Linear-minimum-mean-square-error observer for multi-rate sensor ...

www.ietdl.org Published in IET Control Theory and Applications Received on 30th October 2013 Revised on 22nd February 2014 Accepted on 21st March 2014 doi: 10.1049/iet-cta.2013.0972

ISSN 1751-8644

Linear-minimum-mean-square-error observer for multi-rate sensor fusion with missing measurements Hang Geng,Yan Liang, Xiaojing Zhang School of Automation, Northwestern Polytechnical University, Xi’an, People’s Republic of China E-mail: [email protected]

Abstract: This note presents the problem of designing the linear-minimum-mean-square-error observer for a class of multirate sensor fusion systems with missing measurements. Under the casuality constraint because of the multi-rate nature, the covariances of the equivalent noises in the estimation error system are obtained via multi-rate recursive computation. Through minimising the traces of the covariances of the estimation errors, the optimal observer is obtained. Fortunately, all the observer parameters can be calculated off-line. A numerical example is given to show the effectiveness of the proposed observer.

Nomenclature xk xnj k yj,nj k ∗ yj,n jk

wk vnj k ξk ek Wk Pλ A B Cj Dj H Fi,k , FN ,k , FN +1,k Fk j,k

αj I 0 prob{·} diag {·}

subscript j state vector at time k state vector at time nj k kth measurement observed by sensor j at time nj k kth buffered measurement of sensor j at time nj k process noise, meet N (0, Qk ) measurement noise, meet N (0, Rj ) state estimation error at time kN state estimation errors from time kN to kN + N − 1 equivalent noise of the estimation error system operation of E{(λ)(λ)T } system matrix process noise matrix jth sensor measurement matrix jth sensor measurement noise matrix casuality constraint matrix observer gain matrices observer gain matrix with its ith row sub-block being Fi,k a stochastic sequence that takes values on 1 and 0 with Bernoulli distribution using to describe the missing of measurements of sensor j probability of successful packet transmission for the jth channel sensor measurement identity matrix zero matrix probability measure a block diagonal matrix

IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

matrix or variable related to the jth sensor subscript −1, T , + inverse, transpose and Moore–Penrose inverse operation, respectively (·) the same content as that in the previous parenthesis

1

Introduction

In networked control systems and sensor networks, the performance of estimation and control is severely affected by the unavoidable loss nature of the communication channels, such as random packet dropouts and transmission delay, which gives rise to parameter uncertainties [1–4]. To this end, much attention has been paid on estimation and fusion with missing measurements, which can be divided into two categories according to whether the data containing the measurement is time-stamped or not. In the time-stamped case, it is known whether each measurement is lost or not, and the corresponding optimal filter is represented by the stochastic switch between Kalman filter and Kalman predictor. Many researches on filter stability have been made, for example in [5–10]. Considering Bernoulli-distributed packet losses, Sinopoli et al. [5] exploited that there exists a critical probability threshold such that, if the probability of the measurement availability is greater than this threshold, then the expectation of error covariance will be bounded for all initial conditions as time goes to infinity; otherwise, it will be unbounded for some initial conditions. As pointed out by Plarre and Bullo [6], the known lower bound of the critical probability is the exact critical probability in the case that the observable subspace is invertible. Moreover, the multi-description (MD) measurement coding was found increasing the stability region [7]. Based on the concept of peak-covariance 1375 © The Institution of Engineering and Technology 2014

www.ietdl.org based filtering stability, the upper envelop of the covariance of prediction error was derived for an unstable scalar model with two-state Markov-chain packet loss [8]. Besides, the linear-minimum-mean-square-error (LMMSE) was proposed for asynchronous multi-sensor systems with missing measurements [11]. In no time-stamped case, the probability of measurement missing is assumed known a priori, and the matter of measurement missing is formulated as the parameter uncertainty. In [12], stochastic H∞ norms of an estimation error system were defined, and thus H∞ filter was proposed to deal with the possible delay of one sampling period, uncertain observations and measurement missing under a unified framework. Considering the data missing between the sensor and estimator, the LMMSE filter, predictor and smoother were proposed [13], and further extended to the reducedorder implementation [14] and the maximum successive missing number [15]. For linear systems with polytypic uncertain parameters and subject to measurement quantisation, delay and missing, the robust H∞ estimation scheme was established [16]. The H∞ filter was obtained for networked systems with multi-rate measurement missing [17] and the LMMSE filter was proposed in the case that there exist data missing in both channels from the sensors to the estimator or controller and from the estimator or controller to the actuators [18]. In many complex systems, it is often unrealistic or sometimes impossible to guarantee all physical signals operating at one single rate [19], and hence multi-rate sensor fusion with measurement missing becomes an important topic. The multi-rate H∞ filter was derived for a class of networked multi-sensor fusion systems with multi-rate measurement missing and the existence of filtering stability was explored [17]. In [20], a two-stage distributed fusion estimation method was proposed for wireless sensor networks, where the localised state estimates subject to data missing are obtained at a faster time scale and exchanged among neighbour nodes at a slower time scale for the fast-rate data fusion. In general, these estimators are suitable to the no time-stamped case. In some practical applications, the statistics of process noises and measurement noises is given, instead of the parameter bounds. It motivates us to consider an interesting but still open problem, that is, how to design the multi-rate LMMSE filter, as the counterpart of the H∞ filter in [17]. In this paper, the multi-rate LMMSE observer is proposed for multi-rate sensor fusion systems with missing measurements. It is found that the covariances of the equivalent noises in the estimation error system should be calculated via multi-rate recursive computation. The observer parameters are optimised by minimising the traces of the covariances of the estimation errors. It is worth mentioning that all the parameters can be calculated off-line.

2

Problem formulation

Consider the linear multi-rate sensor system with missing measurements as shown in Fig. 1 xk+1 = Axk + Bwk yj,nj k = Cj xnj k + Dj vnj k , ∗ yj,n jk

= j,nj k yj,nj k + (1 −

(1) j = 1, 2, . . . , p ∗ j,nj k )yj,n j k−nj

1376 © The Institution of Engineering and Technology 2014

Fig. 1

Multi-rate sensor fusion with missing measurements.

where xk is the state vector to be estimated evolving with a period h; yj,nj k is the jth channel sensor measurement vector with a sampling period nj h; matrices A, B, Cj and Dj are known with proper dimensions. In (1)–(3), there are p + 1 periods: the state updating and estimation output period h, the sensors operate at the p slow-rate sampling periods from n1 h to np h; wk and vnj k are uncorrelated white noises with zero means and covariance matrices E{wk wkT } = Qk , E{vnj k vnTj k } = Rj . The stochastic parameters j,nj k in (3) are Bernoulli distributed and white satisfying   (4) prob j,nj k = 1 = αj , 0 < αj ≤ 1 ∗ = yj,nj k , that is, the jth sensor meaIf j,nj k = 1, then yj,n jk ∗ surement at time nj k is received. If j,nj k = 0, then yj,n = jk ∗ yj,nj k−nj , that is, the jth sensor measurement at time nj k − nj is used at time nj k, which is possibly applied to any time instant. Hence (3) can describe possible multiple measurements missing. It is assumed that j,k is independent of wk , vnj k and each other; x0 , wk , vnj k , and j,nj k are mutually independent. The object of this work is to design the multi-rate ∗ LMMSE observer based on the received measurements yj,n . jk

(2) (3)

Remark 1: Our work here is different from that in [17] in the following aspects. As the unknown inputs (the noises) in IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

www.ietdl.org ∗ ∗ ∗ yj,m = yj,m − (1 − αj )yj,m − αj Cj AN xkN j,k j,k j,k −1

[17] are norm-bounded, the multi-rate observer obtained is a linear filter with constant period and the resultant estimation error systems is time-invariant. Owing to insufficient design freedom in decoupling the error with the unknown inputs, the filter design is transformed into a stochastic H∞ optimisation problem to minimise the gain from the unknown inputs to the estimation error. Nevertheless, in this paper, the statistics of the noises are known and the observer obtained is time-varying. The resultant estimation error systems is also time-varying and the covariances of its equivalent noises have to be determined. After that, our LMMSE filter can be derived through the orthogonality principle. In general, the introduction of the statistics of the noises brings in more information of the state and hence, the multi-rate filter design can obtain more accurate estimate than the H∞ one in [17].

Remark 2: As shown in (7), to estimate the state at time instant kN + i, we must use the measurements y∗k and y∗d , k ∗ ∗ ∗ ∗ , y , y , . . . , y are that is, measurements yj,kN −nj j,kN j,kN +nj j,kN +N −nj ∗ needed. However, yj,kN +N −nj is expected to come at time kN + N − nj which is larger than kN + i. And hence in order to guarantee that the observer obtained in (7), (8) is a state filter instead of a state smoother, the casuality constraint must be introduced into the design of the filter.

3

Fi,k y∗k =

Multi-rate observer design

Denote mj,k = (k + 1)N /nj , where N is the least common multiple of nj . Owing to the stochastic measurements ∗ missing, the received measurement yj,n has the dynamic jk ∗ ∗ are expected property shown in (3). Hence yj,nj k and yj,n j k−nj to appear simultaneously in the following unknown input observer with time-varying parameters to compensate such dynamics ∗ ∗d  xkN +i = Ei,k xkN + Fi,k y∗k + Fi,k yk ,

 xkN +N = EN ,k xkN +

FN ,k y∗k

+

i = 1, 2, . . . , N − 1

C j = col{Cj , Cj Anj , . . . , Cj AN −nj }

Remark 3: In (7), the estimate of  xkN +i should not utilise measurements after time instant kN + i, that is

+

FN +1,k yk∗

+

=



I ···I Hi,j = diag  

li,j blocks

=

∗ ∗ ∗ col{y1,m , y2,m , . . . , yp,m } 1,k 2,k p,k

 xkN +i =  xkN +N =

i = 1, 2, . . . , N − 1 +

FN +1,k yk∗



N /nj −li,j blocks

(10)

ξk =  xkN ek = col{ xkN , xkN +1 , . . . , xkN +N −1 } Constructing a matrix F k with its ith row sub-block being Fi,k in (9), then the estimation error system of (7), (8) is

∗ where E1,k , . . . , EN ,k , F1,k , . . . , FN +1,k , F1,k , . . . , FN∗ +1,k are time-varying parameter matrices to be determined. Through decoupling estimation error with the state and its measurements, the observer in (5), (6) is equivalent to the following one

xkN + Fi,k y∗k , A xkN + FN ,k y∗k AN

0 · · · 0

 xk xk = xk −

∗ ∗ ∗ yk∗d = col{y1,m , y2,m , . . . , yp,m } 1,k −1 2,k −1 p,k −1

i

(9)



where li,j = i/nj + 1 and i/nj represents the largest integer no larger than i/nj . Denote

∗ ∗ ∗ y∗j,k = col{yj,kN , yj,kN +nj , . . . , yj,kN +N −nj }

=

Ki,jk (Hi,j y∗j,k )

where Ki,jk , i ∈ {1, 2, . . . , N − 1}, j ∈ {i, 2, . . . , p}, are free matrices to be determined; H is the causality constraint matrix with its (i, j)th block being

y∗d = col{y∗d , y∗d , . . . , y∗d } k 1,k 2,k p,k

yk∗

∗ ∗ ∗ − (1 − αj )yj,kN Ki,jk col{yj,kN −nj , . . . , yj,kN +i/nj nj

(6)

y∗k = col{y∗1,k , y∗2,k , . . . , y∗p,k }

∗ ∗ ∗ col{yj,kN −nj , yj,kN , . . . , yj,kN +N −2nj }

p  j=1

FN∗ +1,k yk∗d

with

y∗d j,k

j=1

∗ xkN − (1 − αj )yj,kN , 0, · · · , 0} − αj Hi,j C j +i/nj −1nj

(5)

FN∗ ,k y∗d k

p 

 ξk+1 = Ak ξk + Bk Wk e k = C k ξk + D k W k

(11)

with

(7)

Ak = AN − Fq,k C 1 − Fq+1,k C 2 AN     Bk = B1,k B2,k , Dk = D1,k D2,k

(8)

Ck =  C0 −  D20 F k C 1

with

Wk = col{wk∗ , ∗k }

y∗k = col{y∗1,k , y∗2,k , . . . , y∗p,k } ∗ ∗ ∗ yk∗ = col{y1,m , y2,m , . . . , yp,m } 1,k 2,k p,k

where y∗j,k

=

y∗j,k

− (1 −

αj )y∗d j,k

where wk∗ = col{wkN , wkN +1 , . . . , wkN +N }, ∗k = col{k , k }

− αj C j xkN

IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

k = col{1,k , . . . , p,k }, k = col{1,k , . . . , p,k } 1377 © The Institution of Engineering and Technology 2014

www.ietdl.org  B1,k = B − FN ,k D1 − FN +1,k C 2 B   B2,k = −FN ,k −FN +1,k   D1,k =  D20 F k D1 0 D10 −    D2,k = − D20 F k 0

−FN +1,k D2



4 4 = αj Anj Pxnj k CjT + αj Anj −1 BPxnj k DjT + (1 − αj )Anj Mj,k Mj,k+1 (15) 2 4 Mj,k+1 = Cj Mj,k+1 3 Mj,k+1

3 + (1 − αj )Anj Mj,k

with

 j = Anj −1 B

(18)

· · · AB

 B ,

ϒj = col{(Cj Anj −1 B)T , . . . (Cj AB)T , (Cj B)T } Qj,k = diag{Qnj k , Qnj k+1 , . . . , Qnj k+nj −1 } Proof: See Appendix 1. Denote PWk = E{(Wk )(·)T } Pwk∗ = E{(wk∗ )(·)T }

1 block

P∗k = E{(∗k )(·)T } we have the following theorem.

N −1 blocks

Theorem 1: For the estimation error system in (11), we have wk∗

∗k

It is easily obtained that and are zero-mean and white, and ∗k is independent of ξk and wk∗ . To obtain the multi-rate LMMSE observer, we need to determine the covariance of Wk , and further optimise the observer parameters (i.e. the observer gain matrices) Fi,k , FN ,k and FN +1,k by minimising the traces of the covariances of the estimation errors under casuality constraint.

PWk = diag{Pwk∗ , P∗k }

(19)

with Pwk∗ = diag{QkN , QkN +1 , . . . , QkN +N }

(20)

P∗k = diag{E[1,k (·) , . . . , p,k (·) , 1,k (·) , . . . , p,k (·)T ]} (21) T

T

T

where

Determination of observer parameters

4.1

(17)

1 3 = Cj Mj,k+1 + αj Dj Rj DjT Mj,k+1

∗  D20 = col{0,  D20 } ∗  D20 = diag{I , . . . , I }  

4

= αj A Pxnj k (Cj A )

nj T

+ αj j Qj,k ϒj + αj (1 − αj )Anj −1 BPxnj k DjT

B = [AN −1 B · · · AB B] T  C 1 = α1 C T1 α2 C T2 · · · αp C Tp  T C 2 = α1 C1T α2 C2T · · · αp CpT T  D1 = α1 DT1 α2 DT2 · · · αp DTp T  D2 = α1 D1T α2 D2T · · · αp DpT  T T   C0 = I AT · · · AN −1 ⎤ ⎡ 0 ··· ··· 0 ⎢ B 0 ··· 0 ··· 0 ⎥ ⎥ ⎢ ⎥ ⎢ . . . . . . ⎢  .. .. .. .. .. ⎥ D10 = ⎢ .. ⎥ ⎥ ⎢ N −2 ⎣A B 0 ⎦ ··· B  



N −1 blocks

(16)

nj

E{j,k (·)T } = diag{βkN +tnj }, t = 0, 1, . . . , [N /nj ]

Covariance of Wk

βkN +tnj = (αj −

Denote

αj2 ){Pyj,kN +tnj

−

(22)

1 Mj,kN +tnj

1 T ∗ − (Mj,kN +tnj ) + Pyj,kN +tn −n } j

Pxk = E{(xk )(·) } Pyj,k = E{(yj,nj k )(·) } ∗ Pyj,k∗ = E{(yj,n )(·)T } jk

=

Proof: See Appendix 2.

∗ E{yj,nj k (yj,n )T } jk

2 ∗ Mj,k = E{yj,nj k (yj,n )T } j k−nj 3 ∗ = E{xnj k (yj,n )T } Mj,k jk 4 ∗ = E{xnj k (yj,n )T } Mj,k j k−nj

To obtain the covariance, the following Lemma is given. Lemma 1: The following recursions hold Pxk+1 = APxk AT + BQk BT

(12)

Pyj,k+1 = Cj Pxk+1 CjT + Dj Rj DjT

(13)

∗ Pyj,k+1 = (1 − αj )Pyj,k∗ + αj Pyj,k+1

(14)

1378 © The Institution of Engineering and Technology 2014

} (24)

j,k −1

T

1 Mj,k

(23)

j

2 2 − (Mj,m )T + Py∗j,m E{j,k (·)T } = (αj −αj2 ){Pyj,mj,k − Mj,m j,k j,k

T

As shown in Fig. 2, the computation of PWk consists of two aspects. One is the multiple dynamic recursive loops 4 of Pxk , Mj,k and Pyj,k∗ coming from the induced dynamics with the period h because of measurements missing and buffering (see (3)) by the fact such loops would no longer exist if there were no measurements miss. The other is the 3 multi-rate recursive loops of Mj,k caused by the multi-rate sampling with the period nj h. In general, such computation of PWk reflects the coupling of the multi-rate nature and measurements missing. Remark 4: Though PWk is time-varying, it can be figured out off-line or beforehand by the fact that its computation 4 , Pyj,k∗ is constituted by the dynamical recursions of Pxk , Mj,k 3 and Mj,k (as shown in Lemma 1), which have nothing to do with the measurements. IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

www.ietdl.org

Fig. 2

4.2

Diagram of the computation of covariance matrix PWk

Parameter optimisation in the LMMSE sense

Denote Pξk = E{ξk (·)T }, Pek = E{ek (·)T }, we have Pξk+1 = E{(A¯ k ξk + B¯ k Wk )(·)T } = A¯ k Pξk A¯ Tk + B¯ k PWk B¯ kT ¯ k+1 Wk+1 )(·)T } Pek+1 = E{(C¯ k+1 ξk+1 + D T T ¯ k+1 ¯ k+1 PWk+1 D = C¯ k+1 Pξk+1 C¯ k+1 +D

Hence, the covariances of the estimation errors are  Pξk+1 = A¯ k Pξk A¯ Tk + B¯ k PWk B¯ kT Pek+1

T T ¯ k+1 ¯ k+1 PWk+1 D = C¯ k+1 Pξk+1 C¯ k+1 +D

  A= B 

0 0

 0 ,



T =

1

C C 2 AN  T

  C  D

Pξk (C 1 )T Pξk (C 2 AN ) CT PW k  DT PW k      Pξk  ( D20 H C 1 )T 0 Uk =  D0 C0  0 PWk ( D20 H C 11 )T      Pξk ( D20 H C 1 )T 0 Vk = C 1 C 11 , 0 PWk ( D20 H C 11 )T Sk =

IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

where   C = D1 0 I    D10 0  D0 = , 0 0

 0 , C 11

  D = C 2 B D2   10 0 D = I 0,

0

I



Proof: See Appendix 3. (25)

Theorem 2: The optimal parameters F k (with its ith row sub-block being Fi,k ), FN ,k and FN +1,k are obtained in the LMMSE sense as follows     FN ,k FN +1,k = AN  (26) A Sk (TSk )+ + +  F k = D20 Uk Vk (27) with

Vk+ = VkT (Vk VkT )−1 + T  T  D20 )−1  D20 D20 = ( D20 , (TSk )+ = (TSk )T [(TSk )(TSk )T ]−1

Remark 5: From the above parameter determination we can see that the computation of the filter parameters is independent of the measurements and hence they can be calculated Table 1 Implementation of the proposed algorithm Step 1. Initialisation Given the initial value of Px0 , Q0 , Pξ0 ,and Rj (j = 1, 2, . . . , p) Step 2. Off-line computation (1) Recursion of Lemma 1 Recursively compute Pxk , Pyj ,k , Py ∗ , Mj4,k , Mj2,k , j ,k

Mj3,k , and Mj1,k using (12)–(18) (2) Recursion of PWk Recursively compute PWk using (19)–(24) (3) Recursion of Pξk and the gain matrices Recursively compute Pξk , F k , FN,k and FN+1,k using (25)–(27) Step 3. On-line computation Calculate state estimate xˆ kN+i (i = 1, 2, . . . , N) via (7), (8) Step 4. Set k = k + 1 and go to Step 2

1379 © The Institution of Engineering and Technology 2014

www.ietdl.org

Fig. 3

Method comparison in RMMSE (100 runs)

off-line. Moreover, as the covariances of the equivalent noises are time-varying, such parameter computation should be done for each time instant. As to the parameter computation in [17], it is also off-line and can be calculated only once because its filtering parameters are time-invariant. Up till now, the design of the LMMSE observer has been finished and the implementation of the whole algorithm is described in Table 1.

5

Simulation example

Considering an example about target tracking with multirate sensors. The state xk consists of position, velocity and acceleration. yj,nj k (j = 1, 2, 3) are the sensor measurements. Sensor S1, S2 and S3, respectively, sample the target position, velocity and acceleration with different periods being h, 2h and 3h. The missing of the measurements are independently Bernoulli distributed with parameters being α1 , α2 and α3 , respectively. wk and vnj k are zero-mean normal distributed with truncation interval being [−3, 3]. The truncated variances are Qk = 0.05 , R1 = 0.01, R2 = 0.09 and R3 = 0.25. wk and vnj k are independent of each other. According to above consideration, we have the following parameters in (1)–(4) 



1 h h2 /2 0 1 h A= 0 0 1  C1 = D1 = 1 0  C3 = D2 = 0 1 n2 = 2,

,  0 ,  0 ,

B = Pξ0 = I3 ,  C2 = D3 = 0 p = 3,

0

6

Conclusions

This paper proposes the LMMSE observer for multi-rate sensor fusion problem with missing measurements. The design of the observer consists of two steps. One is to acquire the covariances of the equivalent noises of the estimation error system, which contains multi-rate recursive computation. The other is to optimise the observer parameters using the orthogonality principle. Fortunately, all the parameters can be calculated off-line. The simulation shows the effectiveness of our method.



1

n1 = 1,

n3 = 3

h = 0.5, α1 = 0.7, α2 = 0.7, α3 = 0.8,  T x0 = 10 0 0 , Qx0 = diag{100, 0, 0} For accuracy comparisons, the Monte Carlo simulation results of 100 runs are shown in Fig. 3. Here, two methods are compared with our method. One is the ‘Liang method’ in [17], which is a H∞ filter in pursuit of the best accuracy in the most possible worst situation. The other is ‘Yan method’ 1380 © The Institution of Engineering and Technology 2014

in [11] which exactly knows whether or not the measurement is missing (i.e. the time-stamped case), whereas our method and ‘Liang method’ only know the statistical property of measurements missing (i.e. the no time-stamped case). Hence, the ‘Yan method’ represents the upper bound of the accuracy of our method. As shown in Fig. 3, our method has the similar accuracy as ‘Yan method’ in both position and velocity, and is much better than ‘Liang method’. For 100 Monte Carlo simulations, the averaged on-line computation time in each N (N = 12)) period is 0.1028s for ‘Liang method’, 0.2325 s for ‘Yan method’ and 0.0984 s for our method. In other words, from the aspect of computation cost, our method approximates ‘Liang method’ and is more computationeffective than ‘Yan method’ because the filter parameters in ‘liang method’ and our method can both be calculated beforehand.

7

Acknowledgments

This research is supported by the National Science Foundation Council of China under Grant 61135001, 61074179 and 61203234 and the Scientific and Technological Innovation Foundation of the Northwestern Polytechnical University.

8 1

References Wang, Z., Yang, F., Ho, D., Liu, X.: ‘Robust H∞ filtering for stochastic time-delayed systems with missing measurements’, IEEE Trans. Signal Process., 2006, 54, (7), pp. 2579–2587 IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

www.ietdl.org 2

3 4 5 6 7 8 9 10 11

12 13 14 15

16 17 18 19 20

Zhang, H., Basin, M., Skliar, M.: ‘Ito-volterra optimal state estimation with continuous, multi-rate, randomly sampled, and delayed measurements’, IEEE Trans. Autom. Control, 2007, 52, (3), pp. 401–416 Hespanha, J.P., Naghshtabrizi, P., Xu, Y.: ‘A survey of recent results in networked control systems’, Proc. IEEE, 2007, 95, (1), pp. 138–162 Xia, Y., Fu, M., Liu, G.: ‘Analysis and synthesis of netwoked Control Systems’ (Springer, New York, 2011) Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M., Sastry, S.: ‘Kalman filtering with intermittent observations’, IEEE Trans. Autom. Control, 2004, 49, (9), pp. 1453–1464 Plarre, K., Bullo, F.: ‘On kalman filtering for detectable systems with intermittent observations., IEEE Trans. Autom. Control, 2009, 54, (2), pp. 386–390 Jin, Z., Gupta, V., Murray R.: ‘State estimation over packet dropping networks using multiple-description coding’, Automatica, 2006, 42, (9), pp. 1441–1452 Huang, M., Dey, S.: ‘Stability of Kalman filtering with Markovian packet losses’, Automatica, 2007, 43, (4), pp. 598–607 Kluge, S., Reif, K., Brokate, M.: ‘Stochastic stability of the extended Kalman filter with intermittent observations’, IEEE Trans. Autom. Control, 2010, 55, (2), pp. 514–518 Censi, A.: ‘Kalman filtering with intermittent observations: convergence for semi-Markov chains and instrinsic measure’, IEEE Trans. Autom. Control, 2011, 55, (2), pp. 376–381 Yan, L.P., Zhou, D.H., Fu, M.Y., Xia, Y.Q.: ‘State estimation for asynchronous multirate multisensor dynamic systems with missing measurements’, IET Signal Process., 2010, 4, (6), pp. 728–739 Sahebsara, M., Chen, T., Shah, S.L.: ‘Optimal H∞ filtering in networked control systems with multiple packet dropouts’, Syst., Control Lett., 2008, 57, (9), pp. 696–702 Sun, S., Xie, L., Xiao, W., Soh, Y.C.: ‘Optimal linear estimation for systems with multiple packet dropouts’, Automatica, 2008, 44, (5), pp. 1333–1342 Sun, S., Xie, L., Xiao, W., Soh, Y.C.: ‘Optimal filtering for systems with multiple packet dropouts’, IEEE Trans. Circuits Syst. II: Express Briefs, 2008, 55, (7), pp. 695–699 Sun, S., Xie, L., Xiao, W., Soh, Y.C.: ‘Optimal full-order and reduced-order estimators for discrete-time systems with multiple packet dropouts’, IEEE Trans. Signal Process., 2008, 56, (8), pp. 4031–4038 Gao, H., Chen, T.: ‘H∞ estimation for uncertain systems with limited communication capacity’, IEEE Trans. Autom. Control, 2007, 52, (11), pp. 2070–2084 Liang, Y., Chen, T., Pan, Q.: ‘Multi-rate stochastic H∞ filtering for networked multi-sensor fusion’, Automatica, 2010, 46, (2), pp. 437–444 Liang, Y., Chen, T.: ‘Optimal linear State estimator with multiple packet dropouts’, IEEE Trans. Autom. Control, 2010, 55, (6), pp. 1428–1433 Chen, T., Francis, B.: ‘Optimal sampled-data control systems, (Springer-Verlag, New York, 1995) Zhang, W., Feng, G., Yu, L.: ‘Multi-rate distributed fusion estimation for sensor networks with packet losses’, Automatica, 2012, 48, (9), pp. 2016–2028

Putting (3) into the definition of Pyj,k∗ , we have ∗ ∗ Pyj,k+1 = E{(j,nj k+nj yj,nj k+nj + (1 − j,nj k+nj )(yj,n )(·)T } jk ∗ )(·)T } = E{(j,nj k+nj yj,nj k+nj )(·)T }+E{(1−j,nj k+nj )2 (yj,n jk ∗ + 2E{j,nj k+nj yj,nj k+nj (1 − j,nj k+nj )(yj,n )T } jk

= (1 − αj )Pyj,k∗ + αj Pyj,k+1 that is, (14). From (1) and (3), we have  xnj k+nj = Anj xnj k + Anj B

· · · AB

B



× col{wnj k , wnj k+1 , · · · , wnj k+nj −1 } ∗ ∗ yj,n = j,nj k+nj yj,nj k+nj + (1 − j,nj k + nj )yj,n j k+nj jk

4 3 Putting them into the definition of Mj,k and Mj,k , we obtain (15) and (17). 2 1 Putting (2) into the definitions of Mj,k and Mj,k , we easily obtain (16) and (18).

9.2

Appendix 2: Proof of Theorem 1

From (11), we have Wk = [wk∗ ∗k ]T and note that ∗k is independent of wk∗ , we have (19). As wk∗ = col{wk , wkN +N }, E{wk } = 0, and E{(wk )(·)T } = Qk , replacing them into the left side of (20), we obtain (20). Putting the expression of ∗k into the left side of (21) and note that E{wk } = 0, we obtain (21). Replacing (2) and (3) into the following two equations j,k =  ) j,k (yj,k − y∗d j,k ∗ j,k = (j,kN +N − αj )(yj,mj,k − yj,m ) j,k −1

we arrive at the following expressions ∗ E{(j,k )(·)T } = (αj − αj2 )diagE{(yj,kN − yj,kN −nj ) ∗ T (·)T , . . . , (yj,kN +N −nj − yj,kN +N −2nj )(·) } ∗ E{(j,k )(·)T } = (αj − αj2 )E{(yj,mj,k − yj,m )(·)T } j,k −1

9 9.1

Appendix

Further we have

Appendix 1: Proof of Lemma 1

Proof: Putting (1) into the definition of Pxk , we have Pxk+1 = E{(Axk + Bwk )(·)T } = APxk AT + BQk BT that is, (12). Putting (2) into the definition of Pyj,k , we have Pyj,k+1 = E{(Cj xnj k+nj + Dj vnj k+nj )(·) } T

= Cj Pxk+1 CjT + Dj Rj DjT that is, (13). IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

1 1 E{(j,k )(·)T } = (αj − αj2 )diag{[Pyj,kN − Mj,kN − (Mj,kN )T 1 ∗ + Pyj,kN ], [Pyj,kN −nj − Mj,kN −nj −n j

−

1 T (Mj,kN −nj )

∗ + Pyj,kN ]} −2n j

= diag{βkN +tnj }, t ∈ {0, 1, . . . , [N /nj ]} E{(j,k )(·)T } = (αj − αj2 ){Pyj,mj,k 2 2 − Mj,m − (Mj,m )T + Py∗j,m j,k j,k

j,k −1

}

Therefore (22)–(24) are achieved. 1381 © The Institution of Engineering and Technology 2014

www.ietdl.org 9.3

Appendix 3: Proof of Theorem 2

that is

Transforming Bk into the similar form as Ak in (11), we have



  Bk =  A − FN ,k  C − FN +1,k  D,  A = B 0 0 0] ,      C = D1 0 I 0 ,  D = C 2 B D2 0 I

A¯ k

 1  Pξk (C )T B¯ k PW k  CT

Pξk (C 2 AN )T

 =0

DT PW k 

(32)

Putting the expressions of Ak and Bk into (32), we have Then we have the trace of Pξk+1 as {(A¯ k )ih (Pξk )hl (A¯ k )il + (B¯ k )ih (PWk )hl (B¯ k )il }

tr(Pξk+1 ) =

   A − FN ,k

 N A

  FN +1,k T Sk = 0

i,h,l

with with (A¯ k )ih = ANih −

(FN ,k )im (C 1 )mh −

(FN +1,k )im (C 2 A)mh

m

(A¯ k )il = ANil −

(FN +1,k )im (C 2 A)ml

m

m

C)mh − (FN ,k )im (



PW k  DT

 ,

T =

C1

C 2 AN

  C  D

or its equivalent expression

(FN +1,k )im ( D)mh

    FN ,k FN +1,k TSk = A A Sk

m

m

(B¯ k )il =  Ail −

PW k  CT

Pξk (C 2 AN )T

m

(FN ,k )im (C 1 )ml −

(B¯ k )ih =  Aih −

Sk =

 Pξk (C 1 )T

(FN ,k )im ( C)ml − m

(FN +1,k )im ( D)ml m

Then we have ∂tr(Pξk+1 ) = ∂(FN ,k )ij

{(−C 1 )jh (Pξk )hl (A¯ k )il } h,l

{(A¯ k )ih (Pξk )hl (−C 1 )jl }

+ h,l

{(− Cjh )(PWk )hl (B¯ k )il }

+ h,l

{(B¯ k )ih (PWk )hl (− Cjl )

−

(28)

h,l

It can be seen from the expression of T , Sk that T , Sk are of full column rank, so by making inverse on both sides of (33), we have (26). It is worth to mention that in deriving the optimal parameters in the LMMSE sense above, we just utilise (30), (31). Strictly, we should testify the convexity. Here such testification is omitted by the fact that the improper filter design will lead to filter divergence, that is, the trace approaches to infinity, and hence if we obtain the unique extremum, then it must correspond to the minimum value of the trace. Such a situation is just similar to the derivation of the standard Kalman filter based on the partial derivative of the trace with respective to the gain matrix. Transforming Dk into the similar form as Ck in (11), we have

The equivalent expression of its counterpart FN +1,k is ∂tr(Pξk+1 ) = ∂(FN +1,k )ij

(33)

Dk =  D0 −  D20 F k C 11 ,   C 11 = D1 0 I 0

{(−C 2 A)jl (Pξk )hl (A¯ k )il }

  D0 =  D10

0

0

 0 ,

hl

Then we have the trace of Pek as

{(A¯ k )ih (Pξk )hl (−C 2 A)jl }

+ hl

{(− Djh )(PWk )hl (B¯ k )il }

+

i,h,l

{(B¯ k )ih (PWk )hl (− Djl )}

+

¯ k )ih (PWk )hl (D ¯ k )il } {(C¯ k )ih (Pξk )hl (C¯ k )il } + (D

tr(Pek ) =

hl

(29)

hl

with

∂tr(Pξk+1 ) ∂tr(Pξk+1 ) opt |FN ,k =F opt = 0 and | = 0, N ,k ∂(FN ,k )ij ∂(FN +1,k )ij FN +1,k =FN +1,k we have

Let

CT = 0 A¯ k Pξk (C 1 )T + B¯ k PWk  A¯ k Pξk (C 2 A)T + B¯ k PWk  DT = 0 1382 © The Institution of Engineering and Technology 2014

(C¯ k )ih = ( C0 )ih − ( D20 )is (F k )st (C 1 )th , (C¯ k )il = ( C0 )il − ( D20 )is (F k )st (C 1 )tl

(30)

¯ k )ih = ( D0 )ih − ( D20 )is (F k )st (C 11 )th , (D

(31)

¯ k )il = ( (D D0 )il − ( D20 )is (F k )st (C 11 )tl , (F k )st = Kstk Hst IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

www.ietdl.org Denote

Then, we have ∂tr(Pek ) = ∂Kijk

{(− D20 )H C 1 )jh (Pξk )hl (C¯ k )il }

 Uk =  C0

h,l

 D0

{(C¯ k )ih (Pξk )hl (− D20 H C 1 )jl }

+



h,l

Vk = [C 1 C 11 ]

¯ k )il } {(− D20 H C 11 )jh (PWk )hl (D

+ h,l

¯ k )ih (PWk )hl (− {(D D20 H C 11 )jl }

+ h,l

¯ k PWk ( = C¯ k Pξk ( D20 H C 1 )T + D D20 H C 11 )T

(34)

∂tr(Pek ) |Kijk =(Kijk )opt = 0 and put the expressions of Ck and ∂Kijk Dk into (34), we have

Let

 D0 PWk ( D20 H C 11 )T C0 Pξk ( D20 H C 1 )T +  D20 H C 1 )T +  D20 F k C 11 PWk ( D20 H C 11 )T = D20 F k C 1 Pξk (

IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1375–1383 doi: 10.1049/iet-cta.2013.0972

  Pξk 0

0



( D20 H C 11 )T

PW k

Pξk

0

0

PW k

( D20 H C 1 )T



( D20 H C 1 )T

 ,



( D20 H C 11 )T

It can be seen from the expressions of  D20 and Vk that Vk is of full column rank, and  D20 is of full row rank. Therefore we finally obtain the solution of F k as +  D20 H C 11 )T ] Fk =  D20 [C0 Pξk ( D20 H C 1 )T +  D0 PWk (

 ×

C 1 Pξk ( D20 H C 1 )T

T

 D20 H C 11 )T C11 PWk (

Thus, we obtain (27).

1383 © The Institution of Engineering and Technology 2014