Non-augmented state estimation for nonlinear stochastic coupling networks

Automatica 78 (2017) 119–122

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Technical communique

Non-augmented state estimation for nonlinear stochastic coupling networks✩ Wenling Li a , Yingmin Jia a , Junping Du b a

The Seventh Research Division, Beihang University (BUAA), Beijing 100191, China

b

School of Computer Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

article

info

Article history: Received 28 February 2016 Received in revised form 2 December 2016 Accepted 5 December 2016

Keywords: State estimation Complex network Non-augmented approach

abstract This paper considers the state estimation problem for discrete-time nonlinear stochastic coupling networks. A non-augmented filter is designed for each node to guarantee an optimized upper bound on the state estimation error covariance matrix despite nodes coupling as well as the linearization errors. Compared with the existing augmented filter, the cross-covariance matrices between coupling nodes are not required to be computed and the gain matrix can be obtained separately for each node by solving two Riccati-like difference equations. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction State estimation for complex networks has been widely studied in the theoretical research community and successfully applied in industry (Ata-ur-Rehman, Lyudmila, & Jonathon, 2016; Zia, Tucker, & Frank, 2005) This is partly due to the fact that state estimation is crucial not only because it helps understand the intrinsic structure of the networks but also because it is the first step to realize synchronization (Rodriguez-Angeles & Nijmeijer, 2004). Compared with the state estimation for an isolated node, the state estimation problem for complex networks becomes more difficult due to the nodes coupling. Specifically, the states of the nodes are not only determined by themselves but also by their neighbors. To overcome this difficulty, many strategies have been proposed to develop filters for complex networks including uncertain stochastic complex networks with missing measurements and time-varying delay (Liang, Wang, & Liu, 2009; Liang, Wang, Liu, & Liu, 2014), H∞ filters with uncertain coupling strength and incomplete measurements (Shen, Wang, Ding, & Shu, 2013; Shen, Wang, & Liu, 2011), uncertain complex networks with time-varying delays (Ding, Wang, Shen, & Shu, 2012), and time-varying com-

plex networks with missing measurements (Hu, Wang, Liu, & Gao, 2016). It should be pointed out that all the existing work focus on developing filters using the augmented approach, i.e., all the state estimation errors of the nodes are formulated in a compact form and the gain matrices for all nodes are obtained simultaneously with respect to the overall covariance matrix of the estimation errors. Thus, the disadvantage of the augmented filters is that high computational cost is often required for large number of nodes. In this paper, we attempt to develop a non-augmented filter for a class of discrete-time nonlinear stochastic coupling networks. By using the structure of the extended Kalman filter (EKF), a novel filter is developed by proposing the predicted and updated estimation error systems. To address the coupling features and the linearization errors, upper bound matrices are introduced for the corresponding covariance matrices so that the gain matrices can be derived by minimizing the trace of the upper bound matrix. A distinct feature of the proposed filter is that the cross-covariance matrices between coupling node are not required to be computed and the gain matrix can be derived separately for each node. 2. Problem statement

✩ This work was supported by the National Basic Research Program of China (973

Program, 2012CB821200, 2012CB821201), and the NSFC (61573031, 61134005, 61327807, 61473010, 61532006, 61520106010). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Carlo Fischione under the direction of Editor André L. Tits. E-mail addresses: [email protected] (W. Li), [email protected] (Y. Jia), [email protected] (J. Du). http://dx.doi.org/10.1016/j.automatica.2016.12.033 0005-1098/© 2016 Elsevier Ltd. All rights reserved.

Consider the following nonlinear stochastic network xi,k+1 = f (xi,k ) +

N 

ωij Γ xj,k + wi,k

(1)

j=1

zi,k = h(xi,k ) + vi,k ,

i = 1, 2, . . . , N

(2)

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W. Li et al. / Automatica 78 (2017) 119–122

where xi,k ∈ Rn is the state vector of the ith node and zi,k ∈ Rp is the measurement vector of the ith node. f (·) and h(·) are known nonlinear functions that are assumed to be twice continuously differentiable. Γ is a matrix and W = [ωij ]N ×N is the coupling configuration matrix with ωij ≥ 0. The process noise wi,k and the measurement noise vi,k are assumed to be mutually uncorrelated zero-mean white Gaussian with known covariance matrices Qi,k and Ri,k , respectively. The structure of the EKF is adopted to design the following filter for the ith node N

x¯ i,k = f (ˆxi,k ) +



ωij Γ xˆ j,k

ei,k+1 eTi,k+1

Pi,k+1 = E{

}.

(13)

As shown in Hu et al. (2016), by using the Taylor series expansion technique, the predicted estimation error and the updated estimation error can be represented as follows f

(5) (6)

(7)

Remark 1. In Hu et al. (2016), the augmented approach has been used to derive the state estimation of the stochastic complex networks (1)–(2) with missing measurements. To be specific, the augmented estimation errors of all nodes and the corresponding covariance matrix are defined as

f

+

f Li,k

and Hi,k+1 =

f

f Ωi,k

f

Ωih,k

f

N 

ωij E{(Fi,k + Lfi,k Ωif,k )ei,k eTj,k Γ T

j =1

+ Γ ej,k eTi,k (Fi,k + Lfi,k Ωif,k )T } N  N 

ωij ωil E{Γ ej,k eTl,k Γ T }.

(16)

j =1 l =1

By using the inequality xyT + yxT ≤ xxT + yyT , an upper bounded matrix can be derived for the third term on the right hand side of (16)

ek+1 eTk+1

(9)

j =1

ωij E{(Fi,k + Lfi,k Ωif,k )ei,k eTj,k Γ T + Γ ej,k eTi,k (Fi,k + Lfi,k Ωif,k )T }

The filter is designed by defining a sequence of positive-definite ˜ k+1 satisfying matrices Φ

≤ (10)

The gain matrix is derived by minimizing the trace of the upper ˜ k+1 at each time instant. It can be seen that the bound matrix Φ ˜ k+1 becomes larger as the number of dimension of the matrix Φ the nodes increases and therefore high computational costs are often required to derive the gain matrix by using the augmented approach.

(15) ∂ h(x) . | ∂ x x=¯xi,k

P¯ i,k = (Fi,k + Li,k Ωi,k )Pi,k (Fi,k + Li,k Ωi,k )T + Qi,k

N 

N 

ωij [(Fi,k + Lfi,k Ωif,k )Pi,k (Fi,k + Lfi,k Ωif,k )T

j =1

+ Γ Pj,k Γ T ] = ω¯ i (Fi,k + Lfi,k Ωif,k )Pi,k (Fi,k + Lfi,k Ωif,k )T +

N 

ωij Γ Pj,k Γ T

(17)

j =1

where ω ¯ i = j=1 ωij . Similarly, an upper bounded matrix can be derived for the fourth term on the right hand side of (16)

N

3. Non-augmented filter The following lemma is adopted from the literature to derive the non-augmented filter. Lemma 1 (Xie, Soh, & Souza, 1994). Given matrices A, B, C and D with appropriate dimensions such that CC T ≤ I. Let U be a symmetric positive definite matrix and a > 0 be an arbitrary positive constant such that a−1 I − DUDT > 0. Then the following inequality holds

(A + BCD)U (A + BCD)T ≤ A(U −1 − aDT D)−1 AT + a−1 BBT .

(14)

and Lhi,k are problem-dependent scaling matrices. and denote the unknown time-varying matrix accounting for the linearization erf f rors satisfying Ωi,k (Ωi,k )T ≤ I and Ωih,k (Ωih,k )T ≤ I, respectively (Giuseppe, 2005). In this paper, as in Giuseppe (2005), the scalf ing matrices Li,k and Lhi,k are employed to account for the linearization errors. For more details we refer the reader to Appendix C of Giuseppe (2005). Then, the predicted estimation error covariance can be derived with respect to (14)

(8)

˜ k+1 . P˜ k+1 ≤ Φ

∂ f (x) | ∂ x x=ˆxi,k

where Fi,k =

e˜ k+1 = [eT1,k+1 , . . . , eTN ,k+1 ]T

}.

ωij Γ ej,k + wi,k

− Ki,k+1 vi,k+1

+

˜

N  j =1

The bound Φi,k+1 is obtained by solving a Riccati difference equation and the gain matrix Ki,k+1 is derived by minimizing the trace of the upper bound matrix Φi,k+1 at each time instant. It should be pointed out that the notation X ≥ Y (respectively, X > Y ) means that X − Y is positive semidefinite in the sense of the corresponding quadratic forms (respectively, positive definite).

P˜ k+1 = E{˜

f

e¯ i,k = (Fi,k + Li,k Ωi,k )ei,k +

(4)

The aim of this paper is to design a filter described by (3)–(4), such that there exists a sequence of positive-definite matrices Φi,k+1 satisfying Pi,k+1 ≤ Φi,k+1 .

(12)

P¯ i,k = E{¯ei,k e¯ Ti,k }.

ei,k+1 = (I − Ki,k+1 Hi,k+1 − Ki,k+1 Lhi,k Ωih,k )¯ei,k

where x¯ i,k and xˆ i,k+1 denote the predicted and the updated estimates at time instant k + 1, respectively. Ki,k+1 is the gain matrix to be determined for the ith node at time instant k + 1. The updated estimation error and its corresponding covariance matrix are defined as ei,k+1 = xi,k+1 − xˆ i,k+1

e¯ i,k = xi,k+1 − x¯ i,k

(3)

j =1

xˆ i,k+1 = x¯ i,k + Ki,k+1 [zi,k+1 − h(¯xi,k )]

Now, we define the predicted estimation error and its covariance matrix

(11)

N  N 

ωij ωil E{Γ ej,k eTl,k Γ T }

j=1 l=1

≤

N N 1 

2 j=1 l=1

= ω¯ i

N  j =1

ωij ωil (Γ Pj,k Γ T + Γ Pl,k Γ T )

ωij Γ Pj,k Γ T .

(18)

W. Li et al. / Automatica 78 (2017) 119–122

Substituting (17) and (18) into (16) and applying Lemma 1 to address the linearization errors yields f

f

f

f

P¯ i,k ≤ (1 + ω ¯ i )(Fi,k + Li,k Ωi,k )Pi,k (Fi,k + Li,k Ωi,k )T

+ Qi,k + (1 + ω¯ i )

N 

ωij Γ Pj,k Γ

≤ (1 + ω¯ i )[Fi,k (Pi−,k1 − αi,k I )−1 FiT,k + αi−,k1 Lfi,k (Lfi,k )T ] N 

ωij Γ Pj,k Γ T

(19)

j =1

where αi,k is a positive scalar satisfying αi−,k1 I > Pi,k . The covariance matrix of the updated estimation error can be obtained with respect to (15) Pi,k+1 = (I − Ki,k+1 Hi,k+1 − Ki,k+1 Lhi,k Ωih,k )P¯ i,k

× (I − Ki,k+1 Hi,k+1 − Ki,k+1 Lhi,k Ωih,k )T .

(20)

Applying Lemma 1 to address the linearization errors of the nonlinear measurement yields 1 −1 T Pi,k+1 ≤ (I − Ki,k+1 Hi,k+1 )(P¯ i− ,k − βi,k I ) (I − Ki,k+1 Hi,k+1 )

+ βi−,k1 Ki,k+1 Lhi,k (Lhi,k )T KiT,k+1 + Ki,k+1 Ri,k+1 KiT,k+1

Then the gain matrix can be now determined by setting

=

0 which leads to the solution as shown in (25). This completes the proof.

¯ i,k and Φi,k+1 can be Notice that the upper bound matrices Φ guaranteed to be positive definite if the initial conditions Φi,0 are positive definite and the inequalities (24) are satisfied. In other words, there exist solutions to the Riccati-like difference equations (22)–(23).

T

j =1

+ Qi,k + (1 + ω¯ i )

121

∂ tr(Φi,k+1 ) ∂ Ki,k+1

(21)

1 ¯ where βi,k is a positive scalar satisfying βi− ,k I > Pi,k . So far, we have derived the upper bounded matrices for the predicted and the updated estimation error covariance matrices. The main result of this paper is summarized in the following theorem.

Remark 2. It should be pointed out that the local system with local update might be unobservable by using the non-augmented approach. When the observability analysis of the local system becomes a concern, some additional constraints on the system parameters might be required, which constitutes one of future research topics. Remark 3. The differences between the augmented and nonaugmented approaches are twofold. First, to derive an upper bound matrix of the estimation error covariance matrix, the cross-covariance matrices between coupling nodes are computed recursively in the augmented approach while they are bounded by individual covariance matrices in the non-augmented approach. Second, the cross-covariance matrices are used to compute the gain matrix in the augmented approach and all the gain matrices of the nodes should be computed simultaneously as shown in Hu et al. (2016) while they are not needed in the non-augmented approach. This facilitates the development of distributed filters for nodes. 4. Numerical study

Theorem 1. Consider the nonlinear complex network described by (1)–(2). Let αi,k and βi,k be positive scalars. If the following two Riccati-like difference equations

We consider the visual tracking problem for four mobile robots with interacting behaviors. The kinematic for each robot can be represented by Chen and Jia (2014)

¯ i,k = (1 + ω¯ i )[Fi,k (Φi−,k1 − αi,k I )−1 FiT,k Φ

ξi,k+1 = ξi,k + ϕi,k cos θi,k +

+ αi−,k1 Lfi,k (Lfi,k )T ]

+ (1 + ω¯ i )

N 

T

(22)

ηi,k+1 = ηi,k + ϕi,k sin θi,k +

¯ i−,k1 − βi,k I )−1 (I − Ki,k+1 Hi,k+1 )T (I − Ki,k+1 Hi,k+1 )(Φ + βi−,k1 Ki,k+1 Lhi,k (Lhi,k )T KiT,k+1 + Ki,k+1 Ri,k+1 KiT,k+1 (23)

¯ i,k and Φi,k+1 with initial conditions have positive-define solutions Φ Pi,0 ≤ Φi,0 such that the following inequalities Φi,k <

αi−,k1 I ,

4 

η

ωij (ξj,k + ηj,k ) + wi,k

j =1

j=1

Φi,k+1 =

ξ

ωij (ξj,k + ηj,k ) + wi,k

j =1

+ Qi,k

ωij Γ Φj,k Γ

4 

¯ i,k < βi−,k1 I Φ

θi,k+1 = θi,k + φi,k +

4 

ωij θj,k + wiθ,k

j =1

where (ξi,k , ηi,k ) and θi,k denote the position and the orientation of the ith robot, respectively. (ϕi,k , φi,k ) denotes the velocity vector. η

ξ

wi,k = (wi,k , wi,k , wiθ,k ) is zero mean white Gaussian noise with

hold for all k ≥ 0, then the matrix Φi,k+1 is an upper bound of Pi,k+1 . Moreover, the gain matrix can be determined by minimizing the trace of the upper bound matrix

covariance Qi,k . The trajectories of the mobile robots are shown in Fig. 1. To generate the visual measurement, one camera is assumed to be mounted on each robot and the features are placed on the ceiling. The visual measurement is given by (Chen & Jia, 2014)

¯ i−,k1 − βi,k I )−1 HiT,k+1 [Hi,k+1 (Φ ¯ i−,k1 − βi,k I )−1 Ki,k+1 = (Φ

pi,k =

×

HiT,k+1

+

βi−,k1 Lhi,k (Lhi,k )T

(24)

+ Ri,k+1 ] . −1

j

(25)

Proof. To show the upper bound of the covariance matrix at each time instant, the mathematical induction approach can be used as in Hu et al. (2016). They are omitted here for space considerations. We are now ready to determine the gain matrix by minimizing the trace of the matrix Φi,k+1 . Taking partial derivative of the trace of the matrix Φi,k+1 with respect to the gain matrix leads to

∂ tr(Φi,k+1 ) ¯ i−,k1 − βi,k I )−1 HiT,k+1 = 2Ki,k+1 [Hi,k+1 (Φ ∂ Ki,k+1 + βi−,k1 Lhi,k (Lhi,k )T + Ri,k+1 ] ¯ i−,k1 − βi,k I )−1 Hi,k+1 . + 2(Φ

j

qi,k

γu zfc

[−(sjξ − ξi,k ) sin θi,k

+ (sjη − ηi,k ) cos θi,k − di,2 ] + p0 + vip,k γv = c [−(sjξ − ξi,k ) cos θi,k zf

− (sjη − ηi,k ) sin θi,k + di,1 ] + q0 + viq,k j

j

where (pi,k , qi,k ) denotes the coordinate of the jth feature in the image plane for the ith robot. (di,1 , di,2 ) is the coordinate of the ith camera in the robot frame. zfc is the distance from the optical center of the camera to the ceiling plane. γu and γv are pixel magnification factors. (p0 , q0 ) is the image coordinate of the camera’s principal j point and (sξ , sjη ) is the coordinate of the jth feature in the world p

(26)

q

frame. vi,k = (vi,k , vi,k ) is zero mean white Gaussian noise with covariance Ri,k .

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W. Li et al. / Automatica 78 (2017) 119–122

Fig. 2. RMSE in position versus time instants.

Fig. 1. Trajectories of the mobile robots.

References In the simulations, the parameters of the visual tracking system are adopted from the experiments in our lab (Chen & Jia, 2014): di,1 = −0.0668, di,2 = 0.0536, zfc = 2.1050, γu = 902.13283, γv = 902.50141, p0 = 347.20436, q0 = 284.34705. The coupling parameter ωij = 10−4 for i ̸= j and ωii = 0. The process noise covariance is Qi,k = diag{0.01, 0.01, 0.01} and the measurement noise covariance is Ri,k = diag{25, 25}. The scalars in the calculation of the upper bound matrices are taken to be αi,k = βi,k = 0.01 (i = 1, 2, 3). To compare the tracking performance of the non-augmented filter with the augmented filter in Hu et al. (2016), the averaged root mean square errors (RMSE) over four robots are shown in Fig. 2 and they are derived over 100 Monte Carlo runs. The plots in Fig. 2 show that the augmented filter performs slightly better than the non-augmented filter. Specifically, the averaged RMSE for the augmented filter and the proposed non-augmented filter are 0.0134 and 0.0144, respectively. To evaluate the computational costs, the averaged CPU time is computed in MATLAB 7.0 on a 2.80-GHz 4 CPU Pentium-based computer. The non-augmented filter consumed 0.047 s for each robot and the augmented filter consumed 0.134 s. It can be seen that the non-augmented filter requires lower computational cost than the augmented filter. 5. Conclusion In this paper, we have presented a non-augmented filter for nonlinear stochastic coupling networks. A distinct feature of the proposed filter is that the cross-covariance matrices between coupling node are not required to be computed and the gain matrix can be derived separately for each node. This is the main contribution of this paper with respect to the existing augmented filters.

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