k−1 + Kk ek X s X bk|k + b (τ ) + B k Sk D ¯ T J −1 ek bk+1|k =Ak X Πτ,k X X k k k|k

(20)

b (τ ) + K (τ ) ek =X k k|k−1 s s X s X X (τ ) (τ ) (i) (τ ) (i,j) b b bk|k Xk+1|k = hi,k Ak Xk|k + hi,k ρk Πj,k X i=1 i=1 j=1

(22)

τ =1

b (τ ) X k|k

bk|k−1 + r¯k zbk|k−1 =C¯k X

with corresponding gain matrices

s X

Kk = Mk C¯kT + r¯k

τ =1

(τ ) Kk

(τ ) =Lk Jk−1

(τ )

(τ )

Lk =(C¯k Mk )T + r¯k

s X τ =1

b Qτ,k X k|k−1 (3)

(τ )

(τ,λ)

Mk

(23) (24)

(τ ) (3)T
Mk Qτ,k Jk−1

s X

(21)

(25) (26)

(3)T

(27)

Qλ,k

λ=1 (1) (1)T φk Ck qk Ck

(3)
Jk =C¯k Mk C¯kT + + Γk s X s X (3) (i,j) (3)T (0) (0)T 2 + r¯k Qi,k Mk Qj,k + r¯k Dk Rk Dk i=1 j=1

+ sym{¯ rk C¯k

s X τ =1

(τ )

(3)T

(28)

Mk Qτ,k }

and the corresponding error covariance matrices s X s X (τ,λ) (i,j) (τ )  (i,j) Mk+1 = hi,k A¯k Pk A¯Tk + ρk (Φk i=1 j=1

s X s X  (λ)T (τ,λ) (τ ) (i,j) + Ψk ) hj,k + σk qk+1 + hi,k ρk i=1 j=1


× Πj,k (qk − Pk )

(τ ) Mk+1

=

s X i=1

+

s X s X

(λ) (i,j)

hi,k ρk Πj,k

i=1 j=1


(i) ¯k Sk D ¯ kT K (i)T A¯Tk h(τ )T A¯k Pk − B k i,k

s X s X  i=1 j=1

(i,j)

Πi,k Pk


T

(i,j) (1) (1) A¯Tk + ρk Qi,k qk ϕk Ak

(29)

10

FINAL VERSION

Mk+1

(2)
T (i,j) (2) (1) − θ¯k r¯k Ak + ρk Qi,k qk −θ¯k r¯k Ak (2)
T  (τ )T + φk Ak hj,k s X s X (i,j) T ¯ ¯ =Ak Pk Ak + Ψk + Πi,k Pk ΠTj,k

(30)

i=1 j=1

−

+ +

(1) (1)T ¯k Sk D ¯ T J −1 D ¯ kST B ¯T B k k k k + ϕk [Ak qk Ak s X s X (1) (i,j) (1)T Qi,k ρk qk Qj,k ] i=1 j=1 s X s X (2) (2)T (2) (i,j) (2)T φk [Ak qk Ak + Qi,k ρk qk Qj,k ] i=1 j=1

 (1) (2)T + sym −θ¯k r¯k [Ak qk Ak s X s X (1) (i,j) (2)T + Qi,k ρk qk Qj,k ] i=1 j=1

¯ k SkT B ¯kT − A¯k Kk D s X (τ ) ¯k Sk D ¯ kT K (τ )T ]ΠTτ,k + [A¯k P − B k

τ =1 (τ,λ) (τ ) (λ)T =Mk − Kk J k Kk (τ ) (τ )T =Mk − Kk Jk Kk Pk =Mk − Kk Jk KkT

(τ,λ) Pk (τ ) Pk

k

(31) (32) (33) (34)

Proof: See Appendix D for the proof. Remark 2: Different from the scalar white multiplicative noises in references [16], [21], [26], the TCMNs are considered in the estimator’s design. As stated in Remark 1, these TCMNs can be described by a linear model with white noises via augmenting noise vectors, and a traditional way is then to merge the augmented multiplicative noise into the original measurement noise, forming new measurement noises C˘k ηk xk + vk . By doing so, the statistical properties of new measurement noises would not be difficult to discover because the one-step correlation of ηk as well as the dimensions of ηk and C˘k increase as s increases. In order to avoid the unaffordable computational load, several new recursive terms are introduced in our estimation algorithms which include the estimates for the product of multiplicative measurement b (τ ) as well as X b (τ ) . noise and states X k|k k+1|k Remark 3: It should be pointed out that the network-induced phenomena (e.g. packet losses and random delays) have not been discussed in [18] despite the fact that the estimation algorithms presented in Theorem 1 are developed along the similar lines in the filter design in [18]. On the other hand, network-induced uncertainties have been considered in [22], [34] where the main attention has been devoted to the design of distributed fusion filter with white multiplicative noises or optimal linear estimators. In our paper, both the network-induced phenomena and the TCMNs are taken into account, and our algorithms are therefore more general. For example, rk , θk , ϕk and φk are used to quantify the occurrence probabilities of random (τ,λ) delays and packet dropouts. In particular, ρk characterizes the statistical properties of TCMNs whose (τ ) (τ ) (τ ) (τ ) (τ,λ) b and X b appearance gives rise to terms X , Mk , Pk k|k k+1|k and, accordingly, the error covariances Pk

11

FINAL VERSION

(τ,λ)

and Mk should be computed, which further demonstrate that our algorithms are indeed a non-trivial extension of existing results. So far, the recursive filter has been provided in Theorem 1 . On this basis, multi-step predictor and smoother will be proposed in the following Theorems. Theorem 2: For system (6)-(7), the recursive linear h-step (h > 1) predictor is given by bk+h|k =Ak+h−1 X bk+h−1|k X s X b (τ ) + Πτ,k+h−1 X k+h−1|k

(35)

τ =1

and the corresponding prediction error covariance matrices are calculated recursively as T

Mk+h|k =Ak+h−1 Mk+h−1 Ak+h−1 s X s X (i,j) + Πi,k+h−1 (qk+h−1 − Mk+h−1 )ΠTj,k+h−1 i=1 j=1

+ Φk+h−1 + Ψk+h−1

(36)

Proof: See Appendix E for the proof. Theorem 3: For system (6)-(7), the recursive linear fixed-lag h-step (h < 0) smoother is given by

with the smoothing gain matrix

bk+h|k = X bk+h|k−1 + Tk+h|k ek X

(37)

s i h X T (τ ) (3)T Ωk+h Qτ,k Jk−1 , Tk+h|k = Ωk+h C k + r¯k

(38)

τ =1

for which the smoothing error covariance matrices can be calculated by T Mk+h|k = Mk+h|k−1 − Tk+h|k Jk Tk+h|k

(39)

where e T ], Ω e Ωk+h = E[Xk+h X k|k−1 k+h = E[Xk+h Xk|k−1 ] (τ )

are given as

T

Ωk−1 =Pk−1 Ak−1 +

s X

(τ )T

(τ )

Pk−1 ΠTτ,k−1

τ =1

Ωk+h

¯ k−1 S T B Tk−1 , − Kk−1 D k−1 s X T (τ ) =Ωk+h−1 Ak−1 + Ωk+h−1 ΠTτ,k−1

(40)

τ =1

T

− (Ωk+h−1 C k−1 + r¯k−1 × [Ak−1 Kk−1 + +

s X

s X

(τ )

(3)T

Ωk+h−1 Qτ,k−1 )

τ =1

(τ )

Πτ,k−1 Kk−1

τ =1 T −1 T B k−1 Sk−1 Dk−1 Jk−1 ]

(41)

12

FINAL VERSION

and (τ ) Ωk−1

=

s X

T (τ )T (i) Pk−1 Ak−1 hi,k−1

i=1

(τ )

Ωk+h

(i,j)T

−

s X s X

(qk−1

i=1 j=1

(τ )T

− Pk−1 )ΠTj,k−1 ρk−1 hi,k−1 ], s X T (τ )T (i) = Ωk+h−1 Ak−1 hi,k−1

(42)

i=1

+ (Ωk+h−1 − Fk+h ) T

s X s X i=1 j=1 s X

− [Ωk+h−1 C k−1 + r¯k−1

(i,j)T

(τ )T

ΠTj,k−1 ρk−1 hi,k−1 (τ )

(3)T

Ωk+h−1 Qτ,k−1 ]

τ =1

s X (i)T T (τ )T ×[ Kk−1 Ak−1 hi,k−1 i=1

+

s X s X

(i,j)T

(τ )T

ΠTτ,k−1 ρk−1 hi,k−1 ]

(43)

i=1 j=1

with

Fk+h

  qk−1 , h = −1 |h| = Q  qk+h = Ak−i , h < −1

(44)

i=2

Proof: See Appendix F for the proof. Remark 4: It is worthwhile to note that, the non-correlation of Wk+h (h > 1) and vk has been used in the derivation of multi-step predictors so that we can easily obtain the corresponding results based on the projection theory. Compared with the multi-step predictor, the multi-step smoother seems more difficult ˜ (µ)T ] in Ωk+h and Ω(τ ) ˜ (µ)T ] and E[Xk+h X to be obtained, which is because the existence of E[Xk+h X k+h k|k−1 k|k−1 adds a further complication to our analysis. Also, the step length h must be carefully considered in the calculation of Fk+h . Remark 5: In this paper, we have endeavored to solve the optimal estimation problem for a class of linear discrete-time systems with transmission delays, packet dropouts and TCMNs. Based on the projection theory, the optimal estimators (including filters, multi-step predictors and smoothers) have been provided where some recursive terms (e.g. the estimates of the product of multiplicative measurement noise and state) have been introduced to deal with the time-correlated noises. Our main results exhibit the following distinctive features: 1) the multiple TCMNs are considered in the measurement equation, which are more general than the traditional white multiplicative noises; 2) the one-step random delay and packet dropouts are simultaneously taken into account in the estimator design so as to better reflect the reality; 3) the innovation technique is used to obtain the design algorithms for the optimal filter, multi-step predictor and smoother, all of which are of recursive forms that are suitable for online implementation. Furthermore, the error covariances are dependent on the system matrices, the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. As illustrated in [12]–[14], additional constraints can be made to ensure the boundedness of the obtained error covariances.

13

FINAL VERSION

IV. I LLUSTRATIVE EXAMPLE In this section, a simulation example is provided to illustrate the effectiveness of the presented algorithms. Consider the linear discrete-time system with following matrices: " #   1 0.1   xk+1 = x k + wk   0.2 −0.9    " # 0.8 0.1  yk = η1,k xk + vk   0.2 1      (1) η1,k+1 =h1,k η1,k + β1,k , k = 0, 1, 2, · · ·

where wk and vk are uncorrelated noises with zero means and covariances diag{0.025, 0.1} and diag{10, 10}, respectively. η1,0 and β1,k are assumed to be uncorrelated variables with zero means and covariances 0.1, respectively. In this experiment, the one-step random delay and packet dropout model are described by model (4). rk and ξk are, respectively, Gaussian white noises with zero means and covariances 0.5 and 0.9. Also, it is assumed that h11 = 0.8, x b(0| − 1) = [100, 20]T and P0 = [40, 5; 5, 40]. According to the estimation algorithms developed in Theorem 1, the recursive linear filter x bk|k is provided in Figs. 1-2, which demonstrates that the real curve xk can be tracked well by x bk|k . The filter error variances are given in Figs. 3-4, which further illustrate that the filter has the satisfactory performance. The above simulation results illustrate the effectiveness of our algorithms. 100 True value Filter in this paper

True value and filter

80 60 40 20 0 -20

0

50

100

150

200

250

300

t/step The first state component

Fig. 1. True value and filter for the first state component with θk = 0.5 and rk = 0.9.

V. C ONCLUSIONS In this paper, we have investigated the state estimation problem for linear discrete-time systems subject to random delay, packet dropout and time-correlated multiplicative measurement noises. First, a new state vector has been formed by augmenting the original system’s state with some defined variables, then the

14

FINAL VERSION

20 True value Filter in this paper

True value and filter

15

10

5

0

-5

0

50

100

150

200

250

300

t/step The second state component

Fig. 2. True value and filter for the second state component with θk = 0.5 and rk = 0.9.

0.35 0.3 Filter

variances

0.25 0.2 0.15 0.1 0.05

0

50

100

150

200

250

300

t/step The first state component

Fig. 3. The variance of the first state component.

original system has been transformed into a parameterized one. By introducing recursive terms without augmentation, the estimators (including filter, multi-step predictor and smoother) in the sense of minimum mean-square error have been proposed via innovation analysis approach. Different from most existing (τ ) (τ ) (τ,λ) (τ,λ) estimation algorithms, error covariances Pk , Mk , Pk and Mk have been considered in the process of deriving the estimation algorithms. The factors that influence the performance of state estimators, like random delay, packet dropout and multiplicative noises, have all been reflected in our algorithms. Finally, a numerical example has been provided to demonstrate the effectiveness of our developed algorithms. Our future research topics would include the extension of our developed estimation method for more general time-varying systems subject to different network-induced phenomena. In addition, the stability and steady-state property of our estimators would be analyzed.

15

FINAL VERSION

0.1 Filter

0.095

variances

0.09 0.085 0.08 0.075 0.07 0.065 0.06

0

50

100

150

200

250

300

t/step The second state component

Fig. 4. The variance of the second state component.

A PPENDIX A. P ROOF OF L EMMA 1 T Proof: Substituting (3) into E[ητ,k+1 ηλ,k+1 ] leads to (τ,λ)

(τ ) (1,1) (λ)T (τ ) (1,s) (λ)T h1,k + · · · + h1,k ρk h1,k (τ ) (1,1) (λ)T (τ ) (s,s) (λ)T + · · · + hs,k ρk h1,k + · · · + hs,k ρk hs,k T + E[βτ,k βλ,k ] s s X X (τ ) (i,j) (λ)T (τ,λ) = hi,k ρk hs,k + σk i=1 j=1

ρk+1 =h1,k ρk

(45)

Hence, Lemma 1 can be easily obtained.

A PPENDIX B. P ROOF OF L EMMA 2 (3)

(3)

(3)

(3)T

Proof: Noticing Ak in (10), we substitute Ak into E[Ak Xk XkT Ak

] and obtain

∆A(3) (qk ) k

s s X  X (1) (1) T =E ( ητ,k Qτ,k )Xk Xk ( ητ,k Qτ,k )T

(i,j)

τ =1 τ =1 (1,1) (1) (1)T (1,s) (1) (1)T =ρk Q1,k qk Q1,k + · · · + ρk Q1,k qk Qs,k + (s,1) (1) (1)T (s,s) (1) (1)T + ρk Qs,k qk Q1,k + · · · + ρk Qs,k qk Qs,k s X s X (i,j) (1) (1)T = ρk Qi,k qk Qj,k i=1 j=1

··· (46)

T where E[ηi,k ηj,k ] = ρk and qk = E[Xk XkT ] have been used. Similarly, we can easily obtain (16)-(17), where the details are omitted for brevity. In light of ∆Ak in (11), it is not difficult to derive that (1)

(1)T

∆∆Ak (qk ) =ϕk Ak qk Ak

(1) (2)T − θ¯k r¯k Ak qk Ak

16

FINAL VERSION

(2) (1)T (3) (3)T − θ¯k r¯k Ak qk Ak + θ¯k E[Ak Xk XkT Ak ] (2)

(2)T

+ φk Ak qk Ak

(4)T

(4)

+ r¯k E[Ak Xk XkT Ak

]

(47)

Substituting (15)-(16) into (47) yields (18). VI. A PPENDIX C. P ROOF OF L EMMA 3 Proof: Based on the definition of qk and (6), we have T ek Xk X T A eT qk+1 =E[Xk+1 Xk+1 ] = E[A k k] ek Xk W T B e T ] + E[B ek Wk X T A eT ] + Ψk . + E[A k

k

k

(48)

k

In light of A˜k in (9), we derive

ek Xk X T A eT E[A k k] (0)

(0)T

=Ak qk Ak

(2)

(1)

(1)T

+ θk Ak qk Ak (2)T

+ r k Ak q k Ak

(3)

(3)T

+ θk E[Ak Xk XkT Ak

]

(4)

(4)T

] + sym{E[θk Ak Xk XkT Ak

(0)

(2)T

]}

+ rk E[Ak Xk XkT Ak + E[rk Ak Xk XkT Ak

(0)

(1)T

] (49)

Taking into account the fact that wk is uncorrelated with vk , one has ek Xk W T B e T ] = 0, E[B ek Wk X T A eT E[A k k k k ] = 0, ek Wk W T B e T ] = diag{Qk , θk Rk , rk Rk , 0} Ψk = E[B k

(50)

k

Hence, substituting (15)-(16) and (49)-(50) into (48) leads to (19) .

VII. A PPENDIX D. P ROOF OF T HEOREM 1 Proof: Applying the projection theorem yields bk|k = E{Xk |Zk−1 , zk } = X bk|k−1 + Kk ek X

(51)

˜ k|k−1 eT ]J −1 is the gain matrix, ek is the innovation sequence and Jk is the covariance where Kk = E[X k k matrix of ek . Similarly, we have bk+1|k =E{Xk+1 |Zk−1 , zk } X ˜k Wk |Zk−1 , zk ] =E[A˜k Xk |Zk−1 , zk ] + E[B

(52)

Substituting A˜k into E[A˜k Xk |Zk ], we obtain bk|k E[A˜k Xk |Zk ] =A¯k X s X (1) (2) b (τ ) + (θk Qτ,k + rk Qτ,k )X k|k

(53)

˜k Wk |Zk ] = B ¯k Sk D ¯ kT J −1 ek E[B k

(54)

τ =1

and

17

FINAL VERSION

˜ T } = 0, E{Wk X ˜ (τ )T } = 0 and E{Wk X T } = 0 has been used. Hence, a where the fact that E{Wk X k k|k−1 k|k−1 combination of (51)-(54) yields (20)-(21). Using the projection theorem again, (22) can be readily obtained. From the similar deducing process for (52)-(54), we derive b (τ ) =Ak X k+1|k

s X i=1

b hi,k X k|k (τ )

(i)

s s X X  (τ ) (1) ¯ + θk E ( hi,k ηi,k )( ητ,k Qτ,k )Xk |Zk τ =1

i=1

s s X X  (τ ) (2) + r¯k E ( hi,k ηi,k )( ητ,k Qτ,k )Xk |Zk

Since

(55)

τ =1

i=1

s s X X  (τ ) (1) E ( hi,k ηi,k )( ητ,k Qτ,k )Xk |Zk τ =1

i=1

=

s X s X

(τ ) (i,j)

(1)

ˆ k|k hi,k ρk Qj,k X

(56)

i=1 j=1

and

s s X  X (2) (τ ) ητ,k Qτ,k )Xk |Zk E ( hi,k ηi,k )( τ =1

i=1

=

s X s X

(τ ) (i,j)

(2)

ˆ k|k . hi,k ρk Qj,k X

(57)

i=1 j=1

we substitute (56)-(57) into (55) and obtain (23). For zbk|k−1 , we have

bk|k−1 + r¯k zbk|k−1 = C¯k X

Then, the innovation ek = zk − zbk|k−1 is rewritten as

s X τ =1

(3) b (τ ) Qτ,k X k|k−1

(58)

˜ k|k−1 + ∆rk Σ(3) Xk ek =C¯k X k s X (3) ˜ (τ ) ˜ + r¯k Qτ,k X k|k−1 + Dk vk

(59)

τ =1

(1)

where Ck + follows:

s P

τ =1

(3)

(3)

ητ,k Qτ,k , Σk . Consequently, the innovation covariance matrix Jk can be calculated as  (3) (3)T  Jk =C¯k Mk C¯kT + φk E Σk Xk XkT Σk s s X X  (3) e (τ ) (3) e (τ ) T 2 + r¯k E ( Qτ,k Xk|k−1 )( Qτ,k X ) k|k−1 τ =1

 ek|k−1 ( + sym E[C¯k X

τ =1

s X τ =1

(3) e (τ ) T ¯k ] Qτ,k X k|k−1 ) r



18

FINAL VERSION

˜ k vk v T D ˜T] + E[D k k

(60)

and then, after straightforward calculation, it is not difficult to derive (28). (τ,λ) (τ ) In the following, we concentrate on the computation for the prediction error covariances Mk , Mk , Mk (τ,λ) (τ ) ˜ k+1|k and the filtering error covariances Pk , Pk , Pk . In view of (21) and (23), the prediction errors X ˜ (τ ) are reformulated as and X k+1|k ˜ k+1|k =A¯k X ˜ k|k + B ˜ k Wk + X

s X

˜ (τ ) Πτ,k X k|k

τ =1


¯ k Sk D ¯ T J −1 ek + ∆θk Σ(1) + ∆rk Σ(2) Xk −B k k k k

Similarly, we have

˜ (τ ) = X k+1|k

s X

(61)

(τ ) ˜ (i) + ηi,k B ˜ k Wk hi,k A¯k X k|k

i=1



˜ k Wk + ∆Ak ηi,k Xk + βτ,k A˜k Xk + B s X s X (τ ) (i,j) ˆ k|k − hi,k ρk Πj,k X

(62)

i=1 j=1

(i)

(i)

where Σk = Ak +

s P

(i)

τ =1

Qτ,k ητ,k , i = 1, 2. (τ,λ)

Based on the definitions of Mk

(τ )

, Mk

and Mk , it is immediately obtained that

s s X X (τ,λ) (τ ) (λ) ˜ (i) )( ˜ (i) )T ] Mk+1 =E[( hi,k A¯k X hi,k A¯k X k|k k|k i=1

i=1

s s X X (τ ) (λ) + E[( hi,k ∆Ak ηi,k Xk )( hi,k ∆Ak ηi,k Xk )T ] i=1

i=1

s s X X (τ ) (τ ) ˜ ˜k Wk )T ] + E[( hi,k ηi,k Bk Wk )( hi,k ηi,k B i=1

(τ,λ)

+ σk

i=1

qk+1 +

s X s X

(τ ) (i,j) ˆ k|k X ˆT ] hi,k ρk Πj,k E[X k|k

i=1 j=1

× (τ,λ)

T Combining E[ητ,k ηλ,k ] = ρk (τ ) For Mk+1 , we have

(τ ) Mk+1

s X s X

(λ) (i,j)

hi,k ρk Πj,k

i=1 j=1


T

ˆ k|k X ˆ T ] = qk − Pk leads to (29). , E[∆Ak Xk XkT ∆ATk ] = Φk and E[X k|k ˜ k|k ( =E[A¯k X

s X

(τ ) ˜ (i) )T ] hi,k A¯k X k|k

i=1

s X (τ ) T ˜ ˆ ˜ (i) )T + E[Bk Wk Xk|k ]( hi,k A¯k X k|k i=1

s s X X (τ ) ˜ (τ ) )( ˜ (i) )T ]× + E[( Πτ,k X hi,k A¯k X k|k k|k τ =1

i=1

(63)

19

FINAL VERSION s X s X (τ ) (i,j) T ˜ k Wk X ˆ k|k − E[B ]( hi,k ρk Πj,k )T i=1 j=1

s X s X (τ ) T T −1 ˆ ¯ ¯ hi,k + Bk Sk Dk Jk E[ek Xk|k ]( i=1 j=1

×

×

(i,j) (1) ρk Πj,k )T + E[(∆θk Σk s X (τ ) ( hi,k ∆Ak ηi,k )T ] i=1

(2)

+ ∆rk Σk )Xk XkT (64)

Hence, (30) can be immediately derived. From (61), it follows that

¯ k Sk D ¯ kT J −1 D ¯ k× Mk+1 =A¯k Pk A¯Tk + ψk + B k s s X X (τ ) T ¯T ˜ ˜ (τ ) )T ] Sk Bk + E[( Πτ,k Xk|k )( Πτ,k X k|k +

+ + −

τ =1 τ =1 (1) (2) (1) E[(∆θk Σk + ∆rk Σk )Xk XkT (∆θk Σk  (2) ˜ k|k WkT ]B ¯k ∆rk Σk )T ] + sym A¯k E[X s s X X (τ ) T ¯ ˜ k Wk X ˜ (τ )T ΠTτ,k ] Ak Pk Πτ,k + E[B k|k τ =1 τ =1 ¯k E[Wk eTk ]J −1 D ¯ k SkT B ¯kT . B k

(65)

Rearranging the above formula leads to (31). ˆ (τ ) from ητ,k Xk , the estimation error is computed by X ˜ (τ ) = X ˜ (τ ) − K (τ ) ek . In Next, subtracting X k k|k k|k k|k−1 (τ,λ) light of the definition of Pk , it is not difficult to obtain (32). Since (33)-(34) can be deduced from the similar derivation process of (32), the details are omitted for conciseness. (τ ) (τ ) Now, let us focus on the computation of the gain matrices Kk , Kk , Lk and Lk . Noting ek in (59) ˜ (τ ) in (62), we have and X k|k−1 (τ ) ˜T ¯T ˜ (τ ) X Lk =E[X k|k−1 k|k−1 ]Ck s X (τ ) (3) (λ) ˜ + r¯k E[Xk|k−1 ( Qλ,k Xk|k−1 )T ].

(66)

λ=1

(τ )

(τ,λ)

Then, combining Mk and Mk , (27) is readily derived. (τ ) According to the definitions of Kk and Kk and the expression of ek , (25) and (26) can be easily calculated. The proof is now complete. VIII. A PPENDIX E. P ROOF OF T HEOREM 2 Proof: Replacing k by k + h − 1 in Xk+1 , Xk+h is immediately obtained. Taking projection on both sides of Xk+h , we derive bk+h|k =E[A˜k+h−1 Xk+h−1 |Zk ] X ˜k+h−1 Wk+h−1 |Zk ] + E[B

(67)

20

FINAL VERSION

˜k+h−1 Wk+h−1 |Zk ] = 0. Since Wk+h−1 (h > 1) is uncorrelated with v1 , · · · , vk , it follows that E[B ˜ k+h|k X ˜ T ], we can obtain (36) after simple Hence (35) holds. In this case, by defining Mk+h|k = E[X k+h|k computation. The proof is now complete. A PPENDIX F. P ROOF OF T HEOREM 3 Proof: Utilizing the projection theory again, we have bk+h|k = X bk+h|k−1 + Tk+h|k ek X

(68)

where Tk+h|k = E[Xk+h eTk ]Jk−1 . Recalling ek in (59), it is deduced that

˜ T ]C¯ T Tk+h|k ={E[Xk+h X k|k−1 k s X ˜ (τ )T ]Q(3)T }J −1 + r¯k E[Xk+h X τ,k k k|k−1

(69)

τ =1

˜ (τ )T ] , Ω(τ ) . ˜ T ] , Ωk+h and E[Xk+h X where E[Xk+h X k+h k|k−1 k|k−1 Based on the definition of Mk+h|k (h < 0), the smoothing error covariance matrix can be obtained in (39). ˜ k|k−1 into Ωk , we have Substituting X ˜T ¯T Ωk−1 =E[Xk−1 X k−1|k−1 ]Ak−1 s X ˜ (τ )T ]ΠTτ,k−1 + E[Xk−1 X k−1|k−1 τ =1

T

−1 T − E[Xk−1 eTk−1 ]Jk−1 Dk−1 Sk−1 B k−1 ,

(70)

and then it is easy to obtain the corresponding results in (40). When h < −1, Ωk+h is given as follows: T ˜ k−1|k−1 Ωk+h =E[Xk+h X ]A¯Tk−1 s X ˜ (τ )T ]ΠT E[Xk+h X + τ,k−1 k−1|k−1 τ =1

−1 ¯ T ¯T Dk−1 Sk−1 B − E[Xk+h eTk−1 ]Jk−1 k−1 s X ˜ (τ )T ]ΠTτ,k−1 =Ωk+h−1 A¯Tk−1 + E[Xk+h X k−1|k−2 τ =1

− +

E[Xk+h eTk−1 ][Ak−1 Kk−1

+

s X

(τ )

Πτ,k−1 Kk−1

τ =1

T −1 T ] , B k−1 Sk−1 Dk−1 Jk−1

(71)

and it then follows that (41) holds. (τ ) From the similar derivation process of (40)-(41), Ωk can be obtained. Meanwhile, we should note the T T computation of E[Xk+h Xk−1 ] , Fk+h . If h = −1, then Fk+h = E[Xk−1 Xk−1 ] = qk−1 ; if h < −1, then |h| Q T Fk+h = E[Xk+h Xk−1 ] = qk+h Ak−i . The proof is now complete. i=2

21

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