Abstract This paper investigated the problem of distributed estimation for a class ofdiscrete-time nonlinear systems with unknown inputs in a sensor network. A modi-fication scheme to the derivative-free versions of nonlinear robust two-stage Kalmanfilter (DNRTSKF) is first introduced based on recently developed cubature Kalmanfilter (CKF) technique. Afterward, a novel information filter is proposed by expressingthe recursion in terms of the information matrix based upon DNRTSKF. In the end,distributed DNRTSKF is developed by applying a new information consensus filter todiffuse local statistics over the entire sensor network. In the implementation procedure,each sensor node only fuses the local observation instead of the global informationand updates its local information state and matrix from its neighbors’ estimates usingAverage-Consensus Algorithm. Simulation results illustrate that the proposed distrib-uted filter reveals the performance comparable to centralized DNRTSKF and betterthan distributed CKF.
Keywords Distributed estimation with unknown inputs · Information consensusfilter (ICF) · Cubature Kalman filter (CKF) · Nonlinear estimation
J. Ding (B) · J. Xiao · M. N. IqbalSchool of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, Sichuan,People’s Republic of Chinae-mail: [email protected]
Y. ZhangChengdu Electric Power Bureau, Chengdu 610041, Sichuan, People’s Republic of Chinae-mail: [email protected]
Circuits Syst Signal Process
1 Introduction
With the development of large-scale sensor networks in target tracking, monitoring,and robot navigation [14,24,31], distributed estimation has attracted much attention ofresearchers. Since it is in general, more robustness to sensor failures and might requireless communication than the traditional centralized schemes [3,27]. Many techniqueshave been proposed recently to solve the problems of distributed estimation. Based onKalman filter, Olfati-Saber et al. have developed the distributed Kaman filter (DKF)for discrete-time linear Gaussian systems by employing dynamic average-consensusstrategies [25,26,28]. The filtering framework relies on mathematically equivalent rep-resentation of the Kalman filter, known as information Kalman filter (IKF) [1]. More-over, using the Extended Kalman Filter (EKF), the method can be directly extendedto nonlinear Gaussian systems [21]. However, these methods may exhibit poor per-formance when nonlinearity is severe. Unscented Kalman Filter (UKF) offers a bettersolution with high filtering accuracy and robustness under the same computationalcost [13]. Distributed version of UKF has been derived using statistical linear errorpropagation and introducing the pseudo-observation matrix [17]. Subsequently, dis-tributed Gaussian mixture UKF (GM-UKF) has been proposed for nonlinear systemswith non-Gaussian noise [18]. However, the performance of UKF is easily affected bythe parameters, and cubature Kalman filter (CKF) has been presented in [2], which isbased on spherical radial rule and adopts a set of cubature point with the same weight toapproximate the posterior distribution of the system state. CKF is the direct Gaussianapproximation to the Bayesian filter and provides more precise state estimation thanexisting Gaussian filters. In [22,23], the CKF is explored in an information domain byembedding with EIF architecture (labeled CIF), and the square root versions of thisfilter (SRCIF) is also derived. Furthermore, both CIF and SRCIF are further devel-oped for multi-sensor state estimation in [23]. However, real-world systems are oftensubject to modeling errors and random disturbances which are immeasurable or hardto measure, and are often treated as stochastic process with unknown statistics. In thisscenario, the filters mentioned above for multi-sensor systems may be invalidated.
On the other side, state estimation problem for discrete-time stochastic systemswith arbitrary unknown inputs has been extensively studied. A global optimal fil-ter for unbiased minimum-variance state estimation of linear systems with unknowninputs has been developed [7,9,19]. Distributed information consensus filter (ICF)for simultaneous input and state estimation of linear discrete-time system in sensornetworks is presented in [8]. In a recent literature [11], an approach to solve the prob-lem of estimation for nonlinear systems with unknown inputs is proposed. It is ineffect a nonlinear version of extended recursive three-step filter (ERTSF), denoted asNERTSF. However, this method may not be applicable for the state estimation of somecomplex nonlinear system, since the derivatives of the nonlinear system are needed.To compensate these drawbacks, a derivative-free version of NERTSF (DNERTSF)has been proposed for general nonlinear system with arbitrary unknown inputs [10].Moreover, a robust two-stage form of the DNERTSF, labeled as DNRTSKF, is derivedand applied for traffic state estimation [12]. Most recently, in [20], a nonlinear fil-ter based on UKF and the Least-squares unbiased estimation algorithm is proposedwhich simultaneously estimates the constrained state and unknown input. Besides, the
Circuits Syst Signal Process
simultaneous input and state estimation for nonlinear systems with and without input–output direct feedthrough is developed by taking a Bayesian perspective and appliedto the flow field estimation [6]. It should be noted that the problem of nonlinear sys-tem estimation without available prior knowledge of boundary condition may arise inmulti-sensor fusion. However, to the best of the authors’ knowledge, the distributedestimator has not been investigated yet for “simultaneous input and state estimationof nonlinear systems with arbitrary unknown inputs in a sensor network.”
In this paper, a novel distributed estimation algorithm is studied to solve the problemof state and input estimation for nonlinear systems with unknown inputs in a sensornetwork. First of all, modified DNRTSKF is derived by the sampling scheme of CKFwith statistical linear error propagation. Since the update stage of Information filtersis computationally economic and can be easily extended to multi-sensor fusion, theinformation form of DNRTSKF is developed in terms of the inverse of the covariancematrix and state information estimate. It can be mentioned that distributed algorithmsare often derived together with consensus strategies for data fusion, such as average-consensus algorithm [15]. Then, by virtue of average-consensus protocol, the proposeddistributed filter is developed by incorporating with the new information consensusfilter (ICF). Designated filter is unbiased and only the local observation informationof each sensor node is fused, otherwise the local information state and its associatedinformation matrix are updated from its neighbors’ estimation [4,5]. The proposedalgorithm not only achieves the input and state estimation simultaneously in multi-sensor nonlinear systems, but also accomplishes distributed data fusion with the newconsensus protocol. Simulation results show that the performance of the proposeddistributed filter is closed to the centralized estimate and better than distributed CKFwhich cannot get any knowledge of the unknown inputs.
Remainder of this paper is organized as follows: Sect. 2 describes the problem ofdistributed estimation for nonlinear system with unknown system in the absence of thedirect feedthrough. Section 3 details the proposed information form of the DNRTSKF.In Sect. 4, the distributed filter for the nonlinear system with unknown system isproposed by embedding the ICF. In Sect. 5, numerical simulations are explained, andconclusions are presented in Sect. 6.
2 Problem Formulations
Consider the following discrete-time nonlinear system with unknown inputs:
xk = fk−1(xk−1, dk−1, uk−1) + ωk−1, (1)
where xk ∈ Rn ,dk ∈ R p, and uk ∈ Rq are the state vector, unknown input vector, andknown input vector, respectively. fk is the system transition function at time k . Theprocess noise ωk ∈ Rn is uncorrelated zero mean white Gaussian, with covariancematrices Qk > 0.
In the distributed sensor network, the measurement of agent i at time k is modeledas
zi,k = hi,k(xk, uk) + νi,k i = 1, 2, . . . , N (2)
Circuits Syst Signal Process
where N is the number of sensor nodes, zi,k ∈ Rm are the i th sensor measurementvectors, and hi,k are related measurement functions. The measurement noise νi,k is zeromean white Gaussian noise N (0, Ri,k) with E[νi,k(νi ′,k′)T ] = δkk′δi i ′ Ri,k and Ri,k >
0. At any discrete-time instant k, the communication topology between senor agentsis described by the directed graph G(k) = (V, E(k)), where V = {1, 2, 3, . . . , N }denotes the node set and E(k) ⊂ {(i, j) |i, j ∈ V} denotes the edge set. The setof neighbors of sensor node i is denoted by Ni (k) = { j ∈ V |(i, j) ∈ E(k)}. Theadjacency matrix A(k) = [ai j (k)] ∈ RN×N of the graph G(k) is defined such thatai j (k) = 1 if i �= j and (i, j) ∈ E(k), otherwise ai j (k) = 0. Central measurementzk ∈ RNm , observation noise νk , and observation function hk can be defined as
zk = [z1,k; z2,k; · · · ; zN ,k], (3)
νk = [ν1,k; ν2,k; · · · ; νN ,k], (4)
hk = [h1,k; h2,k; · · · ; hN ,k]. (5)
Hence, multi-sensor observation model can be represented by the following centralizedobservation model:
zk = hk(xk, uk) + νk . (6)
Since measurement noises of the sensors νi,k are uncorrelated, the global covariancematrix Rk can be written as
Rk = diag(R1,k, R2,k, . . . , RN ,k). (7)
The aim of this paper is to develop a distributed filtering algorithm for discrete-timenonlinear system (1) and (2) with unknown inputs, with realization of simultaneousinput and state estimation. To be specific, the filtering effect of the distributed filter isexpected to be as accurate as that of the centralized estimation.
3 Information Filters
An effective approach to solve the state estimation problem of general nonlinear sys-tems with unknown inputs is derivative-free version of NRTSKF, named as DNRTSKF.The main idea of the DNRTSKF is to generate a set of sampled points with mean andvariance approximating the distribution of real state estimation. Moreover, CKF is theclosest known Gaussian approximation to the Bayesian filter, and DNRTSKF can beimproved by applying the sampling strategy of the CKF. Keeping in view that theinformation filters offer simple solution to multi-sensor data fusion, the focus of thissection is on deriving the information filtering framework of the improved DNRTSKF.
3.1 Modified DNRTSKF
The nonlinear system (1)–(2), having no direct feedthrough from unknown input tothe measurement, is a special case of [12]. Employing DNRTSKF, input and state esti-
Circuits Syst Signal Process
mation can be obtained simultaneously. With the help of CKF, the recursive algorithmof DNRTSKF is derived as follows:
We assume that the posterior probability density function p(xk−1 |Dk−1 ) =N (xk−1|k −1, Pk−1|k−1 ) at time step k − 1 is known, where Dk−1 denotes the historyof input-measurement up to k − 1. The predicted state xk|k−1 and its error covariancematrix Pk|k−1 can be given by
xk|k−1 = 1
L
L∑
�=1
χ∗(�)k|k−1 , (8)
Pk|k−1 = 1
L
L∑
�=1
χ∗(�)k|k−1 χ
∗(�)Tk|k−1 − xk|k−1 xT
k|k−1 + Qk−1, (9)
where χ∗(�)k|k−1 = fk−1
(χ
(�)k−1|k−1 , 0, uk−1
)and set of 2n cubature points χ
(�)k−1|k−1
(� = 1, 2, . . ., L) can be calculated by means of Spherical Radial Rule as
χ(�)k−1|k−1 = Sk−1|k−1 ξ � + xk−1|k−1 , (10)
Pk−1|k−1 = Sk−1|k−1 STk−1|k−1 , (11)
where L = 2n, ξ � =√
L2 [1]�, and [1]� is the �-th element of the following set:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎛
⎜⎜⎜⎝
10...
0
⎞
⎟⎟⎟⎠ ,
⎛
⎜⎜⎜⎝
01...
0
⎞
⎟⎟⎟⎠ , · · · ,
⎛
⎜⎜⎜⎝
00...
1
⎞
⎟⎟⎟⎠ ,
⎛
⎜⎜⎜⎝
−10...
0
⎞
⎟⎟⎟⎠ ,
⎛
⎜⎜⎜⎝
0−1...
0
⎞
⎟⎟⎟⎠ , · · · ,
⎛
⎜⎜⎜⎝
00...
−1
⎞
⎟⎟⎟⎠
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
It can be mentioned that NRTSKF may not be suitable for implementation sincethe Jacobians and Hessians of (1)–(2) are difficult or impossible to obtain for somecomplex nonlinear systems, and hence the derivative-free implementation of NRTSKF(DNRTSKF) is incorporated. In order to simplify the calculation, we can also deriveDNRTSKF by statistical linear error propagation methodology [16]. According to theerror propagation, the observation covariance and its cross-correlation covariance areapproximated by
P zz,k|k−1 = E{[zk − zk|k−1 ][zk − zk|k−1 ]T
}
≈ Ck Pk|k −1CTk + Rk,
(12)
P xz,k|k−1 = E{[xk − xk|k−1 ][zk − zk|k−1 ]T
}
≈ Pk|k−1 CTk ,
(13)
and linearization measurement matrix Ck is obtained as
Ck =[(
Pk|k−1)−1 P xz,k|k−1
]T = (P xz,k|k−1
)T (Pk|k−1)−1
. (14)
Circuits Syst Signal Process
Consequently, for the nonlinear system, pseudo measurement matrix can be definedas
Ck = (P xz,k|k−1
)T (Pk|k−1)−1
. (15)
It is obvious that the measurement matrix Ck is got according to Eq. (13), and theexpression is different from that in [10,12]. Except for the use of a different samplingstrategy, this is partly due to that the latter is directly approximated by Eq. (12),which is relatively complicated and imprecise because of Cholesky decomposition.The cross-correlation covariance P xz,k|k−1 can be evaluated as
P xz,k|k−1 = 1
L
L∑
�=1
(χ
(�)k|k−1 − xk|k−1
) (Z(�)
k|k−1 − zk|k−1
)T, (16)
and the predicted measurement zk|k−1 is
zk|k−1 = 1
L
L∑
�=1
Z(�)k|k−1 , (17)
where Z(�)k|k−1 = hk(χ
(�)k|k−1 , uk) and set of cubature points χ
(�)k|k−1 are computed by
the predicted state xk|k−1 and its error covariance matrix Pk|k−1 in accordance with(10) and (11).
Based on the above results, the DNRTSKF can be easily described as
xk|k = xk|k + V k dk−1, (18)
Pk|k = P xk|k + V k Pd
k|k V Tk , (19)
where
xk|k = xk|k−1 + K k(zk − zk|k−1
)(20)
P xk|k = Pk|k−1 − K k Rk K T
k (21)
with
Rk = 1
L
L∑
�=1
(Z(�)
k|k−1 − zk|k−1
) (Z(�)
k|k−1 − zk|k−1
)T + Rk, (22)
K k = P xz,k|k−1 R−1k , (23)
dk−1 = S∗k
(zk − zk|k−1
), (24)
Pdk|k = S∗
k Rk(S∗k)
T (25)
where
V k = Gk−1 − K k Sk, (26)
Circuits Syst Signal Process
Sk = Ck Gk−1, (27)
S∗k =
(ST
k R−1k Sk
)−1ST
k R−1k , (28)
and Gk = [G1
k · · · G pk
]is suitable partial derivative matrix such that
G jk =
fk
(xk|k ,� j × e j
p, uk
)− fk
(xk|k , 0, uk
)
� j. (29)
Here, � j is a suitable chosen small value and e jp is the i-th column of the identity
matrix with dimension p.
Remark 1 The main difference of the modified DNRTSKF in this paper is the sam-pling scheme. Comparing with the sampling strategy of DNRTSKF in [10,12], theCKF is closer to the Gaussian approximate of true state estimation. Moreover, themeasurement matrix Ck derived from the statistical linear error propagation Eq. (13)offers more accuracy than that obtained by approximating directly.
Lemma 1 Assuming xk−1|k −1 be an unbiased estimate of xk−1, then dk−1 is anapproximately unbiased estimate of dk−1 if and only if S∗
k satisfies S∗k Ck Gk−1 = I p.
Proof See Appendix. �
Remark 2 We have assumed that for any discrete-time instant k, the existence condi-tion of the optimal filter rankCk Gk−1 = rankGk−1 is satisfied [10].
3.2 Information Form of the DNRTSKF
Using the information filtering framework from [8], the information form of DNRT-SKF, denoted as IDNRTSKF, can be developed by restructuring its recursive algorithmin terms of information state vector and information matrix.
Now, we need to rewrite the state estimation of DNRTSKF into a recursive three-step filter as
xk|k = x∗k|k + K k
[(I − Sk S∗
k)(zk − zk|k−1 )]
(30)
x∗k|k = xk|k−1 + Gk−1 dk−1, (31)
where x∗k|k is the state prediction with the input information, and the unknown input
estimator dk−1 is given by (24). Then, the error covariance matrix can be defined as
P∗k|k = E[x∗
k|k x∗Tk|k ], (32)
Circuits Syst Signal Process
where x∗k|k = xk − x∗
k|k . From (31) and (24), the error covariance matrix P∗k|k can be
expressed as
P∗k|k = Pk|k −1 − P xz,k|k−1 S∗T
k GTk−1 − Gk−1S∗
k PTxz,k|k−1 + Gk−1 S∗
k Rk S∗Tk GT
k−1.
(33)It follows from (30) and (33) that the error covariance matrix Pk|k can be rewritten as
Pk|k = P∗k|k + K k R
∗k K T
k − K kU Tk − Uk K T
k , (34)
where
R∗k = Ck P∗
k|k CTk + Rk + Ck T k + T T
k CTk (35)
Uk = P∗k|k C
Tk + T k (36)
T k = −Gk−1 S∗k Rk . (37)
In order to obtain the information filter based on DNRTSKF, one can calculate thegain matrix K k by minimizing the trace of the estimation error covariance matrix (35)under the unbiased condition S∗
k Ck Gk−1 = I p. The Lagrangian is
T r[
P∗k|k + K k R
∗k K T
k − K kU Tk − Uk K T
k
]− 2T r
[(I p − S∗
k Ck Gk−1)�Tk
], (38)
where �k is an n×p matrix of Lagrange multipliers, and Tr denote the trace of a matrix.The coefficient “2” is included for notational convenience. Taking the derivatives withrespect to K k and equaling to zero yields
2 R∗k K T
k − 2Uk = 0. (39)
Then, the gain matrix K k is given by
K k = Uk
(R
∗k
)−1 =(
P∗k|k C
Tk + T k
) (R
∗k
)−1. (40)
Considering the Eq. (35), the gain matrix K k can be rewritten as
K k = P∗k|k
[C
Tk +
(P∗
k|k)−1
T k
] (R
∗k
)−1
= P∗k|k
[Ck + T T
k
(P∗
k|k)−1
]T
[(Ck + T T
k
(P∗
k|k)−1
)P∗
k|k(
Ck + T Tk
(P∗
k|k)−1
)T
+ Rk − T Tk
(P∗
k|k)−1
T k
]−1
= P∗k|k C
Tk
(Ck P∗
k|k CTk + Rk
)−1,
(41)
Circuits Syst Signal Process
where
Ck = Ck + T Tk
(P∗
k|k)−1
Rk = Rk − T Tk
(P∗
k|k)−1
T k .
Substituting (40) in (34) and applying (41), the error covariance matrix Pk|k can berepresented by the following update form:
Pk|k = P∗k|k − K k
(P∗
k|k CTk + T k
)T
= P∗k|k − K k
(C
Tk +
(P∗
k|k)−1
T k
)T
P∗k|k
= P∗k|k − P∗
k|k CTk
(Ck P∗
k|k CTk + Rk
)−1Ck P∗
k|k .
(42)
According to the matrix inversion lemma, we deduce
Pk|k =[(
P∗k|k)−1 + C
Tk R
−1k Ck
]−1
. (43)
Thus, we can further express the gain matrix K k as
K k = Pk|k[(
P∗k|k)−1 + C
Tk R
−1k Ck
]P∗
k|k CTk
(Ck P∗
k|k CTk + Rk
)−1
= Pk|k[C
Tk + C
Tk R
−1k Ck P∗
k|k CTk
] (Ck P∗
k|k CTk + Rk
)−1
= Pk|k CTk R
−1k .
(44)
Let us definite the predicted information state vector y∗k|k and the predicted information
matrix Y∗k|k as
y∗k|k =
(P∗
k|k)−1
x∗k|k = Y∗
k|k x∗k|k (45)
Y∗k|k =
(P∗
k|k)−1
. (46)
Employing Eq. (43), the update information state vector yk|k and the update informa-tion matrix Y k|k are given as
yk|k = (Pk|k
)−1 xk|k = y∗k|k + C
Tk R
−1k
[(I − Sk S∗
k
) (zk − zk|k−1
)+ Ck x∗k|k]
= y∗k|k + j k
(47)
Y k|k = (Pk|k
)−1 =(
P∗k|k)−1 + C
Tk R
−1k Ck
= Y∗k|k + Jk,
(48)
Circuits Syst Signal Process
where the information state contribution and its associated information matrix aregiven by
jk = CTk R
−1k
[(I − Sk S∗
k
) (zk − zk|k−1
)+ Ck x∗k|k]
(49)
Jk = CTk R
−1k Ck . (50)
Summing up, the IDNRTSKF algorithm can be described as
where Nt denotes the number of total time step.
4 Distributed IDNRTSKF (DF-DNRTSKF)
In this section, distributed IDNRTSKF is derived by dynamic version of the average-consensus algorithm.
Circuits Syst Signal Process
4.1 Consensus Protocol
Consensus algorithms are the main methods for accomplishing distributed data fusionin networked multi-agent systems [15,29]. In a consensus filter, each sensor nodeonly exchanges message with its neighbors, and agents can come into agreementconcerning the consensus state asymptotically. In this paper, information consensusprotocol is employed for distributed estimation, with each node updating its state ζ ifrom neighbors’ states according to the following rule [15]:
ζ i (τ + 1) =∑
j∈Ni
αi j [τ ]ζ j (τ ), (51)
where αi j [τ ] is the linear weight on ζ j (τ ) at node i and τ is the consensus iteratingstep.
Lemma 2 (Proposition 2 in [15]) Setting∑N
j=1 αi j [τ ] = 1,∑N
i=1 αi j [τ ] =1,αi j [τ ] ≥ 0, and αi j [τ ] = 0for j /∈ Ni (τ ), if there exists T ≥ 0such that for everyinterval [τ , τ+T] the union of the interaction graph across the interval is stronglyconnected, then each node approaches the averages ζ i (τ ) → (1/N )
∑Nj=1 ζ j (0) as
τ → ∞ and average consensus is asymptotically achieved in a not fully connectednetwork.
Obviously, the Eq. (51) can be further equivalent to
ζ i (τ + 1) = ζ i (τ ) −N∑
j=1
αi j [τ ](ζ i (τ ) − ζ j (τ )). (52)
In this paper, weight αi j [τ ] is adapted by the following Metropolis weights, since itwill guarantee convergence of average consensus:
αi j [τ ] =
⎧⎪⎪⎨
⎪⎪⎩
1/(1 + max{di (τ ), d j (τ )}) if(i, j) ∈ E(τ )
1 − ∑(i,s)∈E(τ )
αis[τ ] if i = j
0 otherwise
(53)
where di (τ ) is the diagonal element of degree matrix D(τ ) = diag{d1(τ ), d2(τ ), . . . , dn(τ )}. Furthermore, in Metropolis weights scheme, each nodeonly needs knowledge of the degree for their neighbors and is well suited for distrib-uted implementation [29].
4.2 Distributed IDNRTSKF Based on ICF
For multiple sensor systems with a fully connected network in which each node cancommunicate with any other node, the estimates obtained by each sensor node are
Circuits Syst Signal Process
identical by (47) and (48) if the nodes begin a common initial information state. Then,the optimal centralized version can be exactly implemented by the following fullydecentralized information filter:
yi,k|k = y∗i,k|k +
N∑
i=1
j i,k (54)
Y i,k|k = Y∗i,k|k +
N∑
i=1
J i,k, (55)
where
j i,k = CTi,k R
−1i,k
[(I − Si,k S∗
i,k
) (zi,k − zi,k|k−1
)+ C i,k x∗i,k|k
](56)
J i,k = CTi,k R
−1i,k C i,k . (57)
In this case, each sensor node can be regarded as a fusion central, and optimal estimatescan be obtained by anyone. However, when the sensor network is not fully connected,issues arise in communicating the observations and fully decentralized informationfilter is no longer equivalent to optimal centralized version.
For distributed information filter, when sensor node i taking measurements zi,k attime step k under the measurement Eq. (2), local estimate of information matrix andinformation state vector can be obtained in terms of (47) and (48) as
Y i,k|k = Y∗i,k|k + J i,k (58)
yi,k|k = y∗i,k|k + j i,k . (59)
In order to estimate the state completely, local estimation accuracy should be improvedby sharing measurements among sensor nodes of the network. As discussed inSect. 4.1, the commutation between agents can be carried by a consensus filter with-out requirement of a fully connected network. Additionally, the estimate in ICF isunbiased, assuming that communication and prediction updates are synchronized inthe sensor network and each agent i only uses local observation instead of globalinformation [5]. In measurement update, each sensor node iupdates its local infor-mation state vector and matrix from its neighbors’ estimates according to (52) ratherthan communicating the measurement directly. Then, distributed DNRTSKF can bederived by combining with ICF.
Algorithm 2 describes the distributed estimate for nonlinear system (1)–(2) withunknown input based on ICF.
Circuits Syst Signal Process
Circuits Syst Signal Process
where Tp is the number of consensus iteration steps to achieve average consensus.
Remark 3 In the algorithm 2, one time step k − 1→ k is equivalent to Tp time stepsfor consensus. It means that in the iterative process of consensus filter, there is onlyone update of the local information filter which runs at every first step of the consensusiteration.
Lemma 3 If the ICF undergoes iteration enough for consensus filter to convergebefore the next prediction step (i.e. Tp ≥ 1), the actual covariance of local estimatesin this distributed filter is comparable to the centralized filter.
Proof Suppose the prior estimates y∗i,k|k and Y∗
i,k|k are equal for each sensor node,then the step 1 in consensus update has no effect. Now assume node i makes anobservation and fuses this into its local estimate yielding the local information statevector yτ0
i,k|k = y∗i,k|k + j i,k and information matrix Y τ0
i,k|k = Y∗i,k|k +J i,k (τ 0 indicates
consensus time when the observations are fused). If the conditions of Lemma 2 aresatisfied, then the average consensus will be achieved and the converged values aregiven by
yi,k|k = 1
n
N∑
i=1
yτ0i,k|k = y∗
i,k|k + 1
n
N∑
i=1
j i,k (60)
Y i,k|k = 1
n
N∑
i=1
Y τ0i,k|k = Y∗
i,k|k + 1
n
N∑
i=1
J i,k (61)
for all i =1,…,n. The local state estimate and its covariance estimate can be representedby xi,k|k = (Y i,k|k )−1 yi,k|k and (Y i,k|k )−1. As shown in [5], the ICF estimates areunbiased, and the actual variance of local estimation errors is comparable to that ofthe centralized filter. Therefore, these properties are maintained for the distributedIDNRTSKF, and this is illustrated by simulations in the following section. Comparedto the hypothetical centralized information filter (Eq. 55) which fuses the observationsfrom all sensor nodes, ICF (Eq. 61) scales the new information by the factor 1
n . Then,local covariance matrix estimates to be more conservative. If the network size n isknown and local information state yτ0
i,k|k and matrix Y τ0i,k|k are stored, then the exact
centralized information filter may be recovered by each agent. �
5 Simulation Results
In order to validate and reveal the effectiveness of proposed results, two-dimensionalradar tracking problem is considered, and performance of the proposed distributed filterand the centralized DNRTSKF is compared. The target executes turn in an x − y planwith a turn rate �, and the target dynamics can be described by the nonlinear motionmodel in [16]. To implement the proposed algorithm for the nonlinear system withunknown input, we apply the following extended version of that nonlinear dynamicmodel with the addition of an input dk :
Circuits Syst Signal Process
xk =
⎛
⎜⎜⎝
1 sin �T�
0 − ( 1−cos �T�
)
0 cos �T 0 − sin �T0 1−cos �T
�1 sin �T
�0 sin �T 0 cos �T
⎞
⎟⎟⎠ xk−1 +
⎡
⎢⎢⎣
10.310.3
⎤
⎥⎥⎦ dk−1 +
⎛
⎜⎜⎝
T 2
2 0T 0
0 T 2
20 T
⎞
⎟⎟⎠wk−1,
(62)where state of the target is x = [ ξ ξ η η ]T ; ξ and η represent target position, ξ andη represent target velocity in x and y directions, respectively; and T is the samplingtime period and wk−1 ∼ N (0, Qk−1) is process noise. dk = cos(0.6k) can be seenas an time-varying process noise, and we assume that the input dk is unknown in theprocess of filtering. Obviously, the mode in [16] is obtained by setting dk = 0 inEq. (62). In order to cover the unknown target motion, two models with different turnrate are applied, one with � = 0◦s−1, Qk−1 = 0.01I2, and another with � = 10◦s−1,Qk−1 = 0.04I2.
Using twelve radars fixed at the origin of plan to measure the range and bearing oftarget, measurement equation can be stated as
zi,k =(
ri,k
θ i,k
)=(√
(ξk − xi,0)2 + (ηk − yi,0)2
tan−1((ηk − yi,0)/(ξk − xi,0))
)+ νi,k, (i = 1, 2, . . . , 12),
(63)where (xi,0, yi,0) denotes the position of the i th radar sensor, and the measurementnoise νi,k ∼ N (0, Ri,k) with Ri,k = diag[σ 2
r σ 2θ ] for all sensors. In the following
simulations, we assume parameter values T = 1 s, σr = 0.2 m, and σθ = 0.015rad.True initial state of target and its associated covariance can be described as
x0 = [−40 m 3 ms−1 10 m 1 ms−1 ]T
P0 = diag[ 22 m2 0.12 m2 s−2 22 m2 0.12 m2 s−2 ].
The initial estimate state of target x0|0 is chosen randomly fromN (x0, P0), and dis-tributions of twelve sensors are shown in Fig. 1. The coordinate are (0, 30), (80, 30),(160, 30), (240, 30), (0, 60), (80, 60), (160, 60), (240, 60), (0, 90), (80, 90), (160,90), and (240, 90), respectively. The communication topology diagram between thesensors is shown in Fig. 2, assuming that the communication topology is fixed in thefollowing simulations.
Root-mean square error (RMSE) in position at time k is
RM SEk =[
1
M
M∑
i=1
((ξ i
k − ξ ik)
2 + (ηik − ηi
k)2)]1/2
(k = 1, 2, . . . , Nt ) (64)
where M is the Monte Carlo runs. Similarly, expressions of RMSE in velocity canalso be written. In the following experiments, we analyze quantitatively and comparethe performance of the proposed algorithm in terms of RMSE in position and velocity.
Circuits Syst Signal Process
Y c
oord
inat
e(m
)
X coordinate (m)-50 0 50 100 150 200 250 3000
20
40
60
80
100
120
Fig. 1 Target tracking scenario with 12 sensors
Fig. 2 Communicationtopology between sensors
The consensus iteration steps are Tc = 45, and 100 independent Monte Carlo runswith the same condition are made, whereas simulation steps per run are 100.
Simulation results In the following simulation experiment, we use distributed CKF(DF-CKF) adopting multi-sensor CIF [23] with the consensus protocol proposed inthis paper for emphasizing the performance of the proposed distributed filter in simul-taneous input and state estimation. For simplicity, we denote i th sensor measurementfor CF-CKF and DF-DNRTSKF as CF-CKFi and DF-DNRTSKFi, respectively. InFigs. 3 and 4, it is obvious that the distributed DNRTSKF can accurately estimatethe actual trajectory and the unknown input. In addition, the filtering effect of DF-DNRTSKF for each sensor node is almost identical and very close to centralizedDNRTSKF. It indicates that the consensus has been reached for each local filter andthe tracking performance of the proposed distributed filter is comparable to the cen-tralized filter. However, the estimation error exists for the CF-CKF algorithm, since itcannot accurately estimate the unknown input.
Circuits Syst Signal Process
-50 0X coordinate (m)
Y c
oord
inat
e (m
)
50 100 150 200 250 30010
20
30
40
50
60
70
80
Truth
DF-CKF1DF-CKF2
DF-CKF3
DF-DNRTSKF1
DF-DNRTSKF2DF-DNRTSKF3
CF-DNRTSKF 190 200 210 220 23030
35
40
45
Fig. 3 Actual and estimated trajectories (one sample run)
0 20 40 60 80 100-2
-1
0
1
2
3
4
Truth
DF-DNRTSKF1
DF-DNRTSKF2DF-DNRTSKF3
CF-DNRTSKF
Time (s)
Inpu
t d
Fig. 4 Actual and estimated input (one sample run)
In order to further illustrate the effectiveness of the proposed distributed filter, theperformance comparison with respect to RMSE in position is shown in Figs. 5 and 6. Itcan be observed that the RMSE of DF-DNRTSKFi is close to centralized DNRTSKFand performs better than CF-CKFi. Table 1 compares average and variance of RMSEin position and velocity for the three algorithms in two modes. From Table 1, it can beseen that the proposed distributed filter is comparable to the centralized DNRTSKFboth in filtering precision and stability.
Circuits Syst Signal Process
Time (s)
RM
SE in
Pos
ition
(m
)
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
4
DF-CKF1DF-CKF2
DF-CKF3
DF-DNRTSKF1
DF-DNRTSKF2
DF-DNRTSKF3CF-DNRTSKF
Fig. 5 Performance comparison with respect to RMSE in position (Model 1)
0 10 20 30
Time (s)
RM
SE in
pos
ition
(m
)
40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
DF-CKF1DF-CKF2
DF-CKF3
DF-DNRTSKF1
DF-DNRTSKF2
DF-DNRTSKF3CF-DNRTSKF
Fig. 6 Performance comparison with respect to RMSE in position (Model 2)
6 Conclusions
In this paper, we have proposed a distributed filter for the simultaneous state andinput estimation of nonlinear system with unknown inputs in a sensor network. Theproposed filter is derived from information filter framework of DNRTSKF and ICF.Specifically, DNRTSKF algorithm is improved by utilizing the CKF. In ICF algo-rithm, communication of measurements is handled by a consensus filter rather than
Circuits Syst Signal Process
Table 1 Performance comparison with respect to AMRSE and DMRSE
communicating the observations directly. Validation and usefulness of the proposeddistributed filter are verified via simulation examples. Comparing distributed filter pro-posed in this paper with distributed CKF and centralized estimate, it is shown that ourdistributed filtering strategy has better performance than distributed CKF since latterlacks true dynamic information of the unknown inputs. Moreover, the performanceof the proposed distributed filter is in close agreement with that of the centralizedestimate.
The distributed filter for nonlinear system with direct feedthrough of unknowninput to the measurement is under investigation and will be proposed in future work.Furthermore, due to the fact that agents may only communicate with their neighborsover some disconnected time intervals in some practical applications, the distributedfilter based only on the intermittent measurements will be also considered.
Acknowledgments This work was supported by National Natural Science Foundation of China(51177137, 61134001).
Appendix
As we know, DNRTSKF is the derivative-free implementation of NERTSF. It is possi-ble to derivate the condition of unbiased estimate for dk−1 from NERTSF and extendit to DNRTSKF.
According to NERTSF, a nonlinear system can be approximated in first-order Taylorseries as
where xk−1|k−1 ≈ xk−1 − xk−1|k−1 , xk|k−1 ≈ xk − xk|k−1 , and
Ak = ∂ fk(xk, dk, uk)
∂xk
∣∣xk = xk|k , dk = 0
Circuits Syst Signal Process
Gk = ∂ fk(xk, dk, uk)
∂dk
∣∣xk = xk|k , dk = 0
Ck = ∂hk(xk, uk)
∂xk
∣∣xk = xk|k−1 .
Substituting (66) in (24),
dk−1 ≈ S∗k Ck xk|k−1 + S∗
kνk . (67)
Employing (65), we have
dk−1 ≈ S∗k Ck(Ak−1 xk−1|k−1 + ωk−1) + S∗
k Ck Gk−1dk−1 + S∗kνk
= S∗k Ck Ak−1 xk−1|k−1 + S∗
k Ck Gk−1dk−1 + S∗k Ckωk−1 + S∗
kνk .(68)
Let xk−1|k−1 be unbiased, then E[xk−1|k−1 ] = 0 and it follows from (68) that
E[dk−1 − dk−1] ≈ E[S∗k Ck Ak−1 xk−1|k−1 + (S∗
k Ck Gk−1 − I p)dk−1+S∗
k Ckωk−1 + S∗kνk] = (S∗
k Ck Gk−1 − Ip)E[dk−1]. (69)
It should be noted that dk−1 is an approximately unbiased estimated of dk−1 if andonly if S∗
k Ck Gk−1 = I p.
Thus, in DNRTSKF, for the condition of unbiased estimate dk−1 to hold, it mustsatisfy the constraint
S∗k Ck Gk−1 = I p. (70)
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