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Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises
Accepted Manuscript
Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises Xuemei Wang , Wenqiang Liu , Zili Deng PII: DOI: Reference:
S0165-1684(17)30071-3 10.1016/j.sigpro.2017.02.015 SIGPRO 6410
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
6 September 2016 5 January 2017 24 February 2017
Please cite this article as: Xuemei Wang , Wenqiang Liu , Zili Deng , Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.02.015
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Highlights Multi-model multisensor systems with uncertain noise variances A universal fictitious noise-based Lyapunov equation approach was presented The four robust weighted fusion Kalman estimators were presented A minimax robust fusion Kalman filtering theory was presented
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Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises Xuemei Wanga,b, Wenqiang Liua and Zili Denga, a
Electronic Engineering College, Heilongjiang University, Harbin, China Heilongjiang College of Technology and Business, Harbin, China
b
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Abstract: For multi-model multisensor systems with both the uncertain-variance multiplicative and linearly correlated additive white noises, a universal fictitious noise-based Lyapunov equation approach is presented, by which the original system can be converted into one with only uncertain additive noise variances, and then the local and four weighted fused minimax robust time-varying Kalman estimators (predictor, filter and smoother) of the common state are presented in a unified framework, where the robust Kalman filter and smoother are designed based on the robust Kalman predictor. They include the three fusers weighted by matrices, scalars and diagonal matrices, and a modified Covariance Intersection(CI) fuser. Their robustness is proved in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Their accuracy relations are proved. The corresponding local and fused robust steady-state Kalman estimators also are presented. The convergence analysis is also given. Two simulation examples applied to design the robust fusers for an autoregressive (AR) signal and an uninterruptible power system (UPS) are given to show the effectiveness of the proposed results. Keywords: multi-model; multiplicative noises; uncertain noise variance; minimax robust Kalman estimator; fictitious noise-based Lyapunov equation approach; weighted fusion
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1. Introduction
With the development of high technology fields, the multi-sensor data fusion has received much attention. Its aim is to obtain a fused estimator of system state, whose accuracy is higher than that of each
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local state estimator. The optimal information fusion Kalman filtering theory and methods have been widely used in many fields including military, defense, robotics, unmanned aerial vehicles, target tracking, GPS positioning and remote sensing[1].According to the unbiased linear minimum variance
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(ULMV) optimal estimation rule, there are the three optimal weighted state fusion approaches weighted by scalars, diagonal matrices and matrices respectively [2,3]. The limitation of optimal information fusion
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Kalman filtering is that the model parameters and noise variances are assumed to be exactly known [2,4]. In practical applications, there inevitably exist uncertainties in the system model due to stochastic perturbances, unmodeled dynamics, and so forth. For uncertain systems, the classical filter performance [4]
. Therefore, the study of the robust Kalman filtering for
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will degrade or the filter may be divergent
uncertain systems received extensive attention. The so-called robust Kalman filter is to find a Kalman filter such that its actual filtering error variances yielded by all admissible uncertainties are guaranteed to have a minimal upper bound [4]. Such property is called robustness. There are the linear matrix inequality (LMI) approach and Riccati equation approach
[4]
to design the
robust Kalman filters for systems with norm-bounded uncertain parameters but known noise variances. For systems with uncertain noise variances but known model parameters, based on the minimax robust estimation principle, the robust weighted fusion time-varying and steady-state Kalman estimators * Corresponding author. Tel.: +86 13804583507. E-mail address: [email protected] (Z. L. Deng). 2
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(predictor, filter and smoother) are presented by the Lyapunov equation approach
[5-7]
.Their disadvantage
is that the predictor, filter and smoother are separately designed, and the smoother is designed by the augmented state approach which yields a larger computational burden. The weighted state fusion approaches
[2,3]
need to compute the cross-covariances among the local estimation errors, but in many
practical applications, the cross-covariances are uncertain. For the systems with uncertain variances and cross-covariances, the robust covariance intersection (CI) fusion method was presented by Julier and Uhlmann
[8,9]
, and the CI fuser is formed by weighting the robust local estimators with a convex
combination. The advantage of robust CI fusion method is that the cross-covariances are avoided. Its disadvantages are that the local estimators and the conservative upper bound of their actual error
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variances are assumed to be known, and a larger conservative upper bound of actual fused estimation error variances is given.
For uncertain systems, the parametric uncertainties fall under two classes
[ 10 ]
: One class is
deterministic uncertainties that the uncertain parameters are assumed to lie in some known bounded set. For example, the norm-bounded uncertainty
[4]
can be used to describe the uncertain parameters over
known bounded ranges. The second class is stochastic uncertainties that the parameters contain random
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perturbations which are also called multiplicative noises or state-dependent noises. In uncertain system, the noises include the multiplicative and additive noises, while the additive noises include the process and measurement noises. The uncertainties of noise variances can also be described by deterministic uncertainties. Since the noise variance matrices are positive semi-definite, we can assume that the actual uncertain noise variances have the known conservative upper bounds
[5-9]
. In recent years, the robust
filtering problems for the uncertain systems with multiplicative noises have received much attention and are widely applied to many fields including communication, tracking, aerospace, image processing, and
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stochastic signal processing, and so on. They have important theoretical and application meanings. Theoretically, some uncertain systems with stochastic parameters can be converted into one system with deterministic parameters and state-dependant noises (multiplicative noises) as follows:
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For uncertain system with stochastic parameter matrices[11], by taking the expectation operations, the stochastic parameter matrices can be decomposed as deterministic parameter matrices (means) plus random perturbation matrices with zero means, so that we obtain the corresponding uncertain system
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with deterministic parameter matrices and multiplicative noises. When the multiplicative and additive noise variances are assumed to be exactly known, by introducing fictitious noises to compensate the
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multiplicative noise terms, we can obtain the conventional system with deterministic model parameters and known noise statistics. Hence the standard Kalman filtering algorithm can be applied to obtain the optimal robust Kalman filter. This is called the fictitious noise method, which was early presented by
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Koning in [11].
For sensor network system, the missing measurements frequently appear due to limited bandwidth of
network, the missing rate can be described as Bernoulli white noise taking values 1 or 0 with known probabilities [12]. Taking value 1 denotes that the measurement is received, and taking value 0 denotes the missing measurement, and only the measurement noise signal is received. Hence, multiplying measurement matrix by a Bernoulli white noise yields the system with missing measurements. Taking the expectation operation to random measurement matrix with Bernoulli white noise yields a deterministic measurement matrix plus a state-dependant measurement noise with zero mean. This yields the system with multiplicative noise in the measurement matrix. In signal processing, the deconvolution system for the autoregressive moving average(ARMA) signal with stochastic parameters can be converted into one with the state space model 3
[13]
. Noting that each
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stochastic parameter can be decomposed as its mean plus random perturbation with zero mean, then furthermore the deconvolution system with deterministic parameter matrices and multiplicative noises can be obtained [13]. For single sensor or multisensor uncertain systems with mixed uncertainties including multiplicative noises, missing measurements, randomly delayed measurements, packet dropouts, assuming that the noise statistics (variances, cross-covariances and correlation matrices) are exactly known, some optimal robust or robust fused state estimators innovation analysis method
[17]
[14-16]
were presented by directly using the projection theory (i.e.,
) or the minimum variances estimation rule, and some other optimal robust
or robust fused state estimators
[18-20]
were presented by the fictitious noise method [11] and minimum
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variance estimation rule or projection theory. Their limitation is that the noise statistics are assumed to be exactly known.
Recently, for uncertain multisensor system with multiplicative noises in state matrix and missing measurements, and with uncertain additive noise variances but known multiplicative noise variances, according to minimax robust estimation principle, based on the worst-case system with conservative upper bounds of uncertain noise variances, by the extended fictitious noise method and Lyapunov equation [21]
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approach, the robust centralized and weighted measurement fusion Kalman estimators were presented
in the sense that their actual fused estimation error variances are guaranteed to have the corresponding minimal upper bounds. Koning‟s fictitious noise method
[11]
is extended to systems with known
multiplicative noise variances and uncertain additive noise variances. For uncertain multisensor system with the multiplicative noises in both the state and measurement matrices and with uncertain additive noise variances but known multiplicative noise variances, applying the extended fictitious noise method [21] and the Lyapunov equation approach, the three weighted fusion minimax robust Kalman estimators and a
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modified CI fuser were presented in [22]. Early, applying the fictitious noise-based game-theoretic approach, the minimax robust deconvolution filters were presented for systems with both the uncertain multiplicative and additive noise variances[13], where uncertain variances have know lower and upper
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bounds. Their disadvantages are that the search for minimax filter gains requires expensive computational cost, and the robust fusion filtering problem was not solved. Recently, for uncertain multisensor system with multiplicative noises and norm-bounded uncertain
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parameters in the state transition matrix, assuming that the multiplicative noises variances are uncertain which lie in the bounded intervals with known lower and upper bounds, while the additive noise variances are assumed to be unknown, a distributed robust fusion state estimator was presented based on convex
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optimization by LMI approach [23]. Regarding the optimal or robust fusion Kalman filtering for multisensor systems, the limitation of all the
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above researches is that all sensor subsystems have the common state equation, but the measurement equations of the subsystems are different. Using the augmented state method to solve the fused state estimation problems for multisensor systems, we frequently meet multi-model systems (i.e., different augmented local state equations or different local dynamic models)
[24]
. The original common state
equation is embedded in each augmented local state equation. The state of the original state equation is called the common state of all the augmented local state equations. The augmented state of each subsystem contains the common system state as its partial components. The state of original non-augmented system is the common state of augmented state subsystems. The three optimal weighted fusion state estimators and input white noise deconvolution estimators have been present in [24, 25], for multi-model multisensor systems with known model parameters and noise variances, and a simulation example applied to an infrared target tracking system with three sensors was given in [24]. 4
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For the ARMA signal with multisensor and stochastic parameters, and with coloured measurement noises described by the ARMA models with stochastic parameters, applying the augmented state method also yields the multi-model system with multiplicative noises (See example 1 in section 9). For multisensor system with stochastic parameters and coloured measurement noises which are described by the state-space models with stochastic parameters. Applying the augmented state method can yield the multi-model system with multiplicative noises (See example 2 in section 9). For networked uncertain multisensor systems with transmission delays and packet dropouts, applying the augmented state method with the augmented states containing delayed measurements, we also obtain the multi-model system with multiplicative noises [26,27]. and have important theoretical and applicable meanings.
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Therefore, the multi-model systems with multiplicative noises have widely application backgrounds, Motivated by the above discussion, to the best of the authors‟ knowledge, so far, for multi-model multisensor uncertain systems with uncertain-variance multiplicative noises in both the state and measurement matrices and uncertain-variance linearly correlated additive white noises, the problem of designing the unified robust fusion Kalman estimators was not solved.
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The novelty of the topic is highlighted as follows:
(1) The design of the robust fused Kalman estimators for the multi-model multisensor uncertain systems is considered the first time.
(2) The considered uncertainties include multiplicative noises in both the state and measurement matrices, and uncertain multiplicative and additive noise variances. Specially, the uncertain multiplicative noise variances are seldom considered in references [13,23]. The main contributions of this paper are as follows:
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(1) A universal fictitious noise-based Lyapunov equation approach is presented for the multi-model multisensor uncertain systems with both the uncertain-variance multiplicative and additive noises. Its principle is that by introducing the fictitious noises to compensate multiplicative noises, the original
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system can be converted into one with only uncertain additive noise variances, further, applying the Lyapunov equation approach
[5-7]
, the minimax robust local and fused Kalman estimators are presented.
The Koning‟s fictitious noise method [11] for uncertain system with both known multiplicative and additive
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noise variances is developed. The extended fictitious noise method [21,22] for uncertain system with known multiplicative noise variances and uncertain additive noise variances, is extended. It is different from the fictitious noise-based game-theoretic approach
and the LMI approach
[23]
. The Lyapunov equation
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[5-7]
[13]
approach
without fictitious noises is extended and developed. Specially it overcome the disadvantages
of reference [13]. By the proposed approach, the local and fused robust Kalman estimators can be obtained
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by solving the Lyapunov equations, and to search the minimax optimal solution [13] is avoided. (2) For multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated
additive white noises, based on the worst-case system with conservative upper bounds of uncertain noise variances, by the proposed fictitious noise-based Lyapunov equation approach, the four weighted fusion time-varying and steady-state minimax robust Kalman estimators (predictor, filter and smoother) of the common state are presented in a unified framework, which include the three fusers weighted by matrices, scalars, diagonal matrices and a modified CI fuser. Their robustness is proved. Their accuracy relations, computational complexities and convergence analysis are also given. They constitute a unified and universal robust Kalman filter theory. The minimax robust fusion Kalman filtering theory
[5-7]
for systems only with uncertain additive noise
variances is extended. The optimal robust Kalman filtering 5
[14-20]
for systems with known multiplicative
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[2,24,25]
for multi-model multisensor systems with known model
parameters and noise statistics are developed. The rest of this paper is organized as follows. The problem formulation is given in Section 2. The preliminary knowledge is given in Section 3. Robust local time-varying Kalman predictor is presented in Section 4. Robust local time-varying Kalman filter and smoother are presented in Section 5. Robust weighted fusion time-varying Kalman estimators of the common state are presented in Section 6. Accuracy analysis is presented in Section 7. Local and fused steady-state estimators and convergence
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analysis are presented in Section 8. Two simulation examples are presented in Section 9. Conclusions are presented in Section 10.
Notation: R n denotes the n-dimensional Euclidean space, tr denotes the trace of a matrix, the subscript “ T ” denotes the transpose, E denotes the mathematical expectation, diag denotes the block-diagonal matrix, and A B denotes B A 0 being a positive semi-definite matrix . 2. Problem formulation
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Consider the linear discrete time-varying multi-model multisensor system with uncertain-variance multiplicative and linearly correlated additive white noises pi (1) xi t 1 i t ik t ik t xi t i t wi t k 1 qi (2) yi t H i t ik t H ik t xi t vi t , i 1, , L k 1
vi t Di t wi t i t , i 1, , L
xi t xcT t , iT t , wi t wT t , eiT t
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T
(3)
T
(4)
where t is the discrete time, T denotes the transpose, xi t R ni , yi t R i , wi t R ri and vi t R m
mi
respectively, i t R
n i
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are state, measurement, process noise and measurement noises of ith sensor subsystem, respectively. xc t R n and w t R r are common state and common process white noise of all subsystems, and ei t R rei respectively are different state and different process white
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noises of subsystems. L , pi and qi respectively are the numbers of sensors and multiplicative noises. From (4) xc t and w t can be denoted respectively as
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xc t Cxi xi t , Cxi I n 0nn , w t Cwi wi t , Cwi I r 0r r i
(5)
i
with I n is the n n identity matrix, 0 is the zero matrices with compatible dimensions. i t , i t , H i t , Di t are known matrices with compatible dimensions. The process and measurement noises are
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also called the additive noises. From (3), we call additive noises vi t and wi t to be linearly correlated. Notice that the equations (4) and (5) mean that xc t is the former n components of xi t .
Assumption 1. w t , ei t , i t R mi , ik t R1 and is t R1 are mutually uncorrelated white noises with zero means and actual covariances w t w u T Q t 0 0 Qei t ij ei t e j u 0 Ε i t j u 0 t u 0 0 ik jr is t j u 0 0
0 0 0 R i t ij 0 0 (6) tu 2 0 ik ij kr 0 2 0 0 is ij s where ij is the Kronecker function, ii 1, ij 0 i j , E denotes the mathematical expectation. 6
0
0
0
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is t , and it has known mean value E xi 0 0 and uncertain actual variance T E xi (0) 0 xi (0) 0 P0
(7)
Assumption 3. Q t , Qei t , R i t , P0 , 2ik and 2is have respectively the known conservative upper bounds Q t , Qei t , R i t , P0 , 2ik and 2is , i.e., with i 1, , L , k 1, covariances
, pi , s 1,
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2 2 2 2 Q t Q t , Qei t Qei t , R i t R i t , P0 P0 , ik ik , is is
, qi . Applying (3) and (4) yields the conservative and actual
0 Q t , Qi t Qii t , i, j 1, Qij t E wi t wTj t Qei t ij 0 ri rj Sij t Ε wi t vTj t Qij t DTj t , Si t Sii t
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(9) (10)
t Di t Qij t D t R t ij , Rvi t Rvii t
(11)
Q t 0 Qij t , Qi t Qii t Qei t ij 0 ri rj
(12)
T j
i
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Rvij t Ε vi t v
T j
(8)
Sij t Qij t DTj t , Si t Sii t
Rvij t Di t Qij t D t R i t ij , Rvi t Rvii t T j
(13) (14)
Our aim is to find the local and fused robust Kalman estimators (predictor, filter and smoother) ˆxc t | t N N 1, N 0, N 0 of the common state xc t , such that for all admissible uncertain
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actual variances Q t , Qei t , R i t , P0 , 2ik and 2is satisfying (8), the corresponding actual estimation error variances Pc t | t N are guaranteed to have the corresponding minimal bounds Pc t | t N ,i.e.,
Pc t | t N Pc t | t N
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(15)
where xˆc t | t N with i, m, s, d ,CI denote the local robust estimators, the robust fusers weighted by
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matrices, scalar, diagonal matrices and the robust CI fuser, respectively. 3. Preliminary knowledge
Introducing fictitious noises wai t , which are used to compensate multiplicative noises in (1), then the
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state equation (1) can be rewritten as xi t 1 i t xi t wai t
(16)
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with defining fictitious process noises wai t as pi
wai t ik t ik t xi t i t wi t
(17)
k 1
Introducing another fictitious noises vai t , which is used to compensate multiplicative noises in measurement equation (2), then (2) can be rewritten as (18) yi t Hi t xi t vai t , i 1, , L with defining fictitious measurement noises vai t as qi
vai t ik t H ik t xi t vi t , i 1,
,L
(19)
k 1
Applying Assumptions 1 and 2, it is easily proved that wai t and vai t are white noises with zero means. From (1), we define the conservative state covariances as X i t Ε xi t xiT t , where xi t is the conservative state yielded by the (1) with the conservative upper bounds of noise variances. From 7
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X i t 1 i t X i t iT t 2ik ik t X i t ikT t i t Qi t iT t
(20)
k 1
and with conservative initial values as
X i 0 P0 0 0T
(21)
We define the actual state covariance as X i t Ε xi t xiT t , where xi t is the actual state yielded
by (1) with actual noise variances. From (1), X i t satisfy the actual generalized Lyapunov equation pi
X i t 1 i t X i t iT t 2ik ik t X i t ikT t i t Qi t iT t
(22)
k 1
with actual initial values
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X i 0 P0 0 0T
(23)
We easily obtain the conservative and actual cross-variances and variances E wai t wajT t of wai t are given respectively as pi
Qaij t 2ik ik t X i t ikT t ij i t Qij t jT t , Qai t Qaii t k 1
pi
Qaij t 2ik ik t X i t ikT t ij i t Qij t jT t , Qai t Qaii t
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k 1
(24) (25)
The conservative and actual cross-covariances and variances of vai t are given respectively as qi
Raij t 2ik H ik t X i t H ikT t ij Rvij t , Rai t Raii t k 1
qi
Raij t 2ik H ik t X i t H ikT t ij Rvij t , Rai t Raii t k 1
(26) (27)
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Applying (17) and (19), there are conservative and actual correlation matrices Ε wai t vajT t of fictitious noises wai t and vaj t respectively as (28)
Saij t i t Qij t D t , Sai t Saii t
(29)
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Saij t i t Qij t DTj t , Sai t Sai t T j
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Lemma 1 [5]. Let be the r r positive semi-definite matrix, i.e. 0 , then the following rL rL matrix is also positive semi-definite, i.e.,
0 rLrL
(30)
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Lemma 2 [5]. Let Ri be the mi mi positive semi-definite matrix, i.e. Ri 0 , then the following m0 m0 block-diagonal matrix R is also positive semi-definite, i.e.,
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with m0 m1
R diag R1 ,
RL 0
(31)
mL .
Lemma 3 The actual fictitious noise variances Qai t and Rai t have the conservative upper bounds
Qai t and Rai t , respectively, i.e., Qai t Qai t , Rai t Rai t , i 1,
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(32)
Proof. See Appendix A.
wai (t ) Defining augmented fictitious noises i (t ) , then their conservative and actual cross vai (t ) covariances are given as Qaij (t ) Saij (t ) Qaij (t ) Saij (t ) , ij (t ) T (33) ij (t ) T Saji (t ) Raij (t ) Saji (t ) Raij (t ) and we define conservative and actual variances as i (t ) ii (t ) and i (t ) ii (t ) , respectively. 8
ACCEPTED MANUSCRIPT Lemma 4 For all admissible uncertain actual variances satisfying (8), we have i (t ) i (t ) Proof. See Appendix B. Lemma 5 Define the conservative and actual global covariances respectively as , Qa t Qij t Qa t Qij t rg rg
(35)
rg rg
where the ri rj matrices Qij t and Qij t are defined in (9) and (12), rg r1
i, j th block elements of
(34)
rL , and they are the
Qa t and Qa t , respectively. Then Qa t Qa t
(36)
Q(t ) Q(t ) , Qa (t ) Q(t ) Q(t ) rLrL
Q(t ) Qa (t ) Q(t )
Applying Q t Q t Q t yields Qa t Qa t , i.e., (36) holds. 4. Robust local time-varying Kalman predictor
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Proof. See Appendix C. Remark 1. For the special case with wi (t ) w(t ) , we have Qij t Q t , and
Q(t ) Q(t )
(37)
rL rL
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According to the minimax robust estimation principle[5], for the worst-case system (16) and (18) with the conservative upper bounds (maximum noise variances) Qai t and Rai t of noise variances to design the optimal (minimum variance) Kalman estimators will yield the minimax robust Kalman estimators. The conservative local minimum variance Kalman predictors are given as [17] xˆi t 1| t pi t xˆi t | t 1 K pi t yi t , i 1, , L (38)
pi t =i t K pi t Hi t
t Sai t Q t Q i t Hi t Pi t | t 1 HiT t Rai t and the conservative local prediction error variances Pi t 1| t satisfy the Riccati equations
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K pi t = i t Pi t | t 1 H
(39)
1 i
T i
(40) (41)
Pi t 1| t i t Pi t | t 1iT t i t Pi t | t 1 HiT t Sai t H i t Pi t | t 1 H i t Rai t
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T
i t Pi t | t 1 H iT t Sai t Qai t
with the conservative initial value
T
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xˆi 1| 0 0 , Pi 1| 0 P0
1
(42) (43)
Remark 2. In the conservative local Kalman predictor (38), the conservative measurement yi t is
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unavailable, which is yielded from the multi-model system (16) and (18) with the conservative upper bounds Qai t , Rai t and P0 . Only the actual measurement yi t is available, which is yielded from the actual system (16) and (18) with the actual variances Qai t , Rai t and P0 .Therefore, replacing the
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conservative measurement yi t with the actual measurement yi t , we define the local Kalman predictor (38) with the actual measurement yi t as the actual local Kalman predictor. We define the conservative local Kalman prediction error variances as xi t 1| t xi t 1
xˆi t 1| t , x t 1 is conservative state in (16), ˆx t 1| t is conservative Kalman predictor in (38). From (16), (18), (38) and(39), it is easy derived that the prediction error systems are given as (44) xi t 1| t pi (t ) xi t | t 1 I n , K pi t i (t ) with defining i t waiT t , vaiT t . So, applying (33) and (44) yields that the conservative and actual T
local prediction error cross-covariances respectively satisfy the Lyapunov equations In Pij t 1| t pi t Pij t | t 1 pjT t I n , K pi t ij (t ) T K pj t 9
(45)
ACCEPTED MANUSCRIPT In Pij t 1| t pi t Pij t | t 1 pjT t I n , K pi t ij (t ) T K pj t with the conservative and actual initial values respectively Pij 1| 0 P0
(46)
(47)
Pij 1| 0 P0
(48)
we define conservative and actual variances as Pi t 1| t Pii t 1| t and Pi t 1| t Pii t 1| t .
minimal upper bound Pi t 1| t ,i.e.,
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Theorem 1. For the multi-model uncertain system (1)-(4) with Assumptions 1-3, each actual local time-varying one-step Kalman predictors (38) is robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding actual prediction error variances Pi t 1| t have the
Pi t 1| t Pi t 1| t , i 1, , L , t 0 We call the actual local Kalman predictors as the robust local Kalman predictors. Proof. See Appendix D. 5. Robust local time-varying Kalman filter and smoother
(49)
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In the following, applying a unified approach [22] of designing the robust local Kalman estimators based on the robust local Kalman predictor, we shall give the robust local Kalman filters and smoothers. Theorem 2. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the actual Kalman filter ( N 0 ) and smoother ( N 0 ) are given as
xˆi t | t N xˆi t | t 1 Ki t | t k i t k , N 0 , i 1, k 0
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i t yi t Hi t xˆi t | t 1 with the smoother and filter gains
(50) (51)
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k 1 Ki t | t k Pi t | t 1 piT t j H iT t k Qi1 t k , k 0 j 0 K fi t =Ki t | t =Pi t | t 1 HiT t Qi1 t
(52) (53)
and the conservative estimation error variances are given as N
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Pi t | t N Pi t | t 1 Ki t | t k Q i t k KiT t | t k
(54)
k 0
where xˆi t | t 1 , pi t , Q i t and Pi t | t 1 is given in (38)-(41), and xˆi t | t 1 is the actual Kalman
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predictor with the actual measurement yi t .
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The conservative and actual estimation error variances and cross-covariances are given as T N K wN j t vN Pij t | t N iN t Pij t | t 1 TjN t KiwN t , K i t ij (t + ) vNT 0 K j t T N K wN j t vN Pij t | t N iN t Pij t | t 1 TjN t KiwN t , K t ( t + ) vNT i ij 0 K j t
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Pi t | t N Pii t | t N , Pi t | t N Pii t | t N
(55)
(56) (57)
For N 0 , define that N
iN t I n Ki t | t k H i t k pi t k , t
(58)
pi t k , t pi t k 1 pi t , pi t , t I n
(59)
k 0
KiwN t KivN t
N
K t | t k H t k t k , t 1 , 0, i
k 1
i
pi
, N 1, KiNwN t 0
(60)
N
K t | t k H t k t k , t 1 K t K t | t ,
k 1
i
i
pi
pi
0, , N 1, KiNvN t Ki t | t N 10
i
(61)
ACCEPTED MANUSCRIPT For N 0 , define that Kiw00 t 0 , Kiv00 t Ki t | t and with the initial values Pij 1| 0 P0 ,
Pij 1| 0 P0 . Each actual Kalman filter ( N 0 ) or smoother ( N 0 ) is robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding estimation error variances Pi t | t N have the minimal upper Pi t | t N ,i.e., Pi t | t N Pi t | t N
(62)
The conservative and actual error variances are given as
CR IP T
We call the actual local Kalman filter or smoother as the robust local Kalman filter or smoother. Proof. See Appendix E. Corollary 1. Under the conditions of Theorem 2, for the case with N 0 , we have the robust Kalman filter as xˆi t | t xˆi t | t 1 K fi t i t (63) Pij t | t I n K fi t Hi t Pij t | t 1 I n K fj t H j t K fi t Raij t K Tfj t
(64)
Pij t | t I n K fi t Hi t Pij t | t 1 I n K fj t H j t K fi t Raij t K Tfj t
(65)
T
T
where K fi t is computed by (53).
AN US
Proof. See Appendix F. Corollary 2. Under the conditions of Theorem 2, the robust local Kalman estimators of the common state xc t are given as
xˆci t | t N Cxi xˆi t | t N , Cxi I n 0 , N 1, N 0, N 0 , i 1,
,L
(66)
and their conservative and actual covariances and variances are given respectively as Pcij t | t N Cxi Pij t | t N CxjT , Pcij t | t N Cxi Pij t | t N CxjT ,
M
Pci t | t N Pcii t | t N , Pci t | t N Pcii t | t N They are robust in the sense that Pci t | t N Pci t | t N , i 1, , L Proof. See Appendix G.
(67) (68)
ED
6. Robust weighted fusion time-varying Kalman estimators of the common state
PT
According the minimax robust estimation principle[7],based on the worst-case system (1)-(3) with the conservative upper bounds of uncertain noise and initial value variances, to design the minimum variance (optimal) weighted fusers will yield the robust weighted fusers, i.e., the minimax robust fusers. The four robust weighted fusion Kalman estimators ( N 1, N 0 ) for the common state xc t will
CE
be presented in this section. They have a unified form as [5-7] L
xˆc t | t N i t | t N xˆci t | t N , m, s, d ,CI
(69)
i 1
with the constraint of unbiasedness L
t | t N I
AC
i 1
i
(70)
n
where m, s, d ,CI denote the fusers with matrix, scalar, diagonal matrix weights and CI fuser matrix weights, respectively. xˆci t | t N are the local robust Kalman estimators of the common state. Under the minimum variance optimal fusion rule, the optimal weights are given as follows: The optimal weighted matrices are computed as [2,3]
1m t | t N , , Lm t | t N eT Pc1 t | t N e eT Pc1 t | t N with the definitions eT I n I n , and 1
Pc t | t N Pcij t | t N
(71)
(72)
nLnL
The conservative fused error variance with matrix weights is given as
Pcm t | t N eT Pc1 t | t N e
The optimal scalar weights are computed as
[2]
11
1
(73)
ACCEPTED MANUSCRIPT is t | t N i t | t N I n , i 1,
, L t | t N eT Ptrc1 t | t N e eT Ptrc1 t | t N 1
1 t | t N ,
with e 1 1 , and
(74)
,L
(75)
T
Ptrc t | t N trPcij t | t N
(76)
L L
where the notation tr denotes the trace of matrix. The conservative fused error variance with scalars weights is given as L
L
Pcs t | t N i t | t N j t | t N Pcij t | t N
(77)
i 1 j 1
The optimal diagonal matrix weights are obtained by [2] id t | t N diag i1 t | t N ,
, Li t | t N eT Pc
with e 1 1T , and
Pc
ii
ii
t | t N
1
e
1
CR IP T
1i t | t N ,
, in t | t N
eT Pc
ii
t | t N
t | t N Pcskii t | t N LL
ii
AN US
fused error variance with diagonal matrix weights is given as L
Pcd t | t N id t | t N Pc i 1 j 1
ii
, i 1,
,n
(79)
(80)
where Pcsk t | t N is the i, i th diagonal element of Pcsk t | t N s, k 1, L
1
(78)
, L . The conservative
t | t N jdT t | t N
(81)
The conservative CI fusion matrix weights are computed as [28]
1
iCI t | t N i t | t N PCI t | t N Pci t | t N , i 1,
,L
(82)
1
(83)
M
1 L PCI t | t N i t | t N Pci t | t N i 1 and the optimal weighting coefficients i t | t N are obtained by minimizing
1 L 1 tr i t | t N Pci t | t N (84) 0 i t |t N 1 i 1 1 t |t N L t |t N 1 This needs to solve a L dimensional nonlinear optimization problem, which can be solved by „fimincon‟ function in MATLAB toolbox. In the following, we shall give a unified form of the conservative and actual fused error variances for the above four weighted fusers (69), and we shall prove the robustness of the fusers. Define (85) t | t N 1 t | t N , , L t | t N , m, s, d ,CI
min
PT
ED
min tr PCI t | t N
CE
From (70) it follows that
L
xc t i t | t N xc t
(86)
i 1
AC
Subtracting (69) from (86) yields the fused errors L
xc t | t N i t | t N xci t | t N
(87)
i 1
which yields the conservative and actual fused error variances respectively as L
L
Pc t | t N i t | t N Pcij t | t N j t | t N , m, s, d ,CI
(88)
i 1 j 1 L
L
Pc t | t N i t | t N Pcij t | t N j t | t N , m, s, d ,CI
(89)
i 1 j 1
Theorem 3. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the four actual weighted fusers (69) ( N 1, N 0 ) have respectively the conservative and actual fused error variances as
Pc t | t N t | t N Pc t | t N T t | t N , m, s, d ,CI 12
(90)
ACCEPTED MANUSCRIPT Pc t | t N t | t N Pc t | t N T t | t N , m, s, d ,CI
(91)
with defining the global conservative and actual covariances of the common state xc t respectively as
Pc t | t N Pcij t | t N
Pc t | t N Pcij t | t N
(92)
nLnL
(93)
nL nL
and they are robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding actual fused error variances Pc t | t N have the minimal upper bound Pc t | t N , i.e.,
CR IP T
Pc t | t N Pc t | t N , t 0, m, s, d ,CI (94) We call the actual Kalman fusers (69) as the robust Kalman fusers. Proof. See Appendix H. Remark 3. Taking m, s, d ,CI in (89) or (91), and applying (71), (75) and (79) yield the corresponding actual fused error variances as
Pcm t | t N eT Pc1 t | t N e eT Pc1 t | t N Pc t | t N Pc1 t | t N e e T Pc1 t | t N e (95) 1
1
L
L
Pcs t | t N i (t | t N ) j (t | t N ) Pcij (t | t N ) i 1 j 1
L
L
i 1 j 1
(97)
AN US
Pcd t | t N id t | t N Pcij (t | t N ) jdT t | t N
(96)
1 L L PcCI t | t N PCI t | t N i t | t N j t | t N Pci t | t N i 1 j 1
1 (98) Pcij t | t N Pc j t | t N PCI t | t N Remark 4. Taking CI , from (90) and (91) we can obtain the conservative and actual error variances PcCI t | t N and PcCI t | t N of the robust CI fuser. It was proved [5] that PCI t | t N defined in (83) is
M
an upper bound of PcCI t | t N , then we have
PcCI t | t N PCI t | t N
and from (94), P
(99)
t | t N is the minimal upper bound of P t | t N , then we have PcCI t | t N PcCI t | t N
(100)
PcCI t | t N PCI t | t N
(101)
CI c
ED
CI c
PT
and
t | t N is a minimal upper bound PcCI t | t N . CI
which means that P
CI c
conservative upper bound of P
t | t N comparing
with the
CE
Remark 5. The above robust fusion algorithms are implemented as follows: Step 1. Compute the local robust Kalman predictor xˆci t 1| t of the common state by (38)-(43), (66). Step 2. Compute the local robust Kalman filter and smoother xˆci t | t N by (50)-(54), (66).
AC
Step 3. Compute the conservative and actual cross-covariances and variances Pcij t | t N and
Pci t | t N N 1, N 0 of the local estimation errors by (45)-(48), (55)-(61), (63), (64) and (67).
Step 4. Compute the four robust weighted fused Kalman estimator xˆc t | t N of the common state by (69), (71)-(89). Above each step is implemented at each time. The computation complexities or the total numbers of multiplications and divisions per cycle for the robust local and fused estimation algorithms are analyzed as follows: The complexities of each local robust Kalman estimators are O ni3 mi3 [22] . It is known that the complexity of computing the inverse
of a n n matrix is O n3 . From (71)-(89) we see that complexity of computing the optimal
matrix-weights is O nL
3
,
while the complexities of computing the optimal scalar-weights and 13
ACCEPTED MANUSCRIPT
diagonal matrix-weights are O L3 , and the complexity of computing the CI matrix-weights is O n3 plus the complexity of computing the optimal weighting coefficients i t | t N by MATLAB toolbox. We see that when the number L of sensors is larger, the fuser with scalar or diagonal matrix-weights can significantly reduce the computational burden compared with the fuser with the matrix-weights. 7. Accuracy analysis
CR IP T
From (68) and (94), according to the accuracy definitions in [5-7], the trace of the estimation error variance matrix is defined as the accuracy of a estimator. The smaller trace means the high accuracy. For the robust local or fused estimator, its actual accuracy is defined as the trace of its actual error variance matrix, while its robust accuracy is defined as the trace of the minimal upper bound of its actual error variances. Theorem 4. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the accuracy relations among the robust local and fused robust Kalman fusers are given as tr Pc t | t N tr Pc t | t N , 1, , L, m, s, d ,CI , N 1, N 0 (102)
trPcm t | t N trPcCI t | t N trPCI t | t N trPci t | t N , i 1,
,L
(103)
trP t | t N trP t | t N trP t | t N trP t | t N , i 1, , L (104) Proof. See Appendix I. Remark 6. The robust accuracy is only determined from the conservative upper bounds of the uncertain noise variances and initial state variances, while the actual accuracies are related to actual noise variances and initial state variances. The accuracy relations (102) show that the actual accuracies of each local or fused robust estimator are higher than its robust accuracy. From (103) and (104), the robust accuracy of the fuser with matrix weights is higher than those of the CI fuser and the fuser with diagonal matrix weights, and the robust accuracy of the fuser with diagonal matrix weights is higher than that of the fuser with scalar weights. The robust accuracies of all fusers are higher than that of each local estimator. Remark 7. Here the conservative robust accuracy of the original robust CI fuser is defined as trPCI t | t N , which has larger conservativeness, while the modified robust accuracy in this paper is d c
s c
i c
AN US
m c
M
defined as trPcCI t | t N , which is the minimal upper bound of all admissible trPcCI t | t N . From (103)
ED
the modified robust accuracy is higher than the original conservative robust accuracy. Comparing (83) with (90) and (92) with CI , we see that PCI t | t N only contains the conservative local estimation error variances, while PcCI t | t N not only contains the conservative local estimation error variances,
PT
but also contains information of the conservative cross-covariances. This is an important way for increasing robust accuracy of the original CI fuser. 8. The local and fused steady-state Kalman estimators and convergence analysis
AC
CE
Now we will present the robust steady-state Kalman estimators based on the robust time-varying Kalman estimators by means of considering their limiting case, and will prove the convergence between robust time-varying and steady-state Kalman estimators. Consider the multi-model uncertain time-invariant multisensor system (1)-(4) with Assumptions 1-3, and with constant parameter matrices and constant noise variances. In order to ensure the existence of robust steady-state Kalman estimators, introducing matrices ai as pi
2 2 ai i i 2 ik ik , ik ik , i 1, 2, k 1
ik
,L
(105)
where the notation denote Kronecher product. According to [28] , assuming that the spectrum radius of ai is less than 1, i.e., ai 1 , then the solutions X i t and X i t of the time-varying generalized Lyapunov equations (20) and (22) with arbitrary initial values X i 0 0 and X i 0 0 converge respectively to the unique positive semi-definite solutions X i 0 and X i 0 for the corresponding steady-state generalized Lyapunov equations pi
X i i X i iT 2ik ik X i ikT i Qi iT , k 1
14
ACCEPTED MANUSCRIPT pi
X i i X i iT 2ik ik X i ikT i Qi iT
(106)
k 1
lim X i t X i ,lim X i t X i t
(107)
t
As t , taking the limiting operations to (24)-(28) and (33), replacing X i t and X i t by X i and X i respectively yields the constant noise statistics Qai and Qai , Rai and Rai , S ai and S ai , i and i , and
Qai t Qai , Qai t Qai , Rai t Rai , Rai t Rai , Sai t Sai , Sai t Sai , i t i and
i t i . In the local robust time-varying Kalman predictor (38)-(43), replacing pi t , K pi t and
CR IP T
Pi t | t 1 by their limits pi , K pi and Pi 1 yields the local robust steady-state Kalman predictor as follows. Theorem 5. For multi-model time-invariant multisensor system (1)-(4) with Assumptions 1-3, and ai 1 , assume that for the equivalent system (16) and (18) with constant noise statistics Qai , Rai and S ai , i , H i is a completely detectable pair, and i Sai Rai1 Hi , Gi with Gi GiT Qai Sai Rai1SaiT is a completely stabilizable pair, then there exists the local actual steady-state Kalman predictors as xˆis t 1| t pi xˆis t | t 1 K pi yi t , i 1, , L
pi =i K pi Hi
(108) (109)
1
AN US
K pi = i Pi 1 HiT Sai Qi1
(110)
Q i Hi Pi 1 HiT Rai
(111)
where pi are stable, i.e., the predictors (108) are asymptotically stable. The superscript s denotes “steady-state”. The local conservative steady-state one-step prediction error variances Pi 1 satisfy the steady-state Riccati equations
1
M
(112) Pi 1 i Pi 1iT i Pi 1 HiT Sai Hi Pi 1 HiT Rai i Pi 1 HiT Sai Qai and the conservative and actual steady-state one-step prediction error variances Pi 1 and Pi 1 also T
respectively satisfy the Lyapunov equations
Pi 1 p iPi 1 T
T
I , n K p i Ii , nK
pi
ED
Pi 1 pi Pi 1 piT I n , K pi i I n , K pi
T
pi
(113) (114)
PT
The local actual steady-state one-step Kalman predictors (108) are robust in the sense that for all admissible uncertain actual variances satisfying (8), it follows that Pi 1 Pi 1 , i 1, , L (115) and Pi 1 is the minimal upper bound of Pi 1 .
CE
The solution Pi t | t 1 of the local time-varying Riccati equation (42) with the time-varying noise statistics Qai t , Rai t and Sai t converges to the solution Pi 1 of the local steady-state Riccati
AC
equation (112) with constant noise statistics Qai , Rai and S ai .i.e.,
Pi t | t 1 Pi 1 , as t , i 1,
,L
(116)
If the measurement data yi t are bounded, then the local robust time-varying Kalman predictor of
time-varying system with time-varying noise statistics (38) and the local robust steady-state Kalman predictor (108) of time-invariant system with constant noise statistics have the convergence in a realization (117) xˆi t 1| t xˆis t 1| t 0 , as t , i.a.r. where the notation “ i.a.r.” denotes the convergence in a realization [30]. Proof. By the dynamic error system analysis (DESA) method [30] , the proof is completely similar to one of Theorem 5 in [22], the details are omitted. In the local robust time-varying Kalman filter and smoother (50)-(61), replacing Pi t | t 1 and
pi t by their limits Pi 1 and pi , respectively, we obtain the local robust steady-state Kalman filters 15
ACCEPTED MANUSCRIPT and smoothers xˆis t | t N N 0 and their conservative and actual steady-state variances and crosscovariances Pi N , Pi N , Pij N and Pij N . Using the dynamic variance error system analysis (DVESA) method [5], similar to the proof in Theorem 7 in [5], we can prove the convergence Pij t | t N Pij N , Pij t | t N Pij N , as t , N 1, N 0
(118)
and applying Theorem 5, we easily prove the convergence between the local time-varying and steady-state Kalman filter and smoother (119) xˆi t | t N xˆis t | t N 0 , as t , i.a.r. , i 1, , L , N 0 From the robust weighted fusion time-varying Kalman estimators xˆc t | t N given by (69)-(94) for
the common state xc t , we can obtain the corresponding steady-state Kalman fusers as L
i 1
(120)
CR IP T
xˆc s t | t N i N xˆcis t | t N , m, s, d ,CI , N 1, N 0
i t | t N i N
(121)
Theorem 6. Under the assumption conditions of Theorem 5, the robust time-varying and steady-state Kalman fusers of the common state xc t have the convergence in a realization as xˆc t | t N xˆc s t | t N 0 , as t , i.a.r.
(122) (123)
and the robust steady Kalman fuser (120) have the robustness in the sense that Pc N Pc N
(124)
AN US
and their estimation error variances have the convergence Pc t | t N Pc N , Pc t | t N Pc N , as t
and Pc N is the minimal upper bound of Pc N .
Proof. The proof is similar to those as shown in [5, 22], which is omitted. Remark 8. As t , taking the limiting operations to Theorems 3 and 4, the conservative and actual fused steady-state error variances are given as
M
Pc N N Pc N T N , Pc N N Pc N T N , m, s, d ,CI
(125)
ED
and the local and fused steady-state robust Kalman estimators have the robust and actual accuracy relations tr Pc N tr Pc N , 1, , L, m, s, d ,CI , N 1, N 0 (126) tr Pcm N tr PcCI N tr PCI N tr Pci N , i 1,
,L
(127)
tr P N tr P N tr P N tr P N , i 1, , L (128) Remark 9. Compared with the references[5-7], in the derivations of main results, there exist two difficulties as follows: one is the proof of robustness for local and fused robust Kalman estimators. In [5-7], the process and measurement white noises are assumed to be mutually uncorrelated, the robustness can simply be proved by the Lyapunov equation approach. While in this paper, they are assumed to be linearly correlated [21,22], the robustness can also be proved by the Lyapunov equation approach ( See Appendix H: the proof of Theorem 3). However, so far, under arbitrary correlation between the process and measurement white noise, the proof of robustness was not solved which is an open difficult problem. The other difficulty is the proof of convergence between the time-varying and steady-state robust Kalman estimators. Here the time-varying robust Kalman estimators are obtained from the time-varying system with time-varying noise statistics, while the steady-state robust Kalman estimators are obtained from the time-invariant system with constant noise statistics. The convergence between both of them is essentially different from that between the time-varying and steady-state robust Kalman estimators for the time-invariant system with constant noise statistics in the classical Kalman filtering theory as shown in the references [5-7, 17]. This difficulty was solved by the convergence of generalized Lyapunov equation [28] , the convergence of Riccati equation, and the dynamic error system analysis (DESA) method [21]. d c
s c
i c
AC
CE
PT
m c
9.Simulation examples Example 1 Consider robust fused steady-state Kalman smoother for the AR signal with 3-sensor, colored and white measurement noises, and uncertain noise variances and stochastic parameters At q 1 s t w t 1
(129)
yi t s t bi t i t , i 1, 2,3
(130)
16
ACCEPTED MANUSCRIPT bi t Bit q 1 ei t At q
1 a t 1 q a t 2 q B q 1 b t q b t q
1
1
1
, a j t a j j t , j 1, 2
(132)
2
, bij t bij ij t , j 1, 2
(133)
2
1
1
it
i1
i2
(131)
2
where yi t is the measurement, bi t is the colored measurement noise, which obeys the moving average (MA) model (131). wi t , ei t and i t are white noises. q 1 is the backward shift operator
q 1s t s t 1 . The a j and bij are known means of stochastic parameters a j t and bij t , respectively, where white noises j t and ij t are the corresponding random perturbations. All the white noises are mutually uncorrelated, and have zero means and uncertain variances satisfying Asumptions 1 and 3. The AR signal s t given by (129) has the state space model with stochastic parameter matrix as (134)
s t H c xc t
(135)
CR IP T
xc t 1 t xc t c w t
a1 t 1 1 , c , H c 1 0 . 0 a2 t 0 where xc t is the common state. s t is the first component of xc t .
t
(137)
AN US
The MA colored noises (131) has the state space models i t 1 Bi i t b ei t
(136)
bi t Hbi t i t ei t with the definitions
(138)
0 0 1 ,b (139) 1 0 0 Hence we obtain the augmented state space model with multi-models (different local models) and with linearly correlated additive noises as xi t 1 ai t xi t i wi t , i 1, 2,3 (140)
M
Hbi t bi1 t bi 2 t , Bi
yi t H ai t xi t vi t
(141)
vi t Di wi t i t , Di 0 1
ED
(142)
where xi t is augmented states, and wi t is augmented noises, with the definitions
PT
x t w t t 0 c 0 xi t c , wi t , ai t ,i , H ai t H c , H bi t (143) Bi 0 b 0 i t ei t Substituting the random parameters a j t and bij t given in (132) and (133) into the augmented
CE
system (140) and (141) yields the multi-model multisensor system (1)-(4) with multiplicative noises ik t and ik t , and with pi 2 , qi 2 , i 1, 2,3 , L 3 and ik t k t , k 1, 2 ,
a1 a i t 2 0 0
1 0 0 1 0 0 0 0 , i1 t 0 0 0 0 0 1 0 0
0 0 0 0 0 0 , 0 0 0 0 0 0 Hi t 1 0 bi1 bi 2 , Hi1 t 0 0 1 0 , Hi 2 t 0 0 0 1 .
AC
0 0 0 0 1 0 0 0 , i 2 t 0 0 0 0 0 0 0 0
(144)
In simulation, we take Q 1.5 , Q 1 ; Qe1 0.012, Qe2 =0.02, Qe3 0.03 , Qei 0.8Qei , i 1, 2,3 ;
R1 0.2 , R 2 0.1 , R 3 0.3 , R1 0.1 , R 2 0.05 , R 3 0.2 ; 21 0.09, 22 =0.04 ; 21 0.08,
2 =0.03 ; 2 0.11 , 2 0.31 , 2 0.21 , 2 0.41 , 2 0.31 , 2 0.18 ; 2 0.1 , 2 0.3 , 2
11
12
21
22
31
32
11
12
2 0.2 , 2 0.4 , 2 0.3 , 2 0.17 ; a1 0.15 , a2 0.3 , b11 0.2 , b12 0.1 , b21 0.4 , 21
22
31
32
b22 0.2 , b31 0.3 , b32 0.3 .
From (135) and (136), the signal s t is the first component xc t , so that the robust filtering problem 17
ACCEPTED MANUSCRIPT of AR signal s t is converted into one of the common state xc t . From (135) applying the projection theory [17], we have the local robust steady-state signal estimators as sˆi t | t N Hc xˆci t | t N and ij T their cross-covariances of the estimation errors as Psij N H c Pc N H c , further, applying the weighted
fusion algorithms in Section 6, we can obtain the local and fused estimators and their conservative and actual error variances. The comparison of robust and actual accuracies for local and fused steady-state Kalman smoothers of signal s t with N 2 are shown in Table 1. We see that the actual accuracy of each local or fused Kalman smoother is higher than its robust accuracy. The robust accuracy of the fuser with scalar weights is higher than that of the modified CI fuser. The robust accuracy of the modified CI fuser is higher than that of the original CI fuser.
s
CI
CR IP T
Table 1 The accuracy comparison of local and fused robust steady-state Kalman smoothers N 2 for s t *
Ps1 (2)
Ps 2 (2)
Ps 3 (2)
Ps (2)
Ps (2)
PCI (2)
0.3063
0.2493
0.4439
0.0710
0.0772
0.1622
Ps1 (2)
Ps 2 (2)
Ps 3 (2)
Ps (2)
Ps (2)
0.1678
0.1471
0.2942
0.0424
0.0442
s
CI
AN US
Fig.1 shows the curves of signal s t and its robust CI fusion estimator sˆCI t | t N N 1, N 0,
N 2 . The solid curves denote the actual signal and the dot curves denote the robust CI fused estimates. We see the CI fuser has the good tracking performance.
M
0
ED
s t , sˆCI t | t 1
2.5
-2.5
t/step
20
40
60
(a) The signal s t and robust CI fused predictor sˆCI t | t 1
AC
s t , sˆCI t | t 2
CE
PT
2.5
0
-2.5
t/step
20
40
(b) The signal s t and robust CI fused filter sˆCI t | t
18
60
ACCEPTED MANUSCRIPT
s t , sˆCI t | t
2.5
0
-2.5
t/step
20
40
60
(c) The signal s t and robust CI fused smoother sˆCI t | t 2
s t
CR IP T
Fig.1. The signal s t and robust steady-state CI fused estimators sˆCI t | t N , N 1, N 0, N 2 ; sˆCI t | t N
The robust and actual standard deviations of the CI fuser sˆCI t | t N ( N 1, N 0, N 2 ) are
N PsCI N and N PsCI N , respectively. The robustness (124) yields N N , which is called the robustness of robust signal fuser. The corresponding estimation error curves, 3 N and 3 N bounds
4 2 0 -2 -4
0
100
AN US
are shown in Fig.2, where the CI fused estimation error curves are denoted by the solid lines, 3 N bounds are denoted by the dashed lines, 3 N bounds are denoted by the dotted lines.
t/step
200
300
M
(a) The actual prediction errors and 3 -actual and robust standard deviations of robust CI fused predictor 1 0.5
-0.5 -1 0
ED
0
100
t/step
200
300
PT
(b) The actual filtering errors and 3 -actual and robust standard deviations of robust CI fused filter 1 0.5 0
CE
-0.5 -1
0
100
t/step
200
300
(c) The actual smoothing errors and 3 -actual and robust standard deviations of robust CI fused smoother
Fig.2. The actual estimation errors and 3 -actual and robust standard deviations of robust steady-state CI fusers
AC
From Fig.2, we can see that over 99 percent of the fused estimation error values lie between 3 N
and 3 N .They also lie between 3 N and 3 N . This verifies the robustness, and the correctness of actual fused variances. [31] Example 2 Consider an uninterruptible power system (UPS) with 3-sensor and with random
parameters and color measurement noises
xc t 1 t xc t c w t
(145)
yi t Hc xc t bi t i t , i 1, 2,3
(146)
0.5 0.9622 1 t 0.633 2 t 0 , 0 , H 23.738 20.287 0 t 1 0 0 c c 0.2 0 1 0 bi t Bit q 1 ei t 19
(147) (148)
ACCEPTED MANUSCRIPT B1t q 1 1 b11 t q 1 ,
B2t q 1 1 b21 t q 1 b22 t q 2 ,
B3t q 1 1 b31 t q 1 b32 t q 2 b33 t q 3 ,
bij t bij ij t .
(149)
where xc t is the common state, yi t is the measurement, random measurement bias bi t is the colored noise which obeys the moving average (MA) model (148). q 1 is the backward shift operator
q1s t s t 1 . The bij are known means of random parameters bij t , while i t and ij t are random perturbations with zero mean and uncertain variances 2i and 2ij , respectively. i t is white
CR IP T
measurement noise. The white noises w t , ei t , i t , i t and ij t satisfy Assumptions 1 and 3. Similar to example 1, bi t i 1, 2,3 ,can be represented by the state-space models. Applying the augmented state method, we can obtain the multi-model multisensor system (1)-(4) with pi 2 , q1 1 ,
q2 2 , q3 3 , L 3 and ik t k t , k 1, 2 .
In simulation, we take Q 1.5 , Q 1 ; Qe1 0.012, Qe2 =0.02, Qe3 0.03 , Qei 0.8Qei , i 1, 2,3 ;
R1 0.2 , R 2 0.4 , R 3 0.3 , R1 0.1 , R 2 0.3 , R 3 0.2 ; 21 0.13, 22 =0.06 ; 21 0.12,
AN US
2 =0.05 ; 211 0.11 , 221 0.21 , 222 0.41 , 231 0.31 , 232 0.18 , 2 0.13 ; 211 0.1 , 2
33
0.2 , 0.04 , 0.3 , 0.17 , 0.12 ; 0.1 , 0.2 , 2 0.04 , 2
2
2
21
2
31
22
2
2
32
11
33
2
21
22
2 0.3 , 2 0.17 , 2 0.12 ; b11 0.01 , b21 0.03 , b22 0.02 , b31 0.03 , b32 0.02 , 31
32
33
ED
M
b31 0.03 . The robust and actual accuracy comparisons of local and fused time-varying Kalman smoothers for the common state xc t with N 2 are shown in Table 2, which verifies the accuracy relations (102)-(104).
Table 2
The accuracy comparison of local and fused robust time-varying Kalman smoothers for xc t with N 2 at t 200 tr Pc t | t 2 tr Pc t | t 2 tr Pc t | t 2 tr Pc 2
0.0454 1 tr Pc
t | t 2
0.0783
2 tr Pc
t | t 2
0.0567
m
0.1265
3 tr Pc
t | t 2
0.0948
t | t 2
tr Pc t | t 2 tr Pc t | t 2 tr Pc d
0.02156 tr
m Pc
t | t 2
s
0.02169 tr
d Pc
0.01369
t | t 2
0.01384
AC
CE
0.0294
3
PT
1
20
0.02174 tr Pc t | t 2 s
0.01390
CI
t | t 2
0.0232 CI tr Pc
t | t 2
0.0145
tr PCI t | t 2 *
0.0351
ACCEPTED MANUSCRIPT
m
x 1 t , xˆc1 t | t 2
4
0
-4
20
t/step
40
60
(a) The component x1 t and its robust fused smoother xˆcm1 t | t 2
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0
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m
x 2 t , xˆc 2 t | t 2
2
-2
20
t/step
40
60
(b) The component x2 t and its robust fused smoother xˆcm2 t | t 2
M
0
ED
m
x 3 t , xˆc 3 t | t 2
2.5
-2.5
20
t/step
40
60
(c) The component x3 t and its robust fused smoother xˆcm3 t | t 2
PT
Fig.3. The common state xc t and robust time-varying fused smoother weighted by matrices xˆcm t | t 2
Fig.3 shows the curves of the components for the common state xc t x1 t , x2 t , x3 t and the T
robust fused smoother weighted by matrix xˆcm t | t 2 xcm1 t | t 2 , xcm2 t | t 2 , xcm3 t | t 2 . The
CE
T
AC
solid curves denote the actual states and the dot curves denote the robust fused smoothing estimates. We see that the fuser has the good tracking performance. Taking 300 Monte Carlo simulation runs, the mean square error (MSE) curves MSE t ,
1, 2,3, m, s, d ,CI of the time-varying local and four weighted fusion robust smoothers xˆc t | t 2 for
the common state xc t are shown in Fig.4, where the solid curves indicate the values of trPc t | t 2 . We see that when t is sufficiently large, the traces trPc t | t 2 of actual time-varying estimation error
variances converge to the corresponding steady-state values trPc 2 , and when is sufficiently large ( 300 ), the values of MSE t are closed to the corresponding values of trPc t | t 2 . This verifies the convergence trPc t | t 2 trPc 2 given in (123), and the consistency of sampled variances.
21
ACCEPTED MANUSCRIPT 0.15 MSE1
MSEs
MSE2
MSECI
MSE3
MSEd
MSEm
MSE
0.1
0
0
20
40
t/step
60
80
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0.05
100
Fig.4. The MSE curves of the time-varying local and fused robust Kalman smoothers xˆc t | t 2 , i, m, s, d ,CI
10. Conclusions
AC
CE
PT
ED
M
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For multi-model multisensor time-varying uncertain systems with both the uncertain-variance multiplicative and linearly correlated additive white noises, the fictitious noise-based Lyapunov equation approach has been presented. Its principle is that by introducing the fictitious noise to compensate multiplicative noises, the original system can be converted into one with only uncertain additive noise variances, and then, according to the minimax robust estimation principle, based on the worst-case system with conservative upper bounds of noise variances, applying the Lyapunov equation approach, the minimax robust local and four weighted fusion time-varying Kalman estimators (predictor, filter and smoother) have been presented in a unified framework. They include the three fusers weighted by matrices, scalar, diagonal matrices and a modified CI fuser. The corresponding robust steady-state Kalman fusers were also presented. The robustness and accuracy relations were proved. The convergence between the robust time-varying and steady-state Kalman fusers have been proved by the dynamic error system analysis (DESA) method. They constitute a unified and universal multi-model multisensor robust fusion Kalman filtering theory. They extended the optimal fusion Kalman filtering for multi-model systems [2,24,25] with known model parameters and noise variances to the multi-model uncertain systems. The robust fused Kalman filtering theory [5-7] only with uncertain additive noise variances but known model parameters have been extended to the multi-model systems with uncertain multiplicative noise variances. The optimal robust Kalman filtering for systems with multiplicative noises (stochastic parameters) but known multiplicative and additive noise variances [14-20] has been developed to the minimax robust fusion Kalman filtering for multi-model multisensor with uncertain multiplicative and additive noise variances. The robust Kalman fusers [21-22] for multisensor systems with known multiplicative noise variances but unknown additive noise variances have been extended to the multi-model multisensor with both the uncertain multiplicative and additive noise variances. The proposed fictitious noise-based Lyapunov equation approach is different from the fictitious noise-based game-theoretic approach[13] and the LMI approach [23]. The future works include the robust Kalman filtering for the multi-model multisensor systems with mixed uncertainties including multiplicative noises, random sensor delays, packet dropouts, and uncertain noise variances, and so on. It is remarkable that in discrete-time Markovian jump systems [32-34], we also meet “multi-model” concept in the sense that at the sequential times, the corresponding system models occur to jump in a finite multi-model set with the transition probabilities described by Markov chain, while “multisensor multi-model” concept in this paper means that each sensor subsystem has different local dynamic model containing a common system state. Recently, the robust filtering problems for Markovian jump systems with uncertain additive noise variances [32], or with the norm-bounded parametric uncertainties were considered [33]. The proposed methods may be applied to Markovian jump systems with uncertain-variance multiplicative and additive noises.
22
ACCEPTED MANUSCRIPT Acknowledgment This work is supported by the National Natural Science Foundation of China under grants NSFC-60874063, NSFC-60374026 and Postgraduate Innovation Project of Heilongjiang Province (YJSCX2015-002HLJU). The authors thank the reviewers and editors for their helpful and constructive comments, which are very valuable for improving quality of the paper. Appendix A. The proof of Lemma 3 Proof. Setting Qai t Qai t Qai t , Qi t Qi t Qi t , X i t X i t X i t and 2ik
2 2 , subtracting (25) from (24) yields ik
ik
pi
pi
Qai t 2 ik X i t ikT 2 ik X i ikT i t Qi t iT t ik
k 1
from (9) we have Q t Q t
Qi t
0
From (8) and Lemma 2, we have
Qei t Qei t ri ri 0
Qi t 0
(150)
CR IP T
ik
k 1
(151)
(152)
(20) yields the Lyapunov equation pi
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From (20) and (21), applying X i 0 0 , iterating (20) yields X i t 0 , t 0 . Subtracting (22) from pi
X i t 1 i t X i t iT t 2 ik t X i t ikT t 2 ik t X i t ikT t ik
k 1
i t Qi t
k 1
T i
ik
t
(153)
From (7), (8), (21) and (23), we have ik t 0 and X i 0 Pi Pi 0 . Applying (152) we have 2
M
i t Qi t iT t 0 . By the induction, from (153) for arbitrary t 0 , we obtain X i t 0
(154)
So, from (150), (152) and (154), we have Qai t 0 , i.e., Qai t Qai t
(155)
ED
Subtracting (27) from (26) yields
Rai t 2 H ik t X i t H ikT t 2 H ik t X i t H ikT t Rvi t qi
k 1
ik
ik
(156)
PT
Setting Rvi t Rvi t Rvi t and R t R t R t , from (11), we have i
i
i
Rvi t Di t Qi t DiT t R i t
(157)
Rvi t 0
(158)
CE
Applying (8), (152), we obtain
Similarly, from (154) and (156), we have Rai t 0 , i.e.,
AC
Rai t Rai t
The proof is completed. Appendix B. The proof of Lemma 4 Proof. Defining i (t ) i (t ) i (t ) , from (11), (25)-(28) and (33)
23
(159)
ACCEPTED MANUSCRIPT
i (t )
can
be
decomposed
i t Qi t DiT t qi 2 T ik H ik t X i t H ik t k 1 2 T ik H ik t X i t H ik t T Di t Qi t Di t R i t
i1 (t )
as
(160)
,
i 2 (t )
and
i3 (t )
,
i.e.,
where
CR IP T
pi 2 T ik ik t X i t ik t k 1 2ik ik t X i t ikT t i t Qi t iT t i (t ) Di t Qi t iT t
2 T ik ik t X i t ik t 0 k 1 2ik ik t X i t ikT t 0 , 3 (t ) 0 i1 (t ) 0 R t , i qi i 2ik H ik t X i t H ikT t k 1 0 2 T ik H ik t X i t H ik t T T t Qi t i t i t Qi t Di t i 2 (t ) i . T T Di t Qi t i t Di t Qi t Di t pi
pi
It
is
obvious
that
t X t t 0 2
k 1
qi
H ik t X i t H ikT t 0 and ik
ik
ik
T ik
i
qi
2 ik
k 1
pi
,
t X t t 0 2
k 1
ik
ik
i
T ik
i1 (t ) 0
(161)
i t
ED
i 2 (t ) can be rewritten i 2 (t ) i t Qi t i T t where i t Q t Qi t Qi t i . Applying (152) and Lemma 1, we have Qi t Qi t
PT
,
H ik t X i t H ikT t 0 . Applying Lemma 2, we have
M
2
k 1
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0
0 Di t
and
i 2 (t ) 0
(162)
i3 (t ) 0
(163)
i (t ) 0
(164)
CE
Applying (8) and Lemma 2 yields So, we have
AC
i.e., (34) holds. The proof is completed. Appendix C. The proof of Lemma 5 Proof. Define Qa t Qa t Qa t , Qij t Qij t Qij t , and applying (9) and (12) yields 0 Q t Qij t 0 Q t ri rj ei ij
and we have with rg r1
Qa t Qij t
rg rg
(165)
(166)
rL . Decompose Qij t as
Qij t Qij* t Qeij* t with ri r rei , and 24
(167)
ACCEPTED MANUSCRIPT 0 Q t 0 0 * Qij* t ，Qeij t 0 0 Qei t ij ri rj 0 ri rj
(168)
Defining Q* t Qij* t
from (166) and (167) we have
rg rg
* , Qe* t Qeij t
rg rg
Qa t Q* t Qe* t
(169) (170)
Applying (168) yields
From (6) and (8), then
, Qe*LL t
Qe* t 0
(171)
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Qe* t diag Qe*11 t ,
(172)
* Hence, from (170) and (172), in order to prove Qa t 0 , we need to prove Q t 0 . * * Interchanging rows and columns of Q t , then Q t can be transformed as
Q(t )
0 Q(t ) 0 rg rg
(173)
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Q(t ) * Q0 t Q(t )
0
* * T i.e., there exists a non-singular elementary trans formation matrix Te such that Q0 t Te Q t Te .
From (8), Q t 0 , applying Lemma 1 yields that
M
Q(t ) Q(t )
Q(t ) 0 Q(t )
(174)
* * * Hence from (173) we have Q0 t 0 . Applying Q0 t 0 yields Q t 0 , and applying (170) and
ED
(172) yields Qa t 0 . The proof is completed. Appendix D. The proof of Theorem 1
Proof. Setting Pi t 1| t Pi t 1| t Pi t 1| t , from (45) and (46) yields the Lyapunov equation
PT
Pi t 1| t pi t Pi t 1| t piT t i t
(175)
where i t I n , K pi t i t I n , K pi t . From (164), we have i t 0 . Subtracting (48)
CE
from (47), we have the initial value
T
Pi 1| 0 P0 P0 0
(176)
By the induction, from (175) and (176) we recursively obtain Pi t 1| t 0 for arbitrary t 0 .i.e., (49)
AC
holds. This means that Pi t 1| t is a upper bound of Pi t 1| t . In (8), selecting the actual variances to equal the corresponding conservative upper bounds, applying (47), (48) and (175) yields Pi t 1| t 0 for arbitrary t 0 , i.e., Pi t | t 1 Pi t | t 1 . For arbitrary other upper bound
Pi* t 1| t of Pi t 1| t , we have Pi t 1| t Pi t 1| t Pi* t 1| t for arbitrary t 0 . This means
that Pi t 1| t is a minimal upper bound of Pi t 1| t . The proof is completed. Appendix E. The proof of Theorem 2 Proof. From (50) and (51), we can obtain the error xi t | t N xi t | t N xˆi t | t N as [22] N
N
0
0
vN xi t | t N iN t xi t | t 1 KiwN t wai t Ki t vai t
where iN t , Ki
t and KivN t are defined in (58)-(61). For N 0 , subtracting (56) from (55) Pi t | t N Pii t | t N Pi t | t N Pi t | t N iN t Pi t | t 1 iNT t i t wN
25
(177) yields (178)
ACCEPTED MANUSCRIPT T N KiwN t vN t , K t ( t ) where i t KiwN . Applying (164) and (49) to (178) yields i vNT i 0 Ki t
Pi t | t N 0 , i.e., the robustness (62) holds for N 0 , which implies that Pi t | t N is a upper
bound of Pi t | t N . Similar to the proof given in Theorem 1, we easily prove that Pi t | t N is the minimal upper bound of Pi t | t N . The proof is completed. Appendix F. The proof of Corollary 1 Proof. Taking N 0 in (50) and applying (53) yields (63)Applying (16), (18), (51) and (63), yields the filtering errors (179) xi t | t I n K fi t Hi t xi t | t 1 K fi t vai t
CR IP T
Applying (179) directly yields (64) and (65). The proof is completed. Appendix G. The proof of Corollary 2 Proof. Taking the projection operation to (5) yields (66). Subtracting (66) from (5) yields
xci t | t N Cxi xi t | t N
(180)
which yields (67), and applying (62) yields (68). The proof is completed. Appendix H. The proof of Theorem 3
Proof. From (85), (87)-(89) we obtain (90) and (91). Setting Pc t | t N Pc t | t N
AN US
Pc t| t N , Pc t | t N Pc t | t N Pc t | t N , and subtracting (91) from (90), it follows that Pc t | t N t | t N Pc t | t N T t | t N Denote ng n1
(181)
nL and defining the global conservative and actual covariances respectively as P t | t N Pij t | t N
(182)
ng ng
P t | t N Pij t | t N
(183)
ng ng
applying (67), (68), (92), (93), (182) and (183) yields
(184)
Pc t | t N CP t | t N C
(185)
M
Pc t | t N CP t | t N C T 0 CxL nLn g
ED
Cx1 C 0
T
(186)
PT
Denoting P t | t N P t | t N P t | t N , and subtracting (185) from (184), we have
Pc t | t N CP t | t N C T
(187)
CE
From (181) and (187), in order to prove Pc t | t N 0 , it is sufficient that to prove
P t | t N 0 .
Taking N 1 , from (45) and (46) we have that P t 1| t and P t 1| t satisfy the global Lyapunov
AC
equation respectively
P t 1| t p t P t | t 1 pT t K p t p t K pT t
(188)
P t 1| t p t P t | t 1
(189)
with the definitions
p t diag p t , 1
T p
t K p t p t K t
T p
, pL t , K p t diag I n , K p1 t ,
11 t p t L1 t with the initial values
1L t
11 t , p t L1 t LL t
26
, I n , K pL t 1L t LL t
(190)
ACCEPTED MANUSCRIPT P0 P0 , P 1| 0 P0 P0
P0 P 1| 0 P0
P0 P0
(191)
Setting P t 1| t P t 1| t P t 1| t and subtracting (189) from (188) yields Lyapunov equations
P t 1| t p t P t | t 1 pT t K p t p t K pT t
(192)
where p t =p t p t . Setting ij (t ) ij (t ) ij (t ) , applying (24)-(29), and (33) yields
i t Qij t D Tj t qi 2ik H ik t X i t H ikT t ij k 1 qi 2ik H ik t X i t H ikT t ij k 1 T Di t Qij t D j t R i t ij
AN US
CR IP T
pi 2 T ik ik t X i t ik t ij k 1 pi 2 T ik ik t X i t ik t ij k 1 t Q t T t i ij j ij (t ) Di t Qij t Tj t Define
ED
M
pi 2 T ik ik t X i t ik t k 1 0 pi 2 T ik ik t X i t ik t k 1 , i1 t ii1 t qi 2ik H ik t X i t H ikT t k 1 0 qi 2 T H t X t H t ik ik i ik k 1 0 t Qij t Tj t i t Qij t D Tj t 0 3 3 ij 2 (t ) i , i t ii t (193) T T Di t Qij t j t Di t Qij t D j t 0 R i t
Then we have p t p t p t p t 2
PT
1
11 t p t 0
11 2 t 2 , p t 2 L1 t L1 t
CE
1L2 t
0
1
AC
3
3
p
13 t 0
, 2 LL t
3 L t 0
(194)
According to (161), (163) and Lemma 2 we have p t 0 and p t 0 . Next, we need to 3
1
prove p t 0 . 2
1 L p 2 can be rewritten p 2 t p t Qp t pT t , where p t diag t , , t ,
i
Q11 t i t 0 t , Qp t Di t 0 L1 Q t
Q
1L
t
Qij t Qij t ij , and Q t Qij t Qij t 2 ri 2 rj LL Q t
Qi t Qii t . Hence the problem is to prove Qp t 0 . From (9) and (12) it follows 27
ACCEPTED MANUSCRIPT 0 Q t Q t 0 . , i j ; Qi t Qii t 0 Qei t 0 r r 0 ri ri i j
Qij t
Decompose Qp t as
Qp t Qb t Qd t QB
1L
Q t 0 , QBi , j t Q t 0
t
LL QB t 2 r 2 r g g
Qd t diag QD
11
t ,
0 0 0 Q t LL ei , QD t , QDi ,i t 0 0 0 Qei t
0 Q t 0 0 0 0 0 Q t 0 0 0 0 2 r 2 r i j 0 Qei t 0 0 0 Qei t 2 r 2 r i i 0
0
CR IP T
QB11 t Qb t L1 QB t
(195)
Applying Lemma 1 yields QDi ,i t 0 , and applying Lemma 2 yields Qd t 0 . Interchanging rows and columns of Qb t yields Q(t )
0 0 Q(t ) 0 2 rg 2 rg
AN US
Q(t ) Qb t Q(t )
0
(196)
Hence Qp t 0 , which yields p t 0 . Therefore it follows that p t 0 . Applying (8), (43), 2
(47), (48) and (191) yields P 1| 0 P 1| 0 P 1| 0 0 , so that iterating (192) yields
M
P t 1| t 0 for arbitrary t.
Taking N 0 , subtracting (56) from (55), we have the global P t | t N as N
ED
P t | t N N t P t | t 1 NT t K N t p t K NT t
(197)
0
N t =diag 1N t ,
, K LwN t , K LvN t
PT
vN K N t =diag K1wN t , K1 t ,
, LN t
(198)
Applying P t | t 1 0 and p t 0 yields P t | t N 0 , so from (187) we have
Pc t | t N 0 , and it follows from (181) that Pc t | t N 0 ,i.e., (94) holds. Similar to the proof
CE
of Theorem 1, we have that Pc t | t N is the minimal upper bound of Pc t | t N . The proof is
AC
completed. Appendix I. The proof of Theorem 4 Proof. Taking the trace operations to (68), (94) and (101) yields (102) and the second inequality of (103). Since the CI fuser is a special fuser with matrix weights, then the first inequality of (103) holds. The third inequality of (103) can easily be proved from (84). In fact, arbitrarily taking i t | t N 1 ,
j t | t N 0 j i yields trPCI t | t N =trPci t | t N , so that minimizing trPCI t | t N with the
constraint 1 t | t N
L t | t N 1 yields the third inequality of (103). The accuracy relations
(104) were proved in [29]. The proof is completed.
28
ACCEPTED MANUSCRIPT References
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[1] M. E. Liggins, D. L. Hall, H. Llinas, Handbook of Multisensor Data:Theory and Practice, Second Edition, New York: CRC Press. 2009. [2] Z. L. Deng, Y. Gao, L. Mao, Y. Li, G. Hao, New approach to information fusion steady-state Kalman filtering, Automatica. 41(10) (2005) 1695-1707. [3] S. L. Sun, Z. L. Deng, Multi-sensor optimal information fusion Kalman filte, Automatica. 40(6) (2004) 1017-1023. [4] F. L. Lewis, L. H. Xie, D. Popa, Optimal and Robust Estimation. Second Edition, CRC Press, New York. 2008. [5] W. J. Qi, P. Zhang, Z. L. Deng, Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances, Signal Processing. 99 (2014) 185-200. [6] W. J. Qi, P. Zhang, G. H. Nie, Z. L. Deng, Robust weighted fusion Kalman predictors with uncertain noise variances, Digital Signal Processing. 30 (2014) 37-54. [7] W. J. Qi, P. Zhang, Z. L. Deng. Robust weighted fusion time-varying Kalman smoothers for multisensor system with uncertain noise variances, Information Sciences. 282 (2014) 15-37. [8] S. J. Julier, J. K. Uhlman, Non-divergent estimation algorithm in the presence of unknown correlations, in: Proceedings of the America Control Conference. 4 (1997) 2369-2373. [9] S. J. Julier, J. K. Uhlman, Using covariance intersection for SLAM. Robot. and Auton. Syst. 55 (1) (2007) 3-20. [10] F. Wang, V. Balakrishnan, Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties, IEEE Transactions on Signal Processing, 50(4) (2002) 803-813. [11] W. L. De Koning, Optimal estimation of linear discrete-time systems with stochastic parameters, Automatica, 20(1) (1984) 113-115. [12] N. E. Nahi, Optimal recursive estimation with uncertain observation, IEEE Transaction on Information Theory 15(4) (1969) 457-462. [13] Y. L. Chen, B.S. Chen, Minimax robust deconvolution filters under stochastic parametric and noise uncertainties, IEEE Transactions on Signal Processing, 42(1) (1994) 32-45. [14] S. Y. Wang, H. J. Fang, X. G. Tian. Minimum variance estimation for linear uncertain systems with one-step correlated noises and incomplete measurements, Digital Signal Processing. 49 (2016) 126-136. [15] T. Tian, S. L. Sun, N. Li, Multi-sensor information fusion estimators for stochastic uncertain systems with correlated noises, Information Fusion. 27 (2016) 126–137. [16] J. X. Feng, Z. D. Wang, M. Zeng, Distributed weighted robust Kalman filter fusion for uncertain systems with autocorrelated and cross-correlated noises, Information Fusion. 14 (1) (2013) 78-86. [17] B. D. O. Anderson, J. B. Moore. Optimal Filtering. Prentice Hall, Englewood Cliffs, New Jersey. 1979. [18] F. Li, J. Zhou, D. Z. Wu, Optimal filtering for systems with finite-step autocorrelated noises and multiple packet dropouts, Aerospace Science and Technology. 24 (2013) 255-263. [19] B. Chen, W. Zhang, G. Hu, L. Yu, Networked fusion kalman filtering with multiple uncertainties, IEEE Transactions on Aerospace and Electronic Systems. 51(3) (2015) 2332-2349. [20] B. Chen, L. Yu, W. A. Zhang, A. Liu, Robust information fusion estimator for multiple delay-tolerant sensors with different failure rates, IEEE Transactions on Circuits and Systems-I: Regular Papers. 60(2) (2013) 401-414. [21] W. Q. Liu, X. M. Wang, Z. L. Deng, Robust centralized and weighted measurement fusion Kalman estimators for uncertain
multisensor systems with linearly correlated white noises, Information Fusion. 35 (2017) 11–25.
CE
PT
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[22] W. Q. Liu, X. M. Wang, Z. L. Deng, Robust weighted fusion Kalman estimators for multisensor systems with multiplicative noises and uncertain covariances linearly correlated white noises, International Journal of Robust and Nonlinear Control (2016) Published online in Wiley Online Library (wileyonlinelibrary). DOI: 10.1002/rnc.3649. [23] B. Chen, G Hu, D. W. C. Ho, W. A. Zhang, L. Yu, Distributed robust fusion estimation with application to state monitoring systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems. (2016) DOI:10.1109/TSMC.2016.2558103. [24] X. J. Sun, Y. Gao, Z. L. Deng, Multi-model information fusion Kalman filtering and white noise deconvolution, Information Fusion. 11 (2010) 163-173. [25] X. J. Sun, Y. Gao, Z. L. Deng, Information fusion white noise deconvolution estimators for time-varying systems, Signal Processing, 88(5) (2008):1233-1247. [26] J. Ma, S. L. Sun, Information fusion estimators for systems with multiple sensors of different packet dropout rates, Information Fusion, 12(3) (2011) 213-222. [27] J. Ma, S. L. Sun, Distributed fusion filter for networked stochastic uncertain systems with transmission delays and packet dropouts . Signal Processing. 130 (2017) 268–278. [28] Z. D. Wang, D. W. C. Ho, X. H. Liu, Robust filtering under randomly varying sensor delay with variance constraints, IEEE Transaction Circuits Systems II：Express Briefs. 51(6) (2004) 320-326. [29] Z. L. Deng, P. Zhang, W. J. Qi, Y. Gao, J. F. Liu, The accuracy comparison of multisensor covariance intersection fuser and three weighting fusers, Information Fusion. 14 (2013) 177-185. [30] Z. L. Deng, Y. Gao, C. B. Li, G. Hao, Self-tuning decoupled information fusion Wiener state component filters and their convergence, Automatica, 44(3) (2008) 685-695. [31] F. W. Yang, Z. D. Wang, Y. S. Hung, M. Gani, H control for networked systems with random communication delays, IEEE
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Transactions on Automatic Control, 51(3) (2006) 511-518. [32] J. Zhu, H. S. Xi, H. B. Ji, B. Wang, Robust Kalman filtering of discrete-time Markovian jump systems based on state estimation performance，Control Theory & Applications, 25(1) (2008) 115-119. [33] F. B. Li, L. G. Wu, P. Shi, C. C. Lim, State estimation and sliding mode control for semi-markovian jump systems with mismatched, Automatica, 51 (2015) 385-393. [34] P. Shi, F. Li, L. Wu, C. C. Lim. Neural network-based passive filtering for delayed neutral-type semi-markovian Jump Systems.
IEEE Transactions on Neural Networks & Learning Systems, DIO:10.1109/TNNLS, 2016. 2573853.
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Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises Xuemei Wang , Wenqiang Liu , Zili Deng PII: DOI: Reference:
S0165-1684(17)30071-3 10.1016/j.sigpro.2017.02.015 SIGPRO 6410
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
6 September 2016 5 January 2017 24 February 2017
Please cite this article as: Xuemei Wang , Wenqiang Liu , Zili Deng , Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.02.015
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Highlights Multi-model multisensor systems with uncertain noise variances A universal fictitious noise-based Lyapunov equation approach was presented The four robust weighted fusion Kalman estimators were presented A minimax robust fusion Kalman filtering theory was presented
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Robust weighted fusion Kalman estimators for multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated additive white noises Xuemei Wanga,b, Wenqiang Liua and Zili Denga, a
Electronic Engineering College, Heilongjiang University, Harbin, China Heilongjiang College of Technology and Business, Harbin, China
b
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Abstract: For multi-model multisensor systems with both the uncertain-variance multiplicative and linearly correlated additive white noises, a universal fictitious noise-based Lyapunov equation approach is presented, by which the original system can be converted into one with only uncertain additive noise variances, and then the local and four weighted fused minimax robust time-varying Kalman estimators (predictor, filter and smoother) of the common state are presented in a unified framework, where the robust Kalman filter and smoother are designed based on the robust Kalman predictor. They include the three fusers weighted by matrices, scalars and diagonal matrices, and a modified Covariance Intersection(CI) fuser. Their robustness is proved in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Their accuracy relations are proved. The corresponding local and fused robust steady-state Kalman estimators also are presented. The convergence analysis is also given. Two simulation examples applied to design the robust fusers for an autoregressive (AR) signal and an uninterruptible power system (UPS) are given to show the effectiveness of the proposed results. Keywords: multi-model; multiplicative noises; uncertain noise variance; minimax robust Kalman estimator; fictitious noise-based Lyapunov equation approach; weighted fusion
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1. Introduction
With the development of high technology fields, the multi-sensor data fusion has received much attention. Its aim is to obtain a fused estimator of system state, whose accuracy is higher than that of each
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local state estimator. The optimal information fusion Kalman filtering theory and methods have been widely used in many fields including military, defense, robotics, unmanned aerial vehicles, target tracking, GPS positioning and remote sensing[1].According to the unbiased linear minimum variance
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(ULMV) optimal estimation rule, there are the three optimal weighted state fusion approaches weighted by scalars, diagonal matrices and matrices respectively [2,3]. The limitation of optimal information fusion
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Kalman filtering is that the model parameters and noise variances are assumed to be exactly known [2,4]. In practical applications, there inevitably exist uncertainties in the system model due to stochastic perturbances, unmodeled dynamics, and so forth. For uncertain systems, the classical filter performance [4]
. Therefore, the study of the robust Kalman filtering for
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will degrade or the filter may be divergent
uncertain systems received extensive attention. The so-called robust Kalman filter is to find a Kalman filter such that its actual filtering error variances yielded by all admissible uncertainties are guaranteed to have a minimal upper bound [4]. Such property is called robustness. There are the linear matrix inequality (LMI) approach and Riccati equation approach
[4]
to design the
robust Kalman filters for systems with norm-bounded uncertain parameters but known noise variances. For systems with uncertain noise variances but known model parameters, based on the minimax robust estimation principle, the robust weighted fusion time-varying and steady-state Kalman estimators * Corresponding author. Tel.: +86 13804583507. E-mail address: [email protected] (Z. L. Deng). 2
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(predictor, filter and smoother) are presented by the Lyapunov equation approach
[5-7]
.Their disadvantage
is that the predictor, filter and smoother are separately designed, and the smoother is designed by the augmented state approach which yields a larger computational burden. The weighted state fusion approaches
[2,3]
need to compute the cross-covariances among the local estimation errors, but in many
practical applications, the cross-covariances are uncertain. For the systems with uncertain variances and cross-covariances, the robust covariance intersection (CI) fusion method was presented by Julier and Uhlmann
[8,9]
, and the CI fuser is formed by weighting the robust local estimators with a convex
combination. The advantage of robust CI fusion method is that the cross-covariances are avoided. Its disadvantages are that the local estimators and the conservative upper bound of their actual error
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variances are assumed to be known, and a larger conservative upper bound of actual fused estimation error variances is given.
For uncertain systems, the parametric uncertainties fall under two classes
[ 10 ]
: One class is
deterministic uncertainties that the uncertain parameters are assumed to lie in some known bounded set. For example, the norm-bounded uncertainty
[4]
can be used to describe the uncertain parameters over
known bounded ranges. The second class is stochastic uncertainties that the parameters contain random
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perturbations which are also called multiplicative noises or state-dependent noises. In uncertain system, the noises include the multiplicative and additive noises, while the additive noises include the process and measurement noises. The uncertainties of noise variances can also be described by deterministic uncertainties. Since the noise variance matrices are positive semi-definite, we can assume that the actual uncertain noise variances have the known conservative upper bounds
[5-9]
. In recent years, the robust
filtering problems for the uncertain systems with multiplicative noises have received much attention and are widely applied to many fields including communication, tracking, aerospace, image processing, and
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stochastic signal processing, and so on. They have important theoretical and application meanings. Theoretically, some uncertain systems with stochastic parameters can be converted into one system with deterministic parameters and state-dependant noises (multiplicative noises) as follows:
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For uncertain system with stochastic parameter matrices[11], by taking the expectation operations, the stochastic parameter matrices can be decomposed as deterministic parameter matrices (means) plus random perturbation matrices with zero means, so that we obtain the corresponding uncertain system
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with deterministic parameter matrices and multiplicative noises. When the multiplicative and additive noise variances are assumed to be exactly known, by introducing fictitious noises to compensate the
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multiplicative noise terms, we can obtain the conventional system with deterministic model parameters and known noise statistics. Hence the standard Kalman filtering algorithm can be applied to obtain the optimal robust Kalman filter. This is called the fictitious noise method, which was early presented by
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Koning in [11].
For sensor network system, the missing measurements frequently appear due to limited bandwidth of
network, the missing rate can be described as Bernoulli white noise taking values 1 or 0 with known probabilities [12]. Taking value 1 denotes that the measurement is received, and taking value 0 denotes the missing measurement, and only the measurement noise signal is received. Hence, multiplying measurement matrix by a Bernoulli white noise yields the system with missing measurements. Taking the expectation operation to random measurement matrix with Bernoulli white noise yields a deterministic measurement matrix plus a state-dependant measurement noise with zero mean. This yields the system with multiplicative noise in the measurement matrix. In signal processing, the deconvolution system for the autoregressive moving average(ARMA) signal with stochastic parameters can be converted into one with the state space model 3
[13]
. Noting that each
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stochastic parameter can be decomposed as its mean plus random perturbation with zero mean, then furthermore the deconvolution system with deterministic parameter matrices and multiplicative noises can be obtained [13]. For single sensor or multisensor uncertain systems with mixed uncertainties including multiplicative noises, missing measurements, randomly delayed measurements, packet dropouts, assuming that the noise statistics (variances, cross-covariances and correlation matrices) are exactly known, some optimal robust or robust fused state estimators innovation analysis method
[17]
[14-16]
were presented by directly using the projection theory (i.e.,
) or the minimum variances estimation rule, and some other optimal robust
or robust fused state estimators
[18-20]
were presented by the fictitious noise method [11] and minimum
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variance estimation rule or projection theory. Their limitation is that the noise statistics are assumed to be exactly known.
Recently, for uncertain multisensor system with multiplicative noises in state matrix and missing measurements, and with uncertain additive noise variances but known multiplicative noise variances, according to minimax robust estimation principle, based on the worst-case system with conservative upper bounds of uncertain noise variances, by the extended fictitious noise method and Lyapunov equation [21]
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approach, the robust centralized and weighted measurement fusion Kalman estimators were presented
in the sense that their actual fused estimation error variances are guaranteed to have the corresponding minimal upper bounds. Koning‟s fictitious noise method
[11]
is extended to systems with known
multiplicative noise variances and uncertain additive noise variances. For uncertain multisensor system with the multiplicative noises in both the state and measurement matrices and with uncertain additive noise variances but known multiplicative noise variances, applying the extended fictitious noise method [21] and the Lyapunov equation approach, the three weighted fusion minimax robust Kalman estimators and a
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modified CI fuser were presented in [22]. Early, applying the fictitious noise-based game-theoretic approach, the minimax robust deconvolution filters were presented for systems with both the uncertain multiplicative and additive noise variances[13], where uncertain variances have know lower and upper
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bounds. Their disadvantages are that the search for minimax filter gains requires expensive computational cost, and the robust fusion filtering problem was not solved. Recently, for uncertain multisensor system with multiplicative noises and norm-bounded uncertain
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parameters in the state transition matrix, assuming that the multiplicative noises variances are uncertain which lie in the bounded intervals with known lower and upper bounds, while the additive noise variances are assumed to be unknown, a distributed robust fusion state estimator was presented based on convex
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optimization by LMI approach [23]. Regarding the optimal or robust fusion Kalman filtering for multisensor systems, the limitation of all the
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above researches is that all sensor subsystems have the common state equation, but the measurement equations of the subsystems are different. Using the augmented state method to solve the fused state estimation problems for multisensor systems, we frequently meet multi-model systems (i.e., different augmented local state equations or different local dynamic models)
[24]
. The original common state
equation is embedded in each augmented local state equation. The state of the original state equation is called the common state of all the augmented local state equations. The augmented state of each subsystem contains the common system state as its partial components. The state of original non-augmented system is the common state of augmented state subsystems. The three optimal weighted fusion state estimators and input white noise deconvolution estimators have been present in [24, 25], for multi-model multisensor systems with known model parameters and noise variances, and a simulation example applied to an infrared target tracking system with three sensors was given in [24]. 4
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For the ARMA signal with multisensor and stochastic parameters, and with coloured measurement noises described by the ARMA models with stochastic parameters, applying the augmented state method also yields the multi-model system with multiplicative noises (See example 1 in section 9). For multisensor system with stochastic parameters and coloured measurement noises which are described by the state-space models with stochastic parameters. Applying the augmented state method can yield the multi-model system with multiplicative noises (See example 2 in section 9). For networked uncertain multisensor systems with transmission delays and packet dropouts, applying the augmented state method with the augmented states containing delayed measurements, we also obtain the multi-model system with multiplicative noises [26,27]. and have important theoretical and applicable meanings.
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Therefore, the multi-model systems with multiplicative noises have widely application backgrounds, Motivated by the above discussion, to the best of the authors‟ knowledge, so far, for multi-model multisensor uncertain systems with uncertain-variance multiplicative noises in both the state and measurement matrices and uncertain-variance linearly correlated additive white noises, the problem of designing the unified robust fusion Kalman estimators was not solved.
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The novelty of the topic is highlighted as follows:
(1) The design of the robust fused Kalman estimators for the multi-model multisensor uncertain systems is considered the first time.
(2) The considered uncertainties include multiplicative noises in both the state and measurement matrices, and uncertain multiplicative and additive noise variances. Specially, the uncertain multiplicative noise variances are seldom considered in references [13,23]. The main contributions of this paper are as follows:
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(1) A universal fictitious noise-based Lyapunov equation approach is presented for the multi-model multisensor uncertain systems with both the uncertain-variance multiplicative and additive noises. Its principle is that by introducing the fictitious noises to compensate multiplicative noises, the original
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system can be converted into one with only uncertain additive noise variances, further, applying the Lyapunov equation approach
[5-7]
, the minimax robust local and fused Kalman estimators are presented.
The Koning‟s fictitious noise method [11] for uncertain system with both known multiplicative and additive
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noise variances is developed. The extended fictitious noise method [21,22] for uncertain system with known multiplicative noise variances and uncertain additive noise variances, is extended. It is different from the fictitious noise-based game-theoretic approach
and the LMI approach
[23]
. The Lyapunov equation
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[5-7]
[13]
approach
without fictitious noises is extended and developed. Specially it overcome the disadvantages
of reference [13]. By the proposed approach, the local and fused robust Kalman estimators can be obtained
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by solving the Lyapunov equations, and to search the minimax optimal solution [13] is avoided. (2) For multi-model multisensor systems with uncertain-variance multiplicative and linearly correlated
additive white noises, based on the worst-case system with conservative upper bounds of uncertain noise variances, by the proposed fictitious noise-based Lyapunov equation approach, the four weighted fusion time-varying and steady-state minimax robust Kalman estimators (predictor, filter and smoother) of the common state are presented in a unified framework, which include the three fusers weighted by matrices, scalars, diagonal matrices and a modified CI fuser. Their robustness is proved. Their accuracy relations, computational complexities and convergence analysis are also given. They constitute a unified and universal robust Kalman filter theory. The minimax robust fusion Kalman filtering theory
[5-7]
for systems only with uncertain additive noise
variances is extended. The optimal robust Kalman filtering 5
[14-20]
for systems with known multiplicative
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[2,24,25]
for multi-model multisensor systems with known model
parameters and noise statistics are developed. The rest of this paper is organized as follows. The problem formulation is given in Section 2. The preliminary knowledge is given in Section 3. Robust local time-varying Kalman predictor is presented in Section 4. Robust local time-varying Kalman filter and smoother are presented in Section 5. Robust weighted fusion time-varying Kalman estimators of the common state are presented in Section 6. Accuracy analysis is presented in Section 7. Local and fused steady-state estimators and convergence
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analysis are presented in Section 8. Two simulation examples are presented in Section 9. Conclusions are presented in Section 10.
Notation: R n denotes the n-dimensional Euclidean space, tr denotes the trace of a matrix, the subscript “ T ” denotes the transpose, E denotes the mathematical expectation, diag denotes the block-diagonal matrix, and A B denotes B A 0 being a positive semi-definite matrix . 2. Problem formulation
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Consider the linear discrete time-varying multi-model multisensor system with uncertain-variance multiplicative and linearly correlated additive white noises pi (1) xi t 1 i t ik t ik t xi t i t wi t k 1 qi (2) yi t H i t ik t H ik t xi t vi t , i 1, , L k 1
vi t Di t wi t i t , i 1, , L
xi t xcT t , iT t , wi t wT t , eiT t
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T
(3)
T
(4)
where t is the discrete time, T denotes the transpose, xi t R ni , yi t R i , wi t R ri and vi t R m
mi
respectively, i t R
n i
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are state, measurement, process noise and measurement noises of ith sensor subsystem, respectively. xc t R n and w t R r are common state and common process white noise of all subsystems, and ei t R rei respectively are different state and different process white
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noises of subsystems. L , pi and qi respectively are the numbers of sensors and multiplicative noises. From (4) xc t and w t can be denoted respectively as
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xc t Cxi xi t , Cxi I n 0nn , w t Cwi wi t , Cwi I r 0r r i
(5)
i
with I n is the n n identity matrix, 0 is the zero matrices with compatible dimensions. i t , i t , H i t , Di t are known matrices with compatible dimensions. The process and measurement noises are
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also called the additive noises. From (3), we call additive noises vi t and wi t to be linearly correlated. Notice that the equations (4) and (5) mean that xc t is the former n components of xi t .
Assumption 1. w t , ei t , i t R mi , ik t R1 and is t R1 are mutually uncorrelated white noises with zero means and actual covariances w t w u T Q t 0 0 Qei t ij ei t e j u 0 Ε i t j u 0 t u 0 0 ik jr is t j u 0 0
0 0 0 R i t ij 0 0 (6) tu 2 0 ik ij kr 0 2 0 0 is ij s where ij is the Kronecker function, ii 1, ij 0 i j , E denotes the mathematical expectation. 6
0
0
0
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is t , and it has known mean value E xi 0 0 and uncertain actual variance T E xi (0) 0 xi (0) 0 P0
(7)
Assumption 3. Q t , Qei t , R i t , P0 , 2ik and 2is have respectively the known conservative upper bounds Q t , Qei t , R i t , P0 , 2ik and 2is , i.e., with i 1, , L , k 1, covariances
, pi , s 1,
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2 2 2 2 Q t Q t , Qei t Qei t , R i t R i t , P0 P0 , ik ik , is is
, qi . Applying (3) and (4) yields the conservative and actual
0 Q t , Qi t Qii t , i, j 1, Qij t E wi t wTj t Qei t ij 0 ri rj Sij t Ε wi t vTj t Qij t DTj t , Si t Sii t
,L
(9) (10)
t Di t Qij t D t R t ij , Rvi t Rvii t
(11)
Q t 0 Qij t , Qi t Qii t Qei t ij 0 ri rj
(12)
T j
i
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Rvij t Ε vi t v
T j
(8)
Sij t Qij t DTj t , Si t Sii t
Rvij t Di t Qij t D t R i t ij , Rvi t Rvii t T j
(13) (14)
Our aim is to find the local and fused robust Kalman estimators (predictor, filter and smoother) ˆxc t | t N N 1, N 0, N 0 of the common state xc t , such that for all admissible uncertain
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actual variances Q t , Qei t , R i t , P0 , 2ik and 2is satisfying (8), the corresponding actual estimation error variances Pc t | t N are guaranteed to have the corresponding minimal bounds Pc t | t N ,i.e.,
Pc t | t N Pc t | t N
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(15)
where xˆc t | t N with i, m, s, d ,CI denote the local robust estimators, the robust fusers weighted by
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matrices, scalar, diagonal matrices and the robust CI fuser, respectively. 3. Preliminary knowledge
Introducing fictitious noises wai t , which are used to compensate multiplicative noises in (1), then the
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state equation (1) can be rewritten as xi t 1 i t xi t wai t
(16)
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with defining fictitious process noises wai t as pi
wai t ik t ik t xi t i t wi t
(17)
k 1
Introducing another fictitious noises vai t , which is used to compensate multiplicative noises in measurement equation (2), then (2) can be rewritten as (18) yi t Hi t xi t vai t , i 1, , L with defining fictitious measurement noises vai t as qi
vai t ik t H ik t xi t vi t , i 1,
,L
(19)
k 1
Applying Assumptions 1 and 2, it is easily proved that wai t and vai t are white noises with zero means. From (1), we define the conservative state covariances as X i t Ε xi t xiT t , where xi t is the conservative state yielded by the (1) with the conservative upper bounds of noise variances. From 7
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X i t 1 i t X i t iT t 2ik ik t X i t ikT t i t Qi t iT t
(20)
k 1
and with conservative initial values as
X i 0 P0 0 0T
(21)
We define the actual state covariance as X i t Ε xi t xiT t , where xi t is the actual state yielded
by (1) with actual noise variances. From (1), X i t satisfy the actual generalized Lyapunov equation pi
X i t 1 i t X i t iT t 2ik ik t X i t ikT t i t Qi t iT t
(22)
k 1
with actual initial values
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X i 0 P0 0 0T
(23)
We easily obtain the conservative and actual cross-variances and variances E wai t wajT t of wai t are given respectively as pi
Qaij t 2ik ik t X i t ikT t ij i t Qij t jT t , Qai t Qaii t k 1
pi
Qaij t 2ik ik t X i t ikT t ij i t Qij t jT t , Qai t Qaii t
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k 1
(24) (25)
The conservative and actual cross-covariances and variances of vai t are given respectively as qi
Raij t 2ik H ik t X i t H ikT t ij Rvij t , Rai t Raii t k 1
qi
Raij t 2ik H ik t X i t H ikT t ij Rvij t , Rai t Raii t k 1
(26) (27)
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Applying (17) and (19), there are conservative and actual correlation matrices Ε wai t vajT t of fictitious noises wai t and vaj t respectively as (28)
Saij t i t Qij t D t , Sai t Saii t
(29)
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Saij t i t Qij t DTj t , Sai t Sai t T j
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Lemma 1 [5]. Let be the r r positive semi-definite matrix, i.e. 0 , then the following rL rL matrix is also positive semi-definite, i.e.,
0 rLrL
(30)
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Lemma 2 [5]. Let Ri be the mi mi positive semi-definite matrix, i.e. Ri 0 , then the following m0 m0 block-diagonal matrix R is also positive semi-definite, i.e.,
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with m0 m1
R diag R1 ,
RL 0
(31)
mL .
Lemma 3 The actual fictitious noise variances Qai t and Rai t have the conservative upper bounds
Qai t and Rai t , respectively, i.e., Qai t Qai t , Rai t Rai t , i 1,
,L
(32)
Proof. See Appendix A.
wai (t ) Defining augmented fictitious noises i (t ) , then their conservative and actual cross vai (t ) covariances are given as Qaij (t ) Saij (t ) Qaij (t ) Saij (t ) , ij (t ) T (33) ij (t ) T Saji (t ) Raij (t ) Saji (t ) Raij (t ) and we define conservative and actual variances as i (t ) ii (t ) and i (t ) ii (t ) , respectively. 8
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(35)
rg rg
where the ri rj matrices Qij t and Qij t are defined in (9) and (12), rg r1
i, j th block elements of
(34)
rL , and they are the
Qa t and Qa t , respectively. Then Qa t Qa t
(36)
Q(t ) Q(t ) , Qa (t ) Q(t ) Q(t ) rLrL
Q(t ) Qa (t ) Q(t )
Applying Q t Q t Q t yields Qa t Qa t , i.e., (36) holds. 4. Robust local time-varying Kalman predictor
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Proof. See Appendix C. Remark 1. For the special case with wi (t ) w(t ) , we have Qij t Q t , and
Q(t ) Q(t )
(37)
rL rL
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According to the minimax robust estimation principle[5], for the worst-case system (16) and (18) with the conservative upper bounds (maximum noise variances) Qai t and Rai t of noise variances to design the optimal (minimum variance) Kalman estimators will yield the minimax robust Kalman estimators. The conservative local minimum variance Kalman predictors are given as [17] xˆi t 1| t pi t xˆi t | t 1 K pi t yi t , i 1, , L (38)
pi t =i t K pi t Hi t
t Sai t Q t Q i t Hi t Pi t | t 1 HiT t Rai t and the conservative local prediction error variances Pi t 1| t satisfy the Riccati equations
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K pi t = i t Pi t | t 1 H
(39)
1 i
T i
(40) (41)
Pi t 1| t i t Pi t | t 1iT t i t Pi t | t 1 HiT t Sai t H i t Pi t | t 1 H i t Rai t
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T
i t Pi t | t 1 H iT t Sai t Qai t
with the conservative initial value
T
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xˆi 1| 0 0 , Pi 1| 0 P0
1
(42) (43)
Remark 2. In the conservative local Kalman predictor (38), the conservative measurement yi t is
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unavailable, which is yielded from the multi-model system (16) and (18) with the conservative upper bounds Qai t , Rai t and P0 . Only the actual measurement yi t is available, which is yielded from the actual system (16) and (18) with the actual variances Qai t , Rai t and P0 .Therefore, replacing the
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conservative measurement yi t with the actual measurement yi t , we define the local Kalman predictor (38) with the actual measurement yi t as the actual local Kalman predictor. We define the conservative local Kalman prediction error variances as xi t 1| t xi t 1
xˆi t 1| t , x t 1 is conservative state in (16), ˆx t 1| t is conservative Kalman predictor in (38). From (16), (18), (38) and(39), it is easy derived that the prediction error systems are given as (44) xi t 1| t pi (t ) xi t | t 1 I n , K pi t i (t ) with defining i t waiT t , vaiT t . So, applying (33) and (44) yields that the conservative and actual T
local prediction error cross-covariances respectively satisfy the Lyapunov equations In Pij t 1| t pi t Pij t | t 1 pjT t I n , K pi t ij (t ) T K pj t 9
(45)
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(46)
(47)
Pij 1| 0 P0
(48)
we define conservative and actual variances as Pi t 1| t Pii t 1| t and Pi t 1| t Pii t 1| t .
minimal upper bound Pi t 1| t ,i.e.,
CR IP T
Theorem 1. For the multi-model uncertain system (1)-(4) with Assumptions 1-3, each actual local time-varying one-step Kalman predictors (38) is robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding actual prediction error variances Pi t 1| t have the
Pi t 1| t Pi t 1| t , i 1, , L , t 0 We call the actual local Kalman predictors as the robust local Kalman predictors. Proof. See Appendix D. 5. Robust local time-varying Kalman filter and smoother
(49)
N
AN US
In the following, applying a unified approach [22] of designing the robust local Kalman estimators based on the robust local Kalman predictor, we shall give the robust local Kalman filters and smoothers. Theorem 2. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the actual Kalman filter ( N 0 ) and smoother ( N 0 ) are given as
xˆi t | t N xˆi t | t 1 Ki t | t k i t k , N 0 , i 1, k 0
,L
i t yi t Hi t xˆi t | t 1 with the smoother and filter gains
(50) (51)
M
k 1 Ki t | t k Pi t | t 1 piT t j H iT t k Qi1 t k , k 0 j 0 K fi t =Ki t | t =Pi t | t 1 HiT t Qi1 t
(52) (53)
and the conservative estimation error variances are given as N
ED
Pi t | t N Pi t | t 1 Ki t | t k Q i t k KiT t | t k
(54)
k 0
where xˆi t | t 1 , pi t , Q i t and Pi t | t 1 is given in (38)-(41), and xˆi t | t 1 is the actual Kalman
PT
predictor with the actual measurement yi t .
CE
The conservative and actual estimation error variances and cross-covariances are given as T N K wN j t vN Pij t | t N iN t Pij t | t 1 TjN t KiwN t , K i t ij (t + ) vNT 0 K j t T N K wN j t vN Pij t | t N iN t Pij t | t 1 TjN t KiwN t , K t ( t + ) vNT i ij 0 K j t
AC
Pi t | t N Pii t | t N , Pi t | t N Pii t | t N
(55)
(56) (57)
For N 0 , define that N
iN t I n Ki t | t k H i t k pi t k , t
(58)
pi t k , t pi t k 1 pi t , pi t , t I n
(59)
k 0
KiwN t KivN t
N
K t | t k H t k t k , t 1 , 0, i
k 1
i
pi
, N 1, KiNwN t 0
(60)
N
K t | t k H t k t k , t 1 K t K t | t ,
k 1
i
i
pi
pi
0, , N 1, KiNvN t Ki t | t N 10
i
(61)
ACCEPTED MANUSCRIPT For N 0 , define that Kiw00 t 0 , Kiv00 t Ki t | t and with the initial values Pij 1| 0 P0 ,
Pij 1| 0 P0 . Each actual Kalman filter ( N 0 ) or smoother ( N 0 ) is robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding estimation error variances Pi t | t N have the minimal upper Pi t | t N ,i.e., Pi t | t N Pi t | t N
(62)
The conservative and actual error variances are given as
CR IP T
We call the actual local Kalman filter or smoother as the robust local Kalman filter or smoother. Proof. See Appendix E. Corollary 1. Under the conditions of Theorem 2, for the case with N 0 , we have the robust Kalman filter as xˆi t | t xˆi t | t 1 K fi t i t (63) Pij t | t I n K fi t Hi t Pij t | t 1 I n K fj t H j t K fi t Raij t K Tfj t
(64)
Pij t | t I n K fi t Hi t Pij t | t 1 I n K fj t H j t K fi t Raij t K Tfj t
(65)
T
T
where K fi t is computed by (53).
AN US
Proof. See Appendix F. Corollary 2. Under the conditions of Theorem 2, the robust local Kalman estimators of the common state xc t are given as
xˆci t | t N Cxi xˆi t | t N , Cxi I n 0 , N 1, N 0, N 0 , i 1,
,L
(66)
and their conservative and actual covariances and variances are given respectively as Pcij t | t N Cxi Pij t | t N CxjT , Pcij t | t N Cxi Pij t | t N CxjT ,
M
Pci t | t N Pcii t | t N , Pci t | t N Pcii t | t N They are robust in the sense that Pci t | t N Pci t | t N , i 1, , L Proof. See Appendix G.
(67) (68)
ED
6. Robust weighted fusion time-varying Kalman estimators of the common state
PT
According the minimax robust estimation principle[7],based on the worst-case system (1)-(3) with the conservative upper bounds of uncertain noise and initial value variances, to design the minimum variance (optimal) weighted fusers will yield the robust weighted fusers, i.e., the minimax robust fusers. The four robust weighted fusion Kalman estimators ( N 1, N 0 ) for the common state xc t will
CE
be presented in this section. They have a unified form as [5-7] L
xˆc t | t N i t | t N xˆci t | t N , m, s, d ,CI
(69)
i 1
with the constraint of unbiasedness L
t | t N I
AC
i 1
i
(70)
n
where m, s, d ,CI denote the fusers with matrix, scalar, diagonal matrix weights and CI fuser matrix weights, respectively. xˆci t | t N are the local robust Kalman estimators of the common state. Under the minimum variance optimal fusion rule, the optimal weights are given as follows: The optimal weighted matrices are computed as [2,3]
1m t | t N , , Lm t | t N eT Pc1 t | t N e eT Pc1 t | t N with the definitions eT I n I n , and 1
Pc t | t N Pcij t | t N
(71)
(72)
nLnL
The conservative fused error variance with matrix weights is given as
Pcm t | t N eT Pc1 t | t N e
The optimal scalar weights are computed as
[2]
11
1
(73)
ACCEPTED MANUSCRIPT is t | t N i t | t N I n , i 1,
, L t | t N eT Ptrc1 t | t N e eT Ptrc1 t | t N 1
1 t | t N ,
with e 1 1 , and
(74)
,L
(75)
T
Ptrc t | t N trPcij t | t N
(76)
L L
where the notation tr denotes the trace of matrix. The conservative fused error variance with scalars weights is given as L
L
Pcs t | t N i t | t N j t | t N Pcij t | t N
(77)
i 1 j 1
The optimal diagonal matrix weights are obtained by [2] id t | t N diag i1 t | t N ,
, Li t | t N eT Pc
with e 1 1T , and
Pc
ii
ii
t | t N
1
e
1
CR IP T
1i t | t N ,
, in t | t N
eT Pc
ii
t | t N
t | t N Pcskii t | t N LL
ii
AN US
fused error variance with diagonal matrix weights is given as L
Pcd t | t N id t | t N Pc i 1 j 1
ii
, i 1,
,n
(79)
(80)
where Pcsk t | t N is the i, i th diagonal element of Pcsk t | t N s, k 1, L
1
(78)
, L . The conservative
t | t N jdT t | t N
(81)
The conservative CI fusion matrix weights are computed as [28]
1
iCI t | t N i t | t N PCI t | t N Pci t | t N , i 1,
,L
(82)
1
(83)
M
1 L PCI t | t N i t | t N Pci t | t N i 1 and the optimal weighting coefficients i t | t N are obtained by minimizing
1 L 1 tr i t | t N Pci t | t N (84) 0 i t |t N 1 i 1 1 t |t N L t |t N 1 This needs to solve a L dimensional nonlinear optimization problem, which can be solved by „fimincon‟ function in MATLAB toolbox. In the following, we shall give a unified form of the conservative and actual fused error variances for the above four weighted fusers (69), and we shall prove the robustness of the fusers. Define (85) t | t N 1 t | t N , , L t | t N , m, s, d ,CI
min
PT
ED
min tr PCI t | t N
CE
From (70) it follows that
L
xc t i t | t N xc t
(86)
i 1
AC
Subtracting (69) from (86) yields the fused errors L
xc t | t N i t | t N xci t | t N
(87)
i 1
which yields the conservative and actual fused error variances respectively as L
L
Pc t | t N i t | t N Pcij t | t N j t | t N , m, s, d ,CI
(88)
i 1 j 1 L
L
Pc t | t N i t | t N Pcij t | t N j t | t N , m, s, d ,CI
(89)
i 1 j 1
Theorem 3. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the four actual weighted fusers (69) ( N 1, N 0 ) have respectively the conservative and actual fused error variances as
Pc t | t N t | t N Pc t | t N T t | t N , m, s, d ,CI 12
(90)
ACCEPTED MANUSCRIPT Pc t | t N t | t N Pc t | t N T t | t N , m, s, d ,CI
(91)
with defining the global conservative and actual covariances of the common state xc t respectively as
Pc t | t N Pcij t | t N
Pc t | t N Pcij t | t N
(92)
nLnL
(93)
nL nL
and they are robust in the sense that for all admissible uncertain actual variances satisfying (8), the corresponding actual fused error variances Pc t | t N have the minimal upper bound Pc t | t N , i.e.,
CR IP T
Pc t | t N Pc t | t N , t 0, m, s, d ,CI (94) We call the actual Kalman fusers (69) as the robust Kalman fusers. Proof. See Appendix H. Remark 3. Taking m, s, d ,CI in (89) or (91), and applying (71), (75) and (79) yield the corresponding actual fused error variances as
Pcm t | t N eT Pc1 t | t N e eT Pc1 t | t N Pc t | t N Pc1 t | t N e e T Pc1 t | t N e (95) 1
1
L
L
Pcs t | t N i (t | t N ) j (t | t N ) Pcij (t | t N ) i 1 j 1
L
L
i 1 j 1
(97)
AN US
Pcd t | t N id t | t N Pcij (t | t N ) jdT t | t N
(96)
1 L L PcCI t | t N PCI t | t N i t | t N j t | t N Pci t | t N i 1 j 1
1 (98) Pcij t | t N Pc j t | t N PCI t | t N Remark 4. Taking CI , from (90) and (91) we can obtain the conservative and actual error variances PcCI t | t N and PcCI t | t N of the robust CI fuser. It was proved [5] that PCI t | t N defined in (83) is
M
an upper bound of PcCI t | t N , then we have
PcCI t | t N PCI t | t N
and from (94), P
(99)
t | t N is the minimal upper bound of P t | t N , then we have PcCI t | t N PcCI t | t N
(100)
PcCI t | t N PCI t | t N
(101)
CI c
ED
CI c
PT
and
t | t N is a minimal upper bound PcCI t | t N . CI
which means that P
CI c
conservative upper bound of P
t | t N comparing
with the
CE
Remark 5. The above robust fusion algorithms are implemented as follows: Step 1. Compute the local robust Kalman predictor xˆci t 1| t of the common state by (38)-(43), (66). Step 2. Compute the local robust Kalman filter and smoother xˆci t | t N by (50)-(54), (66).
AC
Step 3. Compute the conservative and actual cross-covariances and variances Pcij t | t N and
Pci t | t N N 1, N 0 of the local estimation errors by (45)-(48), (55)-(61), (63), (64) and (67).
Step 4. Compute the four robust weighted fused Kalman estimator xˆc t | t N of the common state by (69), (71)-(89). Above each step is implemented at each time. The computation complexities or the total numbers of multiplications and divisions per cycle for the robust local and fused estimation algorithms are analyzed as follows: The complexities of each local robust Kalman estimators are O ni3 mi3 [22] . It is known that the complexity of computing the inverse
of a n n matrix is O n3 . From (71)-(89) we see that complexity of computing the optimal
matrix-weights is O nL
3
,
while the complexities of computing the optimal scalar-weights and 13
ACCEPTED MANUSCRIPT
diagonal matrix-weights are O L3 , and the complexity of computing the CI matrix-weights is O n3 plus the complexity of computing the optimal weighting coefficients i t | t N by MATLAB toolbox. We see that when the number L of sensors is larger, the fuser with scalar or diagonal matrix-weights can significantly reduce the computational burden compared with the fuser with the matrix-weights. 7. Accuracy analysis
CR IP T
From (68) and (94), according to the accuracy definitions in [5-7], the trace of the estimation error variance matrix is defined as the accuracy of a estimator. The smaller trace means the high accuracy. For the robust local or fused estimator, its actual accuracy is defined as the trace of its actual error variance matrix, while its robust accuracy is defined as the trace of the minimal upper bound of its actual error variances. Theorem 4. For the multi-model system (1)-(4) with uncertain-variance linearly correlated white noises, and with Assumptions1-3, the accuracy relations among the robust local and fused robust Kalman fusers are given as tr Pc t | t N tr Pc t | t N , 1, , L, m, s, d ,CI , N 1, N 0 (102)
trPcm t | t N trPcCI t | t N trPCI t | t N trPci t | t N , i 1,
,L
(103)
trP t | t N trP t | t N trP t | t N trP t | t N , i 1, , L (104) Proof. See Appendix I. Remark 6. The robust accuracy is only determined from the conservative upper bounds of the uncertain noise variances and initial state variances, while the actual accuracies are related to actual noise variances and initial state variances. The accuracy relations (102) show that the actual accuracies of each local or fused robust estimator are higher than its robust accuracy. From (103) and (104), the robust accuracy of the fuser with matrix weights is higher than those of the CI fuser and the fuser with diagonal matrix weights, and the robust accuracy of the fuser with diagonal matrix weights is higher than that of the fuser with scalar weights. The robust accuracies of all fusers are higher than that of each local estimator. Remark 7. Here the conservative robust accuracy of the original robust CI fuser is defined as trPCI t | t N , which has larger conservativeness, while the modified robust accuracy in this paper is d c
s c
i c
AN US
m c
M
defined as trPcCI t | t N , which is the minimal upper bound of all admissible trPcCI t | t N . From (103)
ED
the modified robust accuracy is higher than the original conservative robust accuracy. Comparing (83) with (90) and (92) with CI , we see that PCI t | t N only contains the conservative local estimation error variances, while PcCI t | t N not only contains the conservative local estimation error variances,
PT
but also contains information of the conservative cross-covariances. This is an important way for increasing robust accuracy of the original CI fuser. 8. The local and fused steady-state Kalman estimators and convergence analysis
AC
CE
Now we will present the robust steady-state Kalman estimators based on the robust time-varying Kalman estimators by means of considering their limiting case, and will prove the convergence between robust time-varying and steady-state Kalman estimators. Consider the multi-model uncertain time-invariant multisensor system (1)-(4) with Assumptions 1-3, and with constant parameter matrices and constant noise variances. In order to ensure the existence of robust steady-state Kalman estimators, introducing matrices ai as pi
2 2 ai i i 2 ik ik , ik ik , i 1, 2, k 1
ik
,L
(105)
where the notation denote Kronecher product. According to [28] , assuming that the spectrum radius of ai is less than 1, i.e., ai 1 , then the solutions X i t and X i t of the time-varying generalized Lyapunov equations (20) and (22) with arbitrary initial values X i 0 0 and X i 0 0 converge respectively to the unique positive semi-definite solutions X i 0 and X i 0 for the corresponding steady-state generalized Lyapunov equations pi
X i i X i iT 2ik ik X i ikT i Qi iT , k 1
14
ACCEPTED MANUSCRIPT pi
X i i X i iT 2ik ik X i ikT i Qi iT
(106)
k 1
lim X i t X i ,lim X i t X i t
(107)
t
As t , taking the limiting operations to (24)-(28) and (33), replacing X i t and X i t by X i and X i respectively yields the constant noise statistics Qai and Qai , Rai and Rai , S ai and S ai , i and i , and
Qai t Qai , Qai t Qai , Rai t Rai , Rai t Rai , Sai t Sai , Sai t Sai , i t i and
i t i . In the local robust time-varying Kalman predictor (38)-(43), replacing pi t , K pi t and
CR IP T
Pi t | t 1 by their limits pi , K pi and Pi 1 yields the local robust steady-state Kalman predictor as follows. Theorem 5. For multi-model time-invariant multisensor system (1)-(4) with Assumptions 1-3, and ai 1 , assume that for the equivalent system (16) and (18) with constant noise statistics Qai , Rai and S ai , i , H i is a completely detectable pair, and i Sai Rai1 Hi , Gi with Gi GiT Qai Sai Rai1SaiT is a completely stabilizable pair, then there exists the local actual steady-state Kalman predictors as xˆis t 1| t pi xˆis t | t 1 K pi yi t , i 1, , L
pi =i K pi Hi
(108) (109)
1
AN US
K pi = i Pi 1 HiT Sai Qi1
(110)
Q i Hi Pi 1 HiT Rai
(111)
where pi are stable, i.e., the predictors (108) are asymptotically stable. The superscript s denotes “steady-state”. The local conservative steady-state one-step prediction error variances Pi 1 satisfy the steady-state Riccati equations
1
M
(112) Pi 1 i Pi 1iT i Pi 1 HiT Sai Hi Pi 1 HiT Rai i Pi 1 HiT Sai Qai and the conservative and actual steady-state one-step prediction error variances Pi 1 and Pi 1 also T
respectively satisfy the Lyapunov equations
Pi 1 p iPi 1 T
T
I , n K p i Ii , nK
pi
ED
Pi 1 pi Pi 1 piT I n , K pi i I n , K pi
T
pi
(113) (114)
PT
The local actual steady-state one-step Kalman predictors (108) are robust in the sense that for all admissible uncertain actual variances satisfying (8), it follows that Pi 1 Pi 1 , i 1, , L (115) and Pi 1 is the minimal upper bound of Pi 1 .
CE
The solution Pi t | t 1 of the local time-varying Riccati equation (42) with the time-varying noise statistics Qai t , Rai t and Sai t converges to the solution Pi 1 of the local steady-state Riccati
AC
equation (112) with constant noise statistics Qai , Rai and S ai .i.e.,
Pi t | t 1 Pi 1 , as t , i 1,
,L
(116)
If the measurement data yi t are bounded, then the local robust time-varying Kalman predictor of
time-varying system with time-varying noise statistics (38) and the local robust steady-state Kalman predictor (108) of time-invariant system with constant noise statistics have the convergence in a realization (117) xˆi t 1| t xˆis t 1| t 0 , as t , i.a.r. where the notation “ i.a.r.” denotes the convergence in a realization [30]. Proof. By the dynamic error system analysis (DESA) method [30] , the proof is completely similar to one of Theorem 5 in [22], the details are omitted. In the local robust time-varying Kalman filter and smoother (50)-(61), replacing Pi t | t 1 and
pi t by their limits Pi 1 and pi , respectively, we obtain the local robust steady-state Kalman filters 15
ACCEPTED MANUSCRIPT and smoothers xˆis t | t N N 0 and their conservative and actual steady-state variances and crosscovariances Pi N , Pi N , Pij N and Pij N . Using the dynamic variance error system analysis (DVESA) method [5], similar to the proof in Theorem 7 in [5], we can prove the convergence Pij t | t N Pij N , Pij t | t N Pij N , as t , N 1, N 0
(118)
and applying Theorem 5, we easily prove the convergence between the local time-varying and steady-state Kalman filter and smoother (119) xˆi t | t N xˆis t | t N 0 , as t , i.a.r. , i 1, , L , N 0 From the robust weighted fusion time-varying Kalman estimators xˆc t | t N given by (69)-(94) for
the common state xc t , we can obtain the corresponding steady-state Kalman fusers as L
i 1
(120)
CR IP T
xˆc s t | t N i N xˆcis t | t N , m, s, d ,CI , N 1, N 0
i t | t N i N
(121)
Theorem 6. Under the assumption conditions of Theorem 5, the robust time-varying and steady-state Kalman fusers of the common state xc t have the convergence in a realization as xˆc t | t N xˆc s t | t N 0 , as t , i.a.r.
(122) (123)
and the robust steady Kalman fuser (120) have the robustness in the sense that Pc N Pc N
(124)
AN US
and their estimation error variances have the convergence Pc t | t N Pc N , Pc t | t N Pc N , as t
and Pc N is the minimal upper bound of Pc N .
Proof. The proof is similar to those as shown in [5, 22], which is omitted. Remark 8. As t , taking the limiting operations to Theorems 3 and 4, the conservative and actual fused steady-state error variances are given as
M
Pc N N Pc N T N , Pc N N Pc N T N , m, s, d ,CI
(125)
ED
and the local and fused steady-state robust Kalman estimators have the robust and actual accuracy relations tr Pc N tr Pc N , 1, , L, m, s, d ,CI , N 1, N 0 (126) tr Pcm N tr PcCI N tr PCI N tr Pci N , i 1,
,L
(127)
tr P N tr P N tr P N tr P N , i 1, , L (128) Remark 9. Compared with the references[5-7], in the derivations of main results, there exist two difficulties as follows: one is the proof of robustness for local and fused robust Kalman estimators. In [5-7], the process and measurement white noises are assumed to be mutually uncorrelated, the robustness can simply be proved by the Lyapunov equation approach. While in this paper, they are assumed to be linearly correlated [21,22], the robustness can also be proved by the Lyapunov equation approach ( See Appendix H: the proof of Theorem 3). However, so far, under arbitrary correlation between the process and measurement white noise, the proof of robustness was not solved which is an open difficult problem. The other difficulty is the proof of convergence between the time-varying and steady-state robust Kalman estimators. Here the time-varying robust Kalman estimators are obtained from the time-varying system with time-varying noise statistics, while the steady-state robust Kalman estimators are obtained from the time-invariant system with constant noise statistics. The convergence between both of them is essentially different from that between the time-varying and steady-state robust Kalman estimators for the time-invariant system with constant noise statistics in the classical Kalman filtering theory as shown in the references [5-7, 17]. This difficulty was solved by the convergence of generalized Lyapunov equation [28] , the convergence of Riccati equation, and the dynamic error system analysis (DESA) method [21]. d c
s c
i c
AC
CE
PT
m c
9.Simulation examples Example 1 Consider robust fused steady-state Kalman smoother for the AR signal with 3-sensor, colored and white measurement noises, and uncertain noise variances and stochastic parameters At q 1 s t w t 1
(129)
yi t s t bi t i t , i 1, 2,3
(130)
16
ACCEPTED MANUSCRIPT bi t Bit q 1 ei t At q
1 a t 1 q a t 2 q B q 1 b t q b t q
1
1
1
, a j t a j j t , j 1, 2
(132)
2
, bij t bij ij t , j 1, 2
(133)
2
1
1
it
i1
i2
(131)
2
where yi t is the measurement, bi t is the colored measurement noise, which obeys the moving average (MA) model (131). wi t , ei t and i t are white noises. q 1 is the backward shift operator
q 1s t s t 1 . The a j and bij are known means of stochastic parameters a j t and bij t , respectively, where white noises j t and ij t are the corresponding random perturbations. All the white noises are mutually uncorrelated, and have zero means and uncertain variances satisfying Asumptions 1 and 3. The AR signal s t given by (129) has the state space model with stochastic parameter matrix as (134)
s t H c xc t
(135)
CR IP T
xc t 1 t xc t c w t
a1 t 1 1 , c , H c 1 0 . 0 a2 t 0 where xc t is the common state. s t is the first component of xc t .
t
(137)
AN US
The MA colored noises (131) has the state space models i t 1 Bi i t b ei t
(136)
bi t Hbi t i t ei t with the definitions
(138)
0 0 1 ,b (139) 1 0 0 Hence we obtain the augmented state space model with multi-models (different local models) and with linearly correlated additive noises as xi t 1 ai t xi t i wi t , i 1, 2,3 (140)
M
Hbi t bi1 t bi 2 t , Bi
yi t H ai t xi t vi t
(141)
vi t Di wi t i t , Di 0 1
ED
(142)
where xi t is augmented states, and wi t is augmented noises, with the definitions
PT
x t w t t 0 c 0 xi t c , wi t , ai t ,i , H ai t H c , H bi t (143) Bi 0 b 0 i t ei t Substituting the random parameters a j t and bij t given in (132) and (133) into the augmented
CE
system (140) and (141) yields the multi-model multisensor system (1)-(4) with multiplicative noises ik t and ik t , and with pi 2 , qi 2 , i 1, 2,3 , L 3 and ik t k t , k 1, 2 ,
a1 a i t 2 0 0
1 0 0 1 0 0 0 0 , i1 t 0 0 0 0 0 1 0 0
0 0 0 0 0 0 , 0 0 0 0 0 0 Hi t 1 0 bi1 bi 2 , Hi1 t 0 0 1 0 , Hi 2 t 0 0 0 1 .
AC
0 0 0 0 1 0 0 0 , i 2 t 0 0 0 0 0 0 0 0
(144)
In simulation, we take Q 1.5 , Q 1 ; Qe1 0.012, Qe2 =0.02, Qe3 0.03 , Qei 0.8Qei , i 1, 2,3 ;
R1 0.2 , R 2 0.1 , R 3 0.3 , R1 0.1 , R 2 0.05 , R 3 0.2 ; 21 0.09, 22 =0.04 ; 21 0.08,
2 =0.03 ; 2 0.11 , 2 0.31 , 2 0.21 , 2 0.41 , 2 0.31 , 2 0.18 ; 2 0.1 , 2 0.3 , 2
11
12
21
22
31
32
11
12
2 0.2 , 2 0.4 , 2 0.3 , 2 0.17 ; a1 0.15 , a2 0.3 , b11 0.2 , b12 0.1 , b21 0.4 , 21
22
31
32
b22 0.2 , b31 0.3 , b32 0.3 .
From (135) and (136), the signal s t is the first component xc t , so that the robust filtering problem 17
ACCEPTED MANUSCRIPT of AR signal s t is converted into one of the common state xc t . From (135) applying the projection theory [17], we have the local robust steady-state signal estimators as sˆi t | t N Hc xˆci t | t N and ij T their cross-covariances of the estimation errors as Psij N H c Pc N H c , further, applying the weighted
fusion algorithms in Section 6, we can obtain the local and fused estimators and their conservative and actual error variances. The comparison of robust and actual accuracies for local and fused steady-state Kalman smoothers of signal s t with N 2 are shown in Table 1. We see that the actual accuracy of each local or fused Kalman smoother is higher than its robust accuracy. The robust accuracy of the fuser with scalar weights is higher than that of the modified CI fuser. The robust accuracy of the modified CI fuser is higher than that of the original CI fuser.
s
CI
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Table 1 The accuracy comparison of local and fused robust steady-state Kalman smoothers N 2 for s t *
Ps1 (2)
Ps 2 (2)
Ps 3 (2)
Ps (2)
Ps (2)
PCI (2)
0.3063
0.2493
0.4439
0.0710
0.0772
0.1622
Ps1 (2)
Ps 2 (2)
Ps 3 (2)
Ps (2)
Ps (2)
0.1678
0.1471
0.2942
0.0424
0.0442
s
CI
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Fig.1 shows the curves of signal s t and its robust CI fusion estimator sˆCI t | t N N 1, N 0,
N 2 . The solid curves denote the actual signal and the dot curves denote the robust CI fused estimates. We see the CI fuser has the good tracking performance.
M
0
ED
s t , sˆCI t | t 1
2.5
-2.5
t/step
20
40
60
(a) The signal s t and robust CI fused predictor sˆCI t | t 1
AC
s t , sˆCI t | t 2
CE
PT
2.5
0
-2.5
t/step
20
40
(b) The signal s t and robust CI fused filter sˆCI t | t
18
60
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s t , sˆCI t | t
2.5
0
-2.5
t/step
20
40
60
(c) The signal s t and robust CI fused smoother sˆCI t | t 2
s t
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Fig.1. The signal s t and robust steady-state CI fused estimators sˆCI t | t N , N 1, N 0, N 2 ; sˆCI t | t N
The robust and actual standard deviations of the CI fuser sˆCI t | t N ( N 1, N 0, N 2 ) are
N PsCI N and N PsCI N , respectively. The robustness (124) yields N N , which is called the robustness of robust signal fuser. The corresponding estimation error curves, 3 N and 3 N bounds
4 2 0 -2 -4
0
100
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are shown in Fig.2, where the CI fused estimation error curves are denoted by the solid lines, 3 N bounds are denoted by the dashed lines, 3 N bounds are denoted by the dotted lines.
t/step
200
300
M
(a) The actual prediction errors and 3 -actual and robust standard deviations of robust CI fused predictor 1 0.5
-0.5 -1 0
ED
0
100
t/step
200
300
PT
(b) The actual filtering errors and 3 -actual and robust standard deviations of robust CI fused filter 1 0.5 0
CE
-0.5 -1
0
100
t/step
200
300
(c) The actual smoothing errors and 3 -actual and robust standard deviations of robust CI fused smoother
Fig.2. The actual estimation errors and 3 -actual and robust standard deviations of robust steady-state CI fusers
AC
From Fig.2, we can see that over 99 percent of the fused estimation error values lie between 3 N
and 3 N .They also lie between 3 N and 3 N . This verifies the robustness, and the correctness of actual fused variances. [31] Example 2 Consider an uninterruptible power system (UPS) with 3-sensor and with random
parameters and color measurement noises
xc t 1 t xc t c w t
(145)
yi t Hc xc t bi t i t , i 1, 2,3
(146)
0.5 0.9622 1 t 0.633 2 t 0 , 0 , H 23.738 20.287 0 t 1 0 0 c c 0.2 0 1 0 bi t Bit q 1 ei t 19
(147) (148)
ACCEPTED MANUSCRIPT B1t q 1 1 b11 t q 1 ,
B2t q 1 1 b21 t q 1 b22 t q 2 ,
B3t q 1 1 b31 t q 1 b32 t q 2 b33 t q 3 ,
bij t bij ij t .
(149)
where xc t is the common state, yi t is the measurement, random measurement bias bi t is the colored noise which obeys the moving average (MA) model (148). q 1 is the backward shift operator
q1s t s t 1 . The bij are known means of random parameters bij t , while i t and ij t are random perturbations with zero mean and uncertain variances 2i and 2ij , respectively. i t is white
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measurement noise. The white noises w t , ei t , i t , i t and ij t satisfy Assumptions 1 and 3. Similar to example 1, bi t i 1, 2,3 ,can be represented by the state-space models. Applying the augmented state method, we can obtain the multi-model multisensor system (1)-(4) with pi 2 , q1 1 ,
q2 2 , q3 3 , L 3 and ik t k t , k 1, 2 .
In simulation, we take Q 1.5 , Q 1 ; Qe1 0.012, Qe2 =0.02, Qe3 0.03 , Qei 0.8Qei , i 1, 2,3 ;
R1 0.2 , R 2 0.4 , R 3 0.3 , R1 0.1 , R 2 0.3 , R 3 0.2 ; 21 0.13, 22 =0.06 ; 21 0.12,
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2 =0.05 ; 211 0.11 , 221 0.21 , 222 0.41 , 231 0.31 , 232 0.18 , 2 0.13 ; 211 0.1 , 2
33
0.2 , 0.04 , 0.3 , 0.17 , 0.12 ; 0.1 , 0.2 , 2 0.04 , 2
2
2
21
2
31
22
2
2
32
11
33
2
21
22
2 0.3 , 2 0.17 , 2 0.12 ; b11 0.01 , b21 0.03 , b22 0.02 , b31 0.03 , b32 0.02 , 31
32
33
ED
M
b31 0.03 . The robust and actual accuracy comparisons of local and fused time-varying Kalman smoothers for the common state xc t with N 2 are shown in Table 2, which verifies the accuracy relations (102)-(104).
Table 2
The accuracy comparison of local and fused robust time-varying Kalman smoothers for xc t with N 2 at t 200 tr Pc t | t 2 tr Pc t | t 2 tr Pc t | t 2 tr Pc 2
0.0454 1 tr Pc
t | t 2
0.0783
2 tr Pc
t | t 2
0.0567
m
0.1265
3 tr Pc
t | t 2
0.0948
t | t 2
tr Pc t | t 2 tr Pc t | t 2 tr Pc d
0.02156 tr
m Pc
t | t 2
s
0.02169 tr
d Pc
0.01369
t | t 2
0.01384
AC
CE
0.0294
3
PT
1
20
0.02174 tr Pc t | t 2 s
0.01390
CI
t | t 2
0.0232 CI tr Pc
t | t 2
0.0145
tr PCI t | t 2 *
0.0351
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m
x 1 t , xˆc1 t | t 2
4
0
-4
20
t/step
40
60
(a) The component x1 t and its robust fused smoother xˆcm1 t | t 2
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0
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m
x 2 t , xˆc 2 t | t 2
2
-2
20
t/step
40
60
(b) The component x2 t and its robust fused smoother xˆcm2 t | t 2
M
0
ED
m
x 3 t , xˆc 3 t | t 2
2.5
-2.5
20
t/step
40
60
(c) The component x3 t and its robust fused smoother xˆcm3 t | t 2
PT
Fig.3. The common state xc t and robust time-varying fused smoother weighted by matrices xˆcm t | t 2
Fig.3 shows the curves of the components for the common state xc t x1 t , x2 t , x3 t and the T
robust fused smoother weighted by matrix xˆcm t | t 2 xcm1 t | t 2 , xcm2 t | t 2 , xcm3 t | t 2 . The
CE
T
AC
solid curves denote the actual states and the dot curves denote the robust fused smoothing estimates. We see that the fuser has the good tracking performance. Taking 300 Monte Carlo simulation runs, the mean square error (MSE) curves MSE t ,
1, 2,3, m, s, d ,CI of the time-varying local and four weighted fusion robust smoothers xˆc t | t 2 for
the common state xc t are shown in Fig.4, where the solid curves indicate the values of trPc t | t 2 . We see that when t is sufficiently large, the traces trPc t | t 2 of actual time-varying estimation error
variances converge to the corresponding steady-state values trPc 2 , and when is sufficiently large ( 300 ), the values of MSE t are closed to the corresponding values of trPc t | t 2 . This verifies the convergence trPc t | t 2 trPc 2 given in (123), and the consistency of sampled variances.
21
ACCEPTED MANUSCRIPT 0.15 MSE1
MSEs
MSE2
MSECI
MSE3
MSEd
MSEm
MSE
0.1
0
0
20
40
t/step
60
80
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0.05
100
Fig.4. The MSE curves of the time-varying local and fused robust Kalman smoothers xˆc t | t 2 , i, m, s, d ,CI
10. Conclusions
AC
CE
PT
ED
M
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For multi-model multisensor time-varying uncertain systems with both the uncertain-variance multiplicative and linearly correlated additive white noises, the fictitious noise-based Lyapunov equation approach has been presented. Its principle is that by introducing the fictitious noise to compensate multiplicative noises, the original system can be converted into one with only uncertain additive noise variances, and then, according to the minimax robust estimation principle, based on the worst-case system with conservative upper bounds of noise variances, applying the Lyapunov equation approach, the minimax robust local and four weighted fusion time-varying Kalman estimators (predictor, filter and smoother) have been presented in a unified framework. They include the three fusers weighted by matrices, scalar, diagonal matrices and a modified CI fuser. The corresponding robust steady-state Kalman fusers were also presented. The robustness and accuracy relations were proved. The convergence between the robust time-varying and steady-state Kalman fusers have been proved by the dynamic error system analysis (DESA) method. They constitute a unified and universal multi-model multisensor robust fusion Kalman filtering theory. They extended the optimal fusion Kalman filtering for multi-model systems [2,24,25] with known model parameters and noise variances to the multi-model uncertain systems. The robust fused Kalman filtering theory [5-7] only with uncertain additive noise variances but known model parameters have been extended to the multi-model systems with uncertain multiplicative noise variances. The optimal robust Kalman filtering for systems with multiplicative noises (stochastic parameters) but known multiplicative and additive noise variances [14-20] has been developed to the minimax robust fusion Kalman filtering for multi-model multisensor with uncertain multiplicative and additive noise variances. The robust Kalman fusers [21-22] for multisensor systems with known multiplicative noise variances but unknown additive noise variances have been extended to the multi-model multisensor with both the uncertain multiplicative and additive noise variances. The proposed fictitious noise-based Lyapunov equation approach is different from the fictitious noise-based game-theoretic approach[13] and the LMI approach [23]. The future works include the robust Kalman filtering for the multi-model multisensor systems with mixed uncertainties including multiplicative noises, random sensor delays, packet dropouts, and uncertain noise variances, and so on. It is remarkable that in discrete-time Markovian jump systems [32-34], we also meet “multi-model” concept in the sense that at the sequential times, the corresponding system models occur to jump in a finite multi-model set with the transition probabilities described by Markov chain, while “multisensor multi-model” concept in this paper means that each sensor subsystem has different local dynamic model containing a common system state. Recently, the robust filtering problems for Markovian jump systems with uncertain additive noise variances [32], or with the norm-bounded parametric uncertainties were considered [33]. The proposed methods may be applied to Markovian jump systems with uncertain-variance multiplicative and additive noises.
22
ACCEPTED MANUSCRIPT Acknowledgment This work is supported by the National Natural Science Foundation of China under grants NSFC-60874063, NSFC-60374026 and Postgraduate Innovation Project of Heilongjiang Province (YJSCX2015-002HLJU). The authors thank the reviewers and editors for their helpful and constructive comments, which are very valuable for improving quality of the paper. Appendix A. The proof of Lemma 3 Proof. Setting Qai t Qai t Qai t , Qi t Qi t Qi t , X i t X i t X i t and 2ik
2 2 , subtracting (25) from (24) yields ik
ik
pi
pi
Qai t 2 ik X i t ikT 2 ik X i ikT i t Qi t iT t ik
k 1
from (9) we have Q t Q t
Qi t
0
From (8) and Lemma 2, we have
Qei t Qei t ri ri 0
Qi t 0
(150)
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ik
k 1
(151)
(152)
(20) yields the Lyapunov equation pi
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From (20) and (21), applying X i 0 0 , iterating (20) yields X i t 0 , t 0 . Subtracting (22) from pi
X i t 1 i t X i t iT t 2 ik t X i t ikT t 2 ik t X i t ikT t ik
k 1
i t Qi t
k 1
T i
ik
t
(153)
From (7), (8), (21) and (23), we have ik t 0 and X i 0 Pi Pi 0 . Applying (152) we have 2
M
i t Qi t iT t 0 . By the induction, from (153) for arbitrary t 0 , we obtain X i t 0
(154)
So, from (150), (152) and (154), we have Qai t 0 , i.e., Qai t Qai t
(155)
ED
Subtracting (27) from (26) yields
Rai t 2 H ik t X i t H ikT t 2 H ik t X i t H ikT t Rvi t qi
k 1
ik
ik
(156)
PT
Setting Rvi t Rvi t Rvi t and R t R t R t , from (11), we have i
i
i
Rvi t Di t Qi t DiT t R i t
(157)
Rvi t 0
(158)
CE
Applying (8), (152), we obtain
Similarly, from (154) and (156), we have Rai t 0 , i.e.,
AC
Rai t Rai t
The proof is completed. Appendix B. The proof of Lemma 4 Proof. Defining i (t ) i (t ) i (t ) , from (11), (25)-(28) and (33)
23
(159)
ACCEPTED MANUSCRIPT
i (t )
can
be
decomposed
i t Qi t DiT t qi 2 T ik H ik t X i t H ik t k 1 2 T ik H ik t X i t H ik t T Di t Qi t Di t R i t
i1 (t )
as
(160)
,
i 2 (t )
and
i3 (t )
,
i.e.,
where
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pi 2 T ik ik t X i t ik t k 1 2ik ik t X i t ikT t i t Qi t iT t i (t ) Di t Qi t iT t
2 T ik ik t X i t ik t 0 k 1 2ik ik t X i t ikT t 0 , 3 (t ) 0 i1 (t ) 0 R t , i qi i 2ik H ik t X i t H ikT t k 1 0 2 T ik H ik t X i t H ik t T T t Qi t i t i t Qi t Di t i 2 (t ) i . T T Di t Qi t i t Di t Qi t Di t pi
pi
It
is
obvious
that
t X t t 0 2
k 1
qi
H ik t X i t H ikT t 0 and ik
ik
ik
T ik
i
qi
2 ik
k 1
pi
,
t X t t 0 2
k 1
ik
ik
i
T ik
i1 (t ) 0
(161)
i t
ED
i 2 (t ) can be rewritten i 2 (t ) i t Qi t i T t where i t Q t Qi t Qi t i . Applying (152) and Lemma 1, we have Qi t Qi t
PT
,
H ik t X i t H ikT t 0 . Applying Lemma 2, we have
M
2
k 1
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0
0 Di t
and
i 2 (t ) 0
(162)
i3 (t ) 0
(163)
i (t ) 0
(164)
CE
Applying (8) and Lemma 2 yields So, we have
AC
i.e., (34) holds. The proof is completed. Appendix C. The proof of Lemma 5 Proof. Define Qa t Qa t Qa t , Qij t Qij t Qij t , and applying (9) and (12) yields 0 Q t Qij t 0 Q t ri rj ei ij
and we have with rg r1
Qa t Qij t
rg rg
(165)
(166)
rL . Decompose Qij t as
Qij t Qij* t Qeij* t with ri r rei , and 24
(167)
ACCEPTED MANUSCRIPT 0 Q t 0 0 * Qij* t ，Qeij t 0 0 Qei t ij ri rj 0 ri rj
(168)
Defining Q* t Qij* t
from (166) and (167) we have
rg rg
* , Qe* t Qeij t
rg rg
Qa t Q* t Qe* t
(169) (170)
Applying (168) yields
From (6) and (8), then
, Qe*LL t
Qe* t 0
(171)
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Qe* t diag Qe*11 t ,
(172)
* Hence, from (170) and (172), in order to prove Qa t 0 , we need to prove Q t 0 . * * Interchanging rows and columns of Q t , then Q t can be transformed as
Q(t )
0 Q(t ) 0 rg rg
(173)
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Q(t ) * Q0 t Q(t )
0
* * T i.e., there exists a non-singular elementary trans formation matrix Te such that Q0 t Te Q t Te .
From (8), Q t 0 , applying Lemma 1 yields that
M
Q(t ) Q(t )
Q(t ) 0 Q(t )
(174)
* * * Hence from (173) we have Q0 t 0 . Applying Q0 t 0 yields Q t 0 , and applying (170) and
ED
(172) yields Qa t 0 . The proof is completed. Appendix D. The proof of Theorem 1
Proof. Setting Pi t 1| t Pi t 1| t Pi t 1| t , from (45) and (46) yields the Lyapunov equation
PT
Pi t 1| t pi t Pi t 1| t piT t i t
(175)
where i t I n , K pi t i t I n , K pi t . From (164), we have i t 0 . Subtracting (48)
CE
from (47), we have the initial value
T
Pi 1| 0 P0 P0 0
(176)
By the induction, from (175) and (176) we recursively obtain Pi t 1| t 0 for arbitrary t 0 .i.e., (49)
AC
holds. This means that Pi t 1| t is a upper bound of Pi t 1| t . In (8), selecting the actual variances to equal the corresponding conservative upper bounds, applying (47), (48) and (175) yields Pi t 1| t 0 for arbitrary t 0 , i.e., Pi t | t 1 Pi t | t 1 . For arbitrary other upper bound
Pi* t 1| t of Pi t 1| t , we have Pi t 1| t Pi t 1| t Pi* t 1| t for arbitrary t 0 . This means
that Pi t 1| t is a minimal upper bound of Pi t 1| t . The proof is completed. Appendix E. The proof of Theorem 2 Proof. From (50) and (51), we can obtain the error xi t | t N xi t | t N xˆi t | t N as [22] N
N
0
0
vN xi t | t N iN t xi t | t 1 KiwN t wai t Ki t vai t
where iN t , Ki
t and KivN t are defined in (58)-(61). For N 0 , subtracting (56) from (55) Pi t | t N Pii t | t N Pi t | t N Pi t | t N iN t Pi t | t 1 iNT t i t wN
25
(177) yields (178)
ACCEPTED MANUSCRIPT T N KiwN t vN t , K t ( t ) where i t KiwN . Applying (164) and (49) to (178) yields i vNT i 0 Ki t
Pi t | t N 0 , i.e., the robustness (62) holds for N 0 , which implies that Pi t | t N is a upper
bound of Pi t | t N . Similar to the proof given in Theorem 1, we easily prove that Pi t | t N is the minimal upper bound of Pi t | t N . The proof is completed. Appendix F. The proof of Corollary 1 Proof. Taking N 0 in (50) and applying (53) yields (63)Applying (16), (18), (51) and (63), yields the filtering errors (179) xi t | t I n K fi t Hi t xi t | t 1 K fi t vai t
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Applying (179) directly yields (64) and (65). The proof is completed. Appendix G. The proof of Corollary 2 Proof. Taking the projection operation to (5) yields (66). Subtracting (66) from (5) yields
xci t | t N Cxi xi t | t N
(180)
which yields (67), and applying (62) yields (68). The proof is completed. Appendix H. The proof of Theorem 3
Proof. From (85), (87)-(89) we obtain (90) and (91). Setting Pc t | t N Pc t | t N
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Pc t| t N , Pc t | t N Pc t | t N Pc t | t N , and subtracting (91) from (90), it follows that Pc t | t N t | t N Pc t | t N T t | t N Denote ng n1
(181)
nL and defining the global conservative and actual covariances respectively as P t | t N Pij t | t N
(182)
ng ng
P t | t N Pij t | t N
(183)
ng ng
applying (67), (68), (92), (93), (182) and (183) yields
(184)
Pc t | t N CP t | t N C
(185)
M
Pc t | t N CP t | t N C T 0 CxL nLn g
ED
Cx1 C 0
T
(186)
PT
Denoting P t | t N P t | t N P t | t N , and subtracting (185) from (184), we have
Pc t | t N CP t | t N C T
(187)
CE
From (181) and (187), in order to prove Pc t | t N 0 , it is sufficient that to prove
P t | t N 0 .
Taking N 1 , from (45) and (46) we have that P t 1| t and P t 1| t satisfy the global Lyapunov
AC
equation respectively
P t 1| t p t P t | t 1 pT t K p t p t K pT t
(188)
P t 1| t p t P t | t 1
(189)
with the definitions
p t diag p t , 1
T p
t K p t p t K t
T p
, pL t , K p t diag I n , K p1 t ,
11 t p t L1 t with the initial values
1L t
11 t , p t L1 t LL t
26
, I n , K pL t 1L t LL t
(190)
ACCEPTED MANUSCRIPT P0 P0 , P 1| 0 P0 P0
P0 P 1| 0 P0
P0 P0
(191)
Setting P t 1| t P t 1| t P t 1| t and subtracting (189) from (188) yields Lyapunov equations
P t 1| t p t P t | t 1 pT t K p t p t K pT t
(192)
where p t =p t p t . Setting ij (t ) ij (t ) ij (t ) , applying (24)-(29), and (33) yields
i t Qij t D Tj t qi 2ik H ik t X i t H ikT t ij k 1 qi 2ik H ik t X i t H ikT t ij k 1 T Di t Qij t D j t R i t ij
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pi 2 T ik ik t X i t ik t ij k 1 pi 2 T ik ik t X i t ik t ij k 1 t Q t T t i ij j ij (t ) Di t Qij t Tj t Define
ED
M
pi 2 T ik ik t X i t ik t k 1 0 pi 2 T ik ik t X i t ik t k 1 , i1 t ii1 t qi 2ik H ik t X i t H ikT t k 1 0 qi 2 T H t X t H t ik ik i ik k 1 0 t Qij t Tj t i t Qij t D Tj t 0 3 3 ij 2 (t ) i , i t ii t (193) T T Di t Qij t j t Di t Qij t D j t 0 R i t
Then we have p t p t p t p t 2
PT
1
11 t p t 0
11 2 t 2 , p t 2 L1 t L1 t
CE
1L2 t
0
1
AC
3
3
p
13 t 0
, 2 LL t
3 L t 0
(194)
According to (161), (163) and Lemma 2 we have p t 0 and p t 0 . Next, we need to 3
1
prove p t 0 . 2
1 L p 2 can be rewritten p 2 t p t Qp t pT t , where p t diag t , , t ,
i
Q11 t i t 0 t , Qp t Di t 0 L1 Q t
Q
1L
t
Qij t Qij t ij , and Q t Qij t Qij t 2 ri 2 rj LL Q t
Qi t Qii t . Hence the problem is to prove Qp t 0 . From (9) and (12) it follows 27
ACCEPTED MANUSCRIPT 0 Q t Q t 0 . , i j ; Qi t Qii t 0 Qei t 0 r r 0 ri ri i j
Qij t
Decompose Qp t as
Qp t Qb t Qd t QB
1L
Q t 0 , QBi , j t Q t 0
t
LL QB t 2 r 2 r g g
Qd t diag QD
11
t ,
0 0 0 Q t LL ei , QD t , QDi ,i t 0 0 0 Qei t
0 Q t 0 0 0 0 0 Q t 0 0 0 0 2 r 2 r i j 0 Qei t 0 0 0 Qei t 2 r 2 r i i 0
0
CR IP T
QB11 t Qb t L1 QB t
(195)
Applying Lemma 1 yields QDi ,i t 0 , and applying Lemma 2 yields Qd t 0 . Interchanging rows and columns of Qb t yields Q(t )
0 0 Q(t ) 0 2 rg 2 rg
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Q(t ) Qb t Q(t )
0
(196)
Hence Qp t 0 , which yields p t 0 . Therefore it follows that p t 0 . Applying (8), (43), 2
(47), (48) and (191) yields P 1| 0 P 1| 0 P 1| 0 0 , so that iterating (192) yields
M
P t 1| t 0 for arbitrary t.
Taking N 0 , subtracting (56) from (55), we have the global P t | t N as N
ED
P t | t N N t P t | t 1 NT t K N t p t K NT t
(197)
0
N t =diag 1N t ,
, K LwN t , K LvN t
PT
vN K N t =diag K1wN t , K1 t ,
, LN t
(198)
Applying P t | t 1 0 and p t 0 yields P t | t N 0 , so from (187) we have
Pc t | t N 0 , and it follows from (181) that Pc t | t N 0 ,i.e., (94) holds. Similar to the proof
CE
of Theorem 1, we have that Pc t | t N is the minimal upper bound of Pc t | t N . The proof is
AC
completed. Appendix I. The proof of Theorem 4 Proof. Taking the trace operations to (68), (94) and (101) yields (102) and the second inequality of (103). Since the CI fuser is a special fuser with matrix weights, then the first inequality of (103) holds. The third inequality of (103) can easily be proved from (84). In fact, arbitrarily taking i t | t N 1 ,
j t | t N 0 j i yields trPCI t | t N =trPci t | t N , so that minimizing trPCI t | t N with the
constraint 1 t | t N
L t | t N 1 yields the third inequality of (103). The accuracy relations
(104) were proved in [29]. The proof is completed.
28
ACCEPTED MANUSCRIPT References
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[1] M. E. Liggins, D. L. Hall, H. Llinas, Handbook of Multisensor Data:Theory and Practice, Second Edition, New York: CRC Press. 2009. [2] Z. L. Deng, Y. Gao, L. Mao, Y. Li, G. Hao, New approach to information fusion steady-state Kalman filtering, Automatica. 41(10) (2005) 1695-1707. [3] S. L. Sun, Z. L. Deng, Multi-sensor optimal information fusion Kalman filte, Automatica. 40(6) (2004) 1017-1023. [4] F. L. Lewis, L. H. Xie, D. Popa, Optimal and Robust Estimation. Second Edition, CRC Press, New York. 2008. [5] W. J. Qi, P. Zhang, Z. L. Deng, Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances, Signal Processing. 99 (2014) 185-200. [6] W. J. Qi, P. Zhang, G. H. Nie, Z. L. Deng, Robust weighted fusion Kalman predictors with uncertain noise variances, Digital Signal Processing. 30 (2014) 37-54. [7] W. J. Qi, P. Zhang, Z. L. Deng. Robust weighted fusion time-varying Kalman smoothers for multisensor system with uncertain noise variances, Information Sciences. 282 (2014) 15-37. [8] S. J. Julier, J. K. Uhlman, Non-divergent estimation algorithm in the presence of unknown correlations, in: Proceedings of the America Control Conference. 4 (1997) 2369-2373. [9] S. J. Julier, J. K. Uhlman, Using covariance intersection for SLAM. Robot. and Auton. Syst. 55 (1) (2007) 3-20. [10] F. Wang, V. Balakrishnan, Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties, IEEE Transactions on Signal Processing, 50(4) (2002) 803-813. [11] W. L. De Koning, Optimal estimation of linear discrete-time systems with stochastic parameters, Automatica, 20(1) (1984) 113-115. [12] N. E. Nahi, Optimal recursive estimation with uncertain observation, IEEE Transaction on Information Theory 15(4) (1969) 457-462. [13] Y. L. Chen, B.S. Chen, Minimax robust deconvolution filters under stochastic parametric and noise uncertainties, IEEE Transactions on Signal Processing, 42(1) (1994) 32-45. [14] S. Y. Wang, H. J. Fang, X. G. Tian. Minimum variance estimation for linear uncertain systems with one-step correlated noises and incomplete measurements, Digital Signal Processing. 49 (2016) 126-136. [15] T. Tian, S. L. Sun, N. Li, Multi-sensor information fusion estimators for stochastic uncertain systems with correlated noises, Information Fusion. 27 (2016) 126–137. [16] J. X. Feng, Z. D. Wang, M. Zeng, Distributed weighted robust Kalman filter fusion for uncertain systems with autocorrelated and cross-correlated noises, Information Fusion. 14 (1) (2013) 78-86. [17] B. D. O. Anderson, J. B. Moore. Optimal Filtering. Prentice Hall, Englewood Cliffs, New Jersey. 1979. [18] F. Li, J. Zhou, D. Z. Wu, Optimal filtering for systems with finite-step autocorrelated noises and multiple packet dropouts, Aerospace Science and Technology. 24 (2013) 255-263. [19] B. Chen, W. Zhang, G. Hu, L. Yu, Networked fusion kalman filtering with multiple uncertainties, IEEE Transactions on Aerospace and Electronic Systems. 51(3) (2015) 2332-2349. [20] B. Chen, L. Yu, W. A. Zhang, A. Liu, Robust information fusion estimator for multiple delay-tolerant sensors with different failure rates, IEEE Transactions on Circuits and Systems-I: Regular Papers. 60(2) (2013) 401-414. [21] W. Q. Liu, X. M. Wang, Z. L. Deng, Robust centralized and weighted measurement fusion Kalman estimators for uncertain
multisensor systems with linearly correlated white noises, Information Fusion. 35 (2017) 11–25.
CE
PT
ED
M
[22] W. Q. Liu, X. M. Wang, Z. L. Deng, Robust weighted fusion Kalman estimators for multisensor systems with multiplicative noises and uncertain covariances linearly correlated white noises, International Journal of Robust and Nonlinear Control (2016) Published online in Wiley Online Library (wileyonlinelibrary). DOI: 10.1002/rnc.3649. [23] B. Chen, G Hu, D. W. C. Ho, W. A. Zhang, L. Yu, Distributed robust fusion estimation with application to state monitoring systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems. (2016) DOI:10.1109/TSMC.2016.2558103. [24] X. J. Sun, Y. Gao, Z. L. Deng, Multi-model information fusion Kalman filtering and white noise deconvolution, Information Fusion. 11 (2010) 163-173. [25] X. J. Sun, Y. Gao, Z. L. Deng, Information fusion white noise deconvolution estimators for time-varying systems, Signal Processing, 88(5) (2008):1233-1247. [26] J. Ma, S. L. Sun, Information fusion estimators for systems with multiple sensors of different packet dropout rates, Information Fusion, 12(3) (2011) 213-222. [27] J. Ma, S. L. Sun, Distributed fusion filter for networked stochastic uncertain systems with transmission delays and packet dropouts . Signal Processing. 130 (2017) 268–278. [28] Z. D. Wang, D. W. C. Ho, X. H. Liu, Robust filtering under randomly varying sensor delay with variance constraints, IEEE Transaction Circuits Systems II：Express Briefs. 51(6) (2004) 320-326. [29] Z. L. Deng, P. Zhang, W. J. Qi, Y. Gao, J. F. Liu, The accuracy comparison of multisensor covariance intersection fuser and three weighting fusers, Information Fusion. 14 (2013) 177-185. [30] Z. L. Deng, Y. Gao, C. B. Li, G. Hao, Self-tuning decoupled information fusion Wiener state component filters and their convergence, Automatica, 44(3) (2008) 685-695. [31] F. W. Yang, Z. D. Wang, Y. S. Hung, M. Gani, H control for networked systems with random communication delays, IEEE
AC
Transactions on Automatic Control, 51(3) (2006) 511-518. [32] J. Zhu, H. S. Xi, H. B. Ji, B. Wang, Robust Kalman filtering of discrete-time Markovian jump systems based on state estimation performance，Control Theory & Applications, 25(1) (2008) 115-119. [33] F. B. Li, L. G. Wu, P. Shi, C. C. Lim, State estimation and sliding mode control for semi-markovian jump systems with mismatched, Automatica, 51 (2015) 385-393. [34] P. Shi, F. Li, L. Wu, C. C. Lim. Neural network-based passive filtering for delayed neutral-type semi-markovian Jump Systems.
IEEE Transactions on Neural Networks & Learning Systems, DIO:10.1109/TNNLS, 2016. 2573853.
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